Nonlinear Static and Buckling Analysis of
Piezothermoelastic Composite Plates:
Part 1 – Formulation
Balasubramanian Datchanamourtya and George E. Blandfordb1
a
Staff Engineer, Belcan Engineering Group, Caterpillar Champaign Simulation Center,
1901 S 1st Street, Champaign, IL 61820
b
Department of Civil Engineering, University of Kentucky, Lexington, KY 40506, USA
Abstract
Part 1 of the paper presents the geometric nonlinear finite element formulation for
deformable piezothermoelastic composite laminates using first-order shear deformation
theory. Green-Lagrange strain-displacement equations in the von Karman sense represent
geometric nonlinearity. Numerical approximation of the nonlinear equations uses a mixed
finite element formulation in terms of an independent discretization of the transverse shear
stress resultants in addition to the displacement, rotation, and electric potential variables.
Thermoelastic and pyroelectric effects are part of the constitutive equations. Hierarchic
Lagrangian interpolation functions express the inplane and transverse displacement
variations and electric potential variables while approximation of the transverse shear stress
resultants at the Gauss quadrature points use standard Lagrangian shape functions.
KEYWORDS: mixed finite element, geometric nonlinearity, smart composite, and von
Karman
Corresponding author: Telephone number – (859) 257-1855; Fax number – (859) 257-4404; and e-mail –
gebland@engr.uky.edu
1
1
1. Background
Piezoelectric materials exhibit the property of generating an electric potential when
subjected to mechanical deformations and this phenomenon is the direct piezoelectric effect.
The converse piezoelectric effect by which the material changes shape when an electric
voltage is applied is widely used in the actuation and control of vibration in mechanical
devices. Piezoelectric materials, also known as smart materials, find their application in
aerospace structures, piezoelectric motors, ultrasonic transducers, microphones, etc.
Configuring smart composite structures involves bonding piezoelectric layers to the top and
bottom of a multilayered composite elastic laminate. The piezoelectric layers act as
distributed sensor and actuator to monitor and control the static and dynamic response of the
structure.
Jonnalagadda et al. (1994) derived analytical solutions to piezothermoelastic
composite structures using Reissner-Mindlin plate theory and compared the results with
finite element solutions. Heyliger et al. (1994) analyzed the piezoelectric coupled-field
laminates considering the three translational displacements and the electrostatic potential as
variables. They used a piecewise linear variation of the displacement and potential
unknowns through the thickness with the out of plane displacement either variable or
constant. Mitchell and Reddy (1995) developed a linear composite piezoelectric plate model
using a third-order shear deformation theory and a layerwise variation of electric potential
variables in the thickness direction. Xu et al. (1995) presented three-dimensional solutions
for the coupled thermoelectroelastic behavior of multilayered plates using a mixed
formulation. Carrera (1997) presented a mechanical model of multilayered plates with
piezoelectric layers by including a zig-zag variation of inplane displacements and a parabolic
2
distribution of electric variable through the thickness; von Karman geometric nonlinearity
was also included in the formulation.
Heyliger (1997) derived exact solutions for the static response of composite
piezoelectric plates with simply supported boundary conditions. Saravanos (1997) developed
a new mixed field theory for the analysis of laminated shell structures using the equivalent
single layer assumptions for displacements and a layerwise approximation of the electric
variables. Lee and Saravanos (1997) studied the coupled piezothermomechanical behavior of
smart composite plate structures using a bilinear 4-node 2-D finite element. Blandford et al.
(1999) developed a coupled finite element model to study the static response of composite
beams subjected to electrical and thermal loading. They used the Reissner-Mindlin shear
deformation theory and hierarchical quadratic, cubic and quartic approximations for the field
variables along the longitudinal direction of the beam and a piecewise linear variation for the
electric variables. Lee and Saravanos (2000) extended the multifield laminate theory of
Saravanos (1997) to study the impact of coupled piezoelectric and pyroelectric effects on
piezothermoeletric plates. Kapuria and Dumir (2000) studied the response of cross-ply
laminated, rectangular piezothermoelectric plates utilizing the first-order shear deformation
theory. Varelis and Saravanos (2002) formulated a coupled geometrically nonlinear finite
element formulation for predicting the response of smart beam and plate structures. Their
coupled model for composite piezoelectric plate structures used eight-node serendipity twodimensional finite elements. They predicted buckling of multilayered beams and plates and
studied the effects of electromechanical coupling on the buckling load. Lage et al. (2004)
investigated the behavior of piezolaminated plate structures using a mixed variational
formulation and a layerwise finite element model. They approximated transverse stresses,
3
transverse electrical displacement, translational displacements, and electric potential as
primary variables. Kapuria (2004) presented a zig-zag third-order theory for piezoelectric
hybrid cross-ply plates combined with a layerwise linear zig-zag variation of electric
potential.
In this paper, a mixed finite element formulation for piezothermoelastic composite
laminates based on Reissner-Mindlin plate theory is developed. Geometric nonlinearity in
the von Karman sense is considered. Displacement and electric potential degrees of freedom
are discretized using hierarchic quadratic, cubic, and quartic Lagrangian finite elements.
Element level transverse shear stress resultants, interpolated at the Gauss quadrature points
using standard Lagrangian shape functions, are condensed. Nodal temperatures vary linearly
through the entire depth of the plate while electric potentials change piecewise linearly
through the laminate thickness. Eigenvalue and geometrically nonlinear analysis results for
both mechanical and self-strained loadings are in the companion paper (Datchanamourty and
Blandford 2008). These results demonstrate the piezoelectric coupling effect on the critical
loads.
2. Governing Equations
Constitutive equations for a typical layer k of a multilayered piezothermoelastic
composite laminate in the Reissner-Mindlin sense relative to the plate geometric coordinate
axes x, y and z are (see Figure 1)
{}k  [Q]k {}k  [e]T
k {E}k  {}k 
(1a)
{D}k  [e]k {}k  []k {E}k  {P}k 
(1b)
4
where {} =  x  y  xy  yz zx
T
second Piola-Kirchhoff stress vector which is the work
conjugate of Green-Lagrange strain vector {} =  x  y  xy  yz  zx
Ex E y Ez
T
electric field vector; {} =  x  y  xy 0 0
T
T
; {E} =
= temperature-stress vector;
    R ;  = temperature; R = reference temperature; {D} = D x D y D z
displacement vector; {P } = 0 0 Pz
T
T
= electric
= pyroelectric vector; [Q] = material stiffness matrix;
[e] = piezoelectric material matrix; and [] = electric permittivity matrix. The over bar in
the material coefficient matrices and vectors denote transformation from the principal
material axes to the laminate global Cartesian coordinate system as shown in Appendix A.
For non-piezoelectric lamina, the piezoelectric terms are zero.
Expanding equation (1a) using the results of Appendix A gives
 x 
 Q11 Q12
 

 y
Q12 Q 22
 
 xy   Q16 Q 26

 
0
 0
yz
 
 0
 
0
 xz k 
 0

 0
e
 31
0
0
e32
Q16
0
Q 26
0
Q66
0
0
Q 44
0
Q 45
0
0
e36
e14
e24
0
0 

0 
0 

Q 45 
Q55  k
 x 
 
 y
 
 xy 
 
 yz 
 
 xz 
 1 
 
e15  E x 
 2 
  
e25  E y   6  
0
0   E z 
k  
k
 0 k
T
(2a)
Similarly, the expanded form of the electric displacement equation (1b) is
5
 0
D x 

 
D y    0
 

 D z k  e31
0
0
e14
0
0
e24
e32
e36
0
 11 12

 12 22
 0
0

0 

0 
33 
 x 


e15    y 
 

e25    xy 
0    yz 

k 
 
 xz k
E 
x
 
E
 y 
 
k
 E z k
0
 
0 
 
Pz 
(2b)
k
The plate displacements are
u(x, y, z)  u o (x, y)  z  x (x, y)
v(x, y, z)  vo (x, y)  z  y (x, y)
(3)
w(x, y, z)  w o (x, y)
where u, v, and w are the inplane displacements along the x, y and z axes, respectively;
superscript  denotes midplane displacement; and x and y are rotations about the negative
y-axis and positive x-axis, respectively.
The Green-Lagrange strain vector components in terms of the displacements with the
nonlinear strains included in the von Karman sense (e.g., Reddy 2004) are
2
x 1  w o 
u 1  w 
u o
x 
 


z
 


x 2  x 
x
x 2  x 
2
 y 1  w o 
v 1  w 
vo
y 
 
z
 

 
y 2  y 
y
y 2  y 
 xy 
2
2
(4a)
 y  w o w o
 
u v w w u o vo




z x 

y x x y
y
x
x  x y
 y
6
 yz 
v w
w o

  y 
z y
y
(4b)
u w
w o
 xz 

  x 
z x
x
where  x and  y are the inplane extensional strains;  xy is the inplane shear strain; and  xz
and  yz are the transverse shear strains. Equations (4a, b) in vector form are
    ox 
 
x 
 x   o 






z

 y  y
 y

 



  xy    oxy 
  xy 


 x

0



 y

0
 o
  u 
 
y   vo 
 


x 
 N 
 x 
 N
 y  
 N
  xy 
 

 x

z 0

 

 y
  wo


0 
 x

  x  1 
     0
y   y  2 

o

 w
 
x 
 y

0 
  
o  
 w  x  o
 w
y    
 y
 w o   
x 
(5a)
u o 
x  1
 [D ]    [D  ]    [A  (w)]{D N }w o  {o }  z{}  { N }
 y  2
 vo 
  yz 
 
  xz 

 y


 x
 o
w o 
1  w 
 
 
  x   { w D}  [Is ]  x   {s }



 
1 0   y 


 y

0
(5b)
Combining the strain expressions in equations (5a) – (5b) leads to
N
 L 1 N

{b }  {L
b }  { }  [Db ]  2 [D (w)] {u}
(6a)
{s }  [Ds ] {u}
(6b)
[0]3x2 [A  (w)]{D N } [0]3x2 
 [D ] {0}3x1 [0]3x2 
where [DL
; [D N (w)]  
;
b ] 

[0]
{0}
[D
]
[0]
{0}
[0]
3x2
3x1



3x1
3x2 
 3x2
7
o
T
[Ds ]  [0]2x2 {w D} [Is ] ; { L
b }         = linear strain vector;


{ N }    N   0  T = nonlinear strain vector; and {u}   u o vo w o x y T .
The electric field vector, which is the negative of the potential gradient, is
Ex
Ey
Ez
T
 



x
y
z
where {} =  / x  / y  / z
T
T
  {}  (x, y, z)
(7)
= gradient vector. Electric potential  is assumed to
vary piecewise linearly through the thickness of the laminate. Thus, the potential at any
point within a piezoelectric layer  is
 z1  z  
 z  z
 (x, y, z)  
  b (x, y)  
 t

 t




M1 (z)
M 2 (z)
 
  t (x, y)


 (x, y) 
 b




  M  (z) { (x, y)}


 t (x, y) 

(8)
where subscripts t, b = top, bottom of the piezoelectric layer; b ,  t = electric potential at
the bottom and top of the piezoelectric layer  ; t = thickness of piezoelectric layer  ; and
 z1  z 
 z  z 

M1(z)  
 and M 2 (z)  
 are the depth interpolation functions for
 t 
 t 




electric potential. Substituting equation (8) into equation (7) gives
{E}   {} M (z) { (x, y)}   [Z ][D ]{ (x, y)}
(9)
where
 M 1

[Z ]   0

 0
M 2
0
0
0
M 1
M2
0
0
0
0 
electric potential

0
0  = depth interpolation

1 t  1 t  
matrix; and
0
(10a)
8

 x
[D ]  

0

0

x
T

y
0
0

y

1 0
 =

0 1

electric potential inplane
gradient matrix.
(10b)
Applying thermal loading by specifying the temperature on the top and bottom
surfaces of the laminate induces thermoelastic and pyroelectric effects. Assuming  varies
linearly through the entire depth of the plate
1 z
1 z
(x, y, z)     b (x, y)     t (x, y)
2 h
2 h
 (x, y) 
 M 1 (z) M 2 (z)  b
  M  (z) {(x, y)}
 t (x, y) 
(11)
where b , t = bottom and top surface temperatures of the plate at z = h/2 and z = h/2,
1 z
1 z
respectively; M 1 (z)     and M 2 (z)     are the depth interpolation functions
2 h
2 h
for temperature; and {}  b (x, y) t (x, y) . Using equations (8) and (11), the stress and
electric displacement equations (1a) and (1b) in terms of inplane and transverse components
are

{p }


 
{

}
 s 
k

[Qp ] [0]  
{p }


 
 
{

}
 [0] [Qs ] k 
 s 
k
T
[es ] 
 [0]



 e   0   [Z ][D ]{ (x, y)}
 p
 k
{ p }
 
 M (z) {(x, y)}
{0}

k

{Ds }

D  
 p 


[es ]  {p }
 [0]
 e   0   
 
 p
 {s }
(12a)
 [s ] {0}



 0    [Z ][D ]{ (x, y)}
p


9
 {0} 
   M  (z) {(x, y)}
P
 p 
(12b)
where p, s = inplane, transverse shear components; and Pp  Pz .
The stress resultants per unit width of the plate are
{N},{M},{Q}  h 2 {p }, z{p}, {s } dz
h 2
where h = plate thickness; {N} =  N x
=  Mx
Ny
(13)
N xy T = inplane stress resultant vector; {M}
M xy T = moment resultant vector; and {Q} =  Q y
My
Q x T = transverse
shear stress resultant vector, which are shown in Figure 2. The resulting equations are
{N} 


{M}
T
[A] [B]  {o }  { N } [A e ] 
 {} 



T
{}
 [B] [D] 
 [Be ] 
{Q}  [S]{s }  [S1e ]T
[A  ]
[B ]  {}
  


{}  [Se2 ]T {}
x
y
(14a)
(14b)
Using the strain-displacement equations (5a, b) and (6a, b), the stress resultants are
N
T
1
{N}  [C]([D L
b ]  [D (w)]){u}  [Ce ] {}  [ ]{}
2
   N   M  T  {Nu }  {N }  {N }
(15a)
{Q}  [S][Ds ]{u}  [Se ]T [D ]{}  {Qu }  {Q }
(15b)
where { N  } =   N   M  T ; { N  } =  N
x
 M
x
My
T


M
xy  ; { Q } =  Q y
Ny

T
N
xy  ; { M } =
T
Q
x  ; and  = u,  or . The electric potential
inplane gradient matrix and electric potential vector for the laminate are
2
[D ]  diag [D1 ] [D
]

{}   1b 1t    2b  2t 
NP 
[D
]

  bNP  tNP 
(16a)
T
(16b)
10
Depth integrated material coefficient matrices for the laminate are
T

[A] [B] 
T
T [A e ] 
; [Ce ] 
; []  [A ] [B ]
[C]  

[Be ]T 
[B] [D]
NL
[A], [B], [D]    [Qp ]k  (z k 1  z k ),

k 1
(17a, b, c)
1 2
1

(z k 1  z k2 ), (z3k 1  z3k ) 
2
3

(17d)
[Ae ]T  [Ae ]T [Ae ]T
1
2

[Ae ]T 

NP 
(17e)
[Be ]T  [Be ]T [Be ]T
1
2

[Be ]T 

NP 
(17f)
1 
1
[Ae ]     ep  ; [Be ]  (z1  z )[Ae ]
2
1
(17g, h)
1
1
1
1

[A  ]   {A  }  {B }
{A  }  {B }
h
2
h
2

(17i)
1
1
1
1

[B ]   {B }  {D }
{B }  {D }
h
2
h
2

(17j)
NL
{A }, {B }, {D }   { p}k  (z k 1  z k ), 12 (z k2 1  z k2 ), 13 (z3k 1  z3k ) 
(17k)
k 1
NL
[S]  [K s ]  [Qs ]k (z k 1  z k )[K s ]
(17l)
k 1
[Se ]T  [Se ]1T [Se ]T
2

[S1e ] 
t   e14

2  e14
[Se ]TNP  ; [Se ]T  [S1e ]T [Se2 ]T [0]



t
e15 
; [Se2 ]k 

e15 
2
 e24
e
 24
e25 
e25 
(17m, n)
(17o, p)
where NP = number of piezoelectric layers; and NL = total number of layers. Since the
actual variation of transverse shear stresses in a plate is not constant through the depth,
11
k
Reissner-Mindlin theory introduces a shear correction matrix [Ks ]   s2
 0
0 
in the
ks1 
depth integrated transverse shear coefficient matrix. Coefficients k s1 and ks2 are shear
correction factors. Electric potential between adjacent layers is continuous. This paper
assumes that a grounded interface exists between a piezoelectric layer and a structural layer,
i.e., electric potential is zero.
3. Coupled Mixed Variational Principle
Using the modified Hellinger-Reissner functional facilitates independent interpolation
of displacement, electric potential and transverse shear stress resultant variables. The
mechanical energy functional for a mixed variational formulation is
M =
1
1

 ε p  {pu }dV +   ε p  {
p }dV + V  ε p  { p }dV

2 V
2 V
(18)
1
M
   εs  {s }dV    s  [Qs ]1{s }dV   ext
V
V
2
where  M = modified Hellinger-Reissner functional that represents mechanical (elastic)
M
energy;  ext
= potential energy due to externally applied mechanical loads; V = volume of

the plate; { up } , {
p } and { p } = elastic, piezoelectric and thermoelastic contributions to
the inplane stress vector {p } , respectively; and the other symbols are as previously defined.
Separating the plate volume integral in the above equation into integrations through the depth
and over the plate area (A) and introducing the stress resultants from equations (15a, b) to
replace the depth integrated stresses, then substituting the stress resultants with the material
equations given in equations (14a, b), and finally the strain-displacement relationships of
equations (6a, b) gives
12

 

 
T
1
L
1 [D N (w)] {u} [C] [D L ]  1 [D N (w)] {u} dA
[D
]

b
b
2
2
2 A
M =
1
 Q  [S]1{Q}dA

A
2
+
A [Ds ]{u}

A [Db ]  12 [D

T
1
L
1 [D N (w)]{u} [C ]T {}dA
[D
]

b
e
2
2 A
T
L
{Q}dA 
N

(w)]{u}

T
(19)
[] {θ}dA



M
   Q  [S]-1 [Se ]T [D ]{} dA  ext
A
The piezoelectric energy functional is
P 

1
1
   {D u } dV      {D  } dV

2 V
2 V
V   
{D  }dV
(20)
P
 ext
P
where P = piezoelectric energy functional;  ext
= external work due to applied electric
potential; and {Du } , {D  } and {D} = piezoelectric, dielectric and pyroelectric
contributions to the electric displacement vector {D}, respectively. The depth-integrated
piezoelectric energy functional is
P 
1
1
   [Ce ]{b }dA   {[D ]{}}T [Se ]{s }dA

2 A
2 A
(21)
1
P
  {[D ]{}}T [C ][D ]{}dA   {[D ]{}}T [P ]{}dA  ext
A
A
2
where the depth integrated dielectric and pyroelectric matrices for the laminate are
[C ]  diag  [C ]1 [C ]2
[C ] 
z1
z

[C ]NP 
(22a)
[11] [12 ]
 T

[Z ] [] [Z ]dz  [21] [22 ]
 [0]
[0]
[0] 
[0] 
[33 ]
(22b)

13
[ij ] 
(ij ) t   2 1 
(33 )
(i,
j
=
1,
2);
[

]

33 
1 2
6
t


[P ]  diag  [P ]1 [P ]2
[P ] 
z1
z
 1 1
 1 1 


[P ]NP 
(22e)
[0]2x2 
0


 T 
[Z ]  0   M   dz  [0]2x2 
P 
 z 
(22c, d)
(22f)
 [P] 


z1  z

1


(Pz )
h

[P]  
z1  z
2 
 1 

h
z1  z 

h

z1  z 
1 

h

1
(22g)
4. Finite Element Approximation
The displacements, electric potentials, and transverse shear stress resultants in
equations (19) and (21) are functions of the plate inplane coordinates. Thus, discretization
uses two-dimensional finite elements. Hierarchic Lagrangian shape functions interpolate
displacement and electric potential variables (see Figure 3). The discretized element
variables are
{u e }   N1 [I5 ] N 2 [I5 ]
 {uˆ e } 
1


 {uˆ e }2 
e
N nen [I5 ] 
  [N u ]{uˆ }


 e

{uˆ }nen 
(23a)
ˆ ] [N
ˆ ]
{ e }  [N
1
2
 {
ˆ e }1 


e

ˆ
{ }2 
e
ˆ
[N
  [N  ]{ˆ }
nen ] 


 e

ˆ }nen 
{
(23b)
14
 Ni
0

ˆ ]0
where [N
i


 0
0
Ni
Ni
0
0
0 
= electric potential shape function matrix
0


Ni  2NP x (NP 1)
which enforces continuity of the variables between adjacent layers; Ni = ith node shape
function; [I5] = 5 x 5 identity matrix; {uˆ e }i and {ˆ e }i = ith node displacement and potential
vectors of element e; and nen = number of element nodes. For i = 1 to 4, the shape functions
correspond to the corner nodes of the element and the nodal variables associated with these
shape functions are the degrees of freedom. For i = 5 to nen, nodal shape functions are
hierarchic and variables associated with these shape functions represent higher order
derivatives of the degrees of freedom (Zienkiewicz and Taylor 2000).
Interpolation of the transverse shear stress resultants Q y and Q x at the Gauss
integration points (see Figure 4) uses standard Lagrangian shape functions
{Qe }   N1 [I 2 ] N 2 [I 2 ]
ˆ e} 
 {Q
1


ˆ e} 
 {Q
2
ˆ e}
N nens [I 2 ] 
  [N Q ]{Q


 ˆe

{Q }nens 
(23c)
where Ni = shape function corresponding to the ith transverse shear interpolation point; nens
ˆ e } = ith
= number of element transverse shear stress resultant interpolation points; and {Q
i
node transverse shear stress resultants vector of element e.
Under thermal loading, top and bottom surface temperatures are specified at the
corner node points and the intermediate values within an element are interpolated as
15
{ˆ e } 
1


e
{ˆ }2 
e
 e   [N  ]{ˆ }
{ˆ }3 
 e 
{ˆ }4 
{e }   N1 [I 2 ] N 2 [I 2 ] N 3 [I 2 ] N 4 [I 2 ]
(24d)
where Ni = corner node bilinear shape function; {ˆ e }i = ith node temperature vector of
element e.
Using the shape functions of equation (23a, b) in equations (6a, b) and (7), the
element strain-displacement and electric displacement-electric potential matrices are
L
L
L
[BL
b ]  [D b ][N u ]  [[Bb1] [Bb1]
L
[Bbnen
]]
 [Bi ] {0}3x1 [0]3x2 
[BL
bi ]  

[0]3x2 {0}3x1 [Bi ] 
(25a)
(25b)
[Bi ]  [D ][I Ni ] ;
[Bi ]  [D ][I Ni ]
(25c, d)
[Bi ]  [D ][I Ni ] ;
N
[I Ni ]   i
0
(25e, f)
[Bs ]  [Ds ][Nu ]  [[Bs1] [Bs2 ]
[Bsi ]  [0]2x2 {w Bi } Ni [Is ] ;


[B N ]  [D N ][N u ]  [[B1N ] [B2N ]
[A  (w)]
[BiN (w)]  
 [G]i
 [0]3x2 
0
Ni 
[Bsnen ]]
{w Bi }  {w D}Ni
N
[Bnen
]]
(25g)
(25h, i)
(25j)
(25k)
16
  N 

{w}
0


Ni

 x

0
0



N
x
[A  (w)]  
0
{w} ; [G]i  
y
0 0 Ni



y
  N 

N
{w}
{w}


x
 y

[B ]  [D ][N ]  [[B1] [B2 ] [B3 ]

0 0

0 0

(25l, m)
(25n)
[Bnen ]]
ˆ ]
[Bi ]  [D ][N
i
(25o)
Combining the mechanical and piezoelectric energy functionals of equations (19) and
(21), using equations (25a) – (25o), the total energy functional is
1
 e
1 N
e   uˆ e    {[BbL ]  1 [B N (w)]}T [C] {[BL
b ]  2 [B (w)]}da  {uˆ }
2
a
2

ˆ Q
ˆ   1 [N ][S]1[N ]T da  {Q
ˆ e}
+  uˆ e    [Bs ]T [N Q ]da  {Q}
Q
 2 a Q

 a

T
1 N
 ˆe
  û e    {[BL
b ]  2 [B (w)]} [  ][N  ] da  {θ }
 a
T
T
1 N
 ˆ e}
  û e    {[BL
b ]  2 [B (w)]} [Ce ] [N  ] da  {
 a
(26)
ˆ e   [N ]T [S]-1[S ]T [B ]da  {ˆ e }
Q
e

 a Q



1

  ˆ e    [B ]T [Se ][S]1[Se ]T [B ]  [B ] T[C ][B ] da  {ˆ e }
a
2

  ˆ e    [B ] T[P ][N  ]da  {ˆ e }  eext
 a

where e = total energy functional of element e;  eext = total external work due to
mechanical and electrical loading of element e; and a = area of a typical element.
The structure reaches the state of equilibrium when the energy functional is
stationary. Thus solution to the problem can be obtained by seeking a set of values for the
degrees of freedom that renders the energy functional a stationary which is attained by taking
17
a variation of the energy functional and equating it to zero. Taking the variation of the
energy functional gives


N
T
L
1 N
 e
 e   uˆ e    {[BL
b ]  [B (w)]} [C] [Bb ]  2 [B (w)] da  {uˆ }
 a

ˆ e }+  Q
ˆ e   [N ]T [B ] da  {uˆ e }
+  uˆ e    [Bs ]T [NQ ] da  {Q
s
 a

 a Q

ˆ e   [N ][S]1[N ]T da  {Q
ˆ e}
  Q
Q
 a Q

N
T
 ˆe
  û e    {[BL
b ]  [B (w)]} [  ][N  ] da  {θ }
 a
N
T
T
 ˆ e}
  û e    {[BL
b ]  [B (w)]} [Ce ] [N  ] da  {
 a


ˆ e   [N ]T [S]-1 [S ]T [B ] da  {ˆ e }
  Q
 e   
 a Q
1 N
 e
  ˆ e    [N ]T [Ce ] [BL
b ]  2 [B (w)] da  {uˆ }
 a

ˆ e}
  ˆ e    [B ]T [Se ][S]1[NQ ] da  {Q
 a



  ˆ e    [B ]T [Se ][S]1[Se ]T [B ]  [B ] T[C ][B ] da  {ˆ e }

 a
(27)
  ˆ e    [B ] T[P ][N ]da  {ˆ e }   eext  0
 a

ˆ e } ; variables v (v = u,
ˆ e } ; {Qe }  [NQ ]{Q
where {u e }  [N u ]{uˆ e } ; {e }  [N ]{
, Q) = x,y variation of the dependent variables, variable v̂ = nodal degrees of freedom; and
 = variational operator. Since the variations of the nodal degrees of freedom are arbitrary,
the terms associated with them are individually equal to zero. Grouping the terms
multiplying similar variables leads to the following set of equations

u
 uu
 
  [K uu ] [K u ] [K uQ ]   [K N ] [K N ] [0]   {uˆ e }  {f u }
N
 
  u
   e  

u

Q 

[0]
[0]   {ˆ }   {0} 
 [K ] [K ] [K ]   [K N ]
 
  Qu
   ˆ e   {0} 
Q
QQ
} 
e
  [K ] [K ] [K ]   [0]
[0]
[0]   {Q



e 

e

18
{f u }  {f u} 

 


 

 {f  }  {f  }
 {0}   {0} 

 

e
e
(28a)
where {f u }e = mechanical load vector of element e; and {f  }e = electrical load vector of
element e. The element coefficient matrices and thermal load vectors are
T
L
[Kuu ]e   [BL
b ] [C][Bb ]da
(28b)
T
[KuQ ]e  [KQu ]T
e   [Bs ] [NQ ]da
(28c)
[KQQ ]e   [NQ ][S]1[NQ ]T da
(28d)
[Ku ]e  [Ku ]eT   [BbL ]T [Ce ]T [N ]da
(28e)
a
a
a
a
[KQ ]e = [KQ ]T
e =
a [NQ ]
T


[S]-1 [Se ]T [B ] da
[K ]e    [B ]T [Se ][S]1[Se ]T [B ]da 
a

(28f)
a [B ]
T
[C ][B ]da
(28g)

 {[BL ][C] 1 [BN (w)]  [BN (w)]T [C][BL ] 
b
b 
2

uu
[K N ]e   
 da
a
  [BN (w)]T [C] 1 [BN (w)]

2


(28h)
[KuN ]e   [BN (w)]T [Ce ]T [N ]da
(28i)


a
2

u
T
1 [B N (w)] da
[K 
N ]e   [N ] [Ce ]
(28j)
{f u }e    [BL
] [ ][N ]da  {θˆ e }
 a b

(28k)
u
{f N
}e    [BN (w)]T [ ][N ]da  {θˆ e }
 a

(28l)
a
19
{f  }e    [B ] T[P ][N ]da  {ˆ e }
 a

(28m)
Condensation of the non-continuous element transverse shear stress resultants at the
element level simplifies the element matrix equations of (28a), which leads to
[KL ]e  [K N ]e  {Ue }  {f N }e  {f U }e  {f  }e
(29a)
QQ 1
[K L ]e  [K UU ]e  [K QU ]T
]e [K QU ]e
e [K
(29b)
where
 [K uu ] [K u ] [K uQ ] 

 [K UU ] [K UQ ]  

   [Ku ] [K ] [KQ ] 

 [KQU ] [K QQ ]  

e  [KQu ] [KQ ] [KQQ ] 

e
 [K uu ] [K u ] 
N 
 N
[K N ]e  

u
[0] 
 [K N ]

e
 [K uu ] [K u ] 
 ;
[K L ]e  
 [Ku ] [K  ] 

e
(29c, d)
e

{uˆ } 

{U }  
;
e
ˆ
{

}




{f u }
{f N }e   N 
 {0} e
(29e, f)
{f u } 
{f U }e  
 ;

{f
}

e
{f u } 
{f  }e  


{f }e
(29g, h)
e
The over bar denotes condensed matrices.
Assembly of the element equilibrium equations (29a) uses the direct stiffness method
to obtain the structure equations. Global equilibrium equations are
[K L ]  [K N ] {U}  {FN }  {F}
(30)
where [K L ] =  e [K L ]e = structure linear coefficient matrix; [K N ] =
nonlinear coefficient matrix; {FN } =
e{f N }e
e [K N ]e = structure
= nonlinear component of the thermal load
20
vector; {F} = {FU }  {F} ; {FU } = nodal mechanical and electric load vector; and {F} =
e{f }e
= linear component of thermal and pyroelectric load vector.
6. Epilogue
This paper has focused on the formulation of a hierarchic finite element formulation
for the geometric nonlinear analysis of piezothermoelastic composite plates subjected to both
mechanical and self-strain (thermal and electric field) loadings. Geometric nonlinearity has
been included in the von Karman sense, i.e., large transverse displacements with small
inplane displacements. A mixed variational formulation in which an independent
discretization of the transverse shear stress resultants at the Gauss integration points using
standard Lagrangian interpolation in addition to the displacement, rotation, and electric
potential variables expressed in terms of hierarchic finite elements (quadratic, cubic and
quartic) has been presented to construct the element level algebraic equations. Thermoelastic
and pyroelectric effects are part of the constitutive equations.
Results for mechanically and self-strained loaded plates are in the companion paper
(Datchanamourty and Blandford 2008) as well as the solution strategies. This paper presents
the nonlinear and buckling solution details along with results for mechanically and self-strain
loaded composite plates to demonstrate the impact of piezoelectric coupling on the buckling
load magnitudes. Predicted buckling loads include the piezoelectric effect (coupled) and
exclude the effects (uncoupled).
Acknowledgements
The authors wish to acknowledge the financial support provided by the University of
Kentucky Center for Computational Sciences for partial support of the research reported in
21
this paper. The views contained herein are those of the authors and should not be interpreted
as necessarily representing the official policies or endorsements, either expressed or implied,
of the University of Kentucky, Center for Computational Sciences.
APPENDIX A
CONSTITUTIVE COEFFICIENTS
Elastic stiffness coefficients with respect to the lamina principal axes x1, x 2 , x 3 of
an orthotropic material are
Q11 
E1
2
1  12
(A.1a)
Q12  Q 21  12 Q11
(A.1b)
E2
(A.1c)
Q22 
2
1  12
Q44  G 23
(A.1d)
Q55  G31
(A.1e)
Q66  G12
(A.1f)
where Ei = i-axis elastic modulus (i = 1, 2); 12 = Poisson’s ratio in the 1-2 plane and Gij =
shear modulus in the i-j plane (i refers to the axis normal to the plane and j refers to
direction). Thermoelastic material coefficients in the principal directions are
1  Q11 1  Q12 2
(A.2a)
 2  Q21 1  Q22 2
(A.2b)
22
where 1 and 2 are the coefficients of thermal expansion in the 1-axis and 2-axis,
respectively. Piezoelastic material constants in the principal directions are
e31  Q11 d31  Q12 d32
(A.3a)
e32  Q21 d31  Q22 d32
(A.3b)
e24  Q44 d 24
(A.3c)
e15  Q55 d15
(A.3d)
where dkl are the piezoelastic compliance coefficients. Reduced electric permittivity and
pyroelectric coefficients are
ˆ 11  d15 e15
11  
(A.4a)
ˆ 22  d24 e24
22  
(A.4b)
ˆ 33  d31 e31  d32 e32
33  
(A.4c)
P3  Pˆ3  d31 1  d32 2
(A.4d)
where ̂ii = unreduced i-axis electric permittivity coefficients (i = 1, 2, 3) and P̂3 =
unreduced pyroelectric coefficient.
Equations (A.1 – A.4) are in terms of the lamina principle axes. Composite plate
analysis requires the transformation of these principal axes material coefficients to the
common plate coordinate system x, y, z = x 3 . The angle between the plate x-axis and the
lamina 1-axis measured in the counter-clockwise direction is . Transformed elastic
stiffness coefficients are
Q11  Q11 c4  2  Q12  2Q66  c2 s2  Q22 s4

Q12   Q11  Q22  4 Q66  c 2 s 2  Q12 c 4  s 4
(A.5a)

(A.5b)
23
Q22  Q11 s4  2  Q12  2Q66  c2 s2  Q22 c4
(A.5c)




(A.5d)




(A.5e)
Q 16  Q11 c2   Q12  2 Q66  c2  s 2  Q 22 s 2 c s
Q26  Q11 s 2   Q12  2 Q66  c 2  s 2  c 2 Q 22 c s

Q66   Q11  2Q12  Q22  c2 s2  Q66 c2  s2

2
(A.5f)
Q 44  Q 44 c2  Q55 s 2
(A.5g)
Q45   Q55  Q44  cs
(A.5h)
Q55  Q44 s 2  Q55 c2
(A.5i)
where c = cos ; s = sin ; and subscripts on the transformed material coefficients with the
over bar are 1 = x, 2 = y, 3 = z, 4 = yz, 5 = zx, and 6 = xy. Transformed thermoelastic
coefficients are
1  1 c2   2 s 2
(A.6a)
 2  1 s 2   2 c2
(A.6b)
6   1 2  cs
(A.6c)
Transformed piezoelastic coefficients are
e31  e31 c2  e32 s 2
(A.7a)
e32  e31 s 2  e32 c2
(A.7b)
e36   e31  e32  cs
(A.7c)
e14   e15  e24  cs
(A.7d)
24
e24  e 24 c 2  e15 s 2
(A.7e)
e15  e24 s 2  e15 c2
(A.7f)
e25   e15  e24  cs
(A.7g)
Transformed electric permittivity and pyroelectric coefficients are
11  11 c2  22 s 2
(A.8a)
12   22 11  cs
(A.8b)
22  11 s 2  22 c2
(A.8c)
33  33
(A.8d)
P3  P3
(A.9)
25
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Figure Captions
Figure 1. Layout of an N-layer Composite Laminate
Figure 2. Stress and Moment Resultants on an Elemental Plate Volume
Figure 3. Hierarchic Lagrangian Finite Elements
Figure 4. Transverse Shear Stress Resultant Interpolation Points
28