Strain gradient thermoelasticity of functionally graded...

$: Strain gradient thermoelasticity of functionally graded cylindersscientiairanica.sharif.edu/article_3563_fa97fc339147e9... · 2021. 2. 18. · Scientia Iranica B (2014) 21(4), 1415{1423$
Scientia Iranica B (2014) 21(4), 1415{1423

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Strain gradient thermoelasticity of functionally gradedcylinders

H. Sadeghia;�, M. Baghanib and R. Naghdabadic

a. Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0416, USA.b. School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, P.O. Box 11155-4563, Iran.c. Institute for Nano-Science and Technology, Sharif University of Technology, Tehran, Iran.

Received 14 October 2012; received in revised form 30 October 2013; accepted 14 December 2013

KEYWORDSFunctionally-graded;Thermoelasticthick-walled cylinder;Strain gradientelasticity;Intrinsic lengthparameter.

Abstract. In this paper, strain gradient thermo-elasticity formulation for axisymmetricFunctionally Graded (FG) thick-walled cylinders is presented. For this purpose, theelastic strain energy density function is considered to be a function of gradient of straintensor in addition to the strain tensor. The material properties are assumed to varyaccording to a power law in radial direction. Using the constitutive equations and equationof equilibrium in the cylindrical coordinates, a fourth order non-homogenous governingequation for thermo-elastic analysis of thick-walled FG cylinders subjected to thermal andmechanical loadings is obtained and solved numerically. Results show that the intrinsiclength parameter a�ects the stress distribution in FG thick-walled cylinders greatly andincreasing the intrinsic length parameter reduces the maximum radial and hoop stresses.Also, the e�ects of FG power indices on the radial and hoop stresses are studied.© 2014 Sharif University of Technology. All rights reserved.

1. Introduction

There are evidences that material responses in theelastic region are dependent on some other parametersrather than Lame parameters. Strain gradient elastic-ity has been recently developed to take into accountthis characteristic of materials response [1,2]. In straingradient elasticity new parameters enter the constitu-tive equations through the elastic strain energy densityfunction. Mindlin, in 1964 [3] used the assumptionthat the elastic strain energy density function can be afunction of the gradient of strain tensor in addition tostrain tensor. In this way, new material constants wereintroduced and entered into the constitutive equations.

There are many studies which have used thestrain gradient elasticity in the analysis of behavior ofmaterials [4-8]. Eshel and Rosenfeld [4] analyzed thestress concentration problem of a circular cylindrical

*. Corresponding author. Tel.: +1 858 692 9776E-mail address: [email protected] (H. Sadeghi)

hole in a homogenous, isotropic and centrosymmetricin�nite elastic solid subjected to uniaxial tension.They showed that stress concentration depends on theradius of cavity, Poisson ratio and four new materialparameters. Georgiadis et al. [5] studied free harmonictorsional motion of a homogenous half-space by straingradient elasticity. They focused on the possibility ofexistence of torsional surface waves in gradient-elastichalf-space. They showed the dependency of cut-o�frequencies and the character of dispersion upon thesize of the material unit cell. Paulino et al. [6] usedthe strain gradient elasticity for analysis of mode IIIfracture in Functionally Graded (FG) materials andused Fourier transform to solve the governing partialdi�erential equations. They studied the stress anddisplacement �elds near the crack and found that thestress intensity factor is dependent on intrinsic lengthparameters. Kong, et al. [7] studied the static and dy-namic problems of Bernoulli-Euler beams analyticallyby using strain gradient elasticity. They consideredtwo boundary value problems for cantilever beams and

1416 H. Sadeghi et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1415{1423

analyzed the size e�ects on the beam bending responseas well as its natural frequencies.

In recent years, FG materials, in which proper-ties are dependent on position, have attracted manyresearchers because of their unique features. Specialcharacteristics of FG materials make them a potentialcandidate for many applications such as thick-walledcylinders. Several researchers have studied thermo-elastic deformation of FG cylinders in recent years [9-14]. Yang [9] solved the time dependent thermalstresses in FG cylinders. He found an analyticalsolution for thermo-elastic stresses in the materialswith elastic behavior and an asymptotic solution for thematerials with creep behavior. Tarn [14] considered thethermo-mechanical problem of inhomogeneous, solid orhollow circular FG cylinders. He assumed a powerlaw dependence of the moduli and the cylinder wasassumed to be subjected to an axial force and atorque at the ends. Tutuncu and Ozturk [13] studiedthe thermo-elasticity of FG cylinders by exponentiallyvarying properties. They assumed that Poisson ratiois constant and Young modulus varies exponentiallythrough the thickness. They obtained power seriessolutions for stresses and displacements in FG cylin-drical vessels subjected to internal pressure. Jabbariet al. [11] analyzed mechanical and thermal stresses inFG hollow cylinders due to radially symmetric loads.They assumed that material properties depend on theradial position as a power function. They solved theheat conduction and Navier equations directly andobtained an analytical solution for the stress and straincomponents. Shao [10] considered the problem of FGhollow cylinders with �nite length under mechanicaland thermal loading. He used a multi-layered approachbased on the theory of laminated composited and founda series solution for the components of temperature anddisplacement �elds in the �nite FG cylinders. Jabbariet al. [12] studied the axisymmetric mechanical andthermal stresses in thick short length FG cylinders.They used Fourier series to �nd an exact solution forthe steady-state stress components in two-dimensionalaxisymmetric FG cylinders with material propertiesdescribed with a power law.

In most of the studies on thick-walled FG cylin-ders, classical theory of elasticity has been employed.Thus, there is a need for a study to consider thee�ect of intrinsic length parameter on the thermo-elastic response of FG cylinders. In our previous work,we analytically studied the strain gradient elasticitysolution for analysis of FG micro-cylinders [15]. Thematerial properties were assumed to obey a power lawin radial direction.

In this paper, axisymmetric strain gradientthermo-elasticity formulation of thick-walled FG cylin-ders subjected to thermal and mechanical loadings ispresented. The fourth order non-homogenous govern-

ing equation is obtained and it is solved numericallytogether with four boundary conditions. Results arepresented for di�erent values of the intrinsic lengthparameter and it is compared with classical elasticitysolution. Furthermore, the e�ects of power indices andthe outer radius of the cylinder radius on the stressesand radial displacement are studied.

2. Strain gradient thermo-elasticity

2.1. Equation of heat conductionThe equation governing the temperature �eld in along, thick-walled, FG cylinder in the axisymmetriccondition while the temperatures at the inner and outerradii of the cylinder are Ti and To, respectively, is givenby [11,12]:

ddr

�k(r)

dTdr

�+k(r)r

dTdr

= 0; (1)

T = Ti at r = ri;

T = To at r = ro; (2)

where T is the temperature change, k(r) is thermal heatconduction coe�cient, r is radial position, and ri andro are the inner and outer radii of the cylinder. In thispaper, it is assumed that the thermal heat conductioncoe�cient obeys the power law in a radial direction:

k(r) = k0rm; (3)

where k0 and m are material constants. The solutionof Eq. (1) is:

T (r) = c1 + c2r�m; (4)

where c1 and c2 are found from the boundary condi-tions (Eq. (2)) as:

c1 =Tir�mo � Tor�mir�mo � r�mi ;

c2 =To � Ti

r�mo � r�mi : (5)

2.2. Governing equationIn the strain gradient elasticity, the elastic strainenergy density function is assumed to be a functionof the gradient of strain tensor, in addition to thestrain tensor. Altan and Aifantis [16] presented asimpli�ed form of the elastic strain energy densityfunction introduced by Mindlin and Eshel [17]. Here,the same form is used to obtain the elastic strain energydensity function for thick-walled FG cylinders. For aFG cylinder, elastic strain energy density function, w,

H. Sadeghi et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1415{1423 1417

can be written in the cylindrical coordinates as [15]:

w =12�(r)("rr(r) + "��(r) + "zz(r))

2

+ �(r)�"2rr(r) + "2

��(r) + "2zz(r)

�+

12�(r)l2

ddr"rr(r) +

ddr"��(r) +

ddr"zz(r)

!2

+ �(r)l2��

ddr"rr(r)

�2

+�ddr"��(r)

�2

+�ddr"zz(r)

�2�;

(6)

where l is the intrinsic length parameter, "rr and "��are radial and hoop strains, and �(r) and �r are Lameparameters. In this theory, the components of Cauchystress tensor, �ij , the double stress tensor, �ijk, thestrain gradient tensor, �ijk, and total stress tensor, �ij ,are, respectively, given by:

�ij =@w@"ij

; (7)

�ijk =@w@�ijk

; (8)

�ijk = "ij;k =12

(ui;jk + uj;ik); (9)

�ij = �ij � �ijk;k; (10)

where ui's are components of displacement vector. Totake into account the e�ect of temperature change,the transformation "ij ! "ij � �T�ij should be usedin Eq. (6) [18] where � is the coe�cient of thermalexpansion and �ij is Kronecker delta, which results in:

w =12�(r)

�("rr(r)��(r)T (r))

�+�"��(r)��(r)T (r)

�+�"zz(r)��(r)T (r)

�2

+�(r)�"rr(r)��(r)T (r)

�2

+�"��(r)� �(r)T (r)

�2

+�"zz(r)� �(r)T (r)

�2

+12�(r)l2

�ddr

("rr(r)� �(r)T (r)) +ddr

("��(r)

� �(r)T (r)) +ddr

("zz(r)� �(r)T (r))�2

+ �(r)l2�

[ddr

("rr(r)� �(r)T (r))]2

+[ddr

("��(r)� �(r)T (r))]2+[ddr

("zz(r)� �(r)T (r))]2

+ [ddr

("zz(r)� �(r)T (r))]2�:

(11)

Here, the plane strain condition is assumed ("zz =0). Substituting Eq. (11) into Eqs. (7) and (8), theconstitutive equations for an axisymmetric thick-walledFG cylinder are derived as:

�rr =�(r) ("rr(r) + "��(r)� 3�(r)T (r))

+ 2�(r) ("rr(r)� �(r)T (r)) ; (12)

�� =�(r) ("rr(r) + "��(r)� 3�(r)T (r))

+ 2�(r) ("��(r)� �(r)T (r)) ; (13)

�rrr = (�(r) + 2�(r)) l2d"rr(r)

dr+ �(r)l2

d"��(r)dr

� l2 (3�(r) + 2�(r))d (�(r)T (r))

dr; (14)

�r�� = (�(r) + 2�(r)) l2d"��(r)

dr+ �(r)l2

d"rr(r)dr

� l2 (3�(r) + 2�(r))d (�(r)T (r))

dr: (15)

It can be seen from Eqs. (12)-(15) that the relationsbetween Cauchy stresses and strains in the straingradient elasticity are the same as the classical theoryof elasticity. Also, in the absence of the strain gradiente�ect (l = 0), the double stresses will be zero. In thecylindrical coordinates, radial and hoop strains, "rr and"��, are written in terms of radial displacement, u(r),as:

"rr =du(r)

dr; "�� =

u(r)r: (16)

The equilibrium equation in cylindrical coordinates inaxisymmetric condition is:

d�rr

dr+�rr � ��

r= 0: (17)

In this paper, it is assumed that Lame parameters andcoe�cient of thermal expansion vary along the radiusaccording to the power law:

�(r) = �0rn; �(r) = �0rn; �(r) = �0rp; (18)

in which �0; �0; �0; p and n are material constants.Eqs. (10) and (12)-(16) together with equilibriumEq. (17) result in the governing equation for an axisym-metric thick-walled FG cylinder in the strain gradient


thermo-elasticity in terms of the radial displacement,u(r), as:

[f4(r)d4

dr4 + f3(r)d3

dr3 + f2(r)d2

dr2 + f1(r)ddr

+ f0(r)]u(r) = g(r); (19)

where fi(r), i = 0; :::; 4 and g(r) are known functionsand can be written as:

f0(r)=��0

�4�2n

�+ �0

�n�2

��n� 3

��l2r�4

+��0 (n� 1)�2�0

�r�2;

f1(r)=(�0+2�0)(n+1)r�1 +��0

�n� 2

��n� 3

�+�0

�2n�4

��l2r�3;

f2(r)=(�0+2�0)+�

2�0(1�n2)

� �0

�n2+n� 3

��l2r�2;

f3(r)=�(�0+2�0) (1+2n) l2r�1;

f4(r) = �(�0 + 2�0)l2;

g(r)=A�� B+Cr�m

�rp�1 + l2

�D+Fr�m

�rp�3� ;

(20)

where:

A = (3�0 + 2�0) ; B = c1 (p+ n) ;

C = c2 (p+ n�m) ;

D = c1�p3 + (2n� 3) p2 +

�n2 � 3n+ 2

�p;

F =c2�p3 +

�2n� 3m� 3

�p2 +

�3m2

+�

6� 4n�m+ n2 � 3n+ 2

�p�m3

+�

2n� 3�m2 +

��n2 + 3n� 2

�m�: (21)

It can be seen that the governing equation for thick-walled FG cylinders is a fourth order ODE, whereas, inthe classical elasticity, it is a second order ODE. Also,it is noted that in the absence of strain gradient e�ect(l = 0) this fourth order ODE reduces to the secondorder ODE in the classical elasticity.

2.3. Boundary conditionsLet the FG cylinder be subjected to boundary pressuresand on its inner and outer surfaces, respectively. So,boundary conditions for the proposed problem can begiven by [19]:��rr�l2

�d2�rrdr2 +

1r

(d�rrdr�d��dr

)� 2r2 (�rr��)

��r=ri

=�Pi;��rr�l2

�d2�rrdr2 +

1r

(d�rrdr�d��dr

)� 2r2 (�rr��)

��r=ro

=�Po;

l2�rrrjr=ri = 0;

l2�rrrjr=ro = 0: (22)

The prescribed double stress traction is considered zeroon both the inner and outer surfaces. It can be seenthat in the absence of the strain gradient e�ect (l = 0),boundary conditions Eq. (22) reduce to the boundaryconditions in the classical theory of elasticity.

3. Numerical results and discussion

3.1. Veri�cation of the solutionThe governing equation (Eq. (19)) together with fourboundary conditions (Eq. (22)) has been solved nu-merically and the results are presented in this section.The numerical solution used in this problem is acombination of the base scheme (trapezoid), and amethod enhancement scheme (Richardson extrapola-tion) with the traprich method that have been codedin the MAPLE software. For the detailed discussionon the numerical method, we refer to [20,21]. Forveri�cation of the numerical solution, a homogenousthick-walled cylinder is considered with the inner andouter radii, ri = 1�m and ro = 5�m, respectively. Itis assumed that the cylinder is subjected to boundarypressures, Pi = 10 MPa and Po = 0 MPa, at the innerand outer surfaces, respectively. Gao and Park [19]presented an analytical solution for elastic deformationof homogenous thick-walled cylinders using the straingradient elasticity. A plot of the radial stress forthe homogenous cylinder along the radius using thesolution presented by Gao and Park and our solutionis presented in Figure 1 with the parameter values:l = 0:5�m, E = 135 GPa and � = 0:3. Also, theresults obtained from the classical theory of elasticityare presented in this �gure for comparison. It can beseen that the results are in good agreement with thoseobtained from the analytical solution presented by Gaoand Park [19].

3.2. Case studiesIn this section the e�ect of the intrinsic length pa-rameter as well as FG parameters on thermo-elastic


Figure 1. Comparison of non-dimensional radial stressalong the radius for a homogenous thick-walled cylinder.

Table 1. Parameter values used in the numericalexamples.

l(�m) n m p ro(�m)

Variable 0.4 0.6 0.8 50.1 Variable 0.6 0.8 50.1 0.4 Variable 0.8 50.1 0.4 0.6 Variable 50.1 0.4 0.6 0.8 Variable

deformation of thick-walled FG cylinders are discussed.To this aim, the intrinsic length parameter, powerindices and outer radius of the cylinder are variedaccording to Table 1. Lame parameters and thecoe�cient of thermal expansion at the inner radius areassumed to be �i = 1200 MPa, �i = 800 MPa and�i = 1:6 � 10�6 (1=K). The following geometrical,thermal and mechanical parameters have been usedin the numerical examples Ti = 300 K, To = 400 K,ri = 1�m, ro = 5�m, Pi = 10 MPa and Po =0 MPa.

The distribution of non-dimensional radial dis-placement as well as non-dimensional radial and hoopstresses along the cylinder radius for di�erent valuesof the intrinsic length parameter, l, are presented inFigures 2-4, respectively. It is understood that bydecreasing l to zero, the strain gradient solution ap-proaches the classical elasticity solution. Furthermore,it can be seen that by increasing l, the di�erencebetween the strain gradient solution and that of theclassical elasticity raises, rapidly. It is interestingto see that the di�erence between the stresses andradial displacement for di�erent values of l at smallervalues of r=ri is more than the di�erence betweenthe results at larger values of r=ri. For example,the di�erence between �rr=Pi for l = 0:3�m andfor l = 0 at r=ri = 1 is 42%, whereas, there is

Figure 2. Radial displacement along the FG cylinderthickness for di�erent values of the intrinsic lengthparameter, l.

Figure 3. Radial stress along the FG cylinder thicknessfor di�erent values of the intrinsic length parameter, l.

Figure 4. Hoop stress along the FG cylinder thickness fordi�erent values of the intrinsic length parameter, l.


almost no di�erence between these values at r=ri = 5which shows that the intrinsic length parameter hasmore e�ect at smaller dimensions. Also, from theplotted results we see that increasing l decreases theabsolute value of maximum radial displacement as wellas radial and hoop stresses. For example, there is 72%di�erence between the maximum values of ��=Pi forl = 0:3�m and l = 0�m. In addition, it can beobserved that for di�erent values of l, radial and hoopstresses always attain their maximum values at theinner surface. So, it is concluded that intrinsic lengthparameter has a signi�cant e�ect on the displacementand stress distributions in the thick-walled FG cylin-ders.

The variations of non-dimensional radial displace-ment as well as non-dimensional radial and hoopstresses along the cylinder thickness for di�erent val-ues of the power index n are shown in Figures 5-7,respectively. From the plotted results, we see that thepower index n has a signi�cant e�ect on the stress

Figure 5. Radial displacement along the FG cylinderthickness for di�erent values of the power index, n.

Figure 6. Radial stress along the FG cylinder thicknessfor di�erent values of the power index, n.

distribution along the radial direction. Also, as nincreases from 0 to 3, the maximum radial displacementdecreases �rst and then increases. It can be seen thatas n increases, the maximum radial stress increasessharply and the radius, in which the maximum radialstress occurs, moves to larger values. Furthermore,by increasing n, the maximum hoop stress increasesrapidly, too. So, it is understood that the power indexn can signi�cantly a�ect the radial displacement aswell as radial and hoop stresses. Figures 8-10 showthe non-dimensional radial displacement as well asnon-dimensional radial and hoop stresses along theradial direction for di�erent values of the power indexm. It can be seen from these �gures that the valueof m does not have much e�ect on the results forselected values of m. Figures 11-13 show the non-dimensional radial displacement, radial stress and hoopstress distributions through the thickness for di�erent

Figure 7. Hoop stress along the FG cylinder thickness fordi�erent values of the power index, n.

Figure 8. Radial displacement along the FG cylinderthickness for di�erent values of m.


Figure 9. Radial stress along the FG cylinder thicknessfor di�erent values of m.

Figure 10. Hoop stress along the FG cylinder thicknessfor di�erent values of m.

Figure 11. Radial displacement along the FG cylinderthickness for di�erent values of p.

Figure 12. Radial stress along the FG cylinder thicknessfor di�erent values of p.

Figure 13. Hoop stress along the FG cylinder thicknessfor di�erent values of p.

values of power index p, respectively. It can be seenthat p has an important e�ect on the results. It isnoted that increasing p increases the maximum radialdisplacement and hoop stress whereas it has no e�ecton the maximum radial stress.

To study the e�ect of outer radius of the cylinderon the results, ro has been changed from 2�m to 5�mand the results are shown in Figures 14-16. It can beseen from Figure 14 that as ro increases, the maximumradial displacement decreases until ro = 4�m and then,increasing ro to 5�m increases the maximum radialdisplacement. Also, it can be seen that in the cylinderswith larger outer radii, the results show less lengthdependency and follow the same pattern. The samebehavior can be observed from Figure 16 for the hoopstresses. Figure 15 shows the radial stress distributionalong the radial direction for di�erent values of ro. Itcan be seen that the outer radius does not have anye�ect on the maximum radial stress.


Figure 14. Radial displacement along the FG cylinderthickness for di�erent values of outer radius, ro.

Figure 15. Radial stress along the FG cylinder thicknessfor di�erent values of outer radius, ro.

Figure 16. Hoop stress along the FG cylinder thicknessfor di�erent values of outer radius, ro.

4. Summary and conclusion

Strain gradient thermo-elasticity formulation for anaxisymmetric thick-walled FG cylinders is presented.The power law of distribution is assumed for variationof material properties in radial direction. A fourthorder governing equation is obtained in strain gradientthermo-elasticity and it is solved numerically togetherwith four boundary conditions. Results show thatthe intrinsic length parameter has a signi�cant e�ecton the stress distribution of thick-walled FG cylindersand increasing the intrinsic length parameter reducesthe maximum radial and hoop stresses, rapidly. Also,it is shown that the power index m does not havemuch e�ect on the results whereas the power indicesn and p have signi�cant e�ect on the results. Inaddition, it is shown that by increasing n the maximumradial and hoop stresses increases, rapidly. It isnoted that increasing p increases the maximum radialdisplacement and hoop stress, whereas it has no e�ecton the maximum radial stress. Furthermore, it isshown that increasing the outer radius of the FGcylinder a�ects the distribution of the radial and hoopstresses.

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Biographies

Hossein Sadeghi received his BS and MS degreesin Mechanical Engineering from Sharif University ofTechnology, Iran in 2008 and 2010, respectively. Heis now a PhD student in Mechanical and AerospaceEngineering at University of California, San Diego.His research interests include solid mechanics, mechan-ics of advanced materials, and acoustic/elastic wavepropagation with emphasis on periodic composites andmetamaterials.

Mostafa Baghani received his BS degree in Mechan-ical Engineering from University of Tehran, Iran in2006, and his MS and PhD degrees in MechanicalEngineering from the Department of Mechanical en-gineering at Sharif University of Technology, Tehran,Iran, in 2008 and 2012, respectively. He is now an as-sistant professor in School of Mechanical Engineering,University of Tehran. His research interests includesolid mechanics, nonlinear �nite element method, andshape memory materials constitutive modeling.

Reza Naghdabadi received his PhD in MechanicalEngineering from University of Waterloo, Canada in1993. He is now a Professor in the Departmentof Mechanical Engineering at Sharif University ofTechnology, Iran. His research interests include solidmechanics, nanomechanics, and computational me-chanics.

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