A Spline Finite Point Method for Nonlinear Bending Analysis of FG … · 2020. 7. 10. ·...

Research ArticleA Spline Finite Point Method for Nonlinear Bending Analysis ofFG Plates in Thermal Environments Based on a Locking-freeThinThick Plate Theory

Tian-Jing Mo1 Jun Huang 1 Shuang-Bei Li 123 and Hai Wu1

1College of Civil Engineering and Architecture Guangxi University Nanning 530004 China2e Key Laboratory of Disaster Prevention and Structural Safety of the Education Ministry Guangxi UniversityNanning 530004 China3Guangxi Key Laboratory of Disaster Prevention and Engineering Safety Guangxi University Nanning 530004 China

Correspondence should be addressed to Shuang-Bei Li lsbwh90163com

Received 21 March 2020 Revised 17 May 2020 Accepted 5 June 2020 Published 10 July 2020

Academic Editor Mijia Yang

Copyright copy 2020 Tian-Jing Mo et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A spline finite point method (SFPM) based on a locking-free thinthick plate theory which is suitable for analysis of both thick andthin plates is developed to study nonlinear bending behavior of functionally graded material (FGM) plates with differentthickness in thermal environments In the proposed method one direction of the plate is discretized with a set of uniformlydistributed spline nodes instead of meshes and the other direction is expressed with orthogonal functions determined by theboundary conditions e displacements of the plate are constructed by the linear combination of orthogonal functions and cubicB-spline interpolation functions with high efficiency for modeling e locking-free thinthick plate theory used by the proposedmethod is based on the first-order shear deformation theory but takes the shear strains and displacements as basic unknownsematerial properties of the FG plate are assumed to vary along the thickness direction following the power function distribution Bycomparing with several published research studies based on the finite element method (FEM) the correctness efficiency andgenerality of the newmodel are validated for rectangular plates Moreover uniform and nonlinear temperature rise conditions arediscussed respectivelye effect of the temperature distribution in-plane temperature force and elastic foundation on nonlinearbending under different parameters are discussed in detail

1 Introduction

Functionally Graded Materials (FGMs) a new type ofcomposite material developed in recent years are obtainedby mixing two (such as metal and ceramic) or severalmaterials in a certain volume ratio [1] Unlike traditionalcomposite materials the material composition of FGMschanges continuously along one direction without obviousdelamination so the physical properties mutation betweendifferent materials can be eliminated easily FGMs aretemperature dependent and usually used in high tempera-ture environment and moreover can avoid stress con-centration [2] reduce residual stress and improve bondstrength between material layers effectively ereforeFGMs have been widely applied in many fields nowadays

such as aerospace medicine mechanical and constructionengineering which has high scientific research value andgreat potential for development [3]

At present there are a number of research studies on thedeformation of FG plates and shells based on differenttheories including the classical laminated plate theory(CLPT) the first-order shear deformation plate theory(FSDT) the higher-order shear deformation plate theory(HSDT) and the three-dimensional elasticity theoryBeikMohammadlou and EkhteraeiToussi [4] gave inclusiveforms of governing equations for the elastoplastic bucklingof FG plates based on the CLPT Bagheri et al [5] analyzedasymmetric thermal buckling behavior of elastically sup-ported annular FG plates by the FSDT Alshorbagy et al [6]used the first-order shear deformation plate model to

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 2943705 22 pageshttpsdoiorg10115520202943705

investigate uncoupled thermomechanical behavior of FGplates Based on a novel integral first order shear defor-mation theory Bousahla et al [7] studied buckling andvibrational behavior of the composite beam armed withsingle-walled carbon nanotubes resting on Winkler-Pas-ternak elastic foundation Using an nth-order shear defor-mation theory thermomechanical flexural analysis offunctionally graded material sandwich plates with P-FGMface sheets and E-FGM and symmetric S-FGM core isperformed by Boussoula et al [8] Rahmani et al [9] in-vestigated bending and free vibration behavior of elasticallysupported functionally graded sandwich plates with differ-ent boundary conditions by an original novel high ordershear theory By employing a new type of quasi-3D hy-perbolic shear deformation theory Kaddari et al [10] dis-cussed statics and free vibration of functionally gradedporous plates resting on elastic foundations Zarga et al [11]analyzed thermomechanical bending of FGM sandwichplates and Mahmoudi et al [12] studied the effect of thematerial gradation and elastic foundation on the deflectionsand stresses of functionally graded sandwich plate subjectedto thermomechanical loading

It is worth noting that the above theories have their ownadvantages and limitations e CLPT is simple and easy inmechanical analyzing but will induce great errors wheninvestigating thick plates because the transverse shear de-formation of the plates is ignored which is equivalent of overstiffness Although the transverse shear deformation alongthe thickness direction is considered and the calculationaccuracy is improved in the FSDT it is necessary to use theshear correction factor in the calculation e HSDT andthree-dimensional elasticity theory have sufficient accuracyin the deformation analysis of plates and shells but toomanyindependent unknowns make the solution complicated andincrease the computational cost Moreover the analysis ofthin plate based on the first or higher-order plate theory hasconsidered shear deformation may induce false shear strainnamely shear-locking [13] erefore some researchersdevoted themselves to establish a general element andmethod to calculate both thick and thin plates [14ndash16] and alocking-free thinthick plate theory was proposed In thispaper this theory is based on the FSDT but takes the in-plane displacements deflection and shear strains as basicunknowns which can effectively avoid shear locking phe-nomenon with simplicity and high generality

For the past decades analytical semianalytic experi-mental and numerical methods for FGM structures havebeen proposed [17ndash19]e common numerical approachesincluding the finite element methods (FEM) [20] differentialquadrature methods (DQM) [21] and meshless methodswere quite popular in engineering because of their easyimplementation with computers [22] When using thetraditional FEM to analyze nonlinear behaviors and largedeformation problem not only number of unknowns is largebut also much effort needs to be expended in remeshing ateach step of the problem solving [22] On the contrary whenusing the meshless method to analyze flat plate problem theremeshing can be avoided naturally [23] and the boundaryconditions processing is simple with less unknowns and high

precision Among those the spline finite point method(SFPM) presented by Qin [24] is one of the spline meshlessmethods As an active branch of modern function ap-proximation the B-spline function is widely used Based onthe B-spline function orthogonal function and minimumpotential energy principle the displacement modes of SFPMare constructed by the linear combination for the product oforthogonal functions and cubic B-spline interpolationfunctions In this method one direction of the flat structureis discretized by a set of uniformly distributed spline nodesand the other one is expressed by orthogonal functionsrelated to the corresponding boundary conditions eequilibrium equation is established with the minimumpotential energy principle e SFPM has been proved moreadvantageous than the FEMs in analyzing the nonlinearbehaviors of plates and beams Li et al [25] used a bidi-rectional B-spline QRmethod (BB-sQRM) to study the crackcontrol of reinforced concrete beam in which the planemodel was discretized with a set of spline nodes along twodirections e results proved the high efficiency and lowcomputational cost of this method Liu et al [26] investi-gated the free transverse vibration of FG EulerndashBernoullibeams through the SFPM and found that it is simple formodeling and boundary processing Li et al [27] derived thedisplacement and stress solutions of a piezoelectric lami-nated composite plate based on a bidirectional B-spline finitepoint method (BB-sFPM) e numerical results showedthat the method was applied to the material parameteridentification with a small loss of precision

Existing studies showed that in complex temperaturefield the heat conduction phenomenon and temperaturedependence of FGMs can significantly affect the bendingbuckling and vibration of the plate [28] and the thermalload plays an important role in determining the mechanicalbehaviors of FGM structures Besides the FG plate generallydeforms largely when subjected to the combined action ofthermomechanical loads and its mechanical behavior isusually nonlinear in this situation For the nonlinear me-chanical problems of FG plates in thermal environmentsome relevant works were summarized in [29ndash32] All theseworks proved that solving mechanical problems by simplelinearization cannot meet the precision requirement inimportant structural calculations e nonlinear analysis isvery significant to the practical engineering applicationMeanwhile the temperature rising and lowing situations arevaried some papers have been published about differenttemperature distributions of FGM structures Belbachir et al[33] analyzed the response of antisymmetric cross-plylaminated plates subjected to a uniformly distributednonlinear thermomechanical loading Tounsi et al [34]studied the static behavior of elastically supported advancedfunctionally graded ceramic-metal plates subjected to anonlinear hygro-thermomechanical load Dong and Li [35]proposed a linear temperature rise solution to analyze thenonlinear bending of FG rectangular plates Farzad andErfan [36] investigated the thermal effect on FG nanobeamssubjected to two kinds of thermal loading namely linearand nonlinear temperature rise Considering the uniformlinear and nonlinear temperature distributions across the

2 Mathematical Problems in Engineering

plate Tebboune et al [37] studied the thermoelastic bucklingof thick FG plates All the conclusions showed that thetemperature distribution can affect the thermal deformationbehavior of FGM structures

We found that there are less existing research studies forthe nonlinear behaviors of FG plates using the locking-freethinthick plate theory and most of the models and analysesemploying the traditional FEMs are based on the FSDT andHSDT us in order to enrich the theoretical system of FGplate and shell structures increase the practicability of FGMand the diversity of research methods and the SFPM isadopted in this paper to carry out the nonlinear bendinganalysis of FG plates on WinklerndashPasternak elastic foun-dations in thermal environments by the locking-free thinthick plate theory e correctness efficiency and generalityof the model will be verified by comparing with the pub-lished studies Meanwhile several parameters such asvolume fraction index thickness-length ratio length-widthratio temperature distribution heating-up condition andfoundation stiffness will be discussed about their influenceson the in-plane temperature force and elastic foundation forthe nonlinear bending of FG plates which can supplementthe test methods and provide further analytical means

2 Spline Finite Point Method Formulation

Qin introduced the formulation of SFPM in his work [24]accordingly the n degree B-spine function can be definedby

φn(x) 1113944

n+1

k0(minus 1)

kn + 1

k

⎛⎝ ⎞⎠x minus xk( 1113857

n

+

n (1)

where xk is the spline node with

xk k minusn + 12

n + 1

k

⎛⎜⎜⎝ ⎞⎟⎟⎠ (n + 1)

k(n + 1 minus k)

x minus xk( 1113857n+

x minus xk( 1113857n x minus xk ge 0

0 x minus xk lt 0

⎧⎪⎨

⎪⎩

(2)

e cubic B-spline function can be obtained with n 3ie

φ3(x) 1113944

4

k0(minus 1)

k

4

k

⎛⎜⎜⎝ ⎞⎟⎟⎠x minus xk( 1113857

3+

3

16

(x + 2)3+ minus 4(x + 1)

3+ + 6x

3+ minus 4(x minus 1)

3+ +(x minus 2)

3+1113960 1113961

(3)

Suppose the length of a FG plate is a and the width is band its spline discretization model is shown in Figure 1

For rectangular plates there are no restrictions on theselection of x-axis and y-axis by SFPM As shown in Figure 1the plate structure can be divided by the interval value ax in xdirection and the local coordinates of corresponding nodesare given by

xi x0 + iax i 1 N (4)

where N stands for the number of subdivisions in x di-rection and

ax a

N

0 x0 ltx1 ltx2 lt middot middot middot ltxNminus 1 ltxN a

(5)

e cubic B-spline function φ3(x) is employed in thispaper with the spline node xk 0 as its center which can beexpressed as

φ3(x) 16

(x + 2)3 x isin [minus 2 minus 1]

(x + 2)3 minus 4(x + 1)3 x isin [minus 1 0]

(2 minus x)3 minus 4(1 minus x)3 x isin [0 1]

(2 minus x)3 x isin [1 2]

0 |x|ge 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

e graphs of cubic B-spline function and its derivativescan by expressed as follows

Figure 2 shows that the cubic B-spline function and itsderivatives are piecewise symmetrical and compact

us the displacement function in x direction of theplate can be represented by a linear combination of splineinterpolation functions ϕi(x) and generalized parameters cie displacement in y direction is expressed by the or-thogonal function which can meet the boundary conditionsat both ends For example the transverse displacement canbe obtained by

w(x) 1113944

N+1

iminus 1ciϕi(x)Zm(y) [ϕ] c Zm(y) (7)

z

O

a

b

x

y

h

0 1 2 N ndash 1 N

Figure 1 e spline finite point discretization model of a FGMsquare plate

Mathematical Problems in Engineering 3

in which

c cminus 1 c0 c1 middot middot middot cN+11113858 1113859T

[ϕ] ϕminus 1 ϕ0 ϕ1 middot middot middot ϕN+11113858 1113859(8)

where Zm(y) is the orthogonal function For the two endsrsquosimply-supported plate in y direction w 0 when y 0and b so the corresponding orthogonal function is sug-gested to be Zm sin((mπb)y) e spline interpolationfunctions [ϕ] can be constructed in the matrix form asfollows

[ϕ] φ3i1113858 1113859 Qs1113858 1113859 (9)

Qs1113858 1113859 diag([g] [I] [h]) (10)

where [I] is a (N minus 3) times (N minus 3) identity matrix and

[g]

1 minus 4 1

0 1 minus12

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[h]

1 0 0

minus12

1 0

1 minus 4 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

φ3i1113858 1113859 φ3x

h+ 11113874 1113875 φ3

x

h1113874 1113875 φ3

x

hminus 11113874 1113875 middot middot middot φ3

x

hminus N minus 11113874 11138751113876 1113877

(11)

It is very simple to deal with the displacement boundaryconditions using spline interpolation functions with thefollowing properties

07

06

05

04

03

02

01

ndash3 ndash2 ndash1 0 1 2 3

16 16

(a)

10

05

ndash05

ndash10


12

ndash12

(b)


1

ndash1

ndash2

(c)


ndash2

ndash4

2

4

(d)

Figure 2 e cubic B-spline functions and its derivatives (a) cubic B-spline function ϕ3(x) (b) first-order derivative of the cubic B-splinefunction ϕ3prime(x) (c) second-order derivative of the cubic B-spline function ϕ3Prime(x) (d) third-order derivative of the cubic B-spline functionϕPrimeprime3(x)


ϕi(0) 0(ine minus 1)

ϕi(a) 0(ineN + 1)

ϕiprime(0) 0(ine minus 1 0)

ϕiprime(a) 0(ineN N + 1)

(12)

erefore the large number substitution method can beused to process the boundary conditions of the plate that isto substitute the corresponding row and column elements inthe stiffness matrix with very large numbers in calculationFor instance only elements corresponding to i minus 1 and i

N + 1 need to be substituted for the two endsrsquo simplysupported plate in x direction

3 Nonlinear Bending of FG Plates

31 Physical Properties of FGM e properties of two ma-terials are assumed to vary along the thickness of FG platesand follow the power function distribution then Youngrsquosmodulus E(z) material density ρ(z) thermal expansioncoefficient α(z) and thermal conductivity κ(z) are related tocoordinate z in the thickness direction which can be uni-formly expressed as (Gupta and Talha [38])

P(z) Pt minus Pb( 1113857 middoth + 2z

2h1113890 1113891

k

+ Pb (13)

where h denotes the thickness of the FG plate k denotes thematerial volume fraction index and t and b denote the upperand lower surface of the FGM respectively

e temperature dependence of the FGM should beconsidered in a thermal environment Since Youngrsquosmodulus E and thermal expansion coefficient α vary greatlywith the temperature while the Poisson ratio μ materialdensity ρ and thermal conductivity κ generally are tem-perature independent so only E and α in (13) are assumed tobe related to the temperature and uniformly expressed by(Lal et al [20] and Zhang and Zhou [32])

P(z T) Pt(T) minus Pb(T)1113858 1113859 middoth + 2z

2h1113890 1113891

k

+ Pb(T) (14)

Pj(T) P0 Pminus 1Tminus 1

+ 1 + P1T1

+ P2T2

+ P3T3

1113872 1113873(j b t)

(15)

where T is the corresponding temperature and Pminus 1 P0 P1P2 and P3 are the temperature-dependent coefficients

32 Temperature Distribution In view of that FGMs areoften applied to high temperature environment for simu-lating the stress state and deformation accurately and it isnecessary to determine the temperature distribution insidethe materials and consider the influence of the thermalstress We assume that the upper and lower surface of FGplates are in a steady temperature state and the temperaturedistributes uniformly along any longitudinal section andonly vary along the thickness which is independent ofcoordinates (x y) ere are two temperature rise conditionsto be considered in this paper

321 Uniform Temperature Rise Condition In this situa-tion the temperature is uniformly raised from the lowersurface to upper surface e temperature variation ΔT

T(z) minus T0 is a constant in which T(z) is the temperaturedistribution function along the thickness direction (Javaheriand Eslami [39]) and T0 300K is the reference temperaturewithout considering the thermal strain

322 Nonlinear Temperature Rise Condition e steady-state heat transfer equation of FG plates can by expressed asfollows (Lal et al [20])

minusddz

κ(z)dT(z)

dz1113890 1113891 0 (16)

where κ(z) is given in detail in (13) and T(z) is the tem-perature distribution function along the thickness directionBoth of them are the functions of z

e upper and lower surface temperature of the plate areassumed to be Tc and Tm respectively then the temperaturedistribution function inside FG plates can be determined as(Lal et al [20])

T(z) Tm + Tc minus Tm( 1113857η(z) (17)

where

η(z) 1C

2z + h

2h1113888 1113889 minus

κcm

(n + 1)κm

2z + h

2h1113888 1113889

k+1

+κ2cm

(2n + 1)κ2m

2z + h

2h1113888 1113889

k+1

minusκ3cm

(3n + 1)κ3m

2z + h

2h1113888 1113889

3k+1⎡⎣

+κ4cm

(4n + 1)κ4m

2z + h

2h1113888 1113889

4k+1

minusκ5cm

(5n + 1)κ5m

2z + h

2h1113888 1113889

5k+1⎤⎦

C 1 minusκcm

(n + 1)κm

+κ2cm

(2n + 1)κ2mminus

κ3cm

(3n + 1)κ3m+

κ4cm

(4n + 1)κ4mminus

κ5cm

(5n + 1)κ5m

(18)


in which κcm κc minus κm κc and κm are the material thermalconductivities of the upper and lower surfaces respectively

33 Nonlinear Geometric Equation Using the locking-freethinthick plate theory for thick and thin plates which takesthe in-plane displacements u and v deflection w and shearstrains cxz and cyz as the independent unknowns the shearstrain conditions of thick plate (cne 0) and thin plate (c 0)can be met simultaneously e specific formula derivationand description is given in Appendix B us the dis-placement field can be obtained by

u u0 minus zφx

v v0 minus zφy

w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(19)

where u v and w are the total displacements u0 v0 and w0are the corresponding displacements of a point on theneutral surface and φx and φy are the rotations about the y-axis and x-axis respectively In order to establish the stiffnessequation easily the strain equation of the plate is expressedas

εlowast1113864 1113865

εpx

εpy

cpxy

εbx

εby

cbxy

cxz

cyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε0xε0y

c0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

zε1xzε1y

zc1xy

c1xz

c1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

εNLx

εNLy

cNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(20)

in which the linear strains are

ε01113966 1113967

ε0xε0y

c0xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

u0x

v0y

u0y + v0x

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

ε11113966 1113967

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy

minus φxy minus φyx

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(21)

and the nonlinear strain is

εNL1113966 1113967

εNLx

εNLy

cNLxy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

12

w0x1113872 11138732

12

w0y1113872 11138732

w0xw0y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(22)

34 Constitutive Equations in ermal Environment In athermal environment the constitutive equations of FGplates on WinklerndashPasternak elastic foundation can bedefined by (Lal et al [20])

σ Qf1113960 1113961 ε + Pf minus Qf1113960 1113961 α ΔT

Pf K1w + K2nabla2w

ST α T

Qf1113960 1113961Tε + pΔT

(23)

where [Qf] is the elastic modulus of FG plates consideringtemperature dependence α is the thermal expansion co-efficient array Pf is the foundation reaction per unit area K1and K2 are the linear normal stiffness and shear stiffness ofthe elastic foundation respectivelynabla2 (z2zx2) + (z2zy2) is the second order Laplace dif-ferential operator ST is the entropy density function and p isthe specific heat capacity of materials e matrix and vectorexpressions for the aforementioned physical quantities aredefined as follows

Qf1113960 1113961

Qεf1113876 1113877 [0]

[0] Qc

f1113876 1113877

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qεf1113960 1113961

E(z T)

1 minus μ2

1 μ 0

μ 1 0

0 01 minus μ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qc

f1113876 1113877

5E(z T)

12(1 + μ)0

05E(z T)

12(1 + μ)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

α α(z T) α(z T) 0 0 01113864 1113865T

(24)

e following stress expression is employed to analyzethe geometrically nonlinear problems in the thermalenvironment


σlowast1113864 1113865

σ0x

σ0y

τ0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

σ1x

σ1y

τ1xy

τ1xz

τ1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

σNLx

σNLy

τNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

minus

Nθx

Nθy

Nθxy

Mθx

Mθy

Mθxy

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(25)

For the plate analysis stresses are usually rewritten as thegeneralized forces as follows

σlowast1113864 1113865

N

M

Fs1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ε01113864 1113865 + εNL1113864 1113865

ε11113864 1113865

c1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦minus

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)

in which

N Nx Ny Nxy1113960 1113961T

M Mx My Mxy1113960 1113961T

Fs1113864 1113865 Fsx Fsy1113960 1113961T

(27)

Nθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α ΔT dz

Mθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α zΔT dz

(28)

where Nθ and Mθ are the thermal force and thermalmoment caused by temperature respectively [A] [B][D] and [C] are the stretching stiffness stretching-bending coupling stiffness bending stiffness and shearingstiffness respectively and the specific expressions aregiven by

([A] [B] [D]) 1113946h2

minus (h2)Q

ef1113960 1113961 1 z z

21113872 1113873dz

[C] 1113946h2

minus (h2)Q

c

f1113876 1113877dz

(29)

erefore (26) can be simplified as

σlowast1113864 1113865 Dlowast

1113960 1113961 εlowast1113864 1113865 minus Flowast

1113960 1113961 (30)

in which

Dlowast

1113960 1113961

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Flowast

1113960 1113961

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

εlowast1113864 1113865 εlowast1113864 1113865L + εlowast1113864 1113865NL

ε01113864 1113865

ε11113864 1113865

c1113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+

εNL1113864 1113865

0

0

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

(31)

35 Spline Discretization Modeling e FG plate is dis-cretized uniformly in x direction and the displacementfunctions can be constructed by the cubic B-spline inter-polation functions and orthogonal functions as follows

u0 1113944r

m1[ϕ] a mXm(y)

v0 1113944r

m1[ϕ] b mYm(y)

w0 1113944

r

m1[ϕ] c mZm(y)

cxz 1113944r

m1[ϕprime] f1113864 1113865mFm(y)

cyz 1113944r

m1[ϕ] g1113864 1113865mGm(y)

(32)

where r is the convergent series [ϕ] is the cubic B-splineinterpolation function defined by (9) and Xm(y) Ym(y)Zm(y) Fm(y) and Gm(y) are the orthogonal functionswhich meet the boundary conditions in y direction All FGplates are simple supported at all ends (SSSS) in this study sothe boundary conditions are given by

w0 u0 φy 0 atx 0 a

w0 v0 φx 0 aty 0 b(33)

In order to satisfy the boundary conditions at y 0 and bthe corresponding orthogonal functions are suggested to be

Xm(y) cosmπb

y1113874 1113875

Ym(y) sinmπb

y1113874 1113875

Zm(y) sinmπb

y1113874 1113875

Fm(y) sinmπb

y1113874 1113875

Gm(y) cosmπb

y1113874 1113875

(34)


us (32) can be expressed in thematrix form as follows

V Nc

1113858 1113859 δ (35)

in which V u0 v0 w cxz cyz1113960 1113961Tare the displacements

and shear strains of a point on the neutral surface Howeverin the calculation the boundary conditions can be processedsimply by substituting (19) into (35) ie

V01113864 1113865 [N] δ (36)

in which V01113864 1113865 u0 v0 w φx φy1113960 1113961T φx and φy are the

rotations of normal to the neutral surface about the y-axisand x-axis respectively δ is the generalized displacementparameter and both [Nc] and [N] are the displacementshape functions of the plate ie

Nc

1113858 1113859 Nc[ ]1 Nc[ ]2 Nc[ ]3 middot middot middot Nc[ ]r1113858 1113859

[N] [N]1 [N]2 [N]3 middot middot middot [N]r1113858 1113859(37)

Nc

1113858 1113859m

[ϕ]Xm 0 0 0 00 [ϕ]Ym 0 0 00 0 [ϕ]Zm 0 00 0 0 [ϕprime]Fm 00 0 0 0 [ϕ]Gm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[N]m

[ϕ]Xm 0 0 0 00 [ϕ]Ym 0 0 00 0 [ϕ]Zm 0 00 0 ϕprime1113858 1113859Zm minus ϕprime1113858 1113859Fm 00 0 [ϕ]Zm

prime 0 minus [ϕ]Gm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(38)

δ δ T1 δ

T2 δ

T3 middot middot middot δ

Tr

1113960 1113961 (39)

δ m a Tm b

Tm c T

m f1113864 1113865Tm g1113864 1113865

Tm

1113960 1113961 m 1 2 3 r

(40)

So the linear strain in (31) can be denoted by thegeneralized displacement parameter as follows

εlowast1113864 1113865L Bp

L1113960 1113961 BbL1113960 1113961 B

c

L1113858 11138591113960 1113961Tδ (41)

where [BpL] [B

pL]1 [B

pL]2 middot middot middot [B

pL]r1113960 1113961 which is similar to

[BbL] and [B

c

L] and

BpL1113960 1113961

m

ϕprime1113858 1113859Xm [0] [0] [0] [0]

[0] [ϕ]Ymprime [0] [0] [0]

[ϕ]Xmprime ϕprime1113858 1113859Ym [0] [0] [0]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BbL1113960 1113961

m

[0] [0] minus ϕPrime1113858 1113859Zm ϕPrime1113858 1113859Fm [0]

[0] [0] minus [ϕ]ZmPrime [0] [ϕ]Gm

prime

[0] [0] minus 2 ϕprime1113858 1113859Zmprime ϕprime1113858 1113859Fm

prime ϕprime1113858 1113859Gm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BcL1113858 1113859m

[0] [0] [0] ϕprime1113858 1113859Fm [0]

[0] [0] [0] [0] [ϕ]Gm

⎡⎣ ⎤⎦

(42)

e nonlinear strain can be constructed as

εlowast1113864 1113865NL 12

[Q] Q (43)

in which

[Q]

zw

zx0

0zw

zy

zw

zy

zw

zx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q

zw

zx

zw

zy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(44)

Substitute (37) into (44) ie

εlowast1113864 1113865NL BNL1113858 1113859 δ (45)

in which

BNL1113858 1113859 BPNL1113858 1113859 [0] [0]1113858 1113859

T12 [Q] [0] [0]1113960 1113961

T[J]

[J]m 0 0 ϕprime1113858 1113859Zm 0 0

0 0 [ϕ]Zmprime 0 0

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Q 1113944r

m1[J]m c m [J] δ

(46)

us (31) can be rewritten as

εlowast1113864 1113865 BL1113858 1113859 + BNL1113858 1113859( 1113857 δ (47)

36 Nonlinear Equilibrium Equation Based on the virtualwork principle the virtual work equation of a thermo-mechanical FG plate is given by

d δ TΨ B

Ad εlowast1113864 1113865

T σlowast1113864 1113865dA minus d δ T

F 0 (48)

whereΨ is the vector sum of internal and external forces Fdenotes the transverse external force d δ and ε are thevirtual displacement and virtual strain respectively andσlowast Nx Ny Nxy Mx My Mxy Fsx Fsy1113960 1113961

Tare the

generalized forces and given in detail in (27)e nonlinear equilibrium equation of the FG plate on

WinklerndashPasternak elastic foundation is given by

Ψ KS + KF + Kθ( 1113857 δ minus F minus FR1113864 1113865 0 (49)

where KS stands for the nonlinear bending stiffness KF

denotes the elastic foundation stiffness Kθ is the thermalgeometric stiffness and FR1113864 1113865 is the thermal load caused bythe environment temperature which can be expressed asfollows respectively


KS BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA

+ BA

BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA

(50)

KF BA

[R]T

Kl1113858 1113859[R]dA (51)

Kl1113858 1113859

K1 0 00 K2 00 0 K2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[R]

[0] [0] [ϕ]Zm [0] [0]

[0] [0] [ϕprime]Zm [0] [0]

[0] [0] [ϕ]Zmprime [0] [0]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(52)

Kθ 12B

A[J]

TN

θ1113876 1113877[J]dA N

θ1113876 1113877

Nθx Nθ

xy

Nθxy Nθ

y

⎡⎢⎢⎣ ⎤⎥⎥⎦ (53)

FR1113864 1113865 1113946V

Bp

1113858 1113859T

Qf1113960 1113961 α ΔT + Bb

1113960 1113961T

Qf1113960 1113961 α zΔT1113874 1113875dV (54)

e differential of Ψ can be written as

dΨ K0 + Klowastσ + KNL1113858 1113859dδ K

lowastT dδ (55)

in which

K0 BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA (56)

Klowastσ B[J]

TNf1113960 1113961[J]dx dy + B[R]

TKl1113858 1113859[R]dx dy

+12B[J]

TN

θ1113876 1113877[J]dx dy

(57)

KNL BA2 BNL1113858 1113859

TDlowast

1113960 1113961 BL1113858 1113859 + 2 BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113872

+ 4 BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113873dA

(58)

where [Nf] Nx Nxy

Nxy Ny1113890 1113891 and the third term of (57) is the

effect of the in-plane temperature force It is caused by thetemperature change and boundary constraint and regarded asan axial load in this paper e nonlinear equilibriumequation (50) can be solved by the NewtonndashRaphsonmethod

4 Results and Discussion

41 Material Properties For the common FG plate theupper surface is usually ceramic rich and the lower surface ismetal rich In this paper two kinds of FG plates are studiede first kind is the combination of AlZrO2 and AlZr2O3which is temperature independent e typical values ofYoungrsquos modulus E the thermal expansion coefficient αthermal conductivity κ and the Poisson ratio μ are listed inTable 1 from Valizadeh et al [40] e second kind is

SUS304Si3N4 which is temperature dependent e tem-perature-dependent coefficients are listed in Table 2 from Fuet al [41] e thermal material properties of SUS304 andSi3N4 under different temperatures are calculated in Table 3according to (15) in Section 3

42 Model Comparisons and Validations

421 Correctness and Generality Studies of SFPM ebending of AlAl2O3 FGM square plate with differentthicknesses and volume fraction indexes k are calculated andcompared with some published research studies in Table 4Numerical results of CLPT and FSDT are also included Inthis example the spline discretization level Ntimes r 12times 3e FG plates are subjected to transverse sinusoidal loadq q0 sin((πa)x)sin((πb)y) and q0 is the intensity of thetransverse load at the plate center e nondimensionaltransverse displacement w (10Ech

3q0a4)w whereEc 380GPa and w is the center deflection

It can be noticed that in Table 4 for the bending of FGMthin and thick plates the present approach is very close tothe result of FSDTand other published literature methods Itillustrates that the proposed SFPM based on the locking-freethinthick plate theory is applicable to the plates of variousthickness and the model established in this paper is correctand general

422 Convergence studies of SFPM e nonlinear bendingof AlZrO2 FGM square plates with different discretizationlevels are calculated and compared with the NS-FEM resultsthat are given by Nguyen-Xuan et al [45] e numericalresults are listed in Table 5 with various volume fractionindexes e deflection errors obtained by comparing theSFPM and NS-FEM results under different discretizationlevels are shown in Figure 3 In this example the FG platesare subjected to a transverse uniformly pressure the non-dimensional center transverse displacementw 100wEmb312(1 minus μ2)qa4 (ah) 5 (ab) 1 μ 03and Em 70GPa

It was proved by Nguyen-Xuan et al [45] that the NS-FEM can gain super-accurate and super-convergent stresssolutions and it can produce the stress solution that can bemuch more accurate than that of the FEM As shown inTable 5 and Figure 3 with the increase of the spline sub-divisions in x direction N and convergent series r the resultsof SFPM converge to the NS-FEM results gradually Whenthe discretization level Ntimes r is 12times 3 16times 3 20times 3 12times 5and 12times 7 the maximum error is 080 029 023052 and 021 respectively which are all less than 1 Itmeans that the spline discretization level Ntimes r 12times 3 canreach high precision In order to improve the computationalefficiency and reduce the calculation cost the spline dis-cretization level Ntimes r 12times 3 is used in all of the followinganalyses in which the number of unknowns is (12 + 3)times

3times 5 225 and the results are able to meet the precisionrequirement In Figure 3 we can see that the SFPM cannearly reach the same accurate and convergent as NS-FEMand with less unknowns


Table 1 Material properties of the first kind of FGM (from Valizadeh et al [40])

Material E (GPa) μ κ(Wmk) α(times10minus 6K)

Al 70 03 204 230ZrO2 200 03 209 10Zr2O3 151 03 209 10Al2O3 380 03 104 72

Table 2 Temperature-dependent coefficients for the material properties of the second kind of FGM (from Fu et al [41])

Material Properties P0 Pminus 1 P1 P2 P3

Si3N4

E (Pa) 34843e+ 9 0 minus 3070e minus 4 2160e minus 7 minus 8964e minus 11μ 028 0 0 0 0

α(1K) 58723e minus 6 0 9095e-4 0 0κ(WmK) 1204 0 0 0 0

SUS304

E (Pa) 20104e+ 9 0 3079e minus 4 minus 6534e minus 7 0μ 028 0 0 0 0

α(1K) 12330e minus 6 0 8086e minus 4 0 0κ(WmK) 919 0 0 0 0

Table 3 ermal material properties of Si3N4 and SUS304 under different temperatures (according to (15) in Section 3)

Material Temperature E (Pa) μ α(1K) κ(WmK)

Si3N4

T 300K 32227e+ 9 028 748e minus 6 919T 400K 31569e+ 9 028 801e minus 6 919T 600K 30392e+ 9 028 908e minus 6 919

SUS304T 300K 20779e+ 9 028 1632e minus 6 1204T 400K 20478e+ 9 028 1532e minus 6 1204T 600K 19089e+ 9 028 1831e minus 6 1204

Table 4 Comparisons of the nondimensional center transverse displacement for AlAl2O3 FGM square plates with different thicknesses andvolume fraction indexes

Volume fraction index k Sourcew

ah 4 ah 10 ah 100

1

CLPT 05623 05623 05623FSDT 07291 05889 05625

Carrera et al [42] 07171 05875 05625Neves et al [43] 06997 05845 05624Zenkour [44] 07036 05848 05624

Present 07077 05860 05625

4

CLPT 08281 08281 08281FSDT 11125 08736 08281


Present 11166 08773 08285

10

CLPT 09354 09354 09354FSDT 13178 09966 09360


Present 13226 09964 09399


423 Validation of the Elastically Supported FG Plate inermal Environments e center deflection-load curvesfor SUS304Si3N4 FGM square plates on Win-klerndashPasternak elastic foundations in thermal environ-ments are calculated and compared with the results fromLal et al [20] e FG plates subjected to uniformlydistributed transverse loads q with a nonlinear tem-perature rise the nondimensional pressureqlowast (qb4Emh4) ΔT T(z) minus T0 and T0 300K evalues of Em with different temperature increments arelisted in Table 3 e nondimensional linear normal and

shear stiffness of the foundation are k1 (K1a4Emh3)

and k2 (K2a2Emh3) respectively e volume fraction

index k 2 (ba) 1 and (bh) 20 e model is shownin Figure 4 and the comparisons can be observed inFigure 5

As shown in Figure 5 the results of this paper are ingood agreement with the published results using differentapproaches e model for FG plates on Win-klerndashPasternak elastic foundations in the thermal envi-ronment with a nonlinear temperature rise is correctSince the uniform temperature rise solution can be

Table 5 Comparisons of the nondimensional center transverse displacement for AlZrO2 FGM square plates with different volume fractionindexes and discretization levels

Method Discretization levelVolume fraction index k

05 1 2

SFPM (Ntimes r)

4times 3 02369 02835 033006times 3 02352 02799 032378times 3 02345 02767 0319710times 3 02340 02748 0316412times 3 02333 02731 0313214times 3 02328 02723 0311916times 3 02327 02720 0311312times1 02373 02785 0321012times 5 02329 02726 0312312times 7 02328 02721 03114

NS-FEM [45] (mesh)

8times 8 02332 02723 0311612times12 02326 02716 0310716times16 02324 02713 0310420times 20 02322 02711 0310132times 32 02321 02709 03099

N stands for the number of spline subdivisions in x direction and r is the convergent series

Discretization of SFPM and NS-FEM8 times 38 times 8 12 times 312 times 12 16 times 316 times 16 20 times 320 times 20 32 times 332 times 32

282624222018161412100806040200

ndash02

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 05k = 1k = 2

(a)

Discretization of SFPM and NS-FEM12 times 112 times 12 12 times 312 times 12 12 times 512 times 12 12 times 712 times 12

8

6

4

2

0

ndash2

ndash4

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 2k = 1 Nguyen-Xuan et al [45]

k = 05

SFPM

NS-FEM

(b)

Figure 3 Convergence efficiency of the SFPM results compared with the NS-FEM results (a) under different spline subdivisions N(b) under different convergent series r


obtained by simplifying the nonlinear temperature risefunction the correctness of the model for FG plates with auniform temperature rise also can be verified As thenumber of the foundation parameters increases thenonlinear center deflection decreases is is because thefoundation parameters make the stiffness of the plateincrease Meanwhile the larger the temperature incre-ment ΔT is the larger the nonlinear center deflection isis is because the thermal load FR1113864 1113865 increases with theincrease of ΔT which causes the nonlinear bending de-formation to increase

43 Parameter Studies In this section the following di-mensionless parameters nondimensional uniformly dis-tributed transverse load (qlowast) and foundation stiffness (k1and k2) are defined as

qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)

ere are three heating-up conditions to be considered(unless otherwise stated)

C1 Tm 400K Tc 300K C2 Tm 300K Tc 400Kand C3 Tm 400K Tc 400K

And two kinds of situations for the in-plane temperatureforce

S1 the in-plane temperature force is considered and S2the in-plane temperature force is neglected S1 is the defaultsituation unless otherwise stated

431 Discussion of Temperature Distribution

(1) Effect of Volume Fraction Index and Length-icknessRatio In order to compare the nonlinear center deflection ofFG plates under uniform and nonlinear temperature riseconditions the uniformly distributed transverse load andelastic foundation are not considered temporarily Takingthe first kind of FGM as the research object the centerdeflection for AlZrO2 square plates (a b 1m) undertwo kinds of temperature rise conditions with differentlength-thickness ratios and volume fraction indexes areshown in Table 6

Table 6 presents that the center deflections with theuniform temperature rise are larger than that with thenonlinear temperature rise It illustrates that the effect of thein-plane temperature force is weaker under the nonlineartemperature rise condition with C2 is is because theoverall temperature of the plates is higher with the uniformtemperature rise so the influence of the thermal geometricstiffness Kθ is greater than the nonlinear bending stiffness Ks

in this caseWith the increase of volume fraction index k andthickness h of the plate the relative deviation between twokinds of temperature rise conditions becomes bigger andbigger until the FGM tends to homogeneous the materialagain (k⟶+infin) It shows that the type of temperature risecondition has a greater influence on FGM thick plates

(2) Effect of FGM Combination andWidth-Length Ratio enonlinear bending of two kinds of FG plates with differentwidth-length ratios is calculated and compared in Table 7 Inthis example the volume fraction index k 2 b 02mheating-up condition is C2 and two temperature riseconditions are considered

Table 7 shows that the relative deviation between twokinds of temperature rise conditions increases with width-length ratio ba of the plate It illustrates that the type oftemperature distribution function has a greater influence onFGM rectangle plates Besides the relative deviation ofSi3N4SUS304 is smaller which presents that the effect of

a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)

Figure 4 e model of FGM square plates on WinklerndashPasternakelastic foundations

20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350

Present Lal et al [20]I amp 1I amp 2I amp 3

II amp 1II amp 2II amp 3

I amp 1I amp 2I amp 3


I ΔT = 100KII ΔT = 300K1 (k1 k2) = (0 0)2 (k1 k2) = (100 0)3 (k1 k2) = (100 10)

T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20

Figure 5 e nondimensional transverse deflection-load curvesfor SUS304Si3N4 FG plates on WinklerndashPasternak elastic foun-dations subjected to uniform pressure and nonlinear temperaturerise


temperature rise condition is related to material combina-tion e temperature dependent FGM is less affected be-cause the thermal material properties decrease with theincrease of temperature and the variation of nonlinearbending stiffness Ks is different under two kinds of tem-perature rise conditions e following analyses only con-sider the nonlinear temperature rise condition

432 Discussion of In-Plane Temperature Force

(1) Effect of Volume Fraction Index and Heating-Up Con-dition To compare the nonlinear center deflection of FGplates under in-plane temperature forces the uniformlydistributed transverse load and elastic foundation are stillnot considered e AlZr2O3 square plates are under dif-ferent temperature increments a b 1m and(ha) (120) And the center deflection-volume fractionindex curves are shown in Figure 6

As shown in Figure 6(a) when the lower surface of theplate is heated separately the center deflection is negativeand changes more greatly with the volume fraction indexk 0sim2 and it is basically unchanged after kgt 2 is isbecause the FGM properties change greatly with k 0sim2after that the FGM tends to be homogeneous with k in-creasing In Figure 6(b) the upper surface of the plate isheated separately e bending deformation of the FG plateunder the in-plane temperature force is positive and similarto that of the plate subjected to transverse mechanical loads

in normal temperature e center deflection reaches thesmallest when k 0 and increases with the increase of k isis because the thermal expansion coefficient α and thermalconductivity κ of ceramic are smaller than those of metalerefore the temperature change of the upper surface haslittle influence on the deflection Figure 6(c) shows that thehomogeneous plate (k 0) does not deform and the centerdeflection is 0 when the upper and lower surfaces of the plateare heated simultaneously is is because the temperaturestays the same along the thickness and the thermal momentMθ 0 However for FG plates the temperature distri-bution function and thermal expansion coefficient are notonly the functions of thickness z but also related to k so thedeformation is similar to Figure 6(a)

It can be observed from Figure 6 that the center de-flection increases with the temperature is is because thein-plane temperature force is pressure when the plate isheated up which makes the geometric stiffness matrix inKlowastσ negative and the total stiffness decrease us the in-plane temperature force can soften the structure It has agreater effect on FG plates than on homogeneous plateswhen the bending deformation is small and the effect isrelated to the material properties and environmenttemperature

(2) Coupling Effect with Transverse Loads e uniformlydistributed transverse load is added into the considerationon the basis of (1) e nondimensional center deflection-load curves for FGM square plates with different volume

Table 6e center deflection for AlZrO2 FGM square plates under two kinds of temperature rise conditions and C2 with different volumefraction index curves and length-thickness ratios

Volume fraction index ah Uniform temperature rise w (10minus 3m) Nonlinear temperature rise w (10minus 3m) Relative deviation ()

Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132

Relative deviation is evaluated between the values in columns 3 and 4

Table 7 e center deflection for AlZr2O3 and Si3N4SUS304 FG plates under two kinds of temperature rise conditions and C2 withdifferent width-length ratios ((hb) (120))

ba Uniform temperature rise w (10minus 3m) Nonlinear temperature rise w (10minus 3m) Relative deviation ()

AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521



fraction indexes and heating-up conditions are shown inFigure 7

It can be observed that the nonlinear center deflectionincreases with the volume fraction index k in Figure 7is isbecause the larger k is the more metal and the less ceramicthe plate hase composition changes from pure ceramic toceramic-metal composite which makes the elastic modulusand stiffness of the FG plate decrease Besides under dif-ferent heating-up conditions the center deflection valueswith the same volume fraction index are much differentfrom one another when qlowast is small and they tend to be closewith the increase of qlowast For example for the FG plate withk 2 under three heating-up conditions the nondimen-sional center deflections are minus 0112 0002 and -0004

respectively and the maximum deviation is 0114 whileqlowast 0 However when qlowast 50 the center deflections are0948 1013 and 0980 respectively and the maximumdeviation becomes 0065 It illustrates that the influence ofthe environment temperature and in-plane temperatureforce decreases with the increase of nonlinear bending de-formation of the plate e large deflection is mainly con-trolled by the transverse distributed loads

(3) Effect of Length-Width Ratio andickness-Length Ratioe nondimensional center-load curves for AlZr2O3 FGMsquare plates with various length-width ratios are shown inFigure 8 e nonlinear bending of two kinds of FG plateswith different length-width ratios and thickness-length

0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K

qlowast = 0 k1 = 0 k2 = 0AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)

Figure 6 e center deflection-volume fraction index curves for AlZr2O3 FGM square plates under in-plane temperature force withdifferent temperature increments (a) heating the lower surface of the plate separately (b) heating the upper surface of the plate separately(c) heating the upper and lower surfaces of the plate simultaneously


ratios are calculated and compared in Table 8 In this ex-ample the volume fraction index k 2

As shown in Figure 8 the nonlinear center deflectionvalues increase with the increase of length-width ratio ab ofthe plate e larger the uniformly distributed transverseload qlowast is the more significant the influence of the length-width ratio is Table 8 shows that with the increase of length-width ratio ab and thickness-length ratio ha the relativedeviations between situation S1 and S2 become smaller Itillustrates that the thermal environment effect decreasesgradually and the in-plane temperature force has a greaterinfluence on the thin plate with the small length-width ratioMeanwhile the calculationmethod employed in this paper is

suitable for the analysis of both thick and thin plates and thelocking-free thinthick plate theory has a wide applicability

433 Discussion of Elastic Foundation (1) Effect of Trans-verse Distributed Load Length-Width Ratio and Heating-UpCondition e nondimensional center deflection-loadcurves for AlZr2O3 FG plates are shown in Figure 9 enonlinear bending of AlZrO2 plates under different length-width ratios and heating-up conditions are calculated andcompared in Table 9 In this example the plates are rested onWinklerndashPasternak elastic foundations with a b 1m andvolume fraction index k 2

0 10 20 30 40qlowast = qb4Emh4

50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)

Figure 7 Comparisons of the nondimensional center deflection-load curves for AlZr2O3 FGM square plates with different volume fractionindexes and heating-up conditions (a) C1 (b) C2 (c) C3


Figure 9 shows that the stiffness of the FG plates in-creases and the center deflection decreases obviously afterconsidering the effect of the elastic foundations whichindicates that the elastic foundations can stiffen thestructure and weaken the bending behavior effectively ecenter deflection decreases continuously with the increaseof foundation parameter values It can be observed that theshear stiffness K2 has a more obvious stiffening effect onthe plates than the linear normal stiffness K1 As theuniformly distributed transverse load qlowast increases and thenonlinear deformation becomes larger and larger thestiffening effect caused by K2 becomes more and moreobvious

As shown in Table 9 the relative deviations III are closeto one another under different length-width ratios whichindicates that the influence of K2 is unrelated to the shape

and size of the plate However the relative deviations I and IIdecrease with the increase of the length-width ratio It il-lustrates that the shape and size can affect the stiffening effectcaused byK1e smaller the length-width ratio is the largerthe stiffening effect is Besides the relative deviations I underthe three heating-up conditions become closer to one an-other as the length-width ratio increases It shows that theinfluence of thermal environment decreases gradually butthe stiffening effect of the elastic foundation is unrelated tothe heating-up condition

(2) Effect of Volume Fraction Index and ickness-LengthRatio e AlZr2O3 square plates (a b 1m) are elasti-cally supported e center deflection with different volumefraction indexes and thickness-length ratios are shown inTable 10

20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05

Figure 8 e nondimensional center deflection-load curves for AlZr2O3 FG plates with C2 and various length-width ratios

Table 8 e nondimensional center deflection for two kinds of FG plates under C1and qlowast 25 with different length-width ratios andthickness-length ratios (k1 k2 0)

ab haAlZr2O3 SUS304Si3N4 AlZrO2

wh (S1) wh (S2) Relative deviation () wh (S1) wh (S2) Relativedeviation () wh (S1) wh (S2) Relative deviation ()

05180 minus 06564 minus 02506 16193 minus 08646 minus 03371 15648 minus 06424 minus 02378 17014140 minus 01436 minus 00703 10427 minus 02491 minus 01506 6541 minus 01716 minus 00795 11585120 minus 00209 minus 00109 9174 00496 00419 1838 minus 00329 minus 00167 9701

1180 minus 05611 minus 02446 12939 minus 00281 01177 12387 minus 05760 minus 02291 15142140 05289 04074 2982 07349 06255 1749 04689 03490 3436120 06098 05013 2164 05721 05219 962 07138 05538 2889

15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802

Relative deviation is evaluated between the values in S1 and S2


AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)

Figure 9 e nondimensional center deflection-load curves for elastically supported AlZr2O3 FG plates with (a) C1 (b) C2 (c) C3

Table 9 e center deflection for elastically supported AlZrO2 FG plates with different length-width ratios and heating-up conditionsunder qlowast 25 and (ha) (140)

ab Heating-upconditions (TmTc)

k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)

Relativedeviation I ()

Relative deviationII ()

Relativedeviation III ()

05C1 10740 minus 1297 minus 0524 minus 0097 11208 5960 8149C2 8988 1054 0532 0096 8827 4953 8195C3 1096 minus 0674 minus 0220 minus 0041 16150 6736 8136

1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315

Relative deviation I is evaluated between the values in columns 3 and 4 relative deviation II is evaluated between the values in columns 4 and 5 Relativedeviation III is evaluated between the values in columns 5 and 6


Table 10 shows that for the thin plates (ha 180) therelative deviations I are smaller than that of the thick plates(hage 150)e relative deviations II are larger than and therelative deviations III are close to those of the thick plates Itillustrates that the linear normal stiffness K1 has a greaterinfluence on the thin plates while the influence of shearstiffness K2 and the stiffening effect of the elastic foundationare unrelated to the thickness-length ratio By comparing thethree relative deviations with the same thickness-length ratioand different volume fraction indexes the stiffening effect ofthe elastic foundation on FG plates is found to increase withk

5 Conclusions

is paper proposes a SFPM to study the thick and thin FGplates in thermal environments based on a locking-free thinthick plate theory e material parameters and temperatureare assumed to vary along the thickness direction of theplates By comparing with the published models the newmodel is validated to be correct with high precision and goodconvergence for rectangular plates e numerical resultsshow that the proposed method performs well in analyzingthe nonlinear bending of elastically supported FG plateswithout shear locking e effect of the temperature risecondition in-plane temperature force and elastic foundationon the nonlinear bending under different parameters in-cluding the volume fraction index thickness-length ratiolength-width ratio heating-up condition and foundationstiffness are discussed through several examples and theconclusions are summarized as follows

(1) For the nonlinear bending analysis of FG plates theeffect of thermal environment is related to materialcombination and the temperature independentFGM is more affected And thermal environment hasgreater influence on the plates with larger length-thickness ratios and with larger width-length ratiosTwo kinds of temperature rise conditions are dis-cussed as the length-thickness ratios decrease and

width-length ratios increase the stiffnesses of theplates increase so the relative deviations betweentwo kinds of temperature rise conditions increaseAnd the uniform temperature rise condition hasmore effect on bending behavior than the nonlinearone with C2

(2) e in-plane temperature force is generated whenthe cold shrinkage and thermal expansion defor-mation of the material is constrained by theboundary It is related to the environment temper-ature and the material properties and can soften thestructure and make the bending deformation in-crease with temperature rises because the in-planetemperature force produces bending moments thatdeform the plate e softening effect is obvious onthin plates with small length-width ratio and greateron FGMs than homogeneous materials e phe-nomenon conforms to the general law of mechanicsthe smaller the stiffness is the larger the deformationis

(3) As the nonlinear deformations of FG plates increasethe influence of the in-plane temperature andthermal environment decreases gradually e largedeflection is mainly controlled by external me-chanical loads and obviously effected by the length-width ratio Due to the small thickness of plate thebending moment generated by the in-plane tem-perature force is limited so the influence of the in-plane temperature and thermal environments islimited

(4) e proposed method is applicable not only totheoretical research but also to some engineeringapplication such as the elastic foundation analysise analysis found that the elastic foundations canstiffen the structure and weaken the bending be-havior effectively and the nonlinear bending de-creases with the increase of the foundationparameters e linear normal stiffness K1 has a

Table 10 e center deflection for elastically supported AlZr2O3 FG plates with different volume fraction indexes and thickness-lengthratios under qlowast 25 and C2

ha Volume fraction indexk1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)


Relativedeviation II ()


180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441

Relative deviation I is evaluated between the values in columns 3 and 4 Relative deviation II is evaluated between the values in columns 4 and 5 Relativedeviation III is evaluated between the values in columns 5 and 6


greater influence on thin plates e shear stiffnessK2 has a more obvious stiffening effect than K1 andthe larger themechanical load is themore significantthe effect is e stiffening effect is unrelated to thethickness-length ratio and the heating-up conditionbut it can be affected by the length-width ratio aband volume fraction index ke smaller the value ofab and the larger the k the stronger the effect

Appendix

A Calculation of the Nonlinear Spline Integral

e integral of product of three B-spline interpolationfunctions is needed when using SFPM for nonlinear cal-culation Taking the calculation of 1113938

a

0 [ϕ]T[ϕ] b k[ϕ]dx as anexample

[ϕ]T[ϕ] b k[ϕ] [ϕ]

Tbminus 1( 1113857kφ3

x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x

h+ 11113874 11138751113876 1113877 + middot middot middot + bN+1( 1113857kφ3

x

hminus N minus 11113874 11138751113883[ϕ]

(A1)

According to the matrix form of spline interpolationfunctions equations (32) and (33) equation (A1) can berewritten as

[ϕ]T

[ϕ] b k( 1113857[ϕ] φ3i1113858 1113859 Qs1113858 1113859 b k( 1113857 Qs1113858 1113859T φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)

Substituting equation (A3) into equation (A2) gives

[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859

blowastminus 1( 1113857kφ3

x

h+ 11113874 1113875 + b

lowast0( 1113857kφ3

x

h1113874 1113875 + b


x

hminus 11113874 1113875 + middot middot middot + b

lowastN+1( 1113857kφ3

x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

middot middot middot middot middot middot

φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b

lowastminus 1( 1113857kφ3

x

h+ 11113874 11138751113876

+ blowast0( 1113857kφ3

x

h1113874 1113875 + b


x

hminus 11113874 1113875 + middot middot middot

+ blowastN+1( 1113857kφ3

x

hminus N minus 11113874 11138751113877dx

(A6)

given t (xh) minus i and x 0 t minus i x a Nh t N minus iso equation (A6) can be rewritten as

h 1113946Nminus i

minus iφ3(t)φ3 t minus s1( 1113857 1113944

N+1

kminus 1blowastk( 11138571φ3 t minus s2( 1113857⎡⎣ ⎤⎦dt (A7)

where s1 j minus i and s2 k minus i i j k minus 1 0 1 N N + 1

B The Specific Formula Derivation andDescription of Locking-Free ThinThickPlate Theory

Based on the locking-free thinthick plate theory for thickand thin plates the displacement field can be expressed asfollows


w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)

e bending shear strain energy of thick plates is givenby

U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)

Substituting equation (B4) into equation (B2) gives

U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)

For the thin plates h⟶ 0 λ⟶ 0 ie Vc1113966 1113967 0 Soequation (B5) degenerate to the bending shear strain energyof thin plates ie

U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)

Equation (B2) degenerates to

U 12

1113946Sf

κ T[D] κ dS (B9)

Equations (B9) and (B2) show that there is no falseshear strain for thin plates and the shear deformation isconsidered for thick plates e locking-free thinthick platetheory is suitable for the analysis of different thickness plateswithout shear locking

Data Availability

e results in the paper are all implemented by MATLABand the data used to support the findings of this study arecited within the article at references and are available fromthe corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by the grants awarded by theNational Natural Science Foundation of China (no11562001) Science and Technology Major Project ofGuangxi Province (no AA18118029) and InnovationProject of Guangxi Graduate Education (noYCSW2018047)

References

[1] S Abdelbari A Fekrar H Heireche H Said A Tounsi andE A Adda Bedia ldquoAn efficient and simple shear deformationtheory for free vibration of functionally graded rectangularplates on Winkler-Pasternak elastic foundationsrdquo Wind andStructures vol 22 no 3 pp 329ndash348 2016

[2] H M Zhou X M Zhang and Z Y Wang ldquoermal analysisof 2D FGM beam subjected to thermal loading using meshlessweighted least-square methodrdquo Mathematical Problems inEngineering vol 2019 Article ID 2541707 10 pages 2019

[3] S-C Yi L-Q Yao and B-J Tang ldquoA novel higher-ordershear and normal deformable plate theory for the static freevibration and buckling analysis of functionally graded platesrdquoMathematical Problems in Engineering vol 2017 Article ID6879508 20 pages 2017


[4] H BeikMohammadiou and H EkhteraeiToussi ldquoParametricstudies on elastoplastic buckling of rectangular FGM thinplatesrdquo Aerospace Science and Technology vol 69 pp 513ndash525 2017

[5] H Bagheri Y Kiani and M R Eslami ldquoAsymmetric thermalbuckling of temperature dependent annular FGM plates on apartial elastic foundationrdquo Computers amp Mathematics withApplications vol 11 no 21 pp 1ndash16 2017

[6] A E Alshorbagy S S Alieldin M Shaat and F F MahmoudldquoFinite element analysis of the deformation of functionallygraded plates under thermomechanical loadsrdquo MathematicalProblems in Engineering vol 2013 Article ID 56978113 pages 2013

[7] A A Bousahla F Bourada S R Mahmoud et al ldquoBucklingand dynamic behavior of the simply supported CNT-RCbeams using an integral-first shear deformation theoryrdquoComputers and Concrete vol 25 no 2 pp 155ndash166 2020

[8] A Boussoula B Boucham M Bourada et al ldquoA simple nth-order shear deformation theory for thermomechanicalbending analysis of different configurations of FG sandwichplatesrdquo Smart Structures and Systems vol 25 no 2pp 197ndash218 2020

[9] M C Rahmani A Kaci A A Bousahla et al ldquoInfluence ofboundary conditions on the bending and free vibration be-havior of FGM sandwich plates using a four-unknown refinedintegral plate theoryrdquo Computers and Concrete vol 25 no 3pp 225ndash244 2020

[10] M Kaddari A Kaci A A Bousahla et al ldquoA study on thestructural behaviour of functionally graded porous plates onelastic foundation using a new quasi-3D model bending andfree vibration analysisrdquo Computers and Concrete vol 25no 1 pp 37ndash57 2020

[11] D Zarga A Tounsi A A Bousahla F Bourada andS R Mahmoud ldquoermomechanical bending study forfunctionally graded sandwich plates using a simple quasi-3Dshear deformation theoryrdquo Steel and Composite Structuresvol 32 no 3 pp 389ndash410 2019

[12] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[13] S Yin J S Hale T Yu T Q Bui and S P A BordasldquoIsogeometric locking-free plate element a simple first ordershear deformation theory for functionally graded platesrdquoComposite Structures vol 118 pp 121ndash138 2014

[14] L Ke ldquoKoiter-Newton analysis of thick and thin laminatedcomposite plates using a robust shell elementrdquo CompositeStructures vol 161 pp 530ndash539 2017

[15] S B Li H Wu and C M Mo ldquoDynamic analysis of thickthin piezoelectric FGM plate by spline meshless methodrdquoJournal of Vibration and Shock vol 36 no 7 pp 116ndash1222017 in Chinese

[16] I Katili J-L Batoz I J Maknun A Hamdouni and OMilletldquoe development of DKMQ plate bending element for thickto thin shell analysis based on the NaghdiReissnerMindlinshell theoryrdquo Finite Elements in Analysis and Design vol 100pp 12ndash27 2015

[17] W-Y Jung W-T Park and S-C Han ldquoBending and vi-bration analysis of S-FGM microplates embedded in Pas-ternak elastic medium using the modified couple stresstheoryrdquo International Journal of Mechanical Sciences vol 87pp 150ndash162 2014

[18] S Alexandrov S Usov and T Nguyen-oi ldquoA semi-analyticstress solution for elasticplastic FGM discs subject to externalpressurerdquo Materials Science Engineering vol 371 Article ID012060 2018

[19] H Huang and Q Han ldquoResearch on nonlinear postbucklingof functionally graded cylindrical shells under radial loadsrdquoComposite Structures vol 92 no 6 pp 1352ndash1357 2010

[20] A Lal N L Shegokar and B N Singh ldquoFinite element basednonlinear dynamic response of elastically supported piezo-electric functionally graded beam subjected to moving load inthermal environment with random system propertiesrdquo Ap-plied Mathematical Modelling vol 44 pp 274ndash295 2017

[21] M H Hajmohammad M S Zarei M Sepehr and N AbtahildquoBending and buckling analysis of functionally graded an-nular microplate integrated with piezoelectric layers based onlayerwise theory using DQMrdquo Aerospace Science and Tech-nology vol 79 pp 679ndash688 2018

[22] K M Liew L X Peng and S Kitipornchai ldquoVibrationanalysis of corrugated Reissner-Mindlin plates using a mesh-free Galerkin methodrdquo International Journal of MechanicalSciences vol 51 no 9-10 pp 642ndash652 2009

[23] L X Peng Y Tao N Liang et al ldquoSimulation of a crack instiffened plates via a meshless formulation and FSDTrdquo In-ternational Journal of Mechanical Sciences vol 131-132pp 880ndash893 2017

[24] R Qin Spline Function Method in Structural MechanicsGuangxi Peoplersquos Press Nanning China 1st edition 1985

[25] Y Li T-j Mo S-b Li Q-g Liang J-p Li and R QinldquoNonlinear analysis for the crack control of SMA smartconcrete beam based on a bidirectional B-spline QR methodrdquoMathematical Problems in Engineering vol 2018 Article ID6092479 16 pages 2018

[26] P Liu K Lin H Liu and R Qin ldquoFree transverse vibrationanalysis of axially functionally graded tapered euler-Bernoullibeams through spline finite point methodrdquo Shock and Vi-bration vol 2016 Article ID 5891030 23 pages 2016

[27] S Li L Huang L Jiang and R Qin ldquoA bidirectional B-splinefinite point method for the analysis of piezoelectric laminatedcomposite plates and its application in material parameteridentificationrdquo Composite Structures vol 107 pp 346ndash3622014

[28] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metalplatesrdquo International Journal of Solids and Structures vol 35no 33 pp 4457ndash4476 1998

[29] H-S Shen and H Wang ldquoNonlinear bending and post-buckling of FGM cylindrical panels subjected to combinedloadings and resting on elastic foundations in thermal en-vironmentsrdquo Composites Part B Engineering vol 78pp 202ndash213 2015

[30] H-S Shen Y Xiang and F Lin ldquoNonlinear bending offunctionally graded graphene-reinforced composite lami-nated plates resting on elastic foundations in thermal envi-ronmentsrdquo Composite Structures vol 170 pp 80ndash90 2017

[31] J Zhong Y Fu D Wan and Y Li ldquoNonlinear bending andvibration of functionally graded tubes resting on elasticfoundations in thermal environment based on a refined beammodelrdquo Applied Mathematical Modelling vol 40 no 17-18pp 7601ndash7614 2016

[32] D-G Zhang and H-M Zhou ldquoNonlinear bending analysis ofFGM circular plates based on physical neutral surface andhigher-order shear deformation theoryrdquo Aerospace Scienceand Technology vol 41 pp 90ndash98 2015


[33] N Belbachir K Draich A A Bousahla et al ldquoBendinganalysis of anti-symmetric cross-ply laminated plates undernonlinear thermal and mechanical loadingsrdquo Steel andComposite Structures vol 33 no 1 pp 81ndash92 2019

[34] A Tounsi S U Al-Dulaijan M A Al-Osta et al ldquoA fourvariable trigonometric integral plate theory for hygro-thermo-mechanical bending analysis of AFG ceramic-metalplates resting on a two-parameter elastic foundationrdquo Steeland Composite Structures vol 34 no 4 pp 511ndash524 2020

[35] Y H Dong and Y H Li ldquoA unified nonlinear analyticalsolution of bending buckling and vibration for the temper-ature-dependent FG rectangular plates subjected to thermalloadrdquo Composite Structures vol 159 pp 689ndash701 2017

[36] E Farzad and S Erfan ldquoNonlocal thermo-mechanical vi-bration analysis of functionally graded nanobeams in thermalenvironmentrdquo Acta Astronautica vol 113 pp 29ndash50 2015

[37] W Tebboune M Merdjah K H Benrahou and A Tounsildquoermoelastic buckling response of thick functionally gradedplatesrdquo Journal of Applied Mechanics and Technical Physicsvol 55 no 5 pp 857ndash869 2014

[38] A Gupta and M Talha ldquoLarge amplitude free flexural vi-bration analysis of finite element modeled FGM plates usingnew hyperbolic shear and normal deformation theoryrdquoAerospace Science and Technology vol 67 pp 287ndash308 2017

[39] R Javaheri and M R Eslami ldquoermal buckling of func-tionally graded platesrdquo AIAA Journal vol 40 no 1pp 162ndash169 2002

[40] N Valizadeh S Natarajan O A Gonzalez-EstradaT Rabczuk T Q Bui and S P A Bordas ldquoNURBS-basedfinite element analysis of functionally graded plates staticbending vibration buckling and flutterrdquo Composite Struc-tures vol 99 pp 309ndash326 2013

[41] Y Fu J Wang and Y Mao ldquoNonlinear analysis of bucklingfree vibration and dynamic stability for the piezoelectricfunctionally graded beams in thermal environmentrdquo AppliedMathematical Modelling vol 36 no 9 pp 4324ndash4340 2012

[42] E Carrera S Brischetto M Cinefra and M Soave ldquoEffects ofthickness stretching in functionally graded plates and shellsrdquoComposites Part B Engineering vol 42 no 2 pp 123ndash1332011

[43] A M A Neves A J M Ferreira E Carrera et al ldquoBending ofFGM plates by a sinusoidal plate formulation and collocationwith radial basis functionsrdquo Mechanics Research Communi-cations vol 38 no 5 pp 368ndash371 2011

[44] A M Zenkour ldquoBending of FGM plates by a simplified four-unknown shear and normal dfformations theoryrdquo Interna-tional Journal of Applied Mechanics vol 5 no 2 Article ID1350020 2013

[45] H Nguyen-Xuan L V Tran C Hai and T Nguyen-oildquoAnalysis of functionally graded plates by an efficient finiteelement method with node-based strain smoothingrdquo in-Walled Structures vol 54 pp 1ndash18 2012


investigate uncoupled thermomechanical behavior of FGplates Based on a novel integral first order shear defor-mation theory Bousahla et al [7] studied buckling andvibrational behavior of the composite beam armed withsingle-walled carbon nanotubes resting on Winkler-Pas-ternak elastic foundation Using an nth-order shear defor-mation theory thermomechanical flexural analysis offunctionally graded material sandwich plates with P-FGMface sheets and E-FGM and symmetric S-FGM core isperformed by Boussoula et al [8] Rahmani et al [9] in-vestigated bending and free vibration behavior of elasticallysupported functionally graded sandwich plates with differ-ent boundary conditions by an original novel high ordershear theory By employing a new type of quasi-3D hy-perbolic shear deformation theory Kaddari et al [10] dis-cussed statics and free vibration of functionally gradedporous plates resting on elastic foundations Zarga et al [11]analyzed thermomechanical bending of FGM sandwichplates and Mahmoudi et al [12] studied the effect of thematerial gradation and elastic foundation on the deflectionsand stresses of functionally graded sandwich plate subjectedto thermomechanical loading

It is worth noting that the above theories have their ownadvantages and limitations e CLPT is simple and easy inmechanical analyzing but will induce great errors wheninvestigating thick plates because the transverse shear de-formation of the plates is ignored which is equivalent of overstiffness Although the transverse shear deformation alongthe thickness direction is considered and the calculationaccuracy is improved in the FSDT it is necessary to use theshear correction factor in the calculation e HSDT andthree-dimensional elasticity theory have sufficient accuracyin the deformation analysis of plates and shells but toomanyindependent unknowns make the solution complicated andincrease the computational cost Moreover the analysis ofthin plate based on the first or higher-order plate theory hasconsidered shear deformation may induce false shear strainnamely shear-locking [13] erefore some researchersdevoted themselves to establish a general element andmethod to calculate both thick and thin plates [14ndash16] and alocking-free thinthick plate theory was proposed In thispaper this theory is based on the FSDT but takes the in-plane displacements deflection and shear strains as basicunknowns which can effectively avoid shear locking phe-nomenon with simplicity and high generality

For the past decades analytical semianalytic experi-mental and numerical methods for FGM structures havebeen proposed [17ndash19]e common numerical approachesincluding the finite element methods (FEM) [20] differentialquadrature methods (DQM) [21] and meshless methodswere quite popular in engineering because of their easyimplementation with computers [22] When using thetraditional FEM to analyze nonlinear behaviors and largedeformation problem not only number of unknowns is largebut also much effort needs to be expended in remeshing ateach step of the problem solving [22] On the contrary whenusing the meshless method to analyze flat plate problem theremeshing can be avoided naturally [23] and the boundaryconditions processing is simple with less unknowns and high

precision Among those the spline finite point method(SFPM) presented by Qin [24] is one of the spline meshlessmethods As an active branch of modern function ap-proximation the B-spline function is widely used Based onthe B-spline function orthogonal function and minimumpotential energy principle the displacement modes of SFPMare constructed by the linear combination for the product oforthogonal functions and cubic B-spline interpolationfunctions In this method one direction of the flat structureis discretized by a set of uniformly distributed spline nodesand the other one is expressed by orthogonal functionsrelated to the corresponding boundary conditions eequilibrium equation is established with the minimumpotential energy principle e SFPM has been proved moreadvantageous than the FEMs in analyzing the nonlinearbehaviors of plates and beams Li et al [25] used a bidi-rectional B-spline QRmethod (BB-sQRM) to study the crackcontrol of reinforced concrete beam in which the planemodel was discretized with a set of spline nodes along twodirections e results proved the high efficiency and lowcomputational cost of this method Liu et al [26] investi-gated the free transverse vibration of FG EulerndashBernoullibeams through the SFPM and found that it is simple formodeling and boundary processing Li et al [27] derived thedisplacement and stress solutions of a piezoelectric lami-nated composite plate based on a bidirectional B-spline finitepoint method (BB-sFPM) e numerical results showedthat the method was applied to the material parameteridentification with a small loss of precision

Existing studies showed that in complex temperaturefield the heat conduction phenomenon and temperaturedependence of FGMs can significantly affect the bendingbuckling and vibration of the plate [28] and the thermalload plays an important role in determining the mechanicalbehaviors of FGM structures Besides the FG plate generallydeforms largely when subjected to the combined action ofthermomechanical loads and its mechanical behavior isusually nonlinear in this situation For the nonlinear me-chanical problems of FG plates in thermal environmentsome relevant works were summarized in [29ndash32] All theseworks proved that solving mechanical problems by simplelinearization cannot meet the precision requirement inimportant structural calculations e nonlinear analysis isvery significant to the practical engineering applicationMeanwhile the temperature rising and lowing situations arevaried some papers have been published about differenttemperature distributions of FGM structures Belbachir et al[33] analyzed the response of antisymmetric cross-plylaminated plates subjected to a uniformly distributednonlinear thermomechanical loading Tounsi et al [34]studied the static behavior of elastically supported advancedfunctionally graded ceramic-metal plates subjected to anonlinear hygro-thermomechanical load Dong and Li [35]proposed a linear temperature rise solution to analyze thenonlinear bending of FG rectangular plates Farzad andErfan [36] investigated the thermal effect on FG nanobeamssubjected to two kinds of thermal loading namely linearand nonlinear temperature rise Considering the uniformlinear and nonlinear temperature distributions across the






φn(x) 1113944

n+1

k0(minus 1)

kn + 1

k

⎛⎝ ⎞⎠x minus xk( 1113857

n

+

n (1)


xk k minusn + 12

n + 1

k

⎛⎜⎜⎝ ⎞⎟⎟⎠ (n + 1)

k(n + 1 minus k)



0 x minus xk lt 0

⎧⎪⎨

⎪⎩

(2)


φ3(x) 1113944

4

k0(minus 1)

k

4

k

⎛⎜⎜⎝ ⎞⎟⎟⎠x minus xk( 1113857

3+

3

16

(x + 2)3+ minus 4(x + 1)

3+ + 6x


3+ +(x minus 2)

3+1113960 1113961

(3)



xi x0 + iax i 1 N (4)


ax a

N


(5)


φ3(x) 16





0 |x|ge 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)




w(x) 1113944

N+1


z

O

a

b

x

y

h

0 1 2 N ndash 1 N



in which




[ϕ] φ3i1113858 1113859 Qs1113858 1113859 (9)

Qs1113858 1113859 diag([g] [I] [h]) (10)


[g]

1 minus 4 1

0 1 minus12

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[h]

1 0 0

minus12

1 0

1 minus 4 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

φ3i1113858 1113859 φ3x

h+ 11113874 1113875 φ3

x

h1113874 1113875 φ3

x


x

hminus N minus 11113874 11138751113876 1113877

(11)


07

06

05

04

03

02

01


16 16

(a)

10

05

ndash05

ndash10


12

ndash12

(b)


1

ndash1

ndash2

(c)


ndash2

ndash4

2

4

(d)




ϕi(a) 0(ineN + 1)



(12)






2h1113890 1113891

k

+ Pb (13)




2h1113890 1113891

k

+ Pb(T) (14)


+ 1 + P1T1

+ P2T2

+ P3T3

1113872 1113873(j b t)

(15)






minusddz

κ(z)dT(z)

dz1113890 1113891 0 (16)




where

η(z) 1C

2z + h

2h1113888 1113889 minus

κcm

(n + 1)κm

2z + h

2h1113888 1113889

k+1

+κ2cm

(2n + 1)κ2m

2z + h

2h1113888 1113889

k+1

minusκ3cm

(3n + 1)κ3m

2z + h

2h1113888 1113889

3k+1⎡⎣

+κ4cm

(4n + 1)κ4m

2z + h

2h1113888 1113889

4k+1

minusκ5cm

(5n + 1)κ5m

2z + h

2h1113888 1113889

5k+1⎤⎦

C 1 minusκcm

(n + 1)κm

+κ2cm

(2n + 1)κ2mminus

κ3cm

(3n + 1)κ3m+

κ4cm

(4n + 1)κ4mminus

κ5cm

(5n + 1)κ5m

(18)




u u0 minus zφx

v v0 minus zφy

w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(19)


εlowast1113864 1113865

εpx

εpy

cpxy

εbx

εby

cbxy

cxz

cyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε0xε0y

c0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

zε1xzε1y

zc1xy

c1xz

c1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

εNLx

εNLy

cNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(20)


ε01113966 1113967

ε0xε0y

c0xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

u0x

v0y

u0y + v0x

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

ε11113966 1113967

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(21)


εNL1113966 1113967

εNLx

εNLy

cNLxy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

12

w0x1113872 11138732

12

w0y1113872 11138732

w0xw0y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(22)



Pf K1w + K2nabla2w

ST α T

Qf1113960 1113961Tε + pΔT

(23)


Qf1113960 1113961

Qεf1113876 1113877 [0]

[0] Qc

f1113876 1113877

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qεf1113960 1113961

E(z T)

1 minus μ2

1 μ 0

μ 1 0

0 01 minus μ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qc

f1113876 1113877

5E(z T)

12(1 + μ)0

05E(z T)

12(1 + μ)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

α α(z T) α(z T) 0 0 01113864 1113865T

(24)



σlowast1113864 1113865

σ0x

σ0y

τ0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

σ1x

σ1y

τ1xy

τ1xz

τ1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

σNLx

σNLy

τNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

minus

Nθx

Nθy

Nθxy

Mθx

Mθy

Mθxy

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(25)


σlowast1113864 1113865

N

M

Fs1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ε01113864 1113865 + εNL1113864 1113865

ε11113864 1113865

c1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦minus

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)

in which

N Nx Ny Nxy1113960 1113961T

M Mx My Mxy1113960 1113961T

Fs1113864 1113865 Fsx Fsy1113960 1113961T

(27)

Nθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α ΔT dz

Mθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α zΔT dz

(28)


([A] [B] [D]) 1113946h2

minus (h2)Q

ef1113960 1113961 1 z z

21113872 1113873dz

[C] 1113946h2

minus (h2)Q

c

f1113876 1113877dz

(29)




1113960 1113961 (30)

in which

Dlowast

1113960 1113961

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Flowast

1113960 1113961

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ε01113864 1113865

ε11113864 1113865

c1113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+

εNL1113864 1113865

0

0

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

(31)


u0 1113944r

m1[ϕ] a mXm(y)

v0 1113944r

m1[ϕ] b mYm(y)

w0 1113944

r

m1[ϕ] c mZm(y)

cxz 1113944r

m1[ϕprime] f1113864 1113865mFm(y)

cyz 1113944r

m1[ϕ] g1113864 1113865mGm(y)

(32)


w0 u0 φy 0 atx 0 a

w0 v0 φx 0 aty 0 b(33)


Xm(y) cosmπb

y1113874 1113875

Ym(y) sinmπb

y1113874 1113875

Zm(y) sinmπb

y1113874 1113875

Fm(y) sinmπb

y1113874 1113875

Gm(y) cosmπb

y1113874 1113875

(34)



V Nc

1113858 1113859 δ (35)



V01113864 1113865 [N] δ (36)



Nc



Nc

1113858 1113859m


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[N]m



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(38)

δ δ T1 δ

T2 δ


Tr

1113960 1113961 (39)

δ m a Tm b

Tm c T

m f1113864 1113865Tm g1113864 1113865

Tm

1113960 1113961 m 1 2 3 r

(40)


εlowast1113864 1113865L Bp

L1113960 1113961 BbL1113960 1113961 B

c

L1113858 11138591113960 1113961Tδ (41)

where [BpL] [B

pL]1 [B



[BbL] and [B

c

L] and

BpL1113960 1113961

m

ϕprime1113858 1113859Xm [0] [0] [0] [0]

[0] [ϕ]Ymprime [0] [0] [0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BbL1113960 1113961

m



prime



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BcL1113858 1113859m

[0] [0] [0] ϕprime1113858 1113859Fm [0]

[0] [0] [0] [0] [ϕ]Gm

⎡⎣ ⎤⎦

(42)


εlowast1113864 1113865NL 12

[Q] Q (43)

in which

[Q]

zw

zx0

0zw

zy

zw

zy

zw

zx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q

zw

zx

zw

zy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(44)


εlowast1113864 1113865NL BNL1113858 1113859 δ (45)

in which

BNL1113858 1113859 BPNL1113858 1113859 [0] [0]1113858 1113859

T12 [Q] [0] [0]1113960 1113961

T[J]

[J]m 0 0 ϕprime1113858 1113859Zm 0 0

0 0 [ϕ]Zmprime 0 0

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Q 1113944r

m1[J]m c m [J] δ

(46)


εlowast1113864 1113865 BL1113858 1113859 + BNL1113858 1113859( 1113857 δ (47)


d δ TΨ B

Ad εlowast1113864 1113865


F 0 (48)


Tare the







KS BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA

+ BA

BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA

(50)

KF BA

[R]T

Kl1113858 1113859[R]dA (51)

Kl1113858 1113859

K1 0 00 K2 00 0 K2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[R]

[0] [0] [ϕ]Zm [0] [0]

[0] [0] [ϕprime]Zm [0] [0]

[0] [0] [ϕ]Zmprime [0] [0]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(52)

Kθ 12B

A[J]

TN

θ1113876 1113877[J]dA N

θ1113876 1113877

Nθx Nθ

xy

Nθxy Nθ

y

⎡⎢⎢⎣ ⎤⎥⎥⎦ (53)

FR1113864 1113865 1113946V

Bp

1113858 1113859T

Qf1113960 1113961 α ΔT + Bb

1113960 1113961T

Qf1113960 1113961 α zΔT1113874 1113875dV (54)



lowastT dδ (55)

in which

K0 BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA (56)

Klowastσ B[J]

TNf1113960 1113961[J]dx dy + B[R]

TKl1113858 1113859[R]dx dy

+12B[J]

TN

θ1113876 1113877[J]dx dy

(57)

KNL BA2 BNL1113858 1113859

TDlowast

1113960 1113961 BL1113858 1113859 + 2 BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113872

+ 4 BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113873dA

(58)

where [Nf] Nx Nxy
















Al 70 03 204 230ZrO2 200 03 209 10Zr2O3 151 03 209 10Al2O3 380 03 104 72



Si3N4


α(1K) 58723e minus 6 0 9095e-4 0 0κ(WmK) 1204 0 0 0 0

SUS304





Si3N4





ah 4 ah 10 ah 100

1

CLPT 05623 05623 05623FSDT 07291 05889 05625


Present 07077 05860 05625

4

CLPT 08281 08281 08281FSDT 11125 08736 08281


Present 11166 08773 08285

10

CLPT 09354 09354 09354FSDT 13178 09966 09360


Present 13226 09964 09399









05 1 2

SFPM (Ntimes r)


NS-FEM [45] (mesh)




282624222018161412100806040200

ndash02

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 05k = 1k = 2

(a)


8

6

4

2

0

ndash2

ndash4

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2


k = 05

SFPM

NS-FEM

(b)





qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)











a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)


20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350






T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20












Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References





















































φn(x) 1113944

n+1

k0(minus 1)

kn + 1

k

⎛⎝ ⎞⎠x minus xk( 1113857

n

+

n (1)


xk k minusn + 12

n + 1

k

⎛⎜⎜⎝ ⎞⎟⎟⎠ (n + 1)

k(n + 1 minus k)



0 x minus xk lt 0

⎧⎪⎨

⎪⎩

(2)


φ3(x) 1113944

4

k0(minus 1)

k

4

k

⎛⎜⎜⎝ ⎞⎟⎟⎠x minus xk( 1113857

3+

3

16

(x + 2)3+ minus 4(x + 1)

3+ + 6x


3+ +(x minus 2)

3+1113960 1113961

(3)



xi x0 + iax i 1 N (4)


ax a

N


(5)


φ3(x) 16





0 |x|ge 2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)




w(x) 1113944

N+1


z

O

a

b

x

y

h

0 1 2 N ndash 1 N



in which




[ϕ] φ3i1113858 1113859 Qs1113858 1113859 (9)

Qs1113858 1113859 diag([g] [I] [h]) (10)


[g]

1 minus 4 1

0 1 minus12

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[h]

1 0 0

minus12

1 0

1 minus 4 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

φ3i1113858 1113859 φ3x

h+ 11113874 1113875 φ3

x

h1113874 1113875 φ3

x


x

hminus N minus 11113874 11138751113876 1113877

(11)


07

06

05

04

03

02

01


16 16

(a)

10

05

ndash05

ndash10


12

ndash12

(b)


1

ndash1

ndash2

(c)


ndash2

ndash4

2

4

(d)




ϕi(a) 0(ineN + 1)



(12)






2h1113890 1113891

k

+ Pb (13)




2h1113890 1113891

k

+ Pb(T) (14)


+ 1 + P1T1

+ P2T2

+ P3T3

1113872 1113873(j b t)

(15)






minusddz

κ(z)dT(z)

dz1113890 1113891 0 (16)




where

η(z) 1C

2z + h

2h1113888 1113889 minus

κcm

(n + 1)κm

2z + h

2h1113888 1113889

k+1

+κ2cm

(2n + 1)κ2m

2z + h

2h1113888 1113889

k+1

minusκ3cm

(3n + 1)κ3m

2z + h

2h1113888 1113889

3k+1⎡⎣

+κ4cm

(4n + 1)κ4m

2z + h

2h1113888 1113889

4k+1

minusκ5cm

(5n + 1)κ5m

2z + h

2h1113888 1113889

5k+1⎤⎦

C 1 minusκcm

(n + 1)κm

+κ2cm

(2n + 1)κ2mminus

κ3cm

(3n + 1)κ3m+

κ4cm

(4n + 1)κ4mminus

κ5cm

(5n + 1)κ5m

(18)




u u0 minus zφx

v v0 minus zφy

w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(19)


εlowast1113864 1113865

εpx

εpy

cpxy

εbx

εby

cbxy

cxz

cyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε0xε0y

c0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

zε1xzε1y

zc1xy

c1xz

c1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

εNLx

εNLy

cNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(20)


ε01113966 1113967

ε0xε0y

c0xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

u0x

v0y

u0y + v0x

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

ε11113966 1113967

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(21)


εNL1113966 1113967

εNLx

εNLy

cNLxy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

12

w0x1113872 11138732

12

w0y1113872 11138732

w0xw0y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(22)



Pf K1w + K2nabla2w

ST α T

Qf1113960 1113961Tε + pΔT

(23)


Qf1113960 1113961

Qεf1113876 1113877 [0]

[0] Qc

f1113876 1113877

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qεf1113960 1113961

E(z T)

1 minus μ2

1 μ 0

μ 1 0

0 01 minus μ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qc

f1113876 1113877

5E(z T)

12(1 + μ)0

05E(z T)

12(1 + μ)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

α α(z T) α(z T) 0 0 01113864 1113865T

(24)



σlowast1113864 1113865

σ0x

σ0y

τ0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

σ1x

σ1y

τ1xy

τ1xz

τ1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

σNLx

σNLy

τNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

minus

Nθx

Nθy

Nθxy

Mθx

Mθy

Mθxy

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(25)


σlowast1113864 1113865

N

M

Fs1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ε01113864 1113865 + εNL1113864 1113865

ε11113864 1113865

c1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦minus

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)

in which

N Nx Ny Nxy1113960 1113961T

M Mx My Mxy1113960 1113961T

Fs1113864 1113865 Fsx Fsy1113960 1113961T

(27)

Nθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α ΔT dz

Mθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α zΔT dz

(28)


([A] [B] [D]) 1113946h2

minus (h2)Q

ef1113960 1113961 1 z z

21113872 1113873dz

[C] 1113946h2

minus (h2)Q

c

f1113876 1113877dz

(29)




1113960 1113961 (30)

in which

Dlowast

1113960 1113961

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Flowast

1113960 1113961

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ε01113864 1113865

ε11113864 1113865

c1113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+

εNL1113864 1113865

0

0

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

(31)


u0 1113944r

m1[ϕ] a mXm(y)

v0 1113944r

m1[ϕ] b mYm(y)

w0 1113944

r

m1[ϕ] c mZm(y)

cxz 1113944r

m1[ϕprime] f1113864 1113865mFm(y)

cyz 1113944r

m1[ϕ] g1113864 1113865mGm(y)

(32)


w0 u0 φy 0 atx 0 a

w0 v0 φx 0 aty 0 b(33)


Xm(y) cosmπb

y1113874 1113875

Ym(y) sinmπb

y1113874 1113875

Zm(y) sinmπb

y1113874 1113875

Fm(y) sinmπb

y1113874 1113875

Gm(y) cosmπb

y1113874 1113875

(34)



V Nc

1113858 1113859 δ (35)



V01113864 1113865 [N] δ (36)



Nc



Nc

1113858 1113859m


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[N]m



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(38)

δ δ T1 δ

T2 δ


Tr

1113960 1113961 (39)

δ m a Tm b

Tm c T

m f1113864 1113865Tm g1113864 1113865

Tm

1113960 1113961 m 1 2 3 r

(40)


εlowast1113864 1113865L Bp

L1113960 1113961 BbL1113960 1113961 B

c

L1113858 11138591113960 1113961Tδ (41)

where [BpL] [B

pL]1 [B



[BbL] and [B

c

L] and

BpL1113960 1113961

m

ϕprime1113858 1113859Xm [0] [0] [0] [0]

[0] [ϕ]Ymprime [0] [0] [0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BbL1113960 1113961

m



prime



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BcL1113858 1113859m

[0] [0] [0] ϕprime1113858 1113859Fm [0]

[0] [0] [0] [0] [ϕ]Gm

⎡⎣ ⎤⎦

(42)


εlowast1113864 1113865NL 12

[Q] Q (43)

in which

[Q]

zw

zx0

0zw

zy

zw

zy

zw

zx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q

zw

zx

zw

zy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(44)


εlowast1113864 1113865NL BNL1113858 1113859 δ (45)

in which

BNL1113858 1113859 BPNL1113858 1113859 [0] [0]1113858 1113859

T12 [Q] [0] [0]1113960 1113961

T[J]

[J]m 0 0 ϕprime1113858 1113859Zm 0 0

0 0 [ϕ]Zmprime 0 0

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Q 1113944r

m1[J]m c m [J] δ

(46)


εlowast1113864 1113865 BL1113858 1113859 + BNL1113858 1113859( 1113857 δ (47)


d δ TΨ B

Ad εlowast1113864 1113865


F 0 (48)


Tare the







KS BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA

+ BA

BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA

(50)

KF BA

[R]T

Kl1113858 1113859[R]dA (51)

Kl1113858 1113859

K1 0 00 K2 00 0 K2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[R]

[0] [0] [ϕ]Zm [0] [0]

[0] [0] [ϕprime]Zm [0] [0]

[0] [0] [ϕ]Zmprime [0] [0]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(52)

Kθ 12B

A[J]

TN

θ1113876 1113877[J]dA N

θ1113876 1113877

Nθx Nθ

xy

Nθxy Nθ

y

⎡⎢⎢⎣ ⎤⎥⎥⎦ (53)

FR1113864 1113865 1113946V

Bp

1113858 1113859T

Qf1113960 1113961 α ΔT + Bb

1113960 1113961T

Qf1113960 1113961 α zΔT1113874 1113875dV (54)



lowastT dδ (55)

in which

K0 BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA (56)

Klowastσ B[J]

TNf1113960 1113961[J]dx dy + B[R]

TKl1113858 1113859[R]dx dy

+12B[J]

TN

θ1113876 1113877[J]dx dy

(57)

KNL BA2 BNL1113858 1113859

TDlowast

1113960 1113961 BL1113858 1113859 + 2 BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113872

+ 4 BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113873dA

(58)

where [Nf] Nx Nxy
















Al 70 03 204 230ZrO2 200 03 209 10Zr2O3 151 03 209 10Al2O3 380 03 104 72



Si3N4


α(1K) 58723e minus 6 0 9095e-4 0 0κ(WmK) 1204 0 0 0 0

SUS304





Si3N4





ah 4 ah 10 ah 100

1

CLPT 05623 05623 05623FSDT 07291 05889 05625


Present 07077 05860 05625

4

CLPT 08281 08281 08281FSDT 11125 08736 08281


Present 11166 08773 08285

10

CLPT 09354 09354 09354FSDT 13178 09966 09360


Present 13226 09964 09399









05 1 2

SFPM (Ntimes r)


NS-FEM [45] (mesh)




282624222018161412100806040200

ndash02

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 05k = 1k = 2

(a)


8

6

4

2

0

ndash2

ndash4

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2


k = 05

SFPM

NS-FEM

(b)





qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)











a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)


20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350






T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20












Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References

















































in which




[ϕ] φ3i1113858 1113859 Qs1113858 1113859 (9)

Qs1113858 1113859 diag([g] [I] [h]) (10)


[g]

1 minus 4 1

0 1 minus12

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[h]

1 0 0

minus12

1 0

1 minus 4 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

φ3i1113858 1113859 φ3x

h+ 11113874 1113875 φ3

x

h1113874 1113875 φ3

x


x

hminus N minus 11113874 11138751113876 1113877

(11)


07

06

05

04

03

02

01


16 16

(a)

10

05

ndash05

ndash10


12

ndash12

(b)


1

ndash1

ndash2

(c)


ndash2

ndash4

2

4

(d)




ϕi(a) 0(ineN + 1)



(12)






2h1113890 1113891

k

+ Pb (13)




2h1113890 1113891

k

+ Pb(T) (14)


+ 1 + P1T1

+ P2T2

+ P3T3

1113872 1113873(j b t)

(15)






minusddz

κ(z)dT(z)

dz1113890 1113891 0 (16)




where

η(z) 1C

2z + h

2h1113888 1113889 minus

κcm

(n + 1)κm

2z + h

2h1113888 1113889

k+1

+κ2cm

(2n + 1)κ2m

2z + h

2h1113888 1113889

k+1

minusκ3cm

(3n + 1)κ3m

2z + h

2h1113888 1113889

3k+1⎡⎣

+κ4cm

(4n + 1)κ4m

2z + h

2h1113888 1113889

4k+1

minusκ5cm

(5n + 1)κ5m

2z + h

2h1113888 1113889

5k+1⎤⎦

C 1 minusκcm

(n + 1)κm

+κ2cm

(2n + 1)κ2mminus

κ3cm

(3n + 1)κ3m+

κ4cm

(4n + 1)κ4mminus

κ5cm

(5n + 1)κ5m

(18)




u u0 minus zφx

v v0 minus zφy

w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(19)


εlowast1113864 1113865

εpx

εpy

cpxy

εbx

εby

cbxy

cxz

cyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε0xε0y

c0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

zε1xzε1y

zc1xy

c1xz

c1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

εNLx

εNLy

cNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(20)


ε01113966 1113967

ε0xε0y

c0xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

u0x

v0y

u0y + v0x

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

ε11113966 1113967

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(21)


εNL1113966 1113967

εNLx

εNLy

cNLxy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

12

w0x1113872 11138732

12

w0y1113872 11138732

w0xw0y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(22)



Pf K1w + K2nabla2w

ST α T

Qf1113960 1113961Tε + pΔT

(23)


Qf1113960 1113961

Qεf1113876 1113877 [0]

[0] Qc

f1113876 1113877

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qεf1113960 1113961

E(z T)

1 minus μ2

1 μ 0

μ 1 0

0 01 minus μ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qc

f1113876 1113877

5E(z T)

12(1 + μ)0

05E(z T)

12(1 + μ)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

α α(z T) α(z T) 0 0 01113864 1113865T

(24)



σlowast1113864 1113865

σ0x

σ0y

τ0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

σ1x

σ1y

τ1xy

τ1xz

τ1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

σNLx

σNLy

τNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

minus

Nθx

Nθy

Nθxy

Mθx

Mθy

Mθxy

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(25)


σlowast1113864 1113865

N

M

Fs1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ε01113864 1113865 + εNL1113864 1113865

ε11113864 1113865

c1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦minus

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)

in which

N Nx Ny Nxy1113960 1113961T

M Mx My Mxy1113960 1113961T

Fs1113864 1113865 Fsx Fsy1113960 1113961T

(27)

Nθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α ΔT dz

Mθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α zΔT dz

(28)


([A] [B] [D]) 1113946h2

minus (h2)Q

ef1113960 1113961 1 z z

21113872 1113873dz

[C] 1113946h2

minus (h2)Q

c

f1113876 1113877dz

(29)




1113960 1113961 (30)

in which

Dlowast

1113960 1113961

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Flowast

1113960 1113961

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ε01113864 1113865

ε11113864 1113865

c1113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+

εNL1113864 1113865

0

0

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

(31)


u0 1113944r

m1[ϕ] a mXm(y)

v0 1113944r

m1[ϕ] b mYm(y)

w0 1113944

r

m1[ϕ] c mZm(y)

cxz 1113944r

m1[ϕprime] f1113864 1113865mFm(y)

cyz 1113944r

m1[ϕ] g1113864 1113865mGm(y)

(32)


w0 u0 φy 0 atx 0 a

w0 v0 φx 0 aty 0 b(33)


Xm(y) cosmπb

y1113874 1113875

Ym(y) sinmπb

y1113874 1113875

Zm(y) sinmπb

y1113874 1113875

Fm(y) sinmπb

y1113874 1113875

Gm(y) cosmπb

y1113874 1113875

(34)



V Nc

1113858 1113859 δ (35)



V01113864 1113865 [N] δ (36)



Nc



Nc

1113858 1113859m


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[N]m



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(38)

δ δ T1 δ

T2 δ


Tr

1113960 1113961 (39)

δ m a Tm b

Tm c T

m f1113864 1113865Tm g1113864 1113865

Tm

1113960 1113961 m 1 2 3 r

(40)


εlowast1113864 1113865L Bp

L1113960 1113961 BbL1113960 1113961 B

c

L1113858 11138591113960 1113961Tδ (41)

where [BpL] [B

pL]1 [B



[BbL] and [B

c

L] and

BpL1113960 1113961

m

ϕprime1113858 1113859Xm [0] [0] [0] [0]

[0] [ϕ]Ymprime [0] [0] [0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BbL1113960 1113961

m



prime



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BcL1113858 1113859m

[0] [0] [0] ϕprime1113858 1113859Fm [0]

[0] [0] [0] [0] [ϕ]Gm

⎡⎣ ⎤⎦

(42)


εlowast1113864 1113865NL 12

[Q] Q (43)

in which

[Q]

zw

zx0

0zw

zy

zw

zy

zw

zx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q

zw

zx

zw

zy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(44)


εlowast1113864 1113865NL BNL1113858 1113859 δ (45)

in which

BNL1113858 1113859 BPNL1113858 1113859 [0] [0]1113858 1113859

T12 [Q] [0] [0]1113960 1113961

T[J]

[J]m 0 0 ϕprime1113858 1113859Zm 0 0

0 0 [ϕ]Zmprime 0 0

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Q 1113944r

m1[J]m c m [J] δ

(46)


εlowast1113864 1113865 BL1113858 1113859 + BNL1113858 1113859( 1113857 δ (47)


d δ TΨ B

Ad εlowast1113864 1113865


F 0 (48)


Tare the







KS BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA

+ BA

BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA

(50)

KF BA

[R]T

Kl1113858 1113859[R]dA (51)

Kl1113858 1113859

K1 0 00 K2 00 0 K2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[R]

[0] [0] [ϕ]Zm [0] [0]

[0] [0] [ϕprime]Zm [0] [0]

[0] [0] [ϕ]Zmprime [0] [0]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(52)

Kθ 12B

A[J]

TN

θ1113876 1113877[J]dA N

θ1113876 1113877

Nθx Nθ

xy

Nθxy Nθ

y

⎡⎢⎢⎣ ⎤⎥⎥⎦ (53)

FR1113864 1113865 1113946V

Bp

1113858 1113859T

Qf1113960 1113961 α ΔT + Bb

1113960 1113961T

Qf1113960 1113961 α zΔT1113874 1113875dV (54)



lowastT dδ (55)

in which

K0 BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA (56)

Klowastσ B[J]

TNf1113960 1113961[J]dx dy + B[R]

TKl1113858 1113859[R]dx dy

+12B[J]

TN

θ1113876 1113877[J]dx dy

(57)

KNL BA2 BNL1113858 1113859

TDlowast

1113960 1113961 BL1113858 1113859 + 2 BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113872

+ 4 BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113873dA

(58)

where [Nf] Nx Nxy
















Al 70 03 204 230ZrO2 200 03 209 10Zr2O3 151 03 209 10Al2O3 380 03 104 72



Si3N4


α(1K) 58723e minus 6 0 9095e-4 0 0κ(WmK) 1204 0 0 0 0

SUS304





Si3N4





ah 4 ah 10 ah 100

1

CLPT 05623 05623 05623FSDT 07291 05889 05625


Present 07077 05860 05625

4

CLPT 08281 08281 08281FSDT 11125 08736 08281


Present 11166 08773 08285

10

CLPT 09354 09354 09354FSDT 13178 09966 09360


Present 13226 09964 09399









05 1 2

SFPM (Ntimes r)


NS-FEM [45] (mesh)




282624222018161412100806040200

ndash02

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 05k = 1k = 2

(a)


8

6

4

2

0

ndash2

ndash4

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2


k = 05

SFPM

NS-FEM

(b)





qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)











a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)


20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350






T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20












Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References


















































ϕi(a) 0(ineN + 1)



(12)






2h1113890 1113891

k

+ Pb (13)




2h1113890 1113891

k

+ Pb(T) (14)


+ 1 + P1T1

+ P2T2

+ P3T3

1113872 1113873(j b t)

(15)






minusddz

κ(z)dT(z)

dz1113890 1113891 0 (16)




where

η(z) 1C

2z + h

2h1113888 1113889 minus

κcm

(n + 1)κm

2z + h

2h1113888 1113889

k+1

+κ2cm

(2n + 1)κ2m

2z + h

2h1113888 1113889

k+1

minusκ3cm

(3n + 1)κ3m

2z + h

2h1113888 1113889

3k+1⎡⎣

+κ4cm

(4n + 1)κ4m

2z + h

2h1113888 1113889

4k+1

minusκ5cm

(5n + 1)κ5m

2z + h

2h1113888 1113889

5k+1⎤⎦

C 1 minusκcm

(n + 1)κm

+κ2cm

(2n + 1)κ2mminus

κ3cm

(3n + 1)κ3m+

κ4cm

(4n + 1)κ4mminus

κ5cm

(5n + 1)κ5m

(18)




u u0 minus zφx

v v0 minus zφy

w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(19)


εlowast1113864 1113865

εpx

εpy

cpxy

εbx

εby

cbxy

cxz

cyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε0xε0y

c0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

zε1xzε1y

zc1xy

c1xz

c1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

εNLx

εNLy

cNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(20)


ε01113966 1113967

ε0xε0y

c0xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

u0x

v0y

u0y + v0x

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

ε11113966 1113967

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(21)


εNL1113966 1113967

εNLx

εNLy

cNLxy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

12

w0x1113872 11138732

12

w0y1113872 11138732

w0xw0y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(22)



Pf K1w + K2nabla2w

ST α T

Qf1113960 1113961Tε + pΔT

(23)


Qf1113960 1113961

Qεf1113876 1113877 [0]

[0] Qc

f1113876 1113877

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qεf1113960 1113961

E(z T)

1 minus μ2

1 μ 0

μ 1 0

0 01 minus μ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qc

f1113876 1113877

5E(z T)

12(1 + μ)0

05E(z T)

12(1 + μ)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

α α(z T) α(z T) 0 0 01113864 1113865T

(24)



σlowast1113864 1113865

σ0x

σ0y

τ0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

σ1x

σ1y

τ1xy

τ1xz

τ1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

σNLx

σNLy

τNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

minus

Nθx

Nθy

Nθxy

Mθx

Mθy

Mθxy

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(25)


σlowast1113864 1113865

N

M

Fs1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ε01113864 1113865 + εNL1113864 1113865

ε11113864 1113865

c1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦minus

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)

in which

N Nx Ny Nxy1113960 1113961T

M Mx My Mxy1113960 1113961T

Fs1113864 1113865 Fsx Fsy1113960 1113961T

(27)

Nθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α ΔT dz

Mθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α zΔT dz

(28)


([A] [B] [D]) 1113946h2

minus (h2)Q

ef1113960 1113961 1 z z

21113872 1113873dz

[C] 1113946h2

minus (h2)Q

c

f1113876 1113877dz

(29)




1113960 1113961 (30)

in which

Dlowast

1113960 1113961

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Flowast

1113960 1113961

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ε01113864 1113865

ε11113864 1113865

c1113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+

εNL1113864 1113865

0

0

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

(31)


u0 1113944r

m1[ϕ] a mXm(y)

v0 1113944r

m1[ϕ] b mYm(y)

w0 1113944

r

m1[ϕ] c mZm(y)

cxz 1113944r

m1[ϕprime] f1113864 1113865mFm(y)

cyz 1113944r

m1[ϕ] g1113864 1113865mGm(y)

(32)


w0 u0 φy 0 atx 0 a

w0 v0 φx 0 aty 0 b(33)


Xm(y) cosmπb

y1113874 1113875

Ym(y) sinmπb

y1113874 1113875

Zm(y) sinmπb

y1113874 1113875

Fm(y) sinmπb

y1113874 1113875

Gm(y) cosmπb

y1113874 1113875

(34)



V Nc

1113858 1113859 δ (35)



V01113864 1113865 [N] δ (36)



Nc



Nc

1113858 1113859m


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[N]m



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(38)

δ δ T1 δ

T2 δ


Tr

1113960 1113961 (39)

δ m a Tm b

Tm c T

m f1113864 1113865Tm g1113864 1113865

Tm

1113960 1113961 m 1 2 3 r

(40)


εlowast1113864 1113865L Bp

L1113960 1113961 BbL1113960 1113961 B

c

L1113858 11138591113960 1113961Tδ (41)

where [BpL] [B

pL]1 [B



[BbL] and [B

c

L] and

BpL1113960 1113961

m

ϕprime1113858 1113859Xm [0] [0] [0] [0]

[0] [ϕ]Ymprime [0] [0] [0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BbL1113960 1113961

m



prime



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BcL1113858 1113859m

[0] [0] [0] ϕprime1113858 1113859Fm [0]

[0] [0] [0] [0] [ϕ]Gm

⎡⎣ ⎤⎦

(42)


εlowast1113864 1113865NL 12

[Q] Q (43)

in which

[Q]

zw

zx0

0zw

zy

zw

zy

zw

zx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q

zw

zx

zw

zy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(44)


εlowast1113864 1113865NL BNL1113858 1113859 δ (45)

in which

BNL1113858 1113859 BPNL1113858 1113859 [0] [0]1113858 1113859

T12 [Q] [0] [0]1113960 1113961

T[J]

[J]m 0 0 ϕprime1113858 1113859Zm 0 0

0 0 [ϕ]Zmprime 0 0

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Q 1113944r

m1[J]m c m [J] δ

(46)


εlowast1113864 1113865 BL1113858 1113859 + BNL1113858 1113859( 1113857 δ (47)


d δ TΨ B

Ad εlowast1113864 1113865


F 0 (48)


Tare the







KS BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA

+ BA

BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA

(50)

KF BA

[R]T

Kl1113858 1113859[R]dA (51)

Kl1113858 1113859

K1 0 00 K2 00 0 K2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[R]

[0] [0] [ϕ]Zm [0] [0]

[0] [0] [ϕprime]Zm [0] [0]

[0] [0] [ϕ]Zmprime [0] [0]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(52)

Kθ 12B

A[J]

TN

θ1113876 1113877[J]dA N

θ1113876 1113877

Nθx Nθ

xy

Nθxy Nθ

y

⎡⎢⎢⎣ ⎤⎥⎥⎦ (53)

FR1113864 1113865 1113946V

Bp

1113858 1113859T

Qf1113960 1113961 α ΔT + Bb

1113960 1113961T

Qf1113960 1113961 α zΔT1113874 1113875dV (54)



lowastT dδ (55)

in which

K0 BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA (56)

Klowastσ B[J]

TNf1113960 1113961[J]dx dy + B[R]

TKl1113858 1113859[R]dx dy

+12B[J]

TN

θ1113876 1113877[J]dx dy

(57)

KNL BA2 BNL1113858 1113859

TDlowast

1113960 1113961 BL1113858 1113859 + 2 BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113872

+ 4 BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113873dA

(58)

where [Nf] Nx Nxy
















Al 70 03 204 230ZrO2 200 03 209 10Zr2O3 151 03 209 10Al2O3 380 03 104 72



Si3N4


α(1K) 58723e minus 6 0 9095e-4 0 0κ(WmK) 1204 0 0 0 0

SUS304





Si3N4





ah 4 ah 10 ah 100

1

CLPT 05623 05623 05623FSDT 07291 05889 05625


Present 07077 05860 05625

4

CLPT 08281 08281 08281FSDT 11125 08736 08281


Present 11166 08773 08285

10

CLPT 09354 09354 09354FSDT 13178 09966 09360


Present 13226 09964 09399









05 1 2

SFPM (Ntimes r)


NS-FEM [45] (mesh)




282624222018161412100806040200

ndash02

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 05k = 1k = 2

(a)


8

6

4

2

0

ndash2

ndash4

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2


k = 05

SFPM

NS-FEM

(b)





qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)











a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)


20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350






T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20












Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References



















































u u0 minus zφx

v v0 minus zφy

w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(19)


εlowast1113864 1113865

εpx

εpy

cpxy

εbx

εby

cbxy

cxz

cyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ε0xε0y

c0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

zε1xzε1y

zc1xy

c1xz

c1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

εNLx

εNLy

cNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(20)


ε01113966 1113967

ε0xε0y

c0xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

u0x

v0y

u0y + v0x

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

ε11113966 1113967

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(21)


εNL1113966 1113967

εNLx

εNLy

cNLxy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

12

w0x1113872 11138732

12

w0y1113872 11138732

w0xw0y

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(22)



Pf K1w + K2nabla2w

ST α T

Qf1113960 1113961Tε + pΔT

(23)


Qf1113960 1113961

Qεf1113876 1113877 [0]

[0] Qc

f1113876 1113877

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qεf1113960 1113961

E(z T)

1 minus μ2

1 μ 0

μ 1 0

0 01 minus μ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Qc

f1113876 1113877

5E(z T)

12(1 + μ)0

05E(z T)

12(1 + μ)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

α α(z T) α(z T) 0 0 01113864 1113865T

(24)



σlowast1113864 1113865

σ0x

σ0y

τ0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

σ1x

σ1y

τ1xy

τ1xz

τ1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

σNLx

σNLy

τNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

minus

Nθx

Nθy

Nθxy

Mθx

Mθy

Mθxy

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(25)


σlowast1113864 1113865

N

M

Fs1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ε01113864 1113865 + εNL1113864 1113865

ε11113864 1113865

c1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦minus

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)

in which

N Nx Ny Nxy1113960 1113961T

M Mx My Mxy1113960 1113961T

Fs1113864 1113865 Fsx Fsy1113960 1113961T

(27)

Nθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α ΔT dz

Mθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α zΔT dz

(28)


([A] [B] [D]) 1113946h2

minus (h2)Q

ef1113960 1113961 1 z z

21113872 1113873dz

[C] 1113946h2

minus (h2)Q

c

f1113876 1113877dz

(29)




1113960 1113961 (30)

in which

Dlowast

1113960 1113961

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Flowast

1113960 1113961

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ε01113864 1113865

ε11113864 1113865

c1113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+

εNL1113864 1113865

0

0

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

(31)


u0 1113944r

m1[ϕ] a mXm(y)

v0 1113944r

m1[ϕ] b mYm(y)

w0 1113944

r

m1[ϕ] c mZm(y)

cxz 1113944r

m1[ϕprime] f1113864 1113865mFm(y)

cyz 1113944r

m1[ϕ] g1113864 1113865mGm(y)

(32)


w0 u0 φy 0 atx 0 a

w0 v0 φx 0 aty 0 b(33)


Xm(y) cosmπb

y1113874 1113875

Ym(y) sinmπb

y1113874 1113875

Zm(y) sinmπb

y1113874 1113875

Fm(y) sinmπb

y1113874 1113875

Gm(y) cosmπb

y1113874 1113875

(34)



V Nc

1113858 1113859 δ (35)



V01113864 1113865 [N] δ (36)



Nc



Nc

1113858 1113859m


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[N]m



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(38)

δ δ T1 δ

T2 δ


Tr

1113960 1113961 (39)

δ m a Tm b

Tm c T

m f1113864 1113865Tm g1113864 1113865

Tm

1113960 1113961 m 1 2 3 r

(40)


εlowast1113864 1113865L Bp

L1113960 1113961 BbL1113960 1113961 B

c

L1113858 11138591113960 1113961Tδ (41)

where [BpL] [B

pL]1 [B



[BbL] and [B

c

L] and

BpL1113960 1113961

m

ϕprime1113858 1113859Xm [0] [0] [0] [0]

[0] [ϕ]Ymprime [0] [0] [0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BbL1113960 1113961

m



prime



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BcL1113858 1113859m

[0] [0] [0] ϕprime1113858 1113859Fm [0]

[0] [0] [0] [0] [ϕ]Gm

⎡⎣ ⎤⎦

(42)


εlowast1113864 1113865NL 12

[Q] Q (43)

in which

[Q]

zw

zx0

0zw

zy

zw

zy

zw

zx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q

zw

zx

zw

zy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(44)


εlowast1113864 1113865NL BNL1113858 1113859 δ (45)

in which

BNL1113858 1113859 BPNL1113858 1113859 [0] [0]1113858 1113859

T12 [Q] [0] [0]1113960 1113961

T[J]

[J]m 0 0 ϕprime1113858 1113859Zm 0 0

0 0 [ϕ]Zmprime 0 0

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Q 1113944r

m1[J]m c m [J] δ

(46)


εlowast1113864 1113865 BL1113858 1113859 + BNL1113858 1113859( 1113857 δ (47)


d δ TΨ B

Ad εlowast1113864 1113865


F 0 (48)


Tare the







KS BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA

+ BA

BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA

(50)

KF BA

[R]T

Kl1113858 1113859[R]dA (51)

Kl1113858 1113859

K1 0 00 K2 00 0 K2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[R]

[0] [0] [ϕ]Zm [0] [0]

[0] [0] [ϕprime]Zm [0] [0]

[0] [0] [ϕ]Zmprime [0] [0]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(52)

Kθ 12B

A[J]

TN

θ1113876 1113877[J]dA N

θ1113876 1113877

Nθx Nθ

xy

Nθxy Nθ

y

⎡⎢⎢⎣ ⎤⎥⎥⎦ (53)

FR1113864 1113865 1113946V

Bp

1113858 1113859T

Qf1113960 1113961 α ΔT + Bb

1113960 1113961T

Qf1113960 1113961 α zΔT1113874 1113875dV (54)



lowastT dδ (55)

in which

K0 BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA (56)

Klowastσ B[J]

TNf1113960 1113961[J]dx dy + B[R]

TKl1113858 1113859[R]dx dy

+12B[J]

TN

θ1113876 1113877[J]dx dy

(57)

KNL BA2 BNL1113858 1113859

TDlowast

1113960 1113961 BL1113858 1113859 + 2 BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113872

+ 4 BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113873dA

(58)

where [Nf] Nx Nxy
















Al 70 03 204 230ZrO2 200 03 209 10Zr2O3 151 03 209 10Al2O3 380 03 104 72



Si3N4


α(1K) 58723e minus 6 0 9095e-4 0 0κ(WmK) 1204 0 0 0 0

SUS304





Si3N4





ah 4 ah 10 ah 100

1

CLPT 05623 05623 05623FSDT 07291 05889 05625


Present 07077 05860 05625

4

CLPT 08281 08281 08281FSDT 11125 08736 08281


Present 11166 08773 08285

10

CLPT 09354 09354 09354FSDT 13178 09966 09360


Present 13226 09964 09399









05 1 2

SFPM (Ntimes r)


NS-FEM [45] (mesh)




282624222018161412100806040200

ndash02

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 05k = 1k = 2

(a)


8

6

4

2

0

ndash2

ndash4

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2


k = 05

SFPM

NS-FEM

(b)





qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)











a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)


20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350






T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20












Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References

















































σlowast1113864 1113865

σ0x

σ0y

τ0xy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

0

0

0

σ1x

σ1y

τ1xy

τ1xz

τ1yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+

σNLx

σNLy

τNLxy

0

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

minus

Nθx

Nθy

Nθxy

Mθx

Mθy

Mθxy

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(25)


σlowast1113864 1113865

N

M

Fs1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ε01113864 1113865 + εNL1113864 1113865

ε11113864 1113865

c1113864 1113865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦minus

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(26)

in which

N Nx Ny Nxy1113960 1113961T

M Mx My Mxy1113960 1113961T

Fs1113864 1113865 Fsx Fsy1113960 1113961T

(27)

Nθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α ΔT dz

Mθ

1113966 1113967 1113946h2

minus (h2)Q

ef1113960 1113961 α zΔT dz

(28)


([A] [B] [D]) 1113946h2

minus (h2)Q

ef1113960 1113961 1 z z

21113872 1113873dz

[C] 1113946h2

minus (h2)Q

c

f1113876 1113877dz

(29)




1113960 1113961 (30)

in which

Dlowast

1113960 1113961

[A] [B] [0]

[B] [D] [0]

[0] [0] [C]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Flowast

1113960 1113961

Nθ1113864 1113865

Mθ1113864 1113865

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


ε01113864 1113865

ε11113864 1113865

c1113864 1113865

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

+

εNL1113864 1113865

0

0

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

(31)


u0 1113944r

m1[ϕ] a mXm(y)

v0 1113944r

m1[ϕ] b mYm(y)

w0 1113944

r

m1[ϕ] c mZm(y)

cxz 1113944r

m1[ϕprime] f1113864 1113865mFm(y)

cyz 1113944r

m1[ϕ] g1113864 1113865mGm(y)

(32)


w0 u0 φy 0 atx 0 a

w0 v0 φx 0 aty 0 b(33)


Xm(y) cosmπb

y1113874 1113875

Ym(y) sinmπb

y1113874 1113875

Zm(y) sinmπb

y1113874 1113875

Fm(y) sinmπb

y1113874 1113875

Gm(y) cosmπb

y1113874 1113875

(34)



V Nc

1113858 1113859 δ (35)



V01113864 1113865 [N] δ (36)



Nc



Nc

1113858 1113859m


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[N]m



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(38)

δ δ T1 δ

T2 δ


Tr

1113960 1113961 (39)

δ m a Tm b

Tm c T

m f1113864 1113865Tm g1113864 1113865

Tm

1113960 1113961 m 1 2 3 r

(40)


εlowast1113864 1113865L Bp

L1113960 1113961 BbL1113960 1113961 B

c

L1113858 11138591113960 1113961Tδ (41)

where [BpL] [B

pL]1 [B



[BbL] and [B

c

L] and

BpL1113960 1113961

m

ϕprime1113858 1113859Xm [0] [0] [0] [0]

[0] [ϕ]Ymprime [0] [0] [0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BbL1113960 1113961

m



prime



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BcL1113858 1113859m

[0] [0] [0] ϕprime1113858 1113859Fm [0]

[0] [0] [0] [0] [ϕ]Gm

⎡⎣ ⎤⎦

(42)


εlowast1113864 1113865NL 12

[Q] Q (43)

in which

[Q]

zw

zx0

0zw

zy

zw

zy

zw

zx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q

zw

zx

zw

zy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(44)


εlowast1113864 1113865NL BNL1113858 1113859 δ (45)

in which

BNL1113858 1113859 BPNL1113858 1113859 [0] [0]1113858 1113859

T12 [Q] [0] [0]1113960 1113961

T[J]

[J]m 0 0 ϕprime1113858 1113859Zm 0 0

0 0 [ϕ]Zmprime 0 0

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Q 1113944r

m1[J]m c m [J] δ

(46)


εlowast1113864 1113865 BL1113858 1113859 + BNL1113858 1113859( 1113857 δ (47)


d δ TΨ B

Ad εlowast1113864 1113865


F 0 (48)


Tare the







KS BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA

+ BA

BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA

(50)

KF BA

[R]T

Kl1113858 1113859[R]dA (51)

Kl1113858 1113859

K1 0 00 K2 00 0 K2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[R]

[0] [0] [ϕ]Zm [0] [0]

[0] [0] [ϕprime]Zm [0] [0]

[0] [0] [ϕ]Zmprime [0] [0]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(52)

Kθ 12B

A[J]

TN

θ1113876 1113877[J]dA N

θ1113876 1113877

Nθx Nθ

xy

Nθxy Nθ

y

⎡⎢⎢⎣ ⎤⎥⎥⎦ (53)

FR1113864 1113865 1113946V

Bp

1113858 1113859T

Qf1113960 1113961 α ΔT + Bb

1113960 1113961T

Qf1113960 1113961 α zΔT1113874 1113875dV (54)



lowastT dδ (55)

in which

K0 BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA (56)

Klowastσ B[J]

TNf1113960 1113961[J]dx dy + B[R]

TKl1113858 1113859[R]dx dy

+12B[J]

TN

θ1113876 1113877[J]dx dy

(57)

KNL BA2 BNL1113858 1113859

TDlowast

1113960 1113961 BL1113858 1113859 + 2 BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113872

+ 4 BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113873dA

(58)

where [Nf] Nx Nxy
















Al 70 03 204 230ZrO2 200 03 209 10Zr2O3 151 03 209 10Al2O3 380 03 104 72



Si3N4


α(1K) 58723e minus 6 0 9095e-4 0 0κ(WmK) 1204 0 0 0 0

SUS304





Si3N4





ah 4 ah 10 ah 100

1

CLPT 05623 05623 05623FSDT 07291 05889 05625


Present 07077 05860 05625

4

CLPT 08281 08281 08281FSDT 11125 08736 08281


Present 11166 08773 08285

10

CLPT 09354 09354 09354FSDT 13178 09966 09360


Present 13226 09964 09399









05 1 2

SFPM (Ntimes r)


NS-FEM [45] (mesh)




282624222018161412100806040200

ndash02

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 05k = 1k = 2

(a)


8

6

4

2

0

ndash2

ndash4

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2


k = 05

SFPM

NS-FEM

(b)





qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)











a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)


20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350






T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20












Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References


















































V Nc

1113858 1113859 δ (35)



V01113864 1113865 [N] δ (36)



Nc



Nc

1113858 1113859m


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[N]m



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(38)

δ δ T1 δ

T2 δ


Tr

1113960 1113961 (39)

δ m a Tm b

Tm c T

m f1113864 1113865Tm g1113864 1113865

Tm

1113960 1113961 m 1 2 3 r

(40)


εlowast1113864 1113865L Bp

L1113960 1113961 BbL1113960 1113961 B

c

L1113858 11138591113960 1113961Tδ (41)

where [BpL] [B

pL]1 [B



[BbL] and [B

c

L] and

BpL1113960 1113961

m

ϕprime1113858 1113859Xm [0] [0] [0] [0]

[0] [ϕ]Ymprime [0] [0] [0]


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BbL1113960 1113961

m



prime



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

BcL1113858 1113859m

[0] [0] [0] ϕprime1113858 1113859Fm [0]

[0] [0] [0] [0] [ϕ]Gm

⎡⎣ ⎤⎦

(42)


εlowast1113864 1113865NL 12

[Q] Q (43)

in which

[Q]

zw

zx0

0zw

zy

zw

zy

zw

zx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Q

zw

zx

zw

zy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(44)


εlowast1113864 1113865NL BNL1113858 1113859 δ (45)

in which

BNL1113858 1113859 BPNL1113858 1113859 [0] [0]1113858 1113859

T12 [Q] [0] [0]1113960 1113961

T[J]

[J]m 0 0 ϕprime1113858 1113859Zm 0 0

0 0 [ϕ]Zmprime 0 0

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Q 1113944r

m1[J]m c m [J] δ

(46)


εlowast1113864 1113865 BL1113858 1113859 + BNL1113858 1113859( 1113857 δ (47)


d δ TΨ B

Ad εlowast1113864 1113865


F 0 (48)


Tare the







KS BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA

+ BA

BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA

(50)

KF BA

[R]T

Kl1113858 1113859[R]dA (51)

Kl1113858 1113859

K1 0 00 K2 00 0 K2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[R]

[0] [0] [ϕ]Zm [0] [0]

[0] [0] [ϕprime]Zm [0] [0]

[0] [0] [ϕ]Zmprime [0] [0]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(52)

Kθ 12B

A[J]

TN

θ1113876 1113877[J]dA N

θ1113876 1113877

Nθx Nθ

xy

Nθxy Nθ

y

⎡⎢⎢⎣ ⎤⎥⎥⎦ (53)

FR1113864 1113865 1113946V

Bp

1113858 1113859T

Qf1113960 1113961 α ΔT + Bb

1113960 1113961T

Qf1113960 1113961 α zΔT1113874 1113875dV (54)



lowastT dδ (55)

in which

K0 BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA (56)

Klowastσ B[J]

TNf1113960 1113961[J]dx dy + B[R]

TKl1113858 1113859[R]dx dy

+12B[J]

TN

θ1113876 1113877[J]dx dy

(57)

KNL BA2 BNL1113858 1113859

TDlowast

1113960 1113961 BL1113858 1113859 + 2 BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113872

+ 4 BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113873dA

(58)

where [Nf] Nx Nxy
















Al 70 03 204 230ZrO2 200 03 209 10Zr2O3 151 03 209 10Al2O3 380 03 104 72



Si3N4


α(1K) 58723e minus 6 0 9095e-4 0 0κ(WmK) 1204 0 0 0 0

SUS304





Si3N4





ah 4 ah 10 ah 100

1

CLPT 05623 05623 05623FSDT 07291 05889 05625


Present 07077 05860 05625

4

CLPT 08281 08281 08281FSDT 11125 08736 08281


Present 11166 08773 08285

10

CLPT 09354 09354 09354FSDT 13178 09966 09360


Present 13226 09964 09399









05 1 2

SFPM (Ntimes r)


NS-FEM [45] (mesh)




282624222018161412100806040200

ndash02

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 05k = 1k = 2

(a)


8

6

4

2

0

ndash2

ndash4

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2


k = 05

SFPM

NS-FEM

(b)





qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)











a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)


20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350






T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20












Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References

















































KS BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA

+ BA

BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA + 2BA

BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 1113859dA

(50)

KF BA

[R]T

Kl1113858 1113859[R]dA (51)

Kl1113858 1113859

K1 0 00 K2 00 0 K2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[R]

[0] [0] [ϕ]Zm [0] [0]

[0] [0] [ϕprime]Zm [0] [0]

[0] [0] [ϕ]Zmprime [0] [0]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(52)

Kθ 12B

A[J]

TN

θ1113876 1113877[J]dA N

θ1113876 1113877

Nθx Nθ

xy

Nθxy Nθ

y

⎡⎢⎢⎣ ⎤⎥⎥⎦ (53)

FR1113864 1113865 1113946V

Bp

1113858 1113859T

Qf1113960 1113961 α ΔT + Bb

1113960 1113961T

Qf1113960 1113961 α zΔT1113874 1113875dV (54)



lowastT dδ (55)

in which

K0 BA

BL1113858 1113859T

Dlowast

1113960 1113961 BL1113858 1113859dA (56)

Klowastσ B[J]

TNf1113960 1113961[J]dx dy + B[R]

TKl1113858 1113859[R]dx dy

+12B[J]

TN

θ1113876 1113877[J]dx dy

(57)

KNL BA2 BNL1113858 1113859

TDlowast

1113960 1113961 BL1113858 1113859 + 2 BL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113872

+ 4 BNL1113858 1113859T

Dlowast

1113960 1113961 BNL1113858 11138591113873dA

(58)

where [Nf] Nx Nxy
















Al 70 03 204 230ZrO2 200 03 209 10Zr2O3 151 03 209 10Al2O3 380 03 104 72



Si3N4


α(1K) 58723e minus 6 0 9095e-4 0 0κ(WmK) 1204 0 0 0 0

SUS304





Si3N4





ah 4 ah 10 ah 100

1

CLPT 05623 05623 05623FSDT 07291 05889 05625


Present 07077 05860 05625

4

CLPT 08281 08281 08281FSDT 11125 08736 08281


Present 11166 08773 08285

10

CLPT 09354 09354 09354FSDT 13178 09966 09360


Present 13226 09964 09399









05 1 2

SFPM (Ntimes r)


NS-FEM [45] (mesh)




282624222018161412100806040200

ndash02

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 05k = 1k = 2

(a)


8

6

4

2

0

ndash2

ndash4

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2


k = 05

SFPM

NS-FEM

(b)





qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)











a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)


20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350






T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20












Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References



















































Al 70 03 204 230ZrO2 200 03 209 10Zr2O3 151 03 209 10Al2O3 380 03 104 72



Si3N4


α(1K) 58723e minus 6 0 9095e-4 0 0κ(WmK) 1204 0 0 0 0

SUS304





Si3N4





ah 4 ah 10 ah 100

1

CLPT 05623 05623 05623FSDT 07291 05889 05625


Present 07077 05860 05625

4

CLPT 08281 08281 08281FSDT 11125 08736 08281


Present 11166 08773 08285

10

CLPT 09354 09354 09354FSDT 13178 09966 09360


Present 13226 09964 09399









05 1 2

SFPM (Ntimes r)


NS-FEM [45] (mesh)




282624222018161412100806040200

ndash02

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 05k = 1k = 2

(a)


8

6

4

2

0

ndash2

ndash4

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2


k = 05

SFPM

NS-FEM

(b)





qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)











a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)


20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350






T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20












Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References
























































05 1 2

SFPM (Ntimes r)


NS-FEM [45] (mesh)




282624222018161412100806040200

ndash02

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2

k = 05k = 1k = 2

(a)


8

6

4

2

0

ndash2

ndash4

Erro

r of c

alcu

latio

ns b

etw

een

two

met

hods

()

AlZrO2


k = 05

SFPM

NS-FEM

(b)





qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)











a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)


20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350






T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20












Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References



















































qlowast

qb4

Emh4

k1 K1a

4

Emh3

k2 K2a

2

Emh3

(59)











a

zb

y

h

x

Metal

Ceramic

Pasternak (K2)

Winkler (K1)


20

15

10

05

00

wh

0 50 100 150 200qlowast = qb4Emh4

250 300 350






T0 = 300K SUS304Si3N4k = 2 ba = 1 bh = 20












Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References


























































Ceramic (k 0)10 11256 09292 174540 45128 38579 1451100 110478 99061 1033

k 110 12389 09498 233340 47344 38599 1847100 120037 99700 1694

k 510 16656 10781 352740 63468 43977 3071100 153185 114510 2525

Metal (k⟶+infin)10 19929 15890 202740 77468 65119 1594100 213131 171271 1132




AlZr2O3

1 04295 03781 119725 07505 06190 17525 08862 06450 2722

Si3N4SUS3041 02097 01916 86225 03459 02984 13745 04162 03113 2521







0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References





















































0 2 4 6k

8 10

00

ndash10

ndash20

ndash30

ndash40

ndash50

w (m

)times10ndash3

Tm = 400K Tc = 300KTm = 350K Tc = 300K


(a)

0 2 4 6k

8 1000

05

10

15

20

25

30

w (m

)

times10ndash3

Tm = 300K Tc = 400KTm = 300K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)

0 2 4 6k

8 10

00

ndash05

ndash10

ndash15

ndash20

ndash25

ndash30

w (m

)

times10ndash3

Tm = 400K Tc = 400KTm = 350K Tc = 350K

qlowast = 0 k1 = 0 k2 = 0

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)








50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References






















































50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

ndash02

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(a)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

00

02

04

06

08

10

12

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(b)


50

k = 0k = 1 k = 5

k = 2k1 = 0 k2 = 0

10

08

06

04

02

00

wh

AlZr2O3 T0 = 300Kha = 120 a = b = 1

(c)







20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References





















































20

16

12

08

04

00

wh


50

C2 ha = 140 k1 = 0 k2 = 0 k = 2

AlZr2O3 T0 = 300K

ab = 2ab = 1ab = 05







15180 06530 04882 3376 09058 07202 2577 06542 04675 3994140 11281 09480 1900 10927 09780 1173 10351 08506 2169120 10085 09143 1030 09997 09420 613 09514 08137 1692

2180 10240 08320 2308 13184 11976 1009 06609 05130 2883140 11382 10186 1174 12892 12262 514 11504 10423 1037120 11709 10843 799 12641 12441 161 10734 09937 802



AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References

















































AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2


50

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100

ndash005

000

005

010

015

020

025

030w

h

(a)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50000

005

010

015

020

025

030

035

wh

(b)

AlZr2O3 T0 = 300Kha = 120 a = b = 1k = 2

k1 = 10 k2 = 10k1 = 100 k2 = 10k1 = 100 k2 = 100


50ndash005

000

005

010

015

020

025

035

030

wh

(c)




k1 0k2 0

w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)





1C1 11723 2416 1821 0285 7939 2463 8435C2 20079 4669 3267 0490 7675 3003 8500C3 10169 3550 2482 0351 6509 3008 8586

15C1 22120 9730 7980 1170 5601 1799 8534C2 23925 11402 9361 1383 5234 1790 8523C3 22049 11529 9253 1250 4771 1974 8649

2C1 28761 18494 14506 2527 3570 2156 8258C2 29198 19504 17374 2737 3320 1092 8425C3 28903 17556 15577 2624 3926 1127 8315




5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References


















































5 Conclusions









w (10minus 3m)

k1 10k2 10

w (10minus 3m)

k1 100k2 10

w (10minus 3m)

k1 100k2 100

w (10minus 3m)




180k 05 6283 4160 2620 0329 3379 3702 8744k 2 8796 4396 2703 0331 5002 3851 8775k 3 9023 4505 2742 0332 5007 3913 8789

150K 05 17529 3826 2685 0400 7817 2982 8510k 2 19474 4168 2846 0413 7860 3172 8549k 3 19949 4210 2866 0414 7890 3192 8555

125K 05 25824 6302 4505 0708 7560 2851 8428k 2 26643 6434 4590 0717 7585 2866 8438k 3 27276 6550 4610 0719 7599 2962 8440

110k 05 93084 14748 10547 1698 8416 2849 8390k 2 95220 14953 10655 1710 8430 2874 8395k 3 99585 15540 11004 1716 8440 2919 8441




Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References


















































Appendix



a


[ϕ]T[ϕ] b k[ϕ] [ϕ]


x

h+ 11113874 1113875 + b0( 1113857k φ3

x

h1113874 1113875 minus 4φ3

x

h+ 11113874 11138751113876 11138771113882

+ b1( 1113857k φ3x

hminus 11113874 1113875 minus

12φ3

x

h1113874 1113875 + φ

x


x

hminus N minus 11113874 11138751113883[ϕ]

(A1)


[ϕ]T


T φ3i1113858 1113859 Qs1113858 1113859

(A2)

given

Qs1113858 1113859 b k blowast

1113864 1113865k (A3)


[ϕ]T

[ϕ] b k( 1113857[ϕ] Qs1113858 1113859T φ3i1113858 1113859 b

lowast1113864 1113865k( 1113857 φ3i1113858 1113859

T φ3i1113858 1113859 Qs1113858 1113859

(A4)

in which

φ3i1113858 1113859( 1113857 blowast

1113864 1113865k φ3i1113858 1113859T φ3i1113858 1113859


x

h+ 11113874 1113875 + b


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113876 1113877

middot

φ3x

h+ 11113874 1113875φ3

x

h+ 11113874 1113875 φ3

x

h+ 11113874 1113875φ3

x

h1113874 1113875 φ3

x

h+ 11113874 1113875φ3

x


x

h+ 11113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

h1113874 1113875φ3

x

h1113874 1113875 φ3

x

h1113874 1113875φ3

x


x

h1113874 1113875φ3

x

hminus N minus 11113874 1113875

φ3x

hminus 11113874 1113875φ3

x


x

hminus 11113874 1113875φ3

x

hminus N minus 11113874 1113875


φ3x

hminus N minus 11113874 1113875φ3

x

hminus N minus 11113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A5)

For

1113946a

0φ3

x

hminus i1113874 1113875φ3

x

hminus j1113874 1113875 b


x

h+ 11113874 11138751113876


x

h1113874 1113875 + b


x



x

hminus N minus 11113874 11138751113877dx

(A6)


h 1113946Nminus i


N+1






w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References

















































w w0

cxz zw

zxminus φx

cyz zw

zyminus φy

(B1)


U 12

1113946Sf

κ T[D] κ dS +

12

1113946Sf

c1113864 1113865T[C] c1113864 1113865dS (B2)

in which

κ

ε1xε1y

c1xy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

minus φxx

minus φyy


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

c1113864 1113865 c1

xz

c1yz

⎧⎨

⎩

⎫⎬

⎭ w0x minus φx

w0y minus φy

⎧⎨

⎩

⎫⎬

⎭

(B3)

given

κ minus B11113858 1113859 Vw1113864 1113865 + B21113858 1113859 Vc1113966 1113967

c1113864 1113865 1113957B11113858 1113859 Vc1113966 1113967(B4)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865 minus 2 Vc1113966 1113967

TB21113858 1113859

T[D] B11113858 1113859 Vw1113864 11138651113874 1113875dS

+12

1113946Sf

Vc1113966 1113967T

B21113858 1113859T[D] B21113858 1113859 + 1113957B11113858 1113859

T[C] 1113957B11113858 11138591113874 1113875 Vc1113966 1113967dS

(B5)

For (zUz Vy1113966 1113967) 0 ie

Vc1113966 1113967 λa2 L31113858 1113859

L11113858 1113859 + λa2 L21113858 1113859( 1113857Vw1113864 1113865 (B6)

in which

L11113858 1113859 1113946Sf

1113957B11113858 1113859T 1113957B11113858 1113859dS

L21113858 1113859 1113946Sf

B21113858 1113859T

B21113858 1113859dS

L31113858 1113859 1113946Sf

B11113858 1113859T

B21113858 1113859dS

λ h2

5(1 minus μ)a2

(B7)


U 12

1113946Sf

Vw1113864 1113865T

B11113858 1113859T[D] B11113858 1113859 Vw1113864 1113865dS (B8)


U 12

1113946Sf

κ T[D] κ dS (B9)


Data Availability




Acknowledgments


References

















































A Spline Finite Point Method for Nonlinear Bending Analysis of FG … · 2020. 7. 10. ·...

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