2009 Kapuria Buckling

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Assessment of third order smeared and zigzag theories for buckling and vibration

of flat angle-ply hybrid piezoelectric panels

P.C. Dumir, P. Kumari, S. Kapuria *

Department of Applied Mechanics, IIT Delhi, Hauz Khas, New Delhi 110016, India

a r t i c l e i n f o

Article history:

Available online 29 March 2009

Keywords:

Angle-ply

Hybrid panel

Non-linear

Buckling

Forced vibration

a b s t r a c t

A recently developed improved third order theory (ITOT) for angle-ply hybrid piezoelectric plates incylindrical bending is extended to include geometric non-linearity in the Von Karman sense. The trans-

verse deflection is approximated non-uniformly to explicitly account for the transverse strain due to tem-

perature and electric potential. The coupled non-linear equations of motion and the boundary conditions

are derived using the extended Hamiltons principle. The non-linear theory is used to obtain the buckling

and free vibration response of symmetrically laminated hybrid angle-ply panels under inplane electro-

thermomechanical loading. This theory and the third order zigzag theory with additional layerwise terms

for inplane displacements are assessed in direct comparison with the exact 2D piezothermoelasticity

solutions for forced harmonic response, buckling and free vibration response under initial inplane elec-

tro-thermomechanical loading. The comparison establishes the accuracy of the results of the zigzag the-

ory and its superiority over the ITOT for the dynamic and buckling response of angle-ply hybrid panels.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

Angle-ply composite and sandwich laminates with embedded

or surface-bonded piezoelectric layers acting as sensors and actu-

ators are widely used in adaptive structures for active vibration

suppression, acoustic control, shape control, etc. Reviews of vari-

ous laminate theories developed for thermoelectrical response of

laminated smart composite structures have been presented in Refs.

[1,2]. Analytical coupled 2D piezothermoelasticity solutions for

cylindrical bending of angle-ply hybrid piezoelectric simply sup-

ported flat panels under electro-thermomechanical load has been

presented for static [3], dynamic [4] and buckling [5] response.

These solution have revealed that there is significant effect of

transverse shear stress, layerwise distortion of midplane normal

and coupling due to piezoelectric and pyroelectric effects. The clas-

sical laminate theory (CLT)[6], first order shear deformation theory

(FSDT) [79] and third order theory (TOT) [1013] have been pre-

sented based on single expression of displacements across the

thickness. These theories do not account for the slope discontinu-

ities in inplane displacements, and violate shear stress continuity

conditions at the layer interfaces. This discontinuity has been

incorporated in discrete layerwise theories (DLT) [14], which are

accurate but inefficient as the number of variables increases in pro-

portion to the number of layers. Kapuria and his coworkers have

presented efficient zigzag theories (ZIGT) for dynamic and bucklingof hybrid beams [15] and plates [16,17] in which the expressions

are taken layerwise and the transverse normal strain due to tem-

perature and potential is directly accounted for in the approxima-

tion of deflection. The number of variables are reduced to those of

the smeared TOT by satisfying interface and boundary conditions

on transverse shear stresses. These theories yield very accurate re-

sults for hybrid cross-ply laminates.

Recently, the author presented an improved TOT (ITOT) consid-

ering the deformability of thickness due to thermoelectric field and

assessed this theory and the ZIGT for static and free vibration re-

sponse of angle-ply hybrid plates in cylindrical bending. The ITOT

was found to be an improvement over the conventional TOT based

on uniform approximation of deflection across the thickness, but

the ZIGT is found to be superior to the ITOT for static electrome-

chanical response of angle-ply hybrid composite and sandwich

panels. Neither the results based on the analytical solution of the

ITOT and ZIGT nor the assessment of their accuracy with respect

to the exact 2D piezothermoelasticity solution are available in

the literature for the forced vibration and thermoelectromechani-

cal buckling response of angle-ply hybrid plates. The analytical re-

sults of the ITOT and ZIGT are also needed for validating the finite

element solution of the corresponding theory.

The present study is aimed to fill this void in the literature. The

objective herein is to extend the linear ITOT [19] for hybrid panels

to include geometric non-linearity due to deflection in the sense of

Von Karman and use the theory to obtain the initial buckling

response of symmetrically laminated hybrid angle-ply panels

0263-8223/$ - see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruct.2009.03.019

* Corresponding author. Fax: +91 11 26581119.

E-mail address: [email protected] (S. Kapuria).

Composite Structures 90 (2009) 346362

Contents lists available at ScienceDirect

Composite Structures

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under inplane electro-thermomechanical loading. This theory and

the ZIGT of Kapuria and Achary [18] are assessed by direct compar-

ison with the available 2D exact solution for forced vibration re-

sponse [20] under electromechanical loading and buckling

response [5] under inplane electro-thermomechanical loading of

simply supported hybrid angle-ply flat panels of highly inhomoge-

neous, composite and sandwich laminates.

2. Formulation of ITOT

Consider an angle-ply L-layered hybrid piezoelectric flat panel

(Fig. 1) of span a along axis x, thickness h along zaxis and of infinite

length alongy axis. The midplane of the panel is at z 0. The layers

can be orthotropic elastic with a principal material axis along z, or

piezoelectric with orthorhombic class mm2 symmetry with poling

along principal material axis z. The material symmetry direction 1

of the kth layer from bottom is at an angle hk to the x-axis and the

z-coordinate of its bottom surface is denoted as zk1. Thus, the bot-

tom face of the first layer is at z z0 h=2 and the top face of the

Lth layer is at z zL h=2. The interface between the kth and the

k 1th layer is named as the kth interface with z zk.

2.1. Approximations fore, r, T, /, w, u

Consider cylindrical bending of the panel under electro-

thermomechanical load, wherein all entities are independent of

y. Let uxx;z; t; uyx;z; t be the inplane displacements and

wx;z; t be the transverse displacement. The Lagrangian strains

ex; ey; ez; cxy; cyz; czx are related to the displacements, includinggeometric non-linearity due to the deflection w0x; t w0x; 0; t

of the midplane, by

ex ux;x 1

2w2

0;x; ey 0; ez w;z

cxy uy;x; cyz uy;z; czx ux;z wx1

A subscript comma denotes differentiation. Let / be the electric

potential and Ex /;x; Ey 0; Ez /;z, be the components of

the electric field. Unlike most other studies, the inplane electric

field Ex is not considered as zero, since it may be applied by actu-

ation or may be induced by the piezoelectric coupling. The linear

constitutive equations of orthorhombic piezoelectric material of

class mm2 symmetry with poling along z-direction for the stresses

rx; ry; sxy; syz; szx and electric displacements Dx; Dz, using theassumption of transverse normal stress rz 0, are given by

r Qe eT3/;z

bT; s bQc e/;xry Q12ex Q26cxy e32/;z b2T

Dx eTc g11/;x; Dz e3e g33/;z p3T

2

where

e excxy

" #; c

czxcyz

" #; r

rxsxy

!; s

szxsyz

!

Q

Q11 Q16

Q16 Q66" #; eT3 e31e36 !; b b1

b6" #bQ Q55 Q45

Q45 Q44

" #; e

e15

e14

! 3

Qij; eij; bi and gii for axes x; y, at angle hk to the principal mate-rial axes 1, 2, are related to the reduced modulii Qij, piezoelectric

strain constants eij, stresstemperature coefficients bi and electric

permittivitiesgii for the principal material axes 1, 2. The tempera-ture T and the potential / are approximated as piecewise linear

across the thickness in terms of values Tl at nT points at

zlT; l 1; 2; . . . ; nT; and values /j at n/ points at z

j/, j 1; 2; . . . ; n/,

respectively:

T WlTzTlx; t; / Wj/z/jx; t 4

Nomenclature

A; A; A; Al; Al panel stiffness and thermomechanical matricesa; h; L length, thickness, number of pliesc1 damping coefficientDx; Dy; Dz; Ex; Ey; Ez electric displacements and electric fielddij; eij piezoelectric strain and stress constants

F1; F2 stress resultants

F3; Fj6 mechanical and electric loads

bF3; bFj6 mechanical and electric damping loads

Gj; Hj resultants ofDz; DxGij; Yi; mij; Qij shear modulii, elastic modulii, Poissons ratios, re-

duced stiffenessesI; Il; I; Il inertia matricesk0; u0x; u0y; w0x; w0y mid-surface layer and its displacement

variablesL; L; bL; Ln matrices of differential operatorsK; M; KG stiffness matrix, mass matrix, geometric stiffness ma-

trixNx; Mx; Px; S

jx; Nxy; Pxy stress resultants ofrx and sxy

N0

x

; N0

xy

initial stress resultants

P; bP load vectorsp1z; p

2z force applied per unit area on the top and bottom sur-

facesU; Un vector of primary variables, nth Fourier components of

U

p3; p3 pyroelectric constantsqji charge density on the actuated surface

Rk; Rkj; Rkl 2 2 matrices of cubic layerwise functions

S thickness ratio a=h

u; v; w; T; / displacements, temperature, potentialVx; Qx; Q

jx; V

jx; Qy stress resultants ofszx and syz

zk1 zcoordinate of bottom ofkth layerai; bi thermal expansion coefficients, stresstemperature

coefficientsbj

0

; bj; bjw; bjl; bjl; cl panel electromechanical and electrother-mal matrices

cjl; Ejj0

; Ejj panel pyroelectric and dielectric matricesgii; gii electric permittivitiesr; s; e; c stresses and strainse0x ; e

0y ; c

0xy; T0; /

j00 initial uniform strains, temperature and po-tential

Wj/; WlT; W

j/; W

lT interpolation functions and related integral

functionsxn natural frequency for nth spatial mode

U0s ; U0a unknown output voltages, known input voltages

differentiation w.r.t. time

P.C. Dumir et al./ Composite Structures 90 (2009) 346362 347


3/17

where WlTz; Wj/z are linear interpolation functions for T and /,

and summation convention is used for indices l and j. w is

approximated by integrating constitutive equation for ez, by includ-ing only the predominant thermal and potential contributions due

to expansion coefficient a3 and piezoelectric coefficient d33, i.e.,ez w;z d33/;z a3T )

w w0x; t Wj/z/

jx; t WlTzTlx; t 5

where Wj/z Rz

0d33W

j/;zz dz is a piecewise linear function and

WlTz Rz

0a3W

lTz dz is a piecewise quadratic function.

For the improved third order theory ITOT, the inplane displace-

ments ux and uy are approximated as

u u0x; t zw0d zw0x; t z2nx; t z3gx; t 6

where

u ux

uy

!; u0

u0x

u0y

" #; w0

w0x

w0y

" #; n

nx

ny

!;

g gxgy

" #; w0d

w0;x

0 !7

Using Eqs. (5) and (6) in Eq. (2)2 yields

s bQkw0 2zn 3z2g ekWj/z bQkWj/z/jdbQkWlTzTld 8

The conditions of zero transverse shear stresses at bottom and

top surfaces are imposed to express n and g in terms ofu0 and w0.

w0 2z0n 3z2

0g Dj

1/jd D

l3Tld; w0 2zLn 3z

2

Lg

Dj2/jd

Dl4Tld 9

where

D

j

1 Wj

/z0I2 ~e

1

Wj

/z0; Dj

2 Wj

/zLI2 ~eL

Wj

/zL~e1 bQ11e1; ~eL bQL1eLDl

3 WlTz0I2; D

l4

WlTzLI2

10

I2 is a 2 2 identity matrix. The solution of Eqs. (9) gives n and g as

n C1w0 C/

j1/jd C

Tl1T

ld; g C2w0 C

/

j2/jd C

Tl2T

ld 11

where

C1 z2

0 z2L I2=2D; C2 zL z0I2=3D; D z0zLzL z0

C/j1 z2

LDj1

z20Dj

2=2D; C/j2 z0D

j2

zLDj1

=3D

CTl1 z2

0Dl

4 z2LD

l3

=2D; CTl2 zLDl3

z0Dl4

=3D

12

It results in the following expression ofu for ITOT which is of

the same form as in zigzag theory of Kapuria and Achary [17], viz.,

u u0x; t zw0d x; t Rkzw0x; t R

kjz/jdx; t RklzTldx; t

13

where unlike Ref. [17], the functions Rkz; Rkjz; Rklz for ITOT are

the same for all the layers:

Rkz zI2 z2C1 z

3C2; Rkjz z2C/j1 z

3C/j2;

Rklz z2CTl1 z3CTl2 14

The ITOT and the existing theory ZIGT can henceforth be pre-

sented in unified manner. Eqs. (5) and (13) can be written as

u f1zu1 Rkl1 zTl;x; w f2zu2 WlTzTl 15

with

u1 uT0 w0;x wT

0/j;x

h iT; u2 w0 /

j T

f1z I2 zI1 Rkz Rkj

1z

h i; f2z 1 W

j/z

h i 16where I1; R

kj1 ; R

kl1 are the first column of the matrices I2; R

kj; Rkl:

I1 1

0 !; Rkj1 Rkj11

Rkj21

" #; Rkl1 Rkl11Rkl

21

" # 17The elements with indices j mean a sequence of elements with

j 1 to n/. Substituting u; w from Eq. (15) in Eq. (1) yields

e f1ze1 Rkl1

zTl;xx 1

2w2

0;xI1; c f3ze2 CklzTl;x 18

where Cklz Rkl1;zz WlTzI1 and

e1 u1;x; e2 wT

0/j;x

h iT; f3z R

k;zz R

kj1;zz W

j/zI1

h i19

2.2. Governing non-linear differential equations

Let p1z; p2

z be the forces per unit area applied on the bottom and

top surfaces of the panel. Let the idealised distributed linear vis-

cous resistance acting on the top surface of the panel, possibly

due to viscous resistance of the surrounding medium, be

c1 _wx;zL; t per unit area. Let Aji be an internal surface z zji/, where

/ji is prescribed and qji is the extraneous surface charge density on

this surface. The total number of such prescribed potentials is n/.

Considering unit width of the panel in direction y and using the

notation hi PL

k1

Rzk

zk1

. . .dz, the extended Hamiltons principle

can be expressed asZa0

hqduTu qdw w deTr dcTs Dxd/;x Dzd/;zi

p1

zdwx;z0; t fp2

z c1_

wx;zL; tgdwx;zL; t Dzx;z0; td/

1 Dzx;zL; td/n/ qjid/

ji dx

hduTr szxdw Dxd/ija0

0

8 du0; dw0; dw0; d/j 20

Using Eqs. (15) and (18), the inertia and strain energy terms in

Eq. (20) are expressed as

hqduTu qdw w deTr dcTsi

duT1

Iu1 IlTl;x du

T

2Iu2 I

lTl deT1F1 de

T

2F2

w0;xdw0;xNx 21

The inertia matrices I; Il; I; Il are defined by

I; Il

hqfT

1 zf1z;Rkl1 zi; I; I

l

hqfT

2 zf2z;WlTzi 22

I

I11 I12 I13 I14 I15 Ij0

16

I21 I22 I23 I24 I25 Ij0

26

I31 I32 I33 I34 I35 Ij0

36

I41 I42 I43 I44 I45 Ij0

46

I51 I52 I53 I54 I55 Ij0

56

Ij61

Ij62

Ij63

Ij64

Ij65

Ijj0

66

2666666666666664

3777777777777775; Il

Il1

Il2

Il3

Il4

Il5

Ijl6

266666666666664

377777777777775;

II33 I

j0

36

Ij63 Ijj0

6624 35; Il

Il3

Ijl624 35 23

348 P.C. Dumir et al./ Composite Structures 90 (2009) 346362


4/17

The stress resultants F1 and F2 are defined by

F1 hfT

1zri NT Mx P

T SjT

x

h iT; F2 hf

T

3zsi QT Qj

T

x

h iT24

N NxNxyT

hri; Mx hzrxi; Vx hszxi

P PxPxyT hRkzTri; Sjx hR

kj1

zTri

Q QxQyT hRk;zz

Tsi and

Qjx hfRkj1;zzg

T Wj/zIT

1si

25

Using Eq. (4), the electric field terms in Eq. (20) are expressed in

terms of resultants Hj; Gj as

hDxd/;x Dzd/;zi d/j;xH

jx d/

jGj; Hjx hWj/zDxi;

Gj hWj/;zzDzi 26

The loading terms in Eq. (20) are expressed as

F3 bF3dw0 Fj6 bFj6d/j Nxdu0x Nxydu0y Vxd w0 Mxdw0;x Pxdw0x Pxyd

w0y Sjxd

/j;x Hjx V

j/d

/jja0

27

where an over-bar in this expression means values at the ends,Vj/ hW

j/zszxi. The loads F3; F

j6;

bF3; bFj6, are defined byF3 p

1

z p2

z;bF3 c1 _w0 cj0/ _/j0 clT _Tl;bFj

6 cj/ _w0 c

jj0

/_/j

0

cjlT_Tl;

Fj6

p1zWj/z0 p

2

zWj/zL DzLdjn/ Dz0dj1 qjidjji 28

with

cj/ c1Wj/zL; c

lT c1W

lTzL

cjj0

/ c1Wj/zLW

j0

/zL; cjlT c1W

j/zLW

lTzL

29

and dij is Kroneckers delta. Using Eqs. (21), (26) and (27) in Eq. (20)

yields the following equations of motion

I11u0x I12u0y I13 w0;x I14 w0x

I15w0y Ij0

16/j

0

;x Il1Tl;x Nx;x 0

I21u0x I22u0y I23 w0;x I24 w0x

I25w0y Ij0

26/j

0

;x Il2Tl;x Nxy;x 0

I31u0x ;x I32u0y ;x I33 w0;xx I33 w0 I34 w0x ;x

I35w0y ;x Ij0

36/j

0

;xx Ij0

36/j

0

Il3Tl;xx I

l3Tl

Mx;xx Nx;xw0;x Nxw0;xx F3 bF3 0I41u0x I42u0y I43 w0;x I44 w0x I45w0y

Ij0

46/j

0

;x Il4Tl;x Px;x Qx 0

I51u0x I52u0y I53 w0;x I54w0x I55

w0y

Ij056/j0;x Il5Tl;x Pxy;x Qy 0

Ij61u0x ;x I

j62u0y ;x I

j63

w0;xx Ij63

w0 Ij64

w0x ;x

Ij65

w0y ;x Ijj0

66/j

0

;xx Ijj0

66/j

0

Ijl6Tl;xx I

jl6Tl

Qjx;x Sjx;xx H

jx;x G

j Fj6

bFj6

0

30

and boundary conditions which are prescribed values of one factor

in the following products:

u0xNx; u0yNxy; w0;xMx; w0xPx; w0yPxy; /j;xS

jx

w0Mx;x I31u0x I32u0y I33 w0;x I34w0x

I35w0y Ij0

36/j

0

;x Il3Tl;x Nxw0;x

/jIj

61u0x I

j62u0y I

j63

w0;x Ij64

w0x Ij65

w0y

Ijj066

/j0;x Ijl61Tl;x S

jx;x Q

jx H

jx

31

2.3. Electromechanical dynamic equations in terms of primary field

variables

Using e; c; T; / from Eqs. (18) and (4) into Eq. (2) yieldsr; s; Dx; Dz, which are then substituted in Eqs. (24) and (26) toobtain the following constitutive equations of the panel:

F1Ae

1

bj

0

/j0

AlTl

;xx c

lTl

1

2

Aw2

0;x

F2 Ae2 bj0/j

0

;x AlTl;x

Gj bjT

e1 Ejj0/j

0

bjlTl;xx cjlTl

1

2bjww2

0;x

Hjx bj

T

e2 Ejj0/j

0

;x bjlTl;x

32

where A; A; A are the panel stiffnesses, Al; Al are the panel ther-

momechanical coefficients, bj0

; bj0

; bjw are the panel electrome-

chanical coupling matrices, bjl; bjl; cl are the panel electro-thermal matrices, cjl is the panel pyroelectric matrix, and Ejj

0

; Ejj0

are the panel dielectric elements. These are defined in terms of

the material constants by [21]

A;Al;A hfT

1zQf1z;R

kl1

z; I1i; bjl hWj/ze

TCklzi

A;Al hfT3 zbQf3z;Cklzi; bjl hWj/;zze3Rkl1 zibj0 hfT

1zeT

3Wj

0

/;zzi; cjl hp3W

j/;zzW

lTzi

cl hfT1

zbWlTzi;bj

0

hfT3

zeWj0

/zi

Ejj0

hg33Wj/;zzW

j0

/;zzi; Ejj0 hg11W

j/zW

j0

/zi

bjw he3Wj/;zI1i

33

The panel constitutive Eqs. (32) are substituted into Eq. (30) to

yield the coupled non-linear electromechanical equations in terms

of primary variables u0x ; u0y ; w0; w0x ; w0y and /j:

LU::

bL _U LU LnU PbP 34where

U u0x u0y w0 w0x w0y /1 /2 . . . /n/

h iTP P1 P2 P3 P4 P5 P

1

6P2

6. . . P

n/6

TbP bP1 bP2 bP3 bP4 bP5 bP16 bP26 . . . bPn/6h iT

35

L and L are symmetric matrices of linear differential operators in

x. bL contains damping related terms. L; L and bL are given byL11 A11;xx; L34 A34;xxx A12;x

L12 A12;xx; L35 A35;xxx A13;x

L13 A13;xxx; L44 A22 A44;xx

L14 A14;xx; L45 A23 A45;xx

L15 A15;xx; L55 A33 A55;xx

L22 A22;xx; L1;5j0 Aj0

16;xxx b

j0

1;x

L23 A23;xxx; L2;5j0 Aj0

26;xxx b

j0

2;x

L24 A24;xx; L3;5j0 Aj0

36;xxxx A

j0

14 bj

0

3 bj

0

1;xx

L25 A25;xx; L4;5j0 Aj0

46;xxx A

j0

24 bj

0

2 bj

0

4;x

L33 A33;xxxx A11;xx

L5;5j0 Aj0

56;xxx A

j0

34 bj

0

3 bj

0

5;x

L5j;5j0 Ajj0

66;xxxx A

jj0

44 bjj

0

6 bj

0j6

bjj0

4 bj

0j4

Ejj0

;xx Ejj0

36

for j;j

0

1; . . . ;n/. L

n

U are non-linear terms due to geometric non-linearity, which are given by



5/17

LnU1

1

2A11w

2

0;x;x

LnU3

Nxw0;xx Nx;xw0;x 1

2A31w

2

0;x;xx

LnU2

1

2A21w

2

0;x;x;

Ln

U4 1

2A41w2

0;x;x

LnU5

1

2A51w

2

0;x;x;

LnU5j

1

2bjww2

0;x 1

2Aj61

w20;x;xx

37

The non-zero elements of load vectors P and the bP are given byP1 I

l1Tl;x A

l1Tl;xxx c

l1Tl;x

P2 Il2Tl;x A

l2Tl;xxx c

l2Tl;x

P3 F3 Il3Tl;xx I

l3Tl Al

3Tl;xxxx c

l3Tl;xx A

l1Tl;xx

P4 Il4Tl;x A

l4Tl;xxx c

l4

Al2

Tl;x

P5 Il5Tl;x A

l5Tl;xxx c

l5

Al3

Tl;x

Pj6

Fj6

Ijl6Tl;xx I

jl6Tl Ajl

4 bjl bjl cjl

6Tl;xx

cjlTl Ajl6Tl;xxxxbP3 clT _TlbPj

6 cjlT

_Tl

38

3. Analytical solution for simply supported angle-ply panels

3.1. Linear dynamic response

Analytical solution for the linear dynamic response is obtained

for a simply supported angle-ply panel in cylindrical bending for

the following boundary conditions at x 0; a:

Nx 0; Nxy 0; w0 0; Mx 0; Px 0;

Pxy 0; /j 0; Sjx 0; j 1; . . . ; n/ 39

The solution of Eq. (34), without the non-linear terms, satis-

fying the boundary conditions (39), is expanded in Fourier series

as:

w0;/j; T;Nx;Nxy;Mx; Px;Pxy; S

jx;G

j;piz;qj

X1n1

w0;/j; T;Nx;Nxy;Mx; Px; Pxy; S

jx;G

j;piz; qjn sin nx

u0x ; u0y ;w0x ;w0y ;Vx;Qx;Qy;Qjx;H

jx

X1n1

u0x ;u0y ;w0x ;w0y ;Vx;Qx;Qy;Qjx;Hjxn cos nx

40

with n np=a. Substitution of the above expansions in Eq. (34)yields governing equations of motion for the nth Fourier compo-

nent, which are solved for steady state response under harmonic

electromechanical loading following the procedure discussed in

Ref. [16]. The transverse shear stresses s are calculated by integrat-ing the 3D equations of motion.

Fig. 1. Geometry of a hybrid angle-ply infinite panel.



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3.2. Buckling and free vibration under initial stresses

Consider an angle-ply symmetrically laminated panel subjected

to initial uniform inplane normal strains e0x ; e0

y 0, shear strain c0

xy,

uniform temperature rise T0 and actuation potentials independent

of x, y coordinates. This initial equilibrium state is denoted by

superscript 0

. For the symmetrically laminated panel under sym-metrical loading about x-plane,

w00

0; w0x w0

y 0 41

Considering this, the panel constitutive Eqs. (32) and governing

differential equations (30)6 yield

N0x A11e0

x A12c0

xy bj0

1/j

00 c1T0

N0xy A21e0

x A22c0

xy bj0

2/j

00 c

2T0

Ejj0

/j00 bj

1e0x b

j2c0xy F

j06

cjT0

42

where c1 Pnh

l1cl1; c2

Pnhl1c

l2 and c

j Pnh

l1cjl. Defining

U /1 /2 . . . /n/ T; C c1 c2 . . . cn/

T

bk b1k b2k . . . b

n/k

T; F6 F16 F26 . . . Fn/6 T 43with k 1;2, Eq. (42)4 can be written in matrix form as

EU0 b1e0

x b2c0

xy F0

6 CT0 44

U0 is partitioned into a set of unknown output voltages U0s at zj/s

where / is not prescribed and a set of known input actuation volt-

ages U0a at the actuated surfaces. Accordingly, Eq. (44) is partitioned

and arranged as

Ess Esa

Eas Eaa

!U0s

U0a

" #

b1s

b1a

!e0x

b2s

b2a

!c0xy

F06s

F06a

" #

Cs

Ca

!T0 45

Solving Eq. (45) for U0s and substituting it into Eq.(42) yields the

initial stress resultants N0x ;

N0xy

, in terms of the known loading

parameters

N0x A11 bT

1sE1ss b1se

0

x A12 bT

1sE1ss b2sc

0

xy

bT1a b

T

1sE1ss EsaU

0

a bT

1sE1ss F

0

6s c1 bT

1sE1ss CsT0

N0xy A21 bT

2sE1ss b1se

0

x A22 bT

2sE1ss b2sc

0

xy

bT2a b

T

2sE1ss EsaU

0

a bT

2sE1ss F

0

6s c2 bT

2sE1ss CsT0

46

Let the solution for just after buckling/vibration under initialstresses be denoted by ^ on the entities. The size of the buck-

ling/vibration mode U is described by an arbitrary small parame-ter . Thus, bU U0 Uwith Ugiven by Eq. (35)1. Substituting thissolution into Eq. (34), using Eq. (41), N0xy;y 0, N

0x;x 0, and consid-

ering the first order terms in , yield the following equations for U:

LU::

LU 0 0 N0xw0;xx 0 0 0 T

P 0 0 0 0 0 Fj6

h iT: 47

For a set of zero incremental potential at the actuator locations,

zero incremental electric displacement at the unknown potential

locations and zero incremental temperature, the incremental load

Fj6 is zero for index j corresponding to such surfaces. This yields the

following governing equations for buckling/vibration under initial

stresses:

LU::

LU 0 0 N0xw0;xx 0 0 0 T

0 48

The inertia term is zero for the buckling case. The solution of

Eq. (48) for the nth spatial buckling/vibration mode for the sim-

ply supported panel with boundary conditions Eq. (39) is taken

Fig. 2. Configurations of hybrid panels for numerical study.

Table 1

Exact 2D results [20] for fundamental natural frequency x1 for panels (a)(c).

S Open circuit cond ition Closed circuit condition

(a) (b) (c) (a) (b) (c)

5 5.92813 6.40411 3.05690 5.87107 6.33834 3.02339

10 8.11357 8.03394 4.62089 8.00641 7.93293 4.53374

20 9.25291 8.70043 5.66809 9.11085 8.58158 5.52172



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Fig. 3. Distributions of u; w; rx for panel (a) x=x1 :8;c 0:1.

Fig. 4. Distributions of u; w; rx for panel (c) x=x1 :8;c 0:1.



8/17

Fig. 5. Amplitude wm and phase j for panel (a) under load cases (H1) and (H2).

Fig. 6. Amplitude wm and phase j for panel (b) under load cases (H1) and (H2).



9/17

Fig. 7. Amplitude wm and phase j for panel (c) under load cases (H1) and (H2).

Fig. 8. % error of ZIGT and ITOT for amplitude wm and phase j0 under load case (H1).



10/17

as given in Eq. (40), without the summations over n. Substitutingthem in Eq. (48) yields

MUn KUn KGUn 0 49

where Un is the nth Fourier component ofU and KG is the geometric

stiffness matrix with the only non-zero element KG3;3 n2N0

x . K

and Mare the symmetric stiffness and inertia matrices. For synchro-

nous free vibration of natural frequency xn, considering Un Un cosxnt, and partitioning U into the unknown and known partsUs and Ua, Eq. (49) can be written for eUn u0xn u0yn w0n w0xnw0ynU

ns

T as

eK eKG x2n eMeUn0 eK kKG x2n eMeUn0 0 50For the case of free vibration under initial stresses, k 1;

K

G eKG evaluated for the initial stresses, i.e., N0

x and N0

xy. Eq. (50)represents a generalised eigenvalue problem, whose eigenvalues

yield the natural frequencies xn for the nth spatial mode. For thecase of buckling, x 0; KG is evaluated for the initial stresses cor-responding to unit value of the initial loading parameters and the

eigenvalue k gives the buckling load factor. The eigenvalues and

eigenvectors are obtained by QR algorithm after reducing to Heis-

senberg form using NAG subroutines.

4. Numerical results

4.1. Forced vibration

The accuracy of the ZIGT and ITOT for steady state forced

damped response is established by direct comparison with the ex-act 2D piezoelasticity [20] solution for simply supported flat angle-

ply hybrid panels. Three configurations of panels are considered asshown in Fig. 2. Panel (a) is a test panel with layers having highly

inhomogeneous stiffness in tension and shear. Panels (b) and (c)

are composite and sandwich panels. Unless otherwise stated, the

ply-angle h is taken as 15 and 30 for panels (b) and (c), respec-

tively. The interface between elastic substrate and piezoelectric

layer is electrically grounded.

The material constants Y1; Y2; Y3; G12; G23; G31(GPa); m12; m13; m23;q kg=m3; 106a1;a2;a3 K

1); and thermal conductivities

k1; k2; k3 (W/mK) for the materials are:

Material 1: 6.9, 6.9,6.9,2.76,2.76,2.76; 0.25,0.25,0.25,1578;

35.6, 35.6, 35.6; 0.12, 0.12, 0.12

Material 2: 224.25,6.9,6.9, 56.58,1.38,56.58; 0.25, 0.25, 0.25,

1578; 0.25, 35.6, 35.6; 7.2,1.44,1.44Material 3: 172.5,6.9,6.9, 3.45, 1.38, 3.45; 0.25,0.25,0.25,1578;

0.57, 35.6, 35.6; 1.92, 0.96, 0.96

Material 4: 181.0,10.3,10.3,7.17,2.87,7.17; 0.28,0.28,0.33,

1578; 0.02, 22.5, 22.5; 1.5,0.5,0.5

Face: 131.1,6.9,6.9, 3.588,2.3322, 3.588; 0.32, 0.32, 0.49, 1000;

0.0225,22.5,22.5; 1.5,0.5,0.5

Core: 0:0002208; 0:0002001; 2:760; 0:01656;0:4554; 0:5451; 0:99;

3 105; 3 105;70; 30:6;30:6; 30:6; 3:0; 3:0; 3:0

PZT-5A: 61.0, 61.0, 53.2, 22.6, 21.1, 21.1; 0.35, 0.38, 0.38, 7600;

1.5,1.5,2.0; 1.8,1.8,1.8, and d31; d32; d33;

d15; d24pm=V;g11;g22;g33 (nF/m); p3C=m2K 171; 171;

374; 584; 584; 15:3; 15:3; 15:0; 0:0007

Two harmonic load cases are considered for the steady stateforced response:

Fig. 9. % error of ZIGT and ITOT for amplitude wm and phase j0 under load case (H2).



11/17

(H1) Pressure p2z p0 sinpx=a cosxt on the top surface whichis under open circuit condition with Dz 0.

(H2) Actuation potential /n/ /0 sinpx=a cosxt applied to thetop surface.

The bottom surface is grounded /1 0 for both load cases.

The results for these two load cases are non-dimensionalized with

S a=h; Y0 6:9 GPa for panels (a), (c) and Y0 10:3 GPa for panel

(b); d0 374:0 1012 CN1 for all panels; q0 1578 kg=m

3 for

panels (a), (b) and q0 1000 kg=m3 for panel (c):

H1: u; w 100u; w=SY0=hS3p0; rx rx=S

2p0H2: u; w u=10; w=S=Sd0/0; rx rxh=Y0d0/0

The dimensionless damping parameter c is defined as cc1S=2q0ax1, where x1 is the fundamental natural frequency. Thenatural frequency is non-dimensionalized as xn xnaSq0=Y0

1=2. The exact 2D results are obtained using given methodology

of Ref. [20] for the dimensionless undamped natural frequency x1for the first bending mode is presented for ready reference in Table

1 for hybrid panels (a)(c) for both open and close circuit condi-

tions of the top surface.

The through-the-thickness distributions of the dimensionless

inplane displacement ux, deflection and inplane normal stress rx,for angle-ply hybrid panels (a) and (c) under both load cases are

plotted in Figs. 3 and 4 for n 1; x=x1 :8; c 0:1 and S 10.The ZIGT accurately predicts the zigzag variation of the inplane dis-

placements and also the non-uniform distribution of deflection

across the thickness in case of potential load case (H2) quite accu-rately. The ITOT in spite of inclusion of the transverse piezoelectric

normal strain is unable to predict these distributions of the dis-placements accurately and yields inaccurate results for displace-

ments and stresses.

The amplitude wm and phase lag j of the mid-surface deflectionat the centre of the panel, for the two load cases are presented for

S 10 in Figs. 57 as a function of the forcing frequency x, for spa-tial mode n 1 for panels (a)(c). The undamped case c 0 and

damped cases with c 0:1; 0:2 are considered. The percentage er-

rors of wm and j in the ZIGT and the ITOT are plotted in Figs. 8 and9 for the two load cases for the damped case of c 0:1 for thick

S 5, moderately thick S 10 and thin S 20 panels. It is ob-

served from Figs. 8 and 9 that the ZIGT predicts the amplitude and

phase of the forced harmonic response very accurately for all pan-

els for all load cases for the whole frequency range except for small

errors for thick S 5 panels in the neighbourhood of the naturalfrequency x1. In contrast, the errors in the ITOT for both amplitudeand phase are quite large in the whole range of frequency for mod-

erately thick and thick panels with the error increasing in the

neighbourhood ofx1. The error is especially large for panels (a)and (c) having layers of drastically different material properties.

In the range ofx > x1, the error in the deflection amplitude forthe ITOT is moderately larger for the potential load case 2 com-

pared to the mechanical load case 1. Even for the thin panels with

S 20, the maximum error in the ITOT for wm is large being 41.2%,

49.3% for load case (H1) and 49.3%, 56.4% for load case (H2) for

panels (a) and (c), respectively, for damping parameter c 0:1.

The error in wm in the ITOT is large since it is the cumulative effect

of the errors in predicting the static deflection, the natural fre-

quency and the dynamic magnification factor for the forcedresponse.

Fig. 10. Variations of exact 2D results of wm and % error in ZIGT and ITOT with ply-angle h for hybrid panel (b) x=x1 :8;c 0:1.



12/17

The variation of 2D exact result for amplitude wm and % error ofZIGT and ITOT with the ply-angle h are presented for load cases

(H1) and (H2) in Figs. 10 and 11 for hybrid panels (b) and (c). As

expected, the deflection amplitudexm in both load cases increasesas the ply-angle h increases from 0 to 90, due to reduction in the

stiffness in span direction. It is observed that for the hybrid com-

posite panel, for ply-angle hP 15 (which is mostly the case in

practice) the ZIGT yields more accurate prediction of wm than the

ITOT, the error decreasing with the ply-angle. In contrast, the error

in the ITOT reaches a maximum for ply-angle of around 30 and 40

for load cases (H1) and (H2), respectively and reduces thereafter.

For the hybrid sandwich panel (c), the ZIGT yields very accurate

results for the entire range ofh, while the ITOT results have verylarge error in the whole range ofh.

4.2. Buckling and free vibration

Simply supported hybrid panels of three different symmetric

laminate configurations a1; b1 and (c), as shown in Fig. 2, are

Fig. 11. Variations of exact 2D results of wm and % error in ZIGT and ITOT with ply-angle h for sandwich panel (c) x=x1 :8;c 0:1.

Table 2

Exact 2D results and % errors of ZIGT and ITOT for ecr and Nxcr .

Panel S Exact % error in ecr; Nxcr

ecr Nxcr ZIGT ITOT

a1 5 0.3984 2.9319 0.07 9.84

7.5 0.5052 3.7174 0.06 3.13

10 0.5582 4.1074 0.05 3.28

20 0.6216 4.5740 0.03 0.85

b1 5 0.4127 4.0628 0.57 6.44

7.5 0.5388 5.3048 0.48 4.15

10 0.6068 5.9746 0.34 2.73

20 0.6939 6.8317 0.09 0.79

(c) 5 0.4197 1.6060 0.17 44.96

7.5 0.7079 2.7084 0.05 32.44

10 0.9414 3.6020 0.15 23.04

20 1.3940 5.3337 0.11 7.64

Table 3

Exact 2D results and % errors of ZIGT and ITOT for /cr.

S Panel a1 Panel b1 Panel (c)

Exact % error Exact % error Exact % error

ZIGT ITOT ZIGT ITOT ZIGT ITOT

5 0.22 87 2 3 .00 1 3.21 0.47 07 5 4 .1 8 1 0.26 0.12 77 9 0.88 4 6.487 .5 0.29 32 4 1 .9 9 7 .4 1 0.62 38 5 2 .5 5 6 .3 0 0.21 62 3 0.76 3 3.38

10 0.32611 1.32 4.59 0.70778 1.66 4.08 0.28814 0.67 23.68

20 0.36638 0.35 1.23 0.81684 0.48 1.18 0.42805 0.30 7.84

Table 4

Exact 2D results and % errors of ZIGT and ITOT for Tcr and Tcr for panels S 20.

Load case Panel 2D exact % error

ZIGT ITOT

Tcr Tcr Tcr T

cr Tcr T

cr

T1 b1 14.9297 13.6266 8.50 0.25 7.87 0.94

(c) 23.3025 20.2656 1 2.85 0.21 6.30 7.74

T2 b1 6.99 64 6 .7 01 6 3.91 0.32 3.23 1.03

(c) 5.3246 5.1522 3.02 0.23 4.87 8.38



13/17

Fig. 12. % error of buckling temperature for panels b1 and (c) under load cases (T1) and (T2).

Fig. 13. Distributions of u; rx; szx for critical buckling mode for panel (c) for load case (M).



14/17

considered for comparing the results with exact 2D piezothermo-

elasticity solution [5]. Panel a1 is a highly inhomogeneous flat pa-

nel and panel b1 is a hybrid composite panel. Unless otherwisestated, the ply-angle h is taken as 30 for both panels b1 and

(c). The top and the bottom of substrate are grounded.

Four pre-buckling/pre-vibration load cases are considered:

(M) Inplane strain e0x e0 with c0xy 0. The top and bottom sur-

faces are grounded.

(V) Applied actuation potentials at top and bottom surfaces

/zL /z0 /0 with immovable ends e0x c

0xy 0.

(T1) Uniform temperature rise T0 of the panel with top and

bottom surfaces under closed circuit condition /z0

/zL 0 with immovable ends.(T2) Uniform temperature rise T0 with top and bottom surfaces

under open circuit condition Dzz0 DzzL 0 with

immovable ends.

For load case (M), the lowest value of the strain e0 for bucklingis denoted as ecr and the corresponding inplane forces are denotedas Nxcr . For load case (V), the critical value of potential /

0 for

buckling is denoted as /cr. For load cases (T1) and (T2), the critical

Fig. 14. Distributions of u; rx; szx for thermal buckling mode for panel (c) under load case (T2).

Fig. 15. Effect of the ply-angle on Nxcr and/cr and % error in ZIGT and ITOT for panel b1 .



15/17

value ofT0 for buckling is denoted as Tcr. The results for these cases

are non-dimensionalized as follows with S a=h; d0 374

1012 CN1; a0 22:5 106 K1 Y0 6:9 GPa for laminates a1

and (c) and 10:3 GPa for laminate b1.

For load cases (M) and (V):

Nxcr NxcrS3=Y0a; ecr ecrS

2; /cr /crd0S3=a

rx rxS2h=Y0 maxw; u; w Su;w= maxw;

szx szxS3h=Y0 maxw

For load cases (T1) and (T2):

Tcr a0TcrS2; u; w Su;w= maxw

rx rxS2h=Y0 maxw; szx szxS

3h=Y0 maxw

where maxw denotes the largest value ofw through the thickness.

In all cases, the lowest buckling parameter corresponds to mode

n 1.

The exact 2D results ofecr and Nxcr and % errors of the ZIGT andITOT for load case (M) are presented in Table 2 for hybrid panels

a1; b1 and (c) for S 5; 7:5; 10 and 20. It is observed that the

ZIGT yields highly accurate results for ecr for all laminate configu-rations with a maximum error of 0.6% even for thick panels with

S 5. In contrast, ITOT results have errors upto 9.8%, 6.4% and

45.0% for panels a1, b1 and (c), respectively. Even for thin panels

with S 20, the error in ITOT is 7.6% for sandwich panel (c). The

exact 2D results and % errors of the 1D theories for the critical elec-

tric potential are presented in Table 3 for the three panel configu-

rations. It is revealed that the maximum error in the ZIGT for

moderately thick panels with S 10 is 1.7% whereas the error in

ITOT is 4.6%, 4.1%, 23.7% for panels a1; b1 and (c), respectively.

The governing equations of motion in the 1D theories do not

incorporate the pre-buckling transverse normal strain e0z, whereasthe exact 2D piezothermoelasticity solution of Ref. [5] formulated

based on the second PiolaKirchhoff stresses incorporates this ef-

fect. To ascertain the effect ofe0z, the 2D results for buckling tem-perature are obtained considering as well as neglecting e0z, whichare denoted as Tcr and T

cr, respectively. These results and the % er-

ror of the ZIGT and ITOT results with respect toTcr and Tcr are listed

in Table 4 for panels b1 and (c) for S 20 for both the thermal

load cases (T1) and (T2). It is observed that the initial e0z can have

significant effect on the buckling temperature, depending on thelaminate configuration. Therefore, for ascertaining the error due

to displacement approximations across the laminate thickness in

the 1D theories, the errors in the 1D theories for the buckling tem-

perature should be considered with respect to 2D results, Tcr,

excluding the effect ofe0z.The variation of % error in buckling temperatures with the

thickness parameter h=a, is presented in Fig. 12 for panels b1

and (c) for the two thermal load cases with close and open circuit

conditions. It is revealed that though the buckling temperatures of

panels under close and open circuit conditions are considerably

different (as can be seen from Table 4), the errors in the 1D theories

in the two cases differ only marginally. The maximum error in the

ZIGT for the buckling temperature is 1.6% for the given cases,

whereas the error in the ITOT is 3.5% and 24.6% for moderatelythick S 10 panels b1 and (c), respectively.

Fig. 16. Effect of the ply-angle on Nxcr and/cr and % error in ZIGT and ITOT for panel (c).



16/17

The through-the-thickness distributions of modal inplane dis-

placement u, normal stress rx and transverse shear stress szx forthe critical buckling mode at x locations where they are maximum

are compared in Fig. 13 for the hybrid sandwich panel for load case

(M) for S 5 and 10. Similar distributions for thermal load case

(T2) are presented in Fig. 14 for panel (c) with S 10. It is observed

that the ZIGT distributions are in excellent agreement with the ex-

act modal distributions even for the thick case with S 5. In con-

trast, the distributions obtained from ITOT have large error even

for S 10. The 2D exact results for critical load Nxcr and critical

electric potential /cr and % errors of the 1D theory results are plot-ted against ply-angle h in Figs. 15 and 16 for hybrid composite pa-

nel b1 and sandwich panel (c), respectively. Similar plots for the

buckling temperature Tcr under load case (T1) are presented in

Fig. 17. As expected, Nxcr ;/cr and T

cr for buckling decrease with

the increase in ply-angle. The variations of % error in the 1D theo-

ries for Nxcr ,/cr and Tcr with ply-angle follow similar trend as the

steady state deflection amplitude wm.

The % errors in the ZIGT and the ITOT results for the fundamen-

tal flexural frequency x1 of initially stressed hybrid composite pa-nel b1 and sandwich panel (c) are plotted in Fig. 18 for initial

mechanical load (M) with e0=ecr 0:25; 0:75 and potential loadcase (V) with /0=/cr 0:25; 0:25 and thermal load (T1) with

T0=Tcr 0:25; 0:25. As in the case of Tcr , for the same reasons,

the errors in the 1D theories for the natural frequencies under ini-tial thermal loads are obtained with respect to 2D results, xn,

excluding the effect ofe0z. It is revealed that the ZIGT yields veryaccurate results with a maximum error of 2.7% even for thick pan-

els with S 7:5. In contrast, the error in the ITOT for moderately

thick sandwich panel (c) with S 10 is 13.8%, 13.9% and 16.9%

for load case (M), (V) and (T1), respectively. The error in the 1D

theories for the frequency increases with h=a; e0=ecr;/0=/cr andT0=Tcr.

5. Conclusions

From the numerical study, it is inferred that the efficient ZIGT is

generally the most accurate of the two theories, ZIGT and ITOT,

compared herein, for steady state harmonic, buckling and free

vibration of initially stressed hybrid response for angle-ply panels.

The ZIGT yields very accurate prediction of steady state forced re-

sponse under harmonic electromechanical loads, the critical in-

plane strain, potential and temperature for buckling and the

natural frequencies under initial stresses due to inplane electro-

thermomechanical loads of highly heterogeneous composite as

well as sandwich laminates. In contrast, the ITOT results have quite

large errors even for moderately thick (S 10) and even thinner

panels in case of inhomogeneous composite and sandwich angle-

ply laminates. The inaccuracy in the ITOT results is essentially

due to the absence of layerwise terms in the approximation of in-plane displacement in the theory.

Fig. 17. Effect of the ply-angle on Tcr and % error in ZIGT and ITOT for load case (T1).



17/17

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[17] Kapuria S, Achary GGS. A coupled zigzag theory for the dynamics ofpiezoelectric hybrid cross-ply plates. Arch Appl Mech 2005;75:4257.

[18] Kapuria S, Achary GGS. Nonlinear zigzag theory for electro-thermomechanicalbuckling of piezoelectric composite and sandwich plates. Acta Mech2006;184:6176.

[19] Kumari P, Nath JK, Dumir PC, Kapuria S. An improved third order theory andassessment efficient zigzag theory for angle-ply flat hybrid panels. ComposStruct 2008;83:22636.

[20] Kumari P, Nath JK, Dumir PC, Kapuria S. 2D exact solutions for flat hybridpiezoelectric and magnetoelastic angle-ply panels under harmonic load. SmartMater Struct 2007;16:111.

[21] Kumari P. 2D exact solutions and third order zigzag theory for hybrid angle-ply flat panels, M.Tech. thesis, IIT Delhi, Department of Applied Mechanics;2007.

Fig. 18. Percent error in x1 of initially stressed hybrid panels b1 and (c).


2009 Kapuria Buckling

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