Nonlinear Static and Buckling Analysis of
Piezothermoelastic Laminated Composite Plates
Using Reissner-Mindlin Theory and a Mixed
Finite Element Formulation
Balasubramanian Datchanamourtya and George E. Blandfordb1
a
Belcan Corporation
b
Department of Civil Engineering, University of Kentucky, Lexington, KY 40506, USA
Abstract
This paper is concerned with the analysis of nonlinearly deformable
piezothermoelastic composite laminates using the 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. The incremental/iterative nonlinear analysis scheme follows
the modified Newton-Raphson method with spherical arc-length load steps. Results for
eigenvalue based critical buckling loads and nonlinear buckling responses of laminates
with uncoupled and coupled piezoelectric effects are included.
In addition, the
postbuckling response of these composite plates is included.
1. Background
Piezoelectric materials exhibit the property of generating an electric potential
when subjected to mechanical deformations and this phenomenon is the direct
Corresponding author: Telephone number – (859) 257-1855; Fax number – (859) 257-4404; and e-mail –
gebland@engr.uky.edu
1
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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 structures is by 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
plated 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 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 4node 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
2
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) proposed a coupled geometrically nonlinear finite element formulation
for predicting the response of smart beam and plate structures. Varelis and Saravanos
(2002) formulated a geometric nonlinear coupled mechanics for composite piezoelectric
plate structures using eight-node 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, 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. Solution of the
nonlinear equations uses a modified Newton-Raphson scheme with incremental/iterative
arc-length load steps (Forde and Steimer, 1987). Piezoelectric coupling effects on critical
loads using eigenvalue and nonlinear bucking methods are presented.
2. Plate Kinematics
The following are the assumptions used in the Reissner-Mindlin theory of
moderately thick plates: (i) an elemental plane normal to the midsurface before
3
deformation remains plane but not necessarily normal after deformation; (ii) stresses
normal to the plane of the plate are negligible (i.e., transverse displacement is constant
through the thickness of the plate). In the first-order shear deformation theory, the
inplane displacements are assumed to vary linearly in the depth direction whereas the
transverse displacement is constant.
Since the transverse shear strains are nonzero,
rotations are not equal to the derivative of displacements.
This condition leads to
independent interpolation of rotations requiring only C0 continuity between the elements.
The plate displacements in the Reissner-Mindlin theory are expressed as
u(x, y, z)  u o (x, y)  z  x (x, y)
v(x, y, z)  vo (x, y)  z  y (x, y)
(1)
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; x and y are the rotation about the
negative y-axis and the x-axis, respectively.
3. Strain-Displacement Relationships
The strain measure used in the first-order shear deformation theory is the GreenLagrange strain vector. The nonlinear strains are included in the von Karman sense
(Reddy 2004) which is based on the assumption that higher order derivatives of inplane
displacements are negligible compared to that of transverse displacement. The nonlinear
strain-displacement equations are given by
u 1  w 
x 
 

x 2  x 
v 1  w 
y 
 

y 2  y 
 xy 
2
2
(2a)
u v w w


y x x y
4
v w

z y
u w
 xz 

z x
 yz 
(2b)
where  x and  y are the inplane extensional strains;  xy = inplane shear strain;  xz and
 yz are the transverse shear strains. Using equation (1), equations (2a, b) are written as
    ox 
 x   o 
 y    y  

  o 
  xy    xy 
 
x


z  y  


  xy 
 N 
 x 
 N
o
  y   { }  z{}  { N }
 N
  xy 
  yz 
   {s }
  xz 
(3a)
(3b)
The strain components are expressed as

 
 uo

 

x

  x
o

v
 
{o }  
 0
y

 
  u o  vo   

 

 y
x   y
  x
  

 
x

  x
  y
 
{}   
 0
y

 
  x  y    
 y  x   y

 

0
 o
o
  u 
u 
   [D ]  o 
y   vo 
 v 
 



x 

0 

 x 
   x 

 [D  ]  


y   y 
 y 


 
x 
(4a)
(4b)
5
 
o 2 

w
1
 
 
  wo
 2  x  

 
 
 x
2

o
1   w   1 
{ N }   
   0
 y   2 
2
 
 


o
o
o
 w

w

w


 y
 x y 




  wo
 
 y  

y
 y

{s }  

  wo
 
 x   x

 x


0 
  
o  
 w  x  o 1
   w  [A  (w)]{D N }w o
y    
2

 w o   y 
x 
 o
w o 
1  w 
 
 
  x   { w D}  [Is ]  x 



 
1 0    y 


 y

(4c)
0
(4d)
where {o } = mid-plane strain vector; {} = curvature vector; { N } = nonlinear strain
0 1 
vector; {s } = transverse shear strain vector; [Is ]  
.
1 0 
The strain-displacement
expressions (4a) – (4d) can be expressed in a compact form as
N
 L 1 N

{b }  {L
b }  { }  [Db ]  2 [D (w)] {u}
(5a)
{s }  [Ds ] {u}
(5b)
[0]3x2 [A  (w)]{D N } [0]3x2 
 [D ] {0}3x1 [0]3x2 
N
[D
(w)]

where [DL
;
]


;
b
[0]

{0}3x1
[0]3x2 
 3x2 {0}3x1 [D ] 
[0]3x2
o
T
N
T
[Ds ]  [0]2x2 {w D} [Is ] ; { L
b }         ; { }    N   0   ;


{u}   u o vo w o x y T .
4. Piezothermoelastic Constitutive Equations
The coupled constitutive equations for a typical layer k of a multilayered
piezothermoelastic composite laminate relative to the plate geometric coordinate axes x,
y and z are expressed as
{}k  [Q]k {}k  [e]T
k {E}k  {}k 
(6a)
{D}k  [e]k {}k  []k {E}k  {P}k 
(6b)
6
where {} = second Piola-Kirchhoff stress vector which is the work conjugate of GreenLagrange strain vector {} ; [Q] = material stiffness matrix; [e] = piezoelectric material
matrix; {E} = electric field vector; {} = temperature-stress vector;     R ;  =
temperature; R = reference temperature; {D} = electric displacement vector; [] =
electric permittivity matrix; {P } = pyroelectric vector. 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 a purely
elastic layer, the piezoelectric terms are zero. Equation (6a) is expanded as
 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
(7a)
The expanded form of the electric displacement equation (6b) is
 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 
(7b)
k
The electric field vector which is the negative of the potential gradient is expressed as
7
Ex
Ey
where { } =
Ez
T
 



x
y
z
 / x  / y  / z
T
T
  { }  (x, y, z)
= electric potential differential vector.
(8)
The
electric potential  is assumed to vary piecewise linearly through the thickness of the
laminate. The potential at any point within a piezoelectric layer  is expressed as
 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) 

(9)
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  ;
 z1  z 
 z  z 

M1(z)  
 and M 2 (z)  
 are the depth interpolation functions for
 t 
 t 




electric potential. Substituting equation (9) into equation (8) gives
{E}   { } M (z) { (x, y)}   [Z ][D ]{ (x, y)}
(10)
where
 M 1

[Z ]   0

 0
M 2
0
0
0
M 1
M2
0
0
0
0 

0
0 

1 t  1 t  
0
(11a)
= electric potential depth interpolation matrix;

 x
[D ]  

0

0

x

y
0
0

y

1 0


0 1

T
(11b)
= electric potential inplane gradient matrix.
Thermoelastic and pyroelectric effects can be induced, as expressed in the
constitutive equations, when thermal loading is applied by specifying the temperature on
8
the top and bottom surfaces of the laminate. Temperature (  ) is assumed to vary linearly
through the entire depth of the plate and is expressed as
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) 
(12)
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
2 h
2 h
functions for temperature.
Using equations (10) and (12), the stress and electric
displacement equations (7a) and (7b) are expressed in terms of inplane and transverse
components as

{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

 

(13a)
 [s ] {0}



 0    [Z ][D ]{ (x, y)}
p


 {0} 
   M  (z) {(x, y)}
P
 p 
(13b)
where p, s = inplane, transverse shear components; Pp  Pz .
The stress resultants per unit width of the plate are defined as
{N},{M},{Q}  h 2 {p }, z{p}, {s } dz
h 2
where h = plate thickness; {N} =  N x
{M} =  M x
My
Ny
(14)
N xy T = inplane stress resultant vector ;
M xy T = moment resultant vector and {Q} =  Q y
Q x T =
transverse shear stress resultant vector. The resulting equations are given as
9
{N} 


{M}
T
[A] [B]  {o }  { N } [A e ] 
 {} 



T
{}
 [B] [D] 
 [Be ] 
{Q}  [S]{s }  [S1e ]T
[A  ]
[B ]  {}
  
(15a)


{}  [Se2 ]T {}
x
y
(15b)
Using the strain-displacement equations (5a, b), the stress resultants can be expressed as
N
T
1
{N}  [C]([D L
b ]  [D (w)]){u}  [Ce ] {}  [ ]{}
2
   N   M  T  {Nu }  {N }  {N }
(16a)
{Q}  [S]{s }  [Se ]T [D ]{}  {Qu }  {Q }
where { N  } =   N   M  T ;
 M
x
My
T


M
xy  ; { Q } =  Q y
(16b)
{ N  } =  N
x
Ny

T
N
xy  ; { M } =
T
Q
x  ;  = u,  or . The electric potential
inplane gradient matrix and electric potential vector for the laminate are given by
2
[D ]  diag [D1 ] [D
]

NP 
[D
]

{}   1b 1t    2b  2t 
(17)
  bNP  tNP 
T
(18)
The depth integrated material coefficient matrices for the laminate are expressed as
T

[A] [B] 
T
T [A e ] 
[C
]

[C]  
;
; []  [A ] [B ]
e

T
[Be ] 
[B] [D]
NL
[A], [B], [D]    [Qp ]k  (z k 1  z k ),
k 1

1 2
1

(z k 1  z k2 ), (z3k 1  z3k ) 
2
3

(19a, b, c)
(19d)
[Ae ]T  [Ae ]T [Ae ]T
1
2

[Ae ]T 

NP 
(19e)
[Be ]T  [Be ]T [Be ]T

1
2
[Be ]T 
NP 

(19f)
1 
1
[Ae ]     ep  ; [Be ]  (z1  z )[Ae ]
2
1
1
1
1
1

[A  ]   {A  }  {B }
{A  }  {B }
h
2
h
2

(19g, h)
(19i)
10
1
1
1
1

[B ]   {B }  {D }
{B }  {D }
h
2
h
2

(19j)
NL
{A
},
{B
},
{D
}

      { p}k  (z k 1  z k ), 12 (z k2 1  z k2 ), 13 (z3k 1  z3k )  (19k)
k 1
NL
[S]  [K s ]  [Qs ]k (z k 1  z k )[K s ]
(19l)
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 
2
;
[S
]

e k
e15 
2
 e24
e
 24
e25 
e25 
(19m, n)
(19o, 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,
k
shear correction matrix [Ks ]   s2
 0
0 
is introduced in the depth integrated
ks1 
transverse shear coefficient matrix. k s1 and ks2 = shear correction factors. Electric
potential between adjacent layers is continuous. The interface between a piezoelectric
layer and purely elastic layer is grounded, i.e., electric potential is zero.
5. Coupled Mixed Variational Principle
A mixed variational formulation can be achieved by using the modified HellingerReissner functional which facilitates independent interpolation of displacement, electric
potential and transverse shear stress resultant variables.
The mechanical energy
functional for a mixed variational formulation is expressed as
M =
1
1

 ε p  {pu }dV +   ε p  {
p }dV + V  ε p  { p }dV

V
V
2
2
(20)
1
M
   εs  {s }dV    s  [Qs ]1{s }dV   ext
V
2 V
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;  ε p  = inplane strain vector;  εs  = transverse shear strain vector;
11

{ up } , {
p } and { p } are the elastic, piezoelectric and thermoelastic contributions to the
inplane stress vector {p } , respectively; {s } = transverse shear stress vector and [Qs ]
= transverse shear constitutive matrix. 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 (16a) and (16b) to replace the depth integrated
stresses gives
M =
1
1
 ε b  {N u }dA +   ε b  {N  }dA +   ε s  {Q}dA

A
2 A
2 A


(21a)
1
M
   Q  [S]1 Q  [Se ]T [D ]{} dA    ε b  {N  }dA  ext
A
A
2
Replacing the stress resultants by the material coefficient matrices and strains leads to
M 
1
 ε b  [C]{ε b }dA +   εs  {Q}dA
A
2 A

1
1
 Q  [S]-1{Q}dA    ε b  [Ce ]T {}dA

2 A
2 A

1
M
 Q  [S]-1 [Se ]T [D ]{} dA    ε b  [ ]{θ}dA  ext

A
A
2

(21b)

Using the strain-displacement equations (5a) and (5b) in equation (21b) gives
M =

 

 
T
1
L
1 [D N (w)] {u} [C] [D L ]  1 [D N (w)] {u} dA
[D
]

b
b
2
2
2 A
1
 Q  [S]1{Q}dA
2 A
+
A [Ds ]{u}

A 

T
1
L
1 [D N (w)]{u} [C ]T {}dA
[D
]

b
e
2
2 A
T
{Q}dA 

T
1 [D N (w)]{u} [] {θ}dA
[D L
]

b
2

(22)



M
   Q  [S]-1 [Se ]T [D ]{} dA  ext
A
The piezoelectric energy functional is expressed as
12
P 

1
1
   {D u } dV      {D  } dV

2 V
2 V
V    {D
 }dV
(23)
P
 ext
P
where P = piezoelectric energy functional;  ext
= external work due to applied electric
potential;    



x
y
z
T
= electric potential gradient; {Du } , {D  } and
{D} are the piezoelectric, dielectric and pyroelectric contributions to the electric
displacement vector {D}. The volume integral is expressed as depth and area integrals
and the constitutive equations of the laminate are expressed as a sum of the individual
piezoelectric layers through the thickness. Expanding the vectors of electric potential
gradient and electric displacements in equation (23) using equation (13b) leads to

T
 h 2 NP 
 1  [0]



 
[Z ][D ]{ (x, y)}  

e 
A h 2
 2  p
1 
 
P





1
2
[es ]  {p }
 0   {s } 
 

 [s ] {0}



 0    [Z ][D ]{ (x, y)}
p


(24)


P
M (z) {(x, y)} dz dA  ext

 

{0}
 
 Pz 
Since the variations of electric potential and temperature through the laminate
thickness are assumed to be linear, the material coefficient matrices are integrated with
respect to z and the resulting energy functional for the mixed formulation is expressed as
P 
1
1
   [Ce ]{b }dA   {[D ]{}}T [Se ]{s }dA

2 A
2 A
(25a)
1
P
  {[D ]{}}T [C ][D ]{}dA   {[D ]{}}T [P ]{}dA  ext
A
A
2
Using the strain-displacement equations (5a) and (5b) in equation (25a) gives
13
P 

 
1
N
1
   [Ce ] [DL
b ]  2 [D (w)] {u} dA

A
2



1
{[D ]{}}T [Se ][S]1 {Q}  [Se ]T [D ]{} dA

2 A

1
P
{[D ]{}}T [C ][D ]{}dA   {[D ]{}}T [P ]{}dA  ext

A
A
2
(25b)
The depth integrated dielectric and pyroelectric matrices for the laminate are expressed as
[C ]  diag  [C ]1 [C ]2
[C ] 
[ij ] 
z1
z

(26a)
[11] [12 ]
 T

[Z ] [] [Z ]dz  [21] [22 ]
 [0]
[0]
[0] 
[0] 
[33 ]
(ij ) t   2 1 
(33 )
(i,
j
=
1,
2);
[

]

33 
1 2
6
t


[P ]  diag  [P ]1 [P ]2
[P ] 
[C ]NP 
z1
z
(26b)

 1 1
 1 1 


(26c, d)
[P ]NP 
(26e)
[0]2x2 
0


 T 
[Z ]  0   M   dz  [0]2x2 
P 
 z 
z1  z

(Pz )  1 
h

[P]  
z1  z
2 
 1 

h
(26f)
 [P] 


z1  z 

h

z1  z 
1 

h

1
(26g)
6. Finite Element Approximation
The field variables, displacements, electric potential and transverse shear stress
resultants, in equations (22) and (25b) being functions of the plate inplane coordinates are
discretized using two-dimensional finite elements.
Hierarchic Lagrangian shape
functions are used to interpolate displacement and electric potential variables.
The
discretized element variables are expressed as
14
{u e }   N1 [I5 ] N 2 [I5 ]
 {uˆ e } 
1


 {uˆ e }2 
e
N nen [I5 ] 
  [N u ]{uˆ }


 e

{uˆ }nen 
(27a)
ˆ ] [N
ˆ ]
{ e }  [N
1
2
 {
ˆ e }1 


e

ˆ }2 
{
e
ˆ
[N
  [N  ]{ˆ }
nen ] 


 e

ˆ }nen 
{
(27b)
 Ni
0

ˆ ]0
where [N
i


 0
0
Ni
Ni
0
0
0 
= electric potential shape function
0


Ni  2NP x (NP 1)
matrix 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. 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 themselves . For i = 5 to nen, the nodal shape functions are
hierarchic and the variables associated with these shape functions represent higher order
derivatives of the degrees of freedom depending on the order of the polynomial used
(Zienkiewicz and Taylor 1991). The hierarchic Lagrangian shape functions are shown in
Appendix B.
The transverse shear stress resultants Q y and Q x are interpolated at the Gauss
integration points using standard Lagrangian shape functions as follows
{Qe }   N1 [I 2 ] N 2 [I 2 ]
ˆ e} 
 {Q
1


e
ˆ } 
 {Q
2
ˆ e}
N nens [I 2 ] 
  [N Q ]{Q


 ˆe

{Q }nens 
(27c)
15
where Ni = shape function corresponding to the ith transverse shear interpolation point;
nens = number of element transverse shear stress resultant interpolation points and
ˆ e } = ith node transverse shear stress resultants vector of element e. Under thermal
{Q
i
loading, top and bottom surface temperatures are specified at the node points and the
intermediate values within an element are interpolated as
{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 ]
(27d)
where Ni = corner node bilinear shape function; {ˆ e }i = ith node temperature vector of
element e. Using the shape functions of equation (27a, b) in equations (5a, b), the
element strain-displacement and electric displacement-electric potential matrices are
defined as
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 ] 
(28a)
(28b)
[Bi ]  [D ][I Ni ] ;
[Bi ]  [D ][I Ni ]
(28c, d)
[Bi ]  [D ][I Ni ] ;
N
[I Ni ]   i
0
(28e, 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
]]
(28g)
(28h, i)
(28j)
(28k)
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

(28l, m)
(28n)
[Bnen ]]
ˆ ]
[Bi ]  [D ][N
i
(28o)
Combining the mechanical and piezoelectric energy functionals of equations (22)
and (25b) and using equations (28a) – (28o), the total energy functional is expressed as
1

e   uˆ e    {[BbL ]  1 [B N (w)]}T [C] {[BL
]  1 [B N (w)]}da  {uˆ e}
b
2
2
2 a

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
(29)
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 }
2 a

  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; a = area of a typical element.
The structure reaches the state of equilibrium when the energy functional
becomes a 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 a variation of the energy functional and equating it to zero. The
variations of the element degrees of freedom are expressed as
17
{u e }  [N u ]{uˆ e }
(30a)
{u e }  [N u ]{uˆ e }
(30b)
ˆ e}
{Qe }  [NQ ]{Q
(30c)
where  = variational operator. Taking a 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
(31)
  ˆ e    [B ] T[P ][N ]da  {ˆ e }   eext  0
 a

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 

ˆ
[0]
[0]   { }   {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
(32a)
where {f u }e = mechanical load vector of element e; {f  }e = electrical load vector of
element e. The element coefficient matrices and thermal load vectors are defined as
T
L
[Kuu ]e   [BL
b ] [C][Bb ]da
(32b)
T
[KuQ ]e  [KQu ]T
e   [Bs ] [NQ ]da
(32c)
[KQQ ]e   [NQ ][S]1[NQ ]T da
(32d)
[Ku ]e  [Ku ]eT   [BbL ]T [Ce ]T [N ]da
(32e)
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

(32f)
a [B ]
T
[C ][B ]da
(32g)

 {[BL ][C] 1 [BN (w)]  [BN (w)]T [C][BL ] 
b
b 
2

[K uu
]

 da
N e a 
  [BN (w)]T [C] 1 [BN (w)]

2


(32h)
[KuN ]e   [BN (w)]T [Ce ]T [N ]da
(32i)


a
2

u
T
1 [B N (w)] da
[K 
N ]e   [N ] [Ce ]
a
{f u }e    [BL
] [ ][N ]da  {θˆ e }
 a b

u
{f N
}e    [BN (w)]T [ ][N ]da  {θˆ e }
 a

{f  }e    [B ] T[P ][N ]da  {ˆ e }
 a

(32j)
(32k)
(32l)
(32m)
Since the element transverse shear stress resultants are not required to satisfy
interelement continuity conditions, they can be condensed out at the element level. The
linear part of the element coefficient matrix in equation (32a) can be partitioned as
19
 [K uu ] [K u ] [K uQ ] 


 [K UU ] [K UQ ] 
u

Q 


[K L ]e  [K ] [K ] [K ]  


 [KQU ] [KQQ ] 

e
 [KQu ] [KQ ] [K QQ ] 

e
(33a)
 [K uu ] [K u ] 
 ; [KQU ]e  [K UQ ]T
[K UU ]e  
 [KQu ] [KQ ]
e

e

u

 [K ] [K ] 

e
(33b, c)
The modified linear element coefficient matrix after condensing the transverse shear
stress resultants is expressed as
QQ 1
[K L ]e  [K UU ]e  [K QU ]T
]e [K QU ]e
e [K
(33d)
Thus equation (32a) becomes
[KL ]e  [K N ]e  {Ue }  {f N }e  {f U }e  {f  }e
 [K uu ] [K u ] 
 ;
[K L ]e  
 [Ku ] [K  ] 

e
(34a)
 [K uu ] [K u ] 
N 
 N
[K N ]e  

u
[0] 
 [K N ]

e
(34b, c)
e

{uˆ } 

{Ue }  
;
e
ˆ }
{


{f u }
{f N }e   N 
 {0} e
(34d, e)
{f u } 
{f U }e  
 ;
{f  }e
{f u } 
{f  }e  

{f  }e
(34f, g)
The over bar denotes condensed matrices. The element equilibrium equations (34a) are
assembled using the direct stiffness method to obtain the structure equations. The global
equilibrium equations are expressed as
[K L ]  [K N ] {U}  {FN }  {F}
(35)
where [K L ] =  e [K L ]e = structure linear coefficient matrix; [K N ] =
structure nonlinear coefficient matrix; {FN } =
e{f N }e
e [K N ]e =
= nonlinear component of the
thermal load vector; {F} = {FU }  {F} ; {FU } = nodal mechanical and electric load
vector; {F} =
e{f }e
= linear component of thermal and pyroelectric load vector.
20
7. Nonlinear Solution Method
The system of nonlinear equations (35) can be expressed as
(U)  [K(U)]{U}  {FN }  {F}  {0}
(36)
where  (U) = residual force vector; [K(U)] = [K L ] + [K N ] ; {F} = load vector.
Modified Newton-Raphson method with spherical arc-length load steps (Crisfield 1981,
1982), an iterative solution scheme, is used to solve the nonlinear equations (36). An
iterative procedure follows a step by step linearization of the nonlinear structure
equations. The unbalanced force vector (U(i) ) at an intermediate iteration i is nonzero.
Expanding (U(i 1) ) , residual force vector at iteration i+1, in the neighborhood of
{U(i) } using Taylor’s series gives
{(U(i 1) )}  {(U(i) )}  [K T ]{U (i) }
(37)
where {(U(i 1) )} = residual force vector at iteration i+1; {U(i 1) } = nodal degrees of
freedom vector at iteration i+1; {U(i) } = iterative nodal degrees of freedom vector at
iteration i. The structure tangent coefficient matrix is expressed as
 m
  {(U)} 
[KT ] = 
; KTmn 

 Un
 U 
where [KT ] =
e[KT ]e .
(38a)
Expanding equation (38a) using equation (36), the element
tangent coefficient matrix can be written as (Pica et al. 1980)
[KT ] 
e[KT ]e  e [KL ]e  [KNL ]e  [K ]e 
(38b)
where [K L ]e = linear coefficient matrix;
  [BL ][C][B N ]  [B N ]T [C][BL ] 

b 
N T
T
 b
[B ] [Ce ] [N  ] 

 da



[K NL ]e      [B N ]T [C][B N ]


a 




[N  ]T [Ce ] [B N ]
[0]


(38c)
= nonlinear coefficient matrix;
ˆ
 [G]T [N][G]
[0] 
[K  ]e   
 da = geometric stiffness matrix
a
[0]
[0] 

(38d)
21
[G]  [[G1 ] [G 2 ]
N
ˆ  x
[N]
 N xy

[G nen ]]
N xy 
u


 ; N  N  N  N ,  = x, y or xy

Ny 
(38e)
(38f)
The incremental equilibrium equations for the current load step m+1 at iteration i
are expressed as
m
[K T ]{U}(i)  m 1 (i) {F}  m 1{FB }(i 1)
(39)
where m = load level at the beginning of load step m; m [K T ] = tangent coefficient
matrix at load level m; {FB } = balanced force vector;  = load parameter;  denotes
iterative change.
The accumulated incremental nodal degrees of freedom and load
parameter within a load step through iteration i are expressed as
{dU}(i)  {dU}(i 1)  {U}(i)
(40a)
{d}(i)  {d}(i 1)  {}(i)
(40b)
where d denotes incremental change. The incremental nodal variables and incremental
load parameter are zero at i = 0. Accumulated nodal variables and load parameter for
load step m+1 through iteration i are given as
m 1
{U}(i)  m 1{U}(i 1)  {U}(i)
m 1 (i)

 m 1 (i 1)  (i)
m 1
{FB }(i)  m 1{FB }(i 1)  {FB }(i)
m 1
(41a)
(41b)
(41c)
{FR }(i)  m 1(i) {F}  m 1{FB }(i)
(41d)
{FB }(i)  m [K T ]{U}(i)  m 1(i) {FN }(i)
(41e)
where {FR } = residual or unbalanced force vector. Variables at the first iteration of a
new incremental load step are equal to those at the last iteration of previous load step.
The iterative nodal variables in equation can be written in the following for
{U}(i)   (i) {T }  {}(i)
(42a)
where {T } is the tangential nodal degrees of freedom vector and {} is the iterative
nodal variable vector calculated from the unbalanced force
22
{T }  m [K T ]1{F}
(42b)
{}(i)  m [K T ]1 m 1{FR }(i 1)
(42c)
The iterative load parameter for the spherical arc-length method is expressed as
 2 (1  / (i)
n )
  (i)
n
(i 1)
(i 1)
 U w 
{Tw }  d
 (i) 
(43)
where {U w } = structural transverse displacement vector; {Tw } = tangential degrees of
freedom vector with transverse displacement only (Clarke and Hancock 1990) ;
= arc
length; (i)
n = normal plane arc-length given by
(i)  2
(i)
(i)
(i) 2 1/ 2
n     U w n {U w }n  ( n ) 
(44)
where  (i)
n = normal plane iterative load step multiplier (Forde and Steimer 1987,
Ramm 1981) calculated from
  dU w (i 1) { w }(i)
(i)
n 
(45)
 dU w (i 1) {Tw }  d (i 1)
where {w } = iterative transverse displacement vector calculated from unbalanced force.
The details of the arc-length calculation algorithm can be found in Blandford (1996).
Criteria for the convergence of the nonlinear solution method are given by Bathe and
Cimento (1980)

 U (i) m  1  (i) {F}  m  1{BF}(i)


 U (1) m  1  (1) {F}  m{BF }

E
(46a)
2
 {F}  m 1{BF}(i)
m  1 (i)
m  1 (1)

{F} 
m
{BF}
2
2  
F
(46b)
2
where  E (108   E  106 ) and  F (108   F  106 ) are the tolerance values for
errors in energy and force.
8. Buckling Analysis
23
At static equilibrium, i.e., when the internal and external forces are balanced, the
system of nonlinear equations becomes
[KT ]{U}  {(U)}  {0}
(47)
When the plate is subjected to inplane loads only, i.e., when the transverse displacements
are zero, the nonlinear stiffness component in the tangent stiffness matrix doesn’t exist.
If the inplane stresses in the geometric stress matrix are compressive in nature, then an
eigenproblem of the following form can be established.
[K L ]   [K  ] {U}  {0}
(48)
where  is the inplane stress magnification factor. The objective of the problem is to
calculate the values of  that would make the tangent stiffness matrix singular thereby
introducing instability in the plate. Thus critical buckling load and the associated mode
shapes can be predicted by calculating all the eigenvalues and the corresponding
eigenvectors of the problem (48). Expanding equation (48) gives
  [K uu ]

  [K u ]T

 [K u ] [0]   {u} 
[K u ] 

   
 {0}
 [0] [0]   {}
[K  ] 


(49)
where [K uu ] , [K u ] and [K  ] are the elastic, coupled and electric components of the
assembled linear coefficient matrix after condensing the transverse shear stress resultants;
[K u ] is the assembled geometric stiffness matrix associated with the displacement
degrees of freedom. Since buckling is a mechanical phenomenon, the electric potential
degrees of freedom are condensed from the linear structure coefficient matrix.
Condensation needs to be carried out at the global level and not at element level as
electric potential degrees of freedom at common nodes are shared among multiple
elements. Thus equation (49) after condensation becomes
[Ku ]   [K u ] {u}  {0}
 
 L
where [K uL ]  [K uu ]  [K u ]T [K  ]1 [K u ] .
(50)
The geometric stiffness matrix is a
function of inplane stress resultants. As the first step in the buckling analysis, an initial
linear elastic analysis of the plate is conducted under the action of inplane loads to
calculate the inplane stresses and the geometric stiffness matrix is generated using
24
equation (38d).
In the second step, the condensed linear stiffness matrix and the
geometric stiffness matrices are used to calculate the buckling loads and mode shapes as
dictated by the eigenvalue problem of equation (50).
The method discussed above is known as the eigenvalue buckling method.
Another method of computing the critical load is the nonlinear buckling analysis. An
incremental/iterative nonlinear analysis is conducted by subjecting the plate to an inplane
load and a small notional transverse load. The load parameter-deflection curve plotted
for the transverse displacement of an unrestrained node would depict a nonlinear pattern.
At the point where the stiffness matrix approaches singularity, a small perturbation in the
transverse load would lead to an enormous increase in the transverse displacement. Such
a behavior in the load-deflection curve signifies points of instability. The response after
oscillating about unstable points slowly stabilizes into an equilibrium path with
subsequent load steps and iterations.
25