By
Balasubramanian Datchanamourty
and George E. Blandford
University of Kentucky
Lexington, KY
 Assumptions
 Finite Element Equations
 Buckling Analysis
 Numerical Results
 Summary and Conclusions
Each lamina is generally orthotropic
Piecewise linear variation of electromagnetic
potential through the depth of each piezoelectric
lamina
 Piezoelectric surface is grounded where it is in
contact with structural composite material
 Linear variation of temperature through the plate
thickness
 Displacement assumptions consistent with Mindlin
theory
 Nonlinear strains consistent with von Karman
approximation



 [K uu ] [K u ] [0]  
uu
u

uQ
e 

N
  [K ] [K ] [K ]   N

ˆ
{u
} {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 

{f u }  {f u} 

 


 

 {f  }  {f  }
 {0}   {0} 

 

e
e
e
ˆ
{u }i
= ith element node displacement vector;
five displacements per node: u, v, w, x, y
ˆ e }i = ith element node electromagnetic potential
{
ˆ e}
{Q
i
= ith element Gauss point transverse shear stress
resultant vector; two per node: Qx and Qy
{f u }e = mechanical load vector of element e
{f  }e = electrical load vector of element e
{f u }e = temperature-stress load vector
{f  }e = pyroelectric load vector
u
{f N
}e = nonlinear temperature-stress load vector
uu = linear stiffness matrix for {u
ˆ e}
[K]
u = linear coupling matrix between e and
[K]
{uˆ }
uQ
[K]
e
= linear coupling matrix between{u
ˆ } and
e
ˆ
{ }
ˆ e}
{Q
[K] = linear matrix for {
ˆ e}
ˆ e}
ˆ e } and {Q
[K]Q = linear coupling matrix between{
QQ = linear stiffness matrix for ˆ e
{Q }
[K]
uu
[K N ]
u
[K N ]
= nonlinear stiffness matrix consistent with
the von Karman approximation
= nonlinear coupling matrix between displacements and electromagnetic potentials in the
piezoelectric laminae
[K L ]   [K  ] {U}  {0}
 [K uu ] [K u ] 
[K L ]  

[K u ]T [K  ]
[K  ]
[K u ] [0]
 

 [0] [0]
= linear coefficient
matrix
= geometric stiffness matrix
 = inplane stress magnification factor
(U)  [K(U)]{U}  {FN }  {F}  {0}
 (U)
= residual force vector
[K(U)]  [K L ]  [K N ]
[K N ] = nonlinear stiffness matrix consistent
with a total Lagrangian formulation
{F}, {FN }
= linear and nonlinear force vectors
Nonlinear
Solution
Schematic
Thermal Buckling of (0/90/0/90)s Graphite-Epoxy
laminate plus top and bottom piezoelectric lamina –
PVDF or PZT

Simply supported square plate

PVDF
PZT
Graphite-Epoxy
E1 = E2 = E3 =2 GPa
E1 = E2 = E3 = 60 Gpa
E1 =138 GPa, E2 = 8.28 GPa
12 = 13 = 23 = 0.333
12 = 13 = 23 = 0.333
12 = 0.33
G12 = G13 = G23 = 0.75 GPa
G12 = G13 = G23 = 22.5 GPa
G12 = G13 = G23 = 6.9 GPa
1 = 2 = 3 = 1.2x10-4 /0C
1 = 2 = 3 = 1x10-6 /0C
11 = 22 = 33 = 1x10-10 F/m
11 = 22 = 33 = 1.5x10-8 F/m
d31 = d32 = -d24 = -d15
d31 = d32 = -1.75x10-8 0C/N
23x10-12 0C/N
d24 = d15 = 6x10-10 0C/N
p3 = -2.5x10-5 0C/K/m2
p3 = 7.5x10-4 0C/K/m2
1 = 0.18x10-6 /0C
2 = 27x10-6 /0C
-------
T
a
T  0  
h
 FE Mixed
Formulation
2UC  Uncoupled
Piezoelectric
Analysis
3C  Coupled
Piezoelectric
Analysis
1MF
2
a/h
5
10
15
20
25
30
35
40
60
80
100
1000
Analytical
UC2
1.457
1.811
1.898
1.930
1.946
1.954
1.960
1.963
1.969
1.971
1.972
1.973
MF1
UC2
1.457
1.813
1.899
1.932
1.947
1.956
1.961
1.964
1.970
1.972
1.973
1.975
C3
1.502
1.869
1.958
1.992
2.008
2.016
2.022
2.025
2.031
2.034
2.035
2.037
T
MF1
a
T  0  
h
 FE Mixed
Formulation
2UC  Uncoupled
Piezoelectric
Analysis
3C  Coupled
Piezoelectric
Analysis
1MF
2
a/h
5
10
15
20
25
30
35
40
60
80
100
1000
UC2
4.208
5.475
5.799
5.922
5.981
6.013
6.033
6.045
6.069
6.077
6.081
6.088
C3
-6.584
-9.010
-9.675
-9.931
-10.055
-10.123
-10.165
-10.192
-10.242
-10.260
-10.268
-10.283
Results have demonstrated the impact of
piezoelectric coupling on the buckling load
magnitudes by calculating the buckling loads that
include the piezoelectric effect (coupled) and
exclude the effects (uncoupled).

As would be expected, the relatively weak PVDF
layers do not significantly alter the calculated
results when considering piezoelectric coupling.
The net increase is about 3% for the thermal loaded
ten-layer laminate (PVDF/0/90/0/90)s.


However, adding the relatively stiff PZT as the top
and bottom layers produces significant differences
between the uncoupled and coupled results. A
reversal of stress is required to cause buckling in
the coupled analyses due to the sign on the
pyroelectric constant for the PZT material.
Neglecting the sign change, an increase of
approximately 67% is observed in the absolute
buckling load magnitude for the coupled analysis
compared with the uncoupled analysis.




Six layer laminate: (PZT5/0/90)s
Simply supported
a = b = 0.2m
h = 0.001 m