On refined nonlinear theories of laminated composite ...

S&than& Vol. 20, Parts 2-4, April-August 1995, pp. 721-747. © Printed in India.

On refined nonlinear theories of laminated composite structures with piezoelectric laminae

J N REDDY and J A MITCHELL

Department of Mechanical Engineering, Texas A & M University, College Station, TX 78843-3123, USA

Abstract. In this paper geometrically nonlinear theories of laminated composite plates with piezoelectric laminae are developed. The formulations are based on thermopiezoelectricity, and include the coupling between mechanical deformations, temperature changes, and electric displacements. Two different theories are presented: one based on an equivalent-single- layer third-order theory and the other based on the layerwise theory, both of which were developed by the senior author for composite laminates without piezoelectric laminae. In the present study, they are extended to include piezoelectric laminae. In both theories, the electric field is expanded layerwise through the laminate thickness. The dynamic version of the principle of virtual displacements (or Hamilton's principle) is used to derive the equations of motion and associated boundary conditions of the two theories. These theories may be used to accurately determine the response of laminated plate structures with piezoelectric laminae and subjected to thermomechanical loadings.

Keywords. Refined nonlinear theory; laminated composite structures; piezoelectric laminae; Hamilton's principle; thermomechanical loading.

1. Introduction

The study of embedded or surface mounted piezoelectric materials (e.g., conventional ferroelectric polycrystals in the form of ceramics, natural crystals, and polyvinylidene (PVDF)) in the active control of structures has received considerable attention in recent years. One reason for this is the possibility of creating certain types of structures and systems that can be adapted to or corrected for changing operating conditions using piezoelectric materials. The advantage of incorporating these special types of materials into a structure is that the sensing and actuating mechanism becomes part of the structure. These type of mechanisms are referred to as strain sensing and actuating (SSA). This advantage is especially apparent for structures which are deployed in space. Generally, space-borne structures are very flexible because they are not designed for operations in which the force of gravity is present. Most actuator systems other than the strain induced type add considerable weight and

721

722 J N Reddy and J A Mitchell

possibly space to the structure and thereby change its mechanical properties significantly.

A laminated structure with piezoelectric laminae receives actuation through an applied electric field, and the piezoelectric laminae send electric signals that are used to measure the motion or deformation of the laminate. In these problems, the electric field that is applied to actuate a structure provides an additional body force, which couples the stress analysis problem to the electrostatic problem. This coupling is similar to a temperature field inducing a body force that couples the mechanical problem to the thermal problem.

The first nonlinear formulation of an electrically polarized elastic continuum is due to Toupin (1956). Following this work, many investigated the behaviour of continua in the presence of electric and/or magnetic fields (Erigen 1963; Tiersten 1964, 1971, 1981; Nelson 1978). Recently, Tiersten (1993) studied electroded plates undergoing large electric fields. In these works, thin, single-layer plates were considered with small strains and quadratic electric fields. A quadratic approximation of the mechanical displacements and a cubic approximation of the electric potential through the plate thickness were used to develop the plate theory.

In order to utilize the strain sensing and actuating properties of piezoelectric materials, the interaction between the structure and the SSA material must be.well understood. Mechanical models for studying the interaction of piezoelectric patches that are surface mounted to beams have been developed by Crawley & de Luis (1987) and Chandra & Chopra (1993). The SSA lamina can offer both a discrete effect similar to patches as well as a distributed effect. Lee (1990) developed a theory for laminates piezoelectric plates using the classical laminate theory. In this theory, the linear piezoelectric constitutive equations were the only source of coupling between the electric field and the mechanical displacement field. Pai et al (1993) presented a geometrically nonlinear plate theory for the analysis of composite plates with distributed piezoelectric lamina. By using a layerwise type displacement similar to that proposed by Reddy (1987), they have enforced continuity of both displacements and interlaminar stresses. However, their model does not include the charge equations of electrostatics. In contrast, Tiersten (1969, 1993) modelled single-layer piezoelectric plates using the charge equations, but they did not study laminated plates. Tzou & Zhong (1993) developed governing equations for piezoelectric shells using the first-order shear deformation theory, and they included the charge equations of electrostatics. From these equations, classical and first-order shear deformation plate theories were derived for single-layer piezoelectric laminae.

In the present study, two different formulations are developed to study the extensional and flexural motions of plates laminated with orthotropic layers and piezoelectric laminae. In the first model an equivalent single-layer third-order theory (Reddy 1984, 1995) is used, and in the second one the layerwise theory (Reddy 1987) is used to approximate the displacement field. In both models, a layerwise expansion is used for the electric field. In addition, the formulations include geometric nonlinearity through the von K~trm~m strains. Variational statements, which are useful in the development of finite element models, and the governing equations of the two models for piezoelectroelasticity are developed from basic principles. Recently, the authors have used the model based on the third-order theory but with linear strains to study the bending and vibration of composite laminates with piezoelectric laminae (Mitchell & Reddy'1994).

Nonlinear theories o f laminated composite structures 723

2. Equations of thermoelectroelasticity

2.1 Thermodynamic principles

There are two principles of thermodynamics: the first law of thermodynamics and the second law of thermodynamics. The first law of thermodynamics, also known as the principle of conservation of energy, states that the time rate of change of the total energy is equal to the sum of the rate of work done by applied forces and the change of heat content per unit time. The second law of thermodynamics places restrictions on the interconvertibility of heat and work done. For irreversible heat transfer, the second law states that the entropy production is positive.

The principle of conservation of energy leads to the well-known heat conduction equation (see Reddy & Gartling 1994)

dT pc v - ~ = - V ' q + pr + a:Vv, (I)

where Tis the temperature, v is the velocity vector, q is the heat flux vector, r is the internal heat generation per unit mass, cv is the specific heat at constant volume or constant strain and a is the stress tensor. Equation (1) is used to determine the temperature distribution in the body. The viscous dissipation a: Vv couples the thermal problem to the stress problem. Even when the viscous dissipation is neglected, the thermal problem is coupled to the stress problem through stress-strain relations, as explained in the next section.

The thermal problem for the solid requires the temperature or the heat flux to be specified on all parts of the boundary enclosing the heat transfer region:

T = T(s, t) on F r, (2)

fi.q + h e ( T - T~) = ~,(s,t) on Fq, (3)

where F is the total boundary enclosing the heat transfer region, F = F r u F~, hc is the convective heat transfer coefficient, T~ is a reference (or sink) temperature for convective transfer, ~, is the specified boundary flux, and fi is the unit normal to the boundary.

2.2 Thermoelastic constitutive relations

The therr0oelastic problem is governed by the strain-displacement equations, equations of motion, thermodynamic equations (1)-(3), and constitutive equations. The constitutive equation of the thermal problem is given by the well-known Fourier's heat conduction law (Carslaw & Jaeger 1959, Nowinski 1978, Reddy & Gartling 1994)

q = - k.VT, qi = - k~i(dT/dxj) , (4)

where k denotes the thermal conductivity tensor of order two. The constitutive equations of thermoelasticity are derived by assuming the existence

of the f ree-energy function, ~P = ~P(ei~, T), of the form

W(t O, T) - 1 _ #ijF, i jO .31_ ½mO, (5) - - -~C ijkl~,ij~kl

such that (see Carslaw & Jaeger 1959; Nowinski 1978)

a o = d~ /~e o = Cijkle u -- flijO, (6)

724 J N Red@ and J A Mitchell

where 0 = T - T o, T O is the reference temperature, and m and flij are material coefficients. Equation (6) is known as the Duhamel-Neumann law for an anisotropic body. Inverting relations (6), we obtain

eij = SiikttrkZ + CtijO, (7)

where Sij u are the elastic compliances, and ~qj are the thermal coefficients of expansion related to flij by flij = Cokt~tkr

2.3 Constitutive relations of hygrothermal elasticity

Temperature and moisture concentration in fibre-reinforced composites cause reduc- tions of both strength and stiffness (Jost 1952; Shen & Springer 1976; Browning et al 1977). Therefore, it is important to determine the temperature and moisture concentration in composite laminates under given initial and boundary conditions. As described in the previous section, the heat conduction problem described by (1)-(4) can be used to determine the temperature field.

The moisture concentration problem is mathematically similar to the heat transfer problem. The moisture concentration, c, in a solid is described by Fick's second law:

~c t3"--t = - - V ' q f + (I)f , (8a)

qf = - D'Vc, (8b)

where D denotes the mass diffusivity tensor of order two, ql is the flux vector, and ~y is the moisture source in the domain. The negative sign in (8b) indicates that moisture seeps from higher concentration to lower concentration. The boundary conditions involve specifying the moisture concentration or the flux:

c = ~(s,t) on r 1, (9a)

fi'qs = ~lf(S, t) on F 2, (9b)

where F = F1 w F2 and quantities with a hat are specified functions on the respective boundaries.

The moisture-induced strains {e}M are given by

{e}M = {a}~c, (10)

where {~t} M is the vector of coefficients of hyoroscopic expansion. Thus the hygroscopic strains have the same form as the thermal strains [see (7)]. The total strains are given by

{e} = IS] {a} + {~}T(T-- To)+ {~t}U(c- Co) , (11)

where T o and c o are reference values from which the strains and stresses are measured. In view of the similarity between the thermal and moisture strains, we will use only thermal strains to show their contribution to governing equations.

2.4 Electroelasticity

Electroelasticity deals with the phenomena caused by interactions between electric and mechanical fields. The piezoelectric effect is a linear phenomenon of electroelasticity,

Nonl inear theories o f laminated compos i t e s t ruc tures 725

and it is concerned with the effect of the electric field and polarization on the deformation (Nye 1957; Penfield & Hermann 1967; Tierstein 1969; Reddy & Gartling 1994). The effect is described by the polar iza t ion vector P, which represents the electric moment per unit volume or polarization charge per unit area. The relation between the stress tensor and polarization vector is known as the direct effect, and it is given by

P = d:tr or P~ = d~jkajk (12)

where d is the third-order tensor of piezoelectric moduli, tr is the stress tensor, and d~i k and trjk denote their rectangular Cartesian components. The converse effect relates the electric field vector E to the linear strain tensor e by

e = E.d or e i j = d u j E k. (13)

Note that dko is symmetric with respect to indices i and j because of the symmetry of the strain components eij.

The coupling between the mechanical, thermal, and electrical fields can be estab- lished using thermodynamics principles and Maxwelrs relations. Analogous to the strain energy density function U o for elasticity and the free-energy density function ~F o for thermoelasticity, we assume the existence of a density function • o

¢~o (eij, El, T) = U o - E . D - ~l T

= ½Cijk l8 i jF , kl - - eijkF, i j E k - - flijF, i jO

-- ½ ekzEkE l -- (pc J 2 T o)02, (14)

called the electric Gibb's f unc t ion or en tha lpy func t ion , such that

000 0~° t?~° 0~° (15) % = ~e~j' Di = - ~E--~-' ~/= - d--T = 00 '

where Di are the components of the electric displacement vector, and ~/is the entropy (which is governed by the second law of thermodynamics). Use of (14) in (15) gives the linear constitutive equations of a piezoelectric medium:

¢T~i = Ci jk lEk l - - e k o E k - - BoO , (16a)

D k = (kkOeij + £klEt, (16b)

pcv ^ = flij~,ij "~- --~0 O, (16C)

where C~jk~ are the elastic moduli, eOk are the piezoelectric moduli, e o are the dielectric constants, flo are the temperature expansion coefficients, c~ is the specific heat per unit mass, and T o is the reference temperature. In single subscript notation, (16) can be expressed as

t~ i = Co~. j - e j iE j - fliO, (17a)

D k = ekze I + euEl , (17b)

~l = flie~ + ~ o O. (17c)


Equations (17) provide linear constitutive relations for the problem at hand. For a general anisotropic material, there are 21 independent elastic constants, 18 piezoelectric constants, 6 dielectric constants, and 6 thermal expansion coefficients.

In the following derivations, we account for thermal (and hence, moisture) and piezoelectric effects only with the understanding that the material properties are independent of temperature and electricity, and that the temperature T is a known function of position (hence, its variation is zero: 6 T= 0). Thus temperature enters the formulation as a known function through constitutive equations (17). However, since we are primarily interested in developing the coupled equations of piezoelectricity, the electric field E will be treated as a dependent variable (i.e., variable subject to variation). It can be shown that for electrostatics, the electric field E is derivable from a scalar potential function ~O

E = - - V~k . ( 1 8 )

2.5 Approximation of electric field potential

In the interest of representing the electric potential more accurately through the thickness of the laminate, it is approximated using the layerwise expansion (see Nowinski 1978; Reddy 1987; Robbins & Reddy 1993)

M

qJ(x,y,z, t) = ~ qJs(x,Y, t)CpS(z), (19) j = l

where ~0s(x, y, t) denote the nodal values of qJ, M is the number of nodes through thickness, and ~s are the global Lagrange interpolation functions for the discretization of ~ through thickness (see Reddy 1993; Reddy & Gartling 1994). This is equivalent to modelling through-the-thickness variation of ~0 (x, y, z, t) with one-dimensional finite elements (see figure 1). Non-piezoelectric lamina can be modelled by simply setting the piezoelectric constants to zero and retaining the dielectric permittivity constants if necessary. In this way it is possible to model arbitrary applied potential loading conditions as well as more than one material type, including piezoelectric material.

Laminate ~ ~ ~ ~

layer 1 1 u node I

Figure 1. Layerwise approximation of a field variable, u, in a laminated plate.

Nonlinear theories of laminated composite structures 727

3. The third-order theory

3.1 Preliminary comments

The equivalent single-layer (ESL) laminate theories are those in which.a heterogeneous laminated plate is treated as a statically equivalent, single layer having a (possibly) complex constitutive behavior, reducing the 3-D continuum problem to a 2-D problem. The ESL theories are developed by assuming the form of the displacement field as a linear combination of unknown functions and the thickness coordinate (Mitchell & Reddy 1994). Then the governing equations are determined using the principle of virtual displacements (or its dynamic version, when time dependency is to be included). For plate structures, laminated or not, the integration over the domain of the plate is represented, because of the explicit nature of the assumed displacement field in the thickness coordinate, as the (tensor) product of integration over the plane of the plate and integration over the thickness of the plate (volume integral = integral over the plane x integral over the thickness). Since all functions of the theory are explicit functions of the thickness coordinate, the integration over plate thickness is carried out explicitly, reducing the problem to a two-dimensional one. Consequently, the Euler-Lagrange equations will consist of differential relations involving the dependent variables (or generalized displacements) and thickness-averaged stress resultants. The resultants can be written in terms of the generalized displacements with the help of the assumed constitutive equations and kinematic relations. More complete development of this procedure is presented in the coming sections.

The classical and first-order theories of composite laminates are the most commonly used ones. The classical laminate theory does not account for the transverse shear deformation, while the first-order theory accounts for transverse shear strains as constant through the thickness and hence requires shear correction factors. Second and higher-order ESL laminated plate theories use higher-order polynomials in the expansion of the displacement components through the thickness of the laminate. The third-order theory of Reddy accounts for quadratic variation of transverse stresses through the thickness, and thus does not require shear correction factors.

3.2 Displacements and strains

Consider a plate of total thickness h composed of N orthotropic layers, some of which may be piezoelectric, with the principal material coordinates ~ k k (X, X2, X3) of the k-th

Figure 2.

active layers with electrode

Schematic of a laminated plate with piezoelectric laminae.

728 J N R e d d y and J A M i t c h e l l

lamina oriented at an angle O k to the laminate coordinate, x (see figure 2). The xy-plane of the plate coincides with the undeformed midplane ~ of the laminate. The z-axis is taken positive upward from the midplane. The k-th layer is located between the points z = z k and z = z k + 1 in the thickness direction. The total domain ~ of the laminate is the tensor product of f l x ( - h/2,h/2). Different parts ofthe boundary [" = r x ( - h/2,h/2) o f ~ are subjected to different generalized forces or generalized displacements. Here F denotes the boundary of ~ the midplane of the laminate. A discussion of the boundary conditions is given later.

The total displacement components (u,v, w) along the coordinates ( x , y , z ) are approximated in the third-order laminate theory of Reddy (1984, 1995) as,

u = Uo + z(~o,, - Co(OWo/OX) - c l z3q,,,,

v = Vo + z ( % - Co(OWo/Oy) - c x z3~o~,

w = w o, (20)

where u o, vo, Wo, q~, and q~y are unknown functions of position (x, y) and time t, and parameters c o and c~ are introduced to identify the classical laminate theory (c o = 1 and q~x = q~, = 0) or the first-order laminate theory (c o = ca = 0) as special cases of the present theory. Note that (u o, Vo, Wo) denote the displacements of a material point at (x, y, 0) along the three coordinates, and (q~x-(OWo/OX)) and (q~y-(Owo/Oy)) denote rotations of transverse n0rmals at z = 0 about the y and x axes, respectively.

The von Khrmfin strains (see Reddy 1995) associated with the displacement field (20) are (~= =o),

l / xx S,, i+zt¢;l+z l-. I (21a)

where

and

Yyz ~. Z 2 ,Co) { +

e~ aUo l(Owo~ 2

e,o,_ OVo + 1 (OWo~ 2 - ,~y - ~ \ - - & - y / ,

~o~ 0 % . 0 % Ow o Ow o

~x'=-~-y +-~x ~ ex ay '

~o,=_ ca (_~_y-, + ~_~q'x ~) Xy

7 ~ o ) _ a ~ - Ow° yz -- "r y + Co--~"~y ,

e , > = 0%, _ 02Wo ~3~ _ 0~0,, =' Ox Co"~-~x2, * , , , , - - c a O x '

(1) C~(PY 02W0 ^(3) 0~0y ~. = ~y - Co--d7 / , ~. = - cr c~y'

Oq~ x 0~o 02Wo ~(I) __ _.k Y ' ~ ,~, - ~y ~W-,~o~-~y,

:j(2) - 0WO ,(0)_ , = - c 2 % , ','~=~o~,+Co-'~- x , r = - - c 2 ~ o ~ ,

4 4 c~=~, c2=~ , e o = l - c o •

(21b)

(22)


It is easy to verify that the transverse shear strains e4 and e s are zero at the upper and lower surfaces of the plate, and that they vary quadratically through the thickness. Although this particular approximation was developed for laminated plates without piezoelectric lamina, it also serves well when piezoelectric lamina are present since contributions to the shear stress on the major surfaces due to piezoelectric lamina are derived from the gradient of the potential function. Note that only the tangential component of the electric field is required to be continuous across boundaries having material discontinuities. A plate having piezoelectric laminae on its outer surfaces covered with a thin electrode will not have any component ofinplane electric field on these surfaces because the electric field is zero inside the conductor. Further, since transverse shear stresses due to piezoelectric effects result from components of the inplane electric field (this depends upon the particular form of the constitutive relations), it follows that shear stresses on the free surfaces are only due to mechanical effects.

3.3 Equations of motion

Hamilton's principle (or the dynamic version of the principle of virtual displacements) is used to derive the equations of motion for a laminated plate with piezoelectric laminae. Hamilton's principle requires that

0 = (g~ + ~ V-- ~K)dt, (23) 0

where ~} is the virtual Gibb's energy (the volume integral of ~RUo), ~ V is the virtual work done by applied forces and possibly charge, and ~K is the virtual kinetic energy. For the problem at hand, these are given by

r ,o, + | / dfloL

(0) (1) (3) d- N~t~yy + Myy6/~yy + Kwfe)-y

~. (0) (1) (3) + NxyoTxy + M , j ~ + Kxy67~

+ 0 + b'xo~x: + + r . , x - I X= : . , y ~ ' y z - - y - i y z

)} -- ,:lE ~ - - ~ - x + F ; W + P~:f~b, dxdy,

--fFf(h/2) {~nn [~Un+ , Ot~WOX~__ClZ3(~l)n ]

(24)


/ 06Wo'~_ClZS6~p~]+ #.~Wo } dzdr

} +

^ / O 6 W o \ + +

OWo . 6K= f no[(Iofio + T, (ox-col:-~x )O~o + (Iefio + T2(ox-col2 ~'~)6(ox

--( Ilfi° +-~2(Ox-c°I20~°~O'~O---~X ]--~X C1(i3fi + [4(Ox_CoI4_~X ),(O x - ~o

+ (i0130 + ~'1 ~br 0Wo' ' (Ilf)o+12qg_col2._~_y)~q~ r --CoIl-~y )~Vo + - . OWo .

_ (11~o + T2~by 0¢%\06~° { "+I,,(oy_col4__~_y)f(oy _Coi2_~.y )__~y _cl ~13v ° - ~b o

+ I°w°fw° 1 dxdy, (26)

where q = qb + qt, qb is the distributed force at the bottom (z = - h/2) of the laminate, qt is the distributed force at the top (z = h/2) ofthe laminate, (#.., a.s, d.z) are the specified stress components along the normal, tangential, and transverse directions respectively, at a point on the boundary F; (6u., 6us) are the virtual displacements along the normal and tangential directions, respectively; p is the density of the plate material; and superposed dot on a variable indicates time derivative, rio = OUo/Ot. The subscripts n and s refer to normal and tangential directions to the midplane ~, respectively.

Note that in (25)-(28), integration over the laminate thickness is carried out and thickness-averaged resultants are introduced as follows:

N,,# } f , h / 2 ) { lz }dz ' IQ~, t f , h / 2 ) I 1 2 t M.# = a.# = a.. dz, (27) K,# J -(h/2) Z3 S~ d-(h/2)

x_ ['~h/2) d~t

[e;J to,3

fl h/2)

I ,= po(z)'dz, ~=l , -c~I ,+ 2, [ ,=~-ct I~+ 2. (29) J - (h/2)

Note that in (29) i takes the values of 0, 1,2,--.,6, and in (27)-(29), ~ and/~ take the symbols x and y. The same definitions hold for the stress resultants with a hat. Note


should be made of the fact that the stress resultants defined above are the total resultants that contain the mechanical, thermal as well as the piezoelectric parts. We shall separate these parts later.

Substituting for 6~, 6 V, and 6K from (24)-(26) into (23), noting that the virtual strains can be written in terms of the generalized displacements (u o, Vo, Wo, q~, q~, ~'t) using (22), integrating by parts to relieve the virtual generalized displacements in £~ of any differentiation, and using the fundamental lemma of calculus of variations (Reddy 1984), we obtain the following Euler-Lagrange equations:

where

ON,,,, ON,, 02Uo - 02tp,~ O3Wo 6Uo: Ox t---~y = l ° - - ~ T + l t - ' ~ - c ° I ~ o - ~ ' (30)

0Nx~ ~ 02v° - 02tP~ 0aw° (31) 6v°: 0x + 0y =l°--~is-+lc-~-~--c°11OyOt2'

(~e~ (~M~ ~,~ ~(~, ~M,,'~- ! ,wo. + + + t,--D-7-x 0y ) ~t,-~--~ oy )d

02W0 02 (02W0 02W0 ~ + Jff(Wo) + q = I0 Ot 2 c212-~ \ Ox 2 + - ~ y 2 )

o2 (0Uo O~o\ _ 02 ( 0 ~ 0~,~, (32) + c ° l ' ~ t, Ox +~)+c°1~t-g-U+Wy) Ol~Ixx O)~'~x~ = _._ 02u0 A 02(px ^ 03W0

&o,,: Ox + Oy - q " = l ~ - f f i - i - + t 2 ~ - c ° 1 2 O x O - ~ " (33)

ONI,,~, OMIt: - 02v° ^ 02tPY Col 2 03w° (34) 6q)': Ox + Oy - ~ = la-~-~- + I2--~-i~- - OyOt 2,

0P~ pt _, (35) Ox + -~y -- ~ = P '

0 { 0w o dwo/ 0 { 00w o 0Wo\ ~(Wo): ~ t , N ~ + N~ '~ . + ~ t.N~"~-x + N " ~ ) ' O.:=Q:-c2S~, ~t:#=M:#-c~K: B,

(36)

(37)

where 0t and fl take on the letters x and y. It must be recognized that the charge pl can only be applied at a conductor dielectric interface. Therefore, unless node I corresponds to such an interface, pl = 0, in most cases, ~k x (nodal value of the electric potential, @) will be specified at such interfaces, and the corresponding equation will not be there. In these cases, the surface charge density is calculated a posteriori.

The natural boundary conditions of the theory are:

( 6 u . , ~ u , ) : N . . - R . . = O , N . , - R . ~ = O ,

06wo \ 6Wo,__~__n ) : ~ - ( ~ . + 0/~., ' \ Co--~-s ] =0 ,

( < ~ ' ~ . , <~<a,): M.. - (M.. - c ~ g..) = O,

P. - P. - O,

coM..-- ~ . . = O,

~ . , - ( ~ . s - c~R.,) = O,

(38)

732

where

J N Reddy and J A Mitchell

63 M ns Q. - 6o(Qxn ~ + Qyny) + Co---~s +~(Wo)

[- [ ~ M x : t~ M ~ t3 f~ o "X ] n ~ + LCo ~,--~ + --~f + l,-ff~ ) - , , ao - r~ ¢.

/ dWo ., dWo'~ ( dWo ~Wo\

(39a)

(39b)

f t i 2 22.,1{ x /~.. n x ny (40a) = 2 e M , , I(/1.~ -- nxny n~ny n x -- ny _] ~(l xy

( K ~ K.. n~ ny (40b)

. 2 _ 2 / ~ / ~ ' y y ' K n s - - n x n y n x n r n x - - ny _l ( Kxy

I 1 P~ =- Pxnx + Pyny. (41)

Thus the primary variables (i.e., generalized displacements) and secondary variables (i.e., generalized forces) of the theory are:

primary variables: u., us, Wo, dWo/dn, q~., ~o,, ~i, (42a)

second variables: N.., N.s, Q~, M.., M,~, h4t, p~. (42b)

Note that the boundary term involving the tangential derivative ofw o was integrated by parts to obtain the boundary term

- ~ r Mt~t~w°.:-~s dF = ~ r ~ - ~ d w o d r - E M ~ s ' W o ] r .

The term in the square bracket is zero since the end points of a closed curve coincide. This term is then added to Q. (because it is a coefficient of fiWo). This boundary condition, Q. + (dM.Jt~s), is known as the Kirchhoff free-edoe condition (see Reddy 1984, 1995).

The initial conditions of the theory involve specifying the values of the displacements and their first derivatives with respect to time at t = 0:

=. = u ° , us = : , Wo = Wo °, ~ . = ~o , ~ , = ~ o ,

t i , ,= .o -o ~'o f ro, tk,, .o • _ .o (43) Un, U s = U s , = = t p n , q g s - - t P s ,

for all points in Q.

3.4 Special cases

The equations of motion of the third-order theory (c o = 1 and 6o = 0) contain, as special cases, the equations of motion of the classical and first-order laminate theories. These are outlined below.


The classical laminate theory

By setting tpx = 0, tpy = 0, c o = 1, c~ = 0, and c z = 0 in (30)-(44), we obtain the equations of m o t i o n and the boundary condi t ions of the classical laminate theory. The equat ions of motion are:

dNxx dN x ~2Uo d3Wo 3Uo: - ~ x + - - - x y - I - - - - I (44) dy -- o dt 2 l dxdt2 ,

d2Vo daWo 6Vo: ~N__~ + = l ° " - ~ T - l l ~ ' (45)

~Wo: \ ~ + dxdy + dy z J +~(w°)+q

d Wo d 2 d :dUo aVo = I o - - - ~ - - I 2 - - ~ \ dx 2 + dy 2 j +ll-~-i-~--~x +--~y ), (46)

dPt dP~ I 6~,z: ~ + 2 ~ _ p~ = pt. (47)

oy

The first-order laminate theory

By setting c o = 0, c 1 = 0, and c 2 = 0 (hence, Co = 1, ~ = r i = 13 in (30)-(37), we obta in the equations of motion of the first-order laminate theory:

t$Uo: dNxx dNx t32u° d2tpx (48) t3x + ~ = l o - - - ~ - ' + I t ~-~-,

fro: dNx~ dN d2v° detPY (49) dx + ~ y = I ° - ~ - + 11 dt 2 '

dQ~ dQy d2Wo dWo: -~x + ~ + ~(wo) + q = Io ~ , (50)

dMxx ~ - Q x d2UO d2(Px (51) d,x: ~ + ~y = I 1 - - ~ - + I 2 ~ - ? ,

6tpy: dMx~ dM~ _ Q = dZvo dZtpy (52) ax + dy Y 1 t ~ + 12 d - ~ '

de t de{ r tS~bt: ~ + __.1 _ p = pr, (53) dy

wherein tpx and ~py take the meaning of total rotations lthey denote only shear rotations in (30)-(36)].

3.5 Constitutive relations

The theory presented here is valid for laminates with arbitrary lamination scheme. Since the laminate is made of several orthotropic layers, with their material axes oriented arbitrarily with respect to the laminate coordinates, the constitutive equations

734 J N Red@ and J A Mitchell

of each layer must be transformed to the laminate coordinates. The linear constitutive relations for the kth orthotropic lamina are

{0"1} ¢k)O'20" 6 --'-- [~i: ~i: Q660 ](k){ cl - m t A T 0 82 -- ot2AT86 -- [0 0 e31](t) {Et } ( k ) ' 0 0 O0 e320 E 3E2

(54a)

[~ ]{} [ o~,.,f,.},., ,~" .V= . o , , , , . O f . (54b)

"}Lo ]I/ i°t i0000,0],, D 2 = 0 e24 0 0 ~4 + ¢22 0 E 2 , D3 e3t %2 0 0 0 % 0 ¢33 (E3) 86

(54e)

,-,(k) _(k) and (k) where ~u, eu, Cu are the plane stress-reduced stiffnesses, piezoelectric moduli, and dielectric moduli, respectively, for the kth lamina in its material coordinate system, (a~, ~, E~, D~) are the stress, strain, electric field, and electric displacement components, respectively, referred to the material coordinate system (x 1, x2, x3), ~1 and 0% are the coefficients of thermal expansion along the x ~ and x 2 directions, respectively, and AT is the temperature increment from a reference state, AT = T - T o.

The constitutive equations (54), when transformed to the laminate coordinates (x, y, z), take the form

... },., ,. -- [iii o2. l,k, ,..,., _ .T

+ 0 'Z~I I~'/'~Yl ' o e,~ j (a,~/az }

- - l ( ~ ) - - ~ay,"lt'k)-r~,, Q::j ~},,,,}:+r e,4 e2, ta=J -LQ4s tTx, l Lels ezs

I Cxx ) i°l[o o , , o1 , , Dy = 0 e24 e2s ~'Y~ I

D, e31 e32 0 0 e-36 J Y" I ~xy J

- % ta~/ay

(55a)

I a~/ax } o,[ (55b)

(55c)


~(k) where ~.~j are the plane stress-reduced elastic stiffnesses, referred to the laminate coordinates

(~xx = Qtt c°s40 + 2(Q12 + 2Q66) sin2OcOs20 + Q22 sin40,

Q12 = Qa 1 cos20 sin20 + 2Q12( cOs40 + sin40) + (Q22 - 4Q66) sin20cos20,

Q22 = Q11 sin40 + 2(Q12 + 2Q66) sin20 cos20 + Q22 c°s40,

Q16 = (Qll - Qx2 - 2Q66)sin O cos30 + (Q12 - Q22 + 2Q66) sinaOc°s O,

Q26 = (QI 1 - Q12 - 2Q66) sinaO cos 0 + (Qx2 - Q22 + 2Q66)sin 0 cos a O,

Q66 = (Qlt + Q22 - 2Qx2) sin2Oc°s20 + Q66( sin20 - cos20) 2,

Q44 = Q** cOs20 + Qs5 sin20,

{~,5 = (Q55 - Q44) c°s 0sin 0,

(~ss = Q,4 sin20 + Q55 cOs20, (56a)

ax, ay and a~r are the transformed thermal coefficients of expansion,

~ = ~1c0s20 + ~2 sin20,

:ty = a 1 sin20 + a2cos20,

2a~y = 2(~ l - a2)sin 0cos 0, (56b)

ei~ are the transformed piezoelectric moduli, and e~x, ere and ¢xv are transformed dielectric coefficients

~31 = e31 c°s20 + e a2 sin2 0,

e36 = (e31 - ea2)sin 0cos 0,

~24 = e24 cos20 + e15 sin20,

e25 = (el 5 - e2a)sin 0cos 0,

Cry = e l l sin20 + ~22 cOs20'

e32 = e31 sin20 + e32 cOS20,

el4 = (ets - e2,,)sin Ocos O,

e15 = ets cOs20 + e24 sin20,

Cxx = el t COS2 0 + e 22 sin2 O,

¢xy = (el1 -- e22)sin Ocos O.

e33 = e33,

(56c)

3.6 Laminate constitutive relations

The stress resultants in (27) and (28) can be expressed in terms of the strains using the lamina constitutive equations. We have

N.t= ) .dz Nxy j J-h~2 ~ axr

All At2 A16 /~(~o ~) ~#,, = At2 A22 A26 ~%y + Bx2 B22 B26 /-yy

Ax6 A26 A66 (7(x °) B16 B26 B66J [e~J

I ' ' f NL } Et l E12 El6 [~xx ! _~_ [E12 E22 E26 ~.q(3)~ + ~. {.3/2 i#i__ tgyT | :YY [

• ,3, I I : l ( , L ( N L E16 E26 E66 (r : r j

(57)

II II

u II

II It

'~-'

~

~ .~

.~

~ ~'

~,

~

.~

~.

~ ~,

~

~

~.

~ ~

+ ~

+

I I

I ~

l I

II

I I

, ,

~ ~

~ ~, ~

.

~,.

~

+

7, lv

j =

¢h

In~

I I

~n~

n~n

II fl

II

lb.

-F

71",

~e

~ ro

=

I ~

,~

,

I I

Nonlinear theories of laminated composite structures

where {NT}, {MT}, and { K ~ } thermal force resultants

N Z k + l

(KT} = z (. [Q](k)(a)(k)A~z3 dz,

and various laminate stiffnesses are defined as follows

This completes the development of the third-order theory with piezoelectric laminae.

4. Layerwise theory

4.1 Displacements and strains

In contrast to the equivalent-single-layer theories, the layerwise theories are developed by assuming that the displacement field is Cm-continuous through the thickness of each layer of the laminate (see Reddy 1987, 1995; Robbins & Reddy 1993). This amounts to using ESL displacement fields in each (physical or numerical) layer of the laminate. Thus the displacement components are continuous through the laminate thickness but the transverse derivatives of the displacements may be discontinuous at various points through the thickness, thereby allowing for the possibility ofcontinuous

738 J N R e d d y and J A Mi tche l l

transverse stresses at interfaces separating dissimilar material layers. A layerwise displacement field provides a kinematically correct representation of the moderate to severe cross-sectional warping associated with the deformation of moderately thick to very thick laminates.

In the layerwise theory of Reddy (1995), the total displacement field is represented as

N

u(x, y, z, t) = Y~ UAx, y, t)~(z), I = 1

N

VAx, y, t)~(z), v ( x , y , z , t ) = 1 = 1

M

w(x,y,z,t)= Z I = 1

where (Ul , lit, Wt ) denote

W~(x, y, t) Wt (z), (68a)

the values of (u, v, w) at the fih interface through the thickness, N is the number of nodes through the thickness, and ~i and the global interpolation functions for the discretization of the inplane displacements through thickness (see figure 1); M is the number of nodes and ~PJ are the global interpolation functions for discretization of the transverse displacement w through thickness. Note that independent approximations for the inplane and transverse displacements are assumed (~Pt#~) . Consequently, inextensibility of transverse normals (i.e., w = w(x , y ) ) can be imposed, if desired, by simply setting M = 1 and ~p1 = 1 for all z.

The electric potential is also represented using a layerwise approximation

N

~ b ( x , y , z , t ) = ~ ~bt (x ,y , t )~ t (z ) , (68b) I = 1

where ~O1(x, y, t) denote the nodal values of ~O, N is the number of nodes through thickness, and ~ are the global Lagrange interpolation functions for the discretization of q, through thickness.

For linear variation through each numerical layer, the functions ~ t are given by

• ~(z) = ¢~ ' ( z ) , z~ ~< z ~ z~,

¢~'(z) = ~'(2 ' - ')(z), z t_ , <~ z <<, z t (I = 2, 3 . . . . . n), [qJ~(z), z~ <<.z<<.zi÷~

¢9N(z) = ~(2")(Z), ZN- 1 <<" Z <~ ZN, (69a)

~k~ k) = 1 -- ~./hk, d/~ k) = e/hk, 0 <~ ~. <~ hk, (69b)

where N = n + 1, n is the number of numerical subdivisions through the thickness, h k is k denotes the z-coordinate of top of the the thickness of the kth layer, ~ = z - z k, and z,

kth numerical layer. Any desired degree of displacement variation through the thickness is easily obtained by either adding more linear finite element sub-divisions through the thickness or using higher-order elements through the thickness. The layerwise concept is very general in that the number of subdivisions through the thickness can be greater than, equal to, or less than the number of material layers through the thickness. Note that the sublaminate concept can be used (i.e. the number of thickness subdivisions is less than the number of material layers). The layerwise


displacement field (68a) contains, as special cases, the displacement fields of the ESL theories described in § 3.

The yon Kfirmfin nonlinear strains associated with the displacement field (48a) are:

W M

i=i-'~-x - ~ I = i ~---x- I\I=I ~x )

eye, = ~= , ay . ,,, =, - ~ y tF

u dtg~

I = 1

= i= t V:-~-z + 1 = 1 0 y

7xz = ~ dO I

1=1 I=1 ~ X

,=, \-~-y T ; / ,=, ~ / \,~=, ~y /

Note that the transverse normal strain is nonzero, and that the strains are discontinuous at the layer interfaces because of the layerwise definition of the functions ~J and WJ.

4.2 Equations of motion

The governing equations of motion for the present layerwise theory can be derived using Hamilton's principle. The virtual Gibb's energy 6~, virtual work done by applied forces 6 V, and the virtual kinetic energy 6K are given by

Ioo{L [ ~ & ~ + %y~%y a..&~.

+ e~y~7~y + a~.~7~ + ay.~Ty.

-- (D x ~SEx + Dy ~E r + D, ~E z) ] dz } dxdy

+ I = I L Qx--'~X + y-~--y + Q--~Ial~ WI

J = l (~ xx 19X "~- Nxy-~-y ) ~x

+ \ "-g; + "--gf/--~yJ_l

740

where

J N Reddy and J A Mitchell

- [~.~u~ + #~6u~ + #.~6w] dzdF ? d - h i 2

= - - fno [q~W1 + qt6WM + Pt~01]dxdy

] ~1 n ^ I s - - (N.,cSU 1 + N.,,~U,) + E QI. ~W, dr,

I = 1 I = 1

o d - h i 2

= 11.~((:16(_:, + f,,tfg) + ~. "[w i:i,,1c 5 dxdy, o l , J = l l , J = l

(72)

(73)

x'= ,=,~ ~ ~, xx ~x + " ey : +-6y [ N"-T; + "-fly/ ' (74)

= IN,, t= lay, I~l'U/Sdz, (75) N~, I ay, OOldz, ~tJ [ N,, ) 0- h/2 l, a,, ;

Q,,J ..,-,./2 [ % z J d-T ' [Q,J = .,, -,./2 ( % = J

fh/2 d~1 . I - h i 2

i I J : f h / 2 = [ h / 2 po C~1@Jdz ' [Is poWIWZ dz. d - hi 2 d - h~ 2

The Euler-Lagrange equations of the layerwise theory are

(76a)

(76b)

(77)

aN~,, ~ y ~ 1:02Us 8U': Ox + _Qt= I - ~ , (78)

d = l

dV1: dx + ~ - Q' = ~ Iuc32 V~ (79) J = 1 t i t2 '

~WI: -~xdQ~ _Q,O ~I _ ~.i qb611 q,6i M ~ ~ l j ~ 2 W j + ~ 0 ' , + + + = (8o) S = l C~t2 ' vy

c~P~ dP~ ~ t : _~_x +__._z_p =pt. (~y z (81).


The natural (force) boundary conditions of the theory are:

I AI __ I ^I __ N,,-N,,-O, N,~-N,,-O, ( ~ 5 + ~ ' ) ^ ' - - O,, - 0, (82a)

P~ - P~ = 0, (82b)

where (N~., Nt.,) have meaning similar to (N.., N.,) introduced in § 3:

-'n -'n = Qx x 4- Qy y, u 7 = U i n x 4- Vjtn , , usl = - U z n x 4- V lny , (83a)

I-(Rt, aw, +-.,aw,', (-.,aw, -.,ow,)_ q ~a= .,=x ~ L \ ~ dx N~y-~y )n~+ N~y ~---~ + N . d--y- % j. (83b)

Thus, the primary variables (displacements) and secondary variables (forces) of the layerwise theory are:

Primary variables: U~, U~, W I, d~ r, (84a)

Secondary variables: N~., N'.~, 05 + ~I, p~. (84b)

4.3 Laminate constitutive equations

The stresses in the kth layer may be computed from the 3-D stress-strain equations. For the kth (orthotropic) lamina we have

I ~xx O'yy

0"zz • O'y z

( ~xz Gxy

(k) 'Cll C12 C13 0 0 C16 C12 C22 C23 0 0 C26 C13 C23 C33 0 0 C36 0 0 0 C44 C4s 0 0 0 0 C'45 C55 0

Cx6 C26 C36 0 0 C66

X

exx - ~xx AT ey~ -- ~tyxAT

~yz ~2xz

~y - 2%yAT

Ck)

+

0 0 0 0 0 0

g',,, e24 e,s e2s 0 0

e31 (k)

e32e33 ( Ol[I/OX } l ~*/~Y ' o ( aC,/~z

0 e36

(85)

Dy = 0 0 e24 e25 • ~y: Dz e31 e32 e33 0 0 e36 J ~x,

(86)


where the ,~(k) ~,) and p.(k) "~ij, eu , -U are the transformed material coefficients in the (x, y, z) system. If inextensibility of transverse normals is assumed, one may use the plane stress-reduced stiffness in place of the 3-D stiffnesses. Note that the strains at a layer interface depend on the layer, i.e., {e} k # {e} k+ 1 at a point P on the interface of layers k and k + 1.

The laminate constitutive equations for the layerwise theory can be derived using the lamina constitutive equations (56) and (57) and the definitions for the stress resultants in (75) and (76). In the layerwise theory distinction must be made between the number of physical layers (N,) and the number of numerical layers (n) in the laminate. The integrals in the definition of the resultants must be evaluated only over the thickness interval over which the global interpolation functions, Oz and ~F z, are defined.

The relations between the stress resultants and the strains can be expressed as

N / y t = ffYY t~Idz N~rj ~=1 z: -xy

[ A1S AlS2 AI~

= ~ ~AlS2 a is A1S

J = I L A ~ 6 a ~ A ~

or, Ox ov, + Oy

OUs OVj - i f+ ~--;

(B',~) '=xtB~j

+ I " A/Sa/~ Alsr

M ~ 1 2 E AIJK AIJK

~A12 "A22 J,K= 1 AIJK AIJK

. ' ~ 1 6 ' ' 2 6

AIJK 16

AIJK 26

66

ow, 0%, 2 Ox Ox 1 OW s OW K 2 0 y Oy ow, 0%, Ox Oy

+ x I N~(T) } xx NI(T) - - y y

NIIT) --xy

(87)

-iJ

( Nx, ) '% utllU~S d7 ̀

IAmbS ar,s ar,s " ' 1 2 " ' 1 6 ~ 1 KI AKIJAKIJ

= A _ _ "t " '22 "A26 K = I I A KIJ A KIJ A KIJ

L " 1 6 ~ ' 2 6 ~ ' 6 6

OUr Ox

ov,, Oy

OUr OVK --~-y +-~

KIJ

w~

~b

I

! !

I

÷

H

I I

I I

I I

I ~ ~l

II

L~L

~

A

H

I,, I

I I

H

L~ L~

b,q Lt~ b,q ,~"

L~L

~ I

I

H

I I

÷

I I

I I

H

I '

I

! i

IE

÷

~r~ cp~

L~

II


N M Z --IJ ,f~'KJl ~i'KJl ~'KJI~ e:,¢, , + E --36 ) --32 t.--31

J=l J,K=I

1 t3WK t3W, 2 t~x t3x

1 c~W K ~W: 2 tgy tgy ' (92)

~wK ~w+ ~x <~y

N. ['z, ~ dqjl ~'z-- X J " ' - - k=l z~ 6z* dz dz

-- ~ [- . , ~ U d j I O V j + B J I ( ~ U K - =, k . h - ~ ; + n23-&-y 3~ \ -~y +-~x~)]

1 " ( t~W: 3W~ e,:r ~gW: OW K + ~ ,.Xo, _B'~ 0-7 ~-~ + " Vy ~y

M N

D33 W j - Z $! G33~-~ , J= ' J=l

and the thermal stress resultants are defined as

= /£12 C22 C23 C26 / ::: A r O l d z ,

N~T , k=t =:Eel 6 C,26 C36 C66 .2a%

|)/~/'(T)-'YY = ~'~ / ] ( 1 2 C22 C23 ~-'26 / ::: l [K/:(T) k=t 'J zlEC16 C26 C36 C66

2BI:g ~ W: ~ W x + 36 -~x ~ y J

(93)

(94)

AT~Pt~P: dz, (95)

gl(T) = Zft (Cl3(Xxx + C23Qtyy -Jr- C330tzz + 2C36otxy)<k)AT dz. k = I Z~

(96)

In the above equations, Ne denotes the number of physical layers in the laminate, ~ and are coordinates of the bot tom and top of the kth layer, and the laminate stiffnesses are

defined as follows:

IJ __ f --(k) 1 A u - C.U • 4~dz, (97a) k=l

N. [~; --IJ _ --(k) I J A , j - ~ (97b) Ci. / • qJ dz,

k = I Jz~

A.I!K= --~k) I : k Ci j (~ ~ ~l dz, (98) M k=l d k z b

N. Iz, ~ AIJKP __ --(k) I J K P ,j - ~ (99) Cij ~P ~F ~P W dz,

k=1 Jz~


Bi~: = ~ f f - ( k , I d~P: _ (lOOa) k=x I Cu ~ dz d,~,

s. /'z~ cid, s B,']= ~, i :~.~.)~'-- d, (lOOb)

k=l Jz: - ' ~ - dz - - '

s, i"z, ~ clqJl / C79.)---~" qJz ~Pr d~ • (101) B~':K=. ,=~f- Jz:-'~ dz . . . . .

i" -(k)d~ dtI~ DOH__ k=~t Jz: Cu dz dz dz, (102a)

s, fz; _ k . d ~ d ~ S /~//= X " - - - - dz, (102b) k=l z~ Cu dz dz

N. f~,' E, ' ] = Y % w T uz, (103) k = l k z/j j-;

FlS )~I (104a) k= 1 : dz

-(k) d(I)~ J Ff/= X I (104b) k=t. z~ eu dz ~ dz,

N. rz, ~ aa~K F/~ x i j ( IZ, (105)

k = t Jz: dz

- t : ~xJ ~i:~ N. [ 'z f (Exx, -yy, -xy, = ~ (exx, e~,y, ¢xy)(k)O~(1)Sdz, (106a)

k= 1 Jz~

N. fz; k dt~i dtypl "is (106b) E,, = E Tz dz, k = 1 Z~

G ~ = ~ e " - - - - dz. (107) ~=1 z: ~ dz dz

Note that the sum over the number of layers in the definition of the laminate stiffnesses is limited to the numerical layers over which the functions ~ and ~P~ are defined. The functions ~ and ~Pt are defined at the most over two adjacent numerical layers. When the von Kfirmfin nonlinearity is not included, all laminate stiffnesses with three and four superscripts will not enter the governing equations.

5. Closure

In this. paper two theories are presented "for extensional and flexural motion of laminated plates with piezoelectric laminae. The first one is based on a third-order


expansion of the displacement field and the second one is based on a layerwise expansion of the displacement field through the laminate thickness. The first one contains, as special cases, the classical and first-order laminate theories. In both theories, the scalar electric potential is approximated through the laminate thickness to allow independent and higher-order prescription of the electric field. Linear constitutive equations of thermoelectro-elasticity and the yon Khrmfin nonlinear strains in the laminate are used in the development. The theories developed herein can be used to study the response of composite laminates with either surface mounted or embedded piezoelectric laminae.

The model based on the third-order laminate theory is computationally simple because it contains only five dependent variables. It is adequate for predicting global response characteristics or preliminary design of laminated structures. However, the theory does not predict interlaminar stresses accurately in thick composite structures. The model based on the layerwise theory is essentially a three-dimensional model, hence computationally more expensive. However, the model can be used to determine the local stress fields accurately. In practice, a combination of the two models - ESL model in (global) regions of 2-D stress state and layerwise model in (local) regions of 3-D stress state will prove most appropriate. Such hierarchical (or variable kinematic) models with the global-local strategies (Reddy & Robbins 1994) provide the most robust computational methodology for the analysis of practical structures.

The theories presented here can be extended to the case where the plates are subjected to large electric fields (Tiersten 1993). This allows modelling of a larger class of phenomenon - electrostriction and higher-order permeability effects. For quadratic nonlinear constitutive behavior of the electric field, the enthalpy function is assumed to be a quadratic function of strains and a cubic function of the electric field,

1 1 t~ o (eij , Ei, T) = ~ CijklF, ijekl -- eokeijE k -- ~ bijkteoEkE l -- flijeijO

~e , jE ,E i - -12 , j kE iE jEk - - l q , ,E , E j O - - P ~ O 2, (108, 2 210

so that 1

6ij = CijktSkt -- eijkE k - -~bouEkE t - [3o0 , (109)

1 D k = eijkF, ij + ejkE k Jr- ~ i j k E i g j -a t- qjkEk O, (100)

1 pc tl = fl,je~j + ~qoE,Ej + To 0, (111)

where bijkt denote the effective electrostrictive constants, '~ijk a r e the electric susceptibi- lity constants, and q~i are the pyroelectric coefficients.

References

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Chandra R, Chopra 1 1993 Structural modelling of composite beams with induced-strain actuators. AIAA J. 31:1692-1701

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