Thermal Buckling of Piezothermoelastic Composite Plates Using a
Mixed Finite Element Formulation
Balasubramanian Datchanamourty and George E. Blandford
Staff Engineer, Belcan Engineering Group, Caterpillar Champaign Simulation Center,
1901 S 1st Street, Champaign, IL 61820
Department of Civil Engineering, University of Kentucky, Lexington, KY 40506-0281, USA
Introduction
Piezoelectric materials are widely used as sensors and actuators in sensing, actuation, and control
of smart structures and structronic systems. Combining the high strength to weight composite properties and the direct/converse piezoelectric effects,
these smart materials offer a wide range of engineering applications (e.g., [1]).
The objective of this paper is to investigate the
buckling loads of piezothermoelastic composite
plates using an equivalent single layer, ReissnerMindlin plate theory in the context of von Karman
geometric nonlinearity within a mixed finite element representation. Buckling (eigenvalue) analyses are used to determine the thermal stability
loads. A grounded surface is assumed between a
piezoelectric layer and a structural layer.
Buckling Analysis
At static equilibrium, i.e., when the internal
and external forces are balanced, the system of
nonlinear equations becomes
[K T ]{U}  {0}
(1)
where [KT ] = tangent stiffness matrix; and {U}
T
=  u     = incremental vector of unknown displacement and electromagnetic potential
variables. 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 does not exist. If the
inplane stresses can lead to buckling, then an
eigenproblem exists
[K L ]   [K  ] {U}  {0}
(2)
where  is the inplane stress magnification factor;
[K L ] = linear stiffness matrix; and [K  ] = initial
stress matrix that is only a function of inplane
stress resultants and thus do not multiply the nodal
electromagnetic potentials. Consequently, they are
condensed from (2) for the eigen-analyses [2].
Results and Discussion
To investigate the piezoelectric coupling effect
for thermally loaded composite plates, an eightlayer symmetric laminate (0/90/0/90)s, with piezoelectric layers (PVDF or PZT) at the top and bottom thus making it a ten-layer composite, is considered. Material properties of the lamina and the
PVDF and PZT piezoelectric layers are given in
[2].
Validation of the uncoupled results uses the
analytical results of [3]. Additionally, piezoelectric coupling effects are investigated. Table 1
shows the thermal buckling results using a 4 x 4
mesh of quadratic elements for various a/h ratios
of the symmetric composite with PVDF layers on
top and bottom. Critical thermal buckling loads
are nondimensionalized as
 h
T  0 a
2
where T = nondimensionalized temperature;  =
actual temperature; 0 = 1.2 x 10-4 /oC = linear coefficient of thermal expansion; a = side dimension
of square plate; and h = plate thickness. Table 1
shows excellent agreement (errors << 1%) between the finite element and first-order shear deformation theory analytical solutions. For the
PVDF laminate, the piezoelectric coupling effect
increases the buckling load by approximately 3%.
Table 2 shows the thermal buckling load for
the same laminate configuration but with PZT layers on top and bottom. It is interesting to note that
Table 1. Nondimensionalized Thermal Buckling
Loads ( T ) for a Ten-Layer Symmetric Piezoelectric Composite Laminate (PVDF/0/90/0/90)s
MF1
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
a/h
5
10
15
20
25
30
35
40
60
80
100
1000
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
MF  Mixed Formulation (Finite Element)
UC  Uncoupled Piezoelectric Analysis
3
C  Coupled Piezoelectric Analysis
1
2
Table 2. Nondimensionalized Thermal Buckling
Loads ( T ) for a Ten-Layer Symmetric Piezoelectric Composite Laminate (PZT/0/90/0/90)s
MF1
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
MF  Mixed Formulation (Finite Element)
UC  Uncoupled Piezoelectric Analysis
3
C  Coupled Piezoelectric Analysis
1
2
piezoelectric coupling reverses the inplane stresses
induced in the PZT layers, which leads to negative
buckling loads for the various a/h ratios. This is
due to the pyroelectric coefficient of the PZT material being positive as opposed to negative for
PVDF materials. Ignoring the sign change, the
coupled buckling results are 64.6% to 68.9% higher than the corresponding uncoupled buckling
loads for 10 ≤ a/h ≤ 1000 ((-coupled result – uncoupled result)/uncoupled result).
Conclusions
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.
References
1. Tzou, H. S. (1993), Piezoelectric Shells
(Distributed Sensing and Control of Continua),
Kluwer Academic Publishers, Boston, MA.
2. Datchanamourty, Balasubramanian (2008),
“Nonlinear Static, Buckling and Dynamic
Analysis of Piezothermoelastic Composite
Plates Using Reissner-Mindlin Theory Based
on a Mixed Hierarchic Finite Element
Formulation,” Ph.D. Dissertation, University
of Kentucky, Lexington, KY.
3. Jonnalagadda, K. D. (1993), “Development of
Higher-Order Plate Theories and Applications
to Piezothermoelastic Laminates,” Master of
Science Thesis, University of Kentucky, Lexington, KY.