1_ASHIDA.ppt

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Material Design of a Functionally

Graded Piezoelectric Composite Diskfor Control of Thermal Stress

The Third Asian Conference on Mechanics of Functional Materials and Structures

Indian Institute of Technology Delhi, December 5-8, 2012

Fumihiro ASHIDA, Shimane University, Japan

Sei-ichiro SAKATA, Kinki University, Japan

Hikaru SUZUKI, Koito Manufacturing Co. Ltd., Japan

D1-S1.2: Mechanics of Functional Structures-1, Bharti Building, Room 101

11:10-12:50, Tuesday, December 6


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The vision of the Japan Aerospace

Exploration Agency for next 20 years

includes a project for demonstrating a

hypersonic aircraft with the cruising

speed at Mach 5.

It is considered that a body surface of the hypersonic aircraft will be exposed

to a severe thermal environment.

A safety system that controls the maximum thermal stress is required, because

a thermal load beyond the allowable limit may act on a structural member.

1.1 Background

1. Introduction


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Fig. 1 Analytical model of the previous works

1.2 Previous works (1)

The performance of the stress control was evaluated by the suppression ratio.

R = 1

0max

0max

T

100[%]


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F. Ashida, S. Sakata, K. Matsumoto,

Control of Thermal Stress in a Piezo-composite Disk,

Journal of Thermal Stresses, Vol. 30, No.9-10, pp.1025-1040, 2007.

Piezoelectric layers of equal thicknesses had same electrode arrangements.

Electrodes of same widths were arranged at equal intervals.

A nonlinear optimization problem was solved using the BFGS quasi-Newton

method.

The highest suppression ratio was 15.98%.

F. Ashida, S. Sakata, K. Matsumoto,

Structure Design of a Piezoceramic Composite Disk for Control of Thermal Stress,

Journal of Applied Mechanics, Vol. 75, No.6, CID 61009, 2008.

Piezoelectric layers of equal thicknesses had a same electrode arrangement.

Electrodes of various widths were arranged at different intervals

A linear programming problem transformed from the nonlinear optimization

problem was solved using the Simplex method.

The highest suppression ratio was 33.70%.


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A. Elsawaf, F. Ashida, S. Sakata,

Optimum Structure Design of a Multilayer Piezo-composite Disk for Control ofThermal Stress, Journal of Thermal Stresses, Vol. 35, No. 9, pp. 805-819, 2012.

Piezoelectric layers of various thicknesses had different electrode arrangements.

Electrodes of various widths were arranged at different intervals

The nonlinear design problem was solved using a hybrid optimization technique

combining the PSO and Simplex method. The highest suppression ratio was 40.83% and almost saturated.

In order to increase the suppression ratio substantially, a new structure of a

composite disk should be investigated.



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2. A New Analytical ModelLet us consider a composite disk consisting of a transversely isotropic structural

layer and a functionally graded piezoelectric material (FGPM) layer.

Fig.2 Geometry of a functionally graded piezoelectric composite disk

It is assumed that the FGPM layer consists of homogeneous piezoelectric layers

of class 6mm and the material constants vary gradually in the axial direction.

N

T0 f(r)

Number of electrodes

V0v(r) V

k{H(r r

k) H(r r

k w

k)}

k1

M

Applied Voltages


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3. Flow Chart of Analysis

Analysis of the temperature field Analysis of the elastic and electric fields

Thermoelastic problem

Analysis of the elastic andelectric fields

Electro-elastic problem

Superposition

Response due to a thermal load:

, , , ,

T T T T

i i i i iT u D

Response due to an electric load:

, , ,

E E E E

i i i iu D

Resultant response due to both loads:

, , , ,i i i i iT u D

Solution Techniques Proposed by F. Ashida, et al.

Potential function method for transversely isotropic solids

Potential function method for piezoelectric solids of class 6mm


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4. Variations of FGPM Properties

: piezoelectric constants,eij

mcoefficients to be determined,

n : order of the polynomial,

Fig. 3 Variation of material constants

of the FGPM layer

i

position of the th piezoelectric layerizi

where

Yij: Youngs modulus,

ij: thermal conductivities,

ij: coefficients of linear thermal expansion,

It is assumed that ratios of the material constants of the th constituent piezoelectric

layer to those of the first constituent piezoelectric layer are expressed by

( eij, Y

ij,

ij,

ij)

eij

e1j

,Yij

Y1j

,ij

1j

,ij

1j

1 (m

ej

i1

n

,mY

j ,m

j ,

m

j )z

i

m (i 2,3,,N)


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5. Optimum Design Problem of FGPM LayerThe optimization design problem of minimizing the maximum thermal stress in

the structural layer is defined by

find r={

1

r ,

2

r , ,

n

r},

z={

1

z ,

2

z , ,

n

z},

r={

1

r ,

2

r , ,

n

r},

z={

1

z ,

2

z , ,

n

z},

Yr={

1

Yr ,

2

Yr , ,

n

Yr},Y

z={

1

Yz ,

2

Yz , ,

n

Yz},

e1={

1e1 ,

2e1 , ,n

e1 },e3={

1e3 ,

2e3 , ,n

e3 },e4={

1e4 ,

2e4 , ,n

e4 },

V={V1,V

2, ,V

M}

to minimize fobj

(r,

z,

r,

z, Y

r, Y

z, e

1, e

3, e

4, V)

0max

subject to ( ij

, ij

, Yij

, eij

) 1 (m

j ,

m

j ,

m

Yj ,

m

ej )z

i

m

m1

n

(i 2,3, ,N),

{0.5 ( ir

, iz

, ir

, iz

) 2.0, 1.0 ( Yir

, Yiz

) 2.0,

0.5 ( ei1

, ei3

, ei4

) 1.2,

pc

A (irr

,i

,izz

) pt

A, irz ps

A } (i 1,2, ,N)

where , and are allowable tensile, compressive, and shear stresses.pt pc ps

5.1 Definition of optimization problem


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The optimization variables have strong dependence on each other, namely the

optimum variations of FGPM properties can be obtained only when the

optimum voltages are determined accurately

It is hard to obtain the optimum solution, because there are many optimization

variables.

Points at issues

Development of hybrid optimization technique

5.2 Issues to be solved

The nonlinear optimization problem for determining the applied voltages can

be transformed into a linear programming problem and then the optimum

solution is successfully obtained.

The optimum variations of FGPM properties can be determined using PSO

(Particle Swarm Optimization) which is suitable for solving multimodal

optimization problems.


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5.3 Linearization of the optimization problem

ui

E,i

E ,i

E,Di

E

P

k(u

i

E)k,(

i

E )k,(

i

E)k,(D

i

E)k

k1

M

in which is the magnification factor.Pk

Vk PkVu

When a voltage of arbitrary magnitude is applied to the th electrode, it is

expressed as

Vk

Let the discrete response quantities in the th layer be , ,

and , when the unit voltage is applied to the th electrode only.Vu k

(ui

E)k

(i

E )k (i

E)k

(DiE

)k

i

The nonlinear optimization problem for determining the applied voltages can

be transformed into the linear programming problem for determining the

magnification factors .

In the case where an arbitrary voltage is applied to every electrode, the response

quantities are given by

k

Vk

Pk


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Sub-problem (1) for determining the variations of FGPM properties

Sub-problem (2) for determining the applied voltages

6. Hybrid Optimization Techniquefind

r={

1

r ,

2

r , ,

n

r},

z={

1

z ,

2

z , ,

n

z}

r={

1r ,

2r , ,n

r},z={

1z ,

2z , ,n

z},

Yr={

1

Yr ,

2

Yr , ,

n

Yr},Y

z={

1

Yz ,

2

Yz , ,

n

Yz},

e1={

1

e1 ,

2

e1 , ,

n

e1 },e

3={

1

e3 ,

2

e3 , ,

n

e3 },e

4={

1

e4 ,

2

e4 , ,

n

e4 },

to minimize fobj

(r,

z,

r,

z, Y

r, Y

z, e

1, e

3, e

4, P

* ) 0max

subject to ( ij,

ij, Y

ij, e

ij) 1 (

m

j ,

m

j ,

m

Yj ,

m

ej )z

i

m

m1

n

(i 2,3, ,N),

{0.5 ( ir

, iz

, ir

, iz

) 2.0, 1.0 ( Yir

, Yiz

) 2.0,

0.5 ( ei1

, ei3

, ei4

) 1.2} (i 1,2, ,N)

find P* ={P

1

* ,P2

* , ,PM

* }

to minimize fobj

(P* ) Maxr,z

0rr

, 0

, 0zz

, 0rz

subject to pcA

(irr,i,izz) ptA

, irz psA

(i 1,2, ,N)


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7. Conditions for Numerical Results

7.1 Material constants and dimensionless quantities

Material of the first piezoelectric layer: CdSer z 9 Wm

1K1, (1,3 ) (0.621, 0.551)106 NK1m2 ,

(c11, c12 , c13 , c33 , c44 ) (74.1, 45.2, 39.3, 83.6,13.2) 109 Nm2 ,

(e1, e3 , e4 ) (0.160, 0.347 , 0.138) Cm2 , p3 2.94 10

6 CK1m2 ,

(1, 3 ) (82.6, 90.3) 1012 C 2N1m2 , d1 3.92 1012 CN1

1

0 0

( , , , , , )( , , , , , ) , , ,

i i ki i k k i i k k k k i ik

r r r

d Vr z b c q wr z b c q w B ah V

a Y T a T

Dimensionless quantities

( r, z) (1, 0.5)Wm1K

1, ( 1, 3 ) (1.84, 0.40) 10

6NK

1m

2,

(c11, c12 , c13 , c33 , c44 ) (100.2, 49.8, 6.86,10.9, 2.87) 109

Nm2

,

Yr 74.3 109

Nm2

, r 11.3106 K

1

Material of the transversely isotropic structural layer: CFRP


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Bottom surface:

Top surface:

Biots numbers

Bb 1

Bt 0.1

Layers

Thicknesses:

Number of constituent

piezoelectric layers:

c0

c1

~ c10

0.002N 10

7.2 Parameter settings

Fig. 4 Heating temperature distribution

ro 0.5

2 4

2 4

( ) ( ) 1 2o o o

r rf r H r r

r r

Heating temperature

ro: Radius of heating region

ptA 0.004

Allowable stresses

pcA 0.04

psA 0.002

Tensile stress:

Compressive stress:

Shear stress:

w1

~ w5

0.1

Number: M 5

Widths:

Intervals: q1

0, q2

~ q5

0.1

Electrodes


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(0maxT )

HPM : Maximum thermal stress in the case of the homogeneous

piezoelectric layer

RDV

1(

0max)

FGPM

(0max

T )HPM

100[%]Suppression ratio due to FGPMdesign and piezoelectric actuation

(0max

T )FGPM

: Maximum thermal stress in the case of the designed

FGPM layer

RV

1 (0max )FGPM(

0max

T )FGPM

100[%]Suppression ratio due topiezoelectric actuation

8. Performance of Stress ControlThe control performance of the maximum thermal stress in the structural layer is

evaluated by the two suppression ratios.

(0max

)FGPM

: Maximum resultant stress in the case of the designed

FGPM layer subject to the determined applied voltages

where


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(a) Variation of ir (b) Variation of iz

Fig. 5 Design results for coefficients of thermal conductivity

9. Presentation of Numerical Results9.1 Thermal conductivities


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(a) Variation of ir (b) Variation of iz

Fig. 6 Design results for coefficients of linear thermal expansion

9.2 Coefficients of thermal expansion


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(a) Variation of irY

Fig. 7 Design results for Young's moduli

(b) Variation of izY

9.3 Youngs moduli


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Fig. 8 Design results for piezoelectric coefficients

1(a) Variation of ie 3(b) Variation of ie 4(c) Variation of ie

9.4 Piezoelectric coefficients


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RV [%]0max

0max

T

n

V1 103

V2 103

V

3103

V4 103

V5 103

Designed FGPM layer Homogeneous

CdSe layer1 2 3

0.0868 0.0824 0.0699 0.4662

-0.1291 -0.0622 -0.0370 -0.1811

-0.1618 -0.1367 -0.1227 -0.3537

-0.1993 -0.2018 -0.1994 -0.2950

-0.2278 -0.2436 -0.2449 -0.2759

0.0820 0.0720 0.0694 0.1298

0.0749 0.0652 0.0633 0.1202

8.66 9.44 8.89 7.42

42.30 49.80 51.26

Table 1 Numerical results for designs of FGPM layer

RDV

[%]

Note: The layer thicknesses and the electrode dimensions have not been designed.

9.5 Comparison of numerical results


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10. Concluding Remarks

For a two layer composite disk consisting of a structural layer and a FGPM

layer when five electrodes of the same widths are arranged at the equal

intervals, the variations of FGPM properties and the applied voltages have

been determined by employing the hybrid optimization technique so that the

maximum thermal stresses in the structural layer is minimized.

Comparing the maximum stresses before and after applying the determined

voltages for the case of the designed FGPM layer, the maximum suppressionratio is 9.44%.

There may be possibility of obtaining a higher suppression ratio, when

combined with the optimum designs of the electrodes and layer thicknesses.

Comparing the maximum thermal stress for the case of the homogeneous

piezoelectric layer with the maximum resultant stress for the case of the

designed FGPM layer subject to the determined applied voltages, the maximumsuppression ratio is 51.23%.


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Thank you very much for your kind attention!


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- 0.1073 -0.0968 - 0.1182 - 0.1297

- 0.0990 - 0.0895 - 0.1112 - 0.1201

7.73 7.56 5.93 7.41

23.76 31.06 14.35 7.48

z

r

z

r

(

0max

T )FGM

(

0max)

FGM

RV [%]

RDV [%]

- 0.1305 - 0.1298 - 0.1299 - 0.1298 - 0.1298

- 0.1197 - 0.1201 - 0.1193 - 0.1192 - 0.1201

8.29 7.48 8.11 8.19 7.54

7.84 7.48 8.09 8.20 7.53

Yz

Yr

e1

e3

e4

Table 2 Suppression ratios obtained for each material constant


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Fig. 9 Comparison of radial stress distributions before and after applying

the determined voltages in the case of the homogeneous CdSe layer


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Fig. 10 Comparison of radial stress distributions before and after applying

the determined voltages in the case of the designed FGPM layer


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Fig. 9 Comparison between radial thermal stress distributions in the case of

the homogeneous CdSe layer and radial resultant stress distributions

in the case of the designed FGPM layer


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u0r

T l

1J1

(l

r)l1

( F1

F2

) A0l

T coshlz

B

0l

T sinhlz

D0jlT coshlz

j E0jlT sinh

lz

j

j1

2

u0z

E C00

E l

1J0

(l

r)l1

kj

jj1

2

D0jl

E sinhlz

jE0jl

E coshlz

j

u0zT

1 A00T

zB00Tz2

2

C00

T

l1

J0 (lr)l1

k1 F1 k2 F2

A0lT sinhlz

B0lT cosh

lz

kj

jj1

2

D0jlT sinhlz

jE0jlT cosh

lz

j

u0rE l1J1(lr)

l1

D0jlE coshlz

jE0jlE sinh

lz

j

j1

2

The displacements induced by a thermal load are

The displacements induced by an electric load are

Response in the structural layer


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iE Ci0EzGi0E l1J0 (lr)

nj

jj1

3

l1

DijlEsinhlz

jEijlEcosh

lz

j

i

T 2Ai0

Tz Bi0

Tz2

2

Ci0

TzGi0

T l

1J0

(lr) F

3Ail

Tsinhlz

Bil

T coshlz

l1

nj

jDijlT sinh

lz

j EijlT cosh

lz

j

j1

3

uir

T l

1J1

(l

r)l1

(F1

F2

) Ail

Tcoshlz

Bil

Tsinhlz

j DijlT coshlz

jEijlT sinh

lz

j

j1

3

uirE l1J1(lr)

l1

j DijlEcoshlz

jEijlEsinh

lz

j

j1

3

The radial displacement and electric potential induced by a thermal load are

The radial displacement and electric potential induced by an electric load are

Response in the th constituent piezoelectric layeri


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1 2 3

Particle number 400 400 400

Iteration number 100 600 800

n

Table 3 Parameters for PSO


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1rr

r

1rz

z

(1rr

1

)

r 0,

1rz

r

1zz

z

1rz

r 0

Equation of equilibrium

1rr

u

1r

r

, 1

u

1r

r

, 1zz

u

1z

z

, 1rz

u

1z

r

u

1r

z

Relations between the strains and the displacements

E1r

1

r, E

1z 1

z

1 1 1 0r z rD D D

r z r

Relations between the electric field intensities and the electric potential

Equation of electrostatics

Basic equations for a piezoelectric solid of crystal class 6mm (2/2)

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