Thermal Buckling of Piezother-moelastic Composite Plates Using a Mixed Finite Element Formulation

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Thermal Buckling of Piezother-moelastic Composite
Plates Using a Mixed Finite Element Formulation

• By
• Balasubramanian Datchanamourty
• and George E. Blandford
• University of Kentucky
• Lexington, KY

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Outline

• Assumptions
• Finite Element Equations
• Buckling Analysis
• Numerical Results
• Summary and Conclusions

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Laminated Plate
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Assumptions

• 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

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Finite Element Equations
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Unknown Vectors
ith element node displacement vector
five displacements per node u, v, w, ?x, ?y
ith element node electromagnetic
potential ith element Gauss point
transverse shear stress resultant vector two
per node Qx and Qy
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Force Vectors
mechanical load vector of element e
electrical load vector of element e
temperature-stress load vector pyroelectric
load vector nonlinear temperature-stress load
vector
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Linear Matrices
linear stiffness matrix for
linear coupling matrix between and
linear coupling matrix between and
linear matrix for
linear coupling matrix between and
linear stiffness matrix for
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Nonlinear Matrices
nonlinear stiffness matrix consistent with the
von Karman approximation
nonlinear coupling matrix between
displace-ments and electromagnetic potentials in
the piezoelectric laminae
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Hierarchic Lagrangian Nodes
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Stress Resultant Nodes (x)
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Buckling Analysis
linear coefficient matrix
geometric stiffness matrix
? inplane stress magnification factor
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Nonlinear Analysis
residual force vector
nonlinear stiffness matrix consistent with a
total Lagrangian formulation
linear and nonlinear force vectors
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Nonlinear Solution Schematic
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Numerical Results

• Thermal Buckling of (0/90/0/90)s Graphite-Epoxy
  laminate plus top and bottom piezoelectric lamina
  PVDF or PZT
• Simply supported square plate

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Material Property Data
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 ?1 0.18x10-6 /0C ?2 27x10-6 /0C
?11 ?22 ?33 1x10-10 F/m ?11 ?22 ?33 1.5x10-8 F/m ---
d31 d32 -d24 -d15 23x10-12 0C/N d31 d32 -1.75x10-8 0C/N d24 d15 6x10-10 0C/N ---
p3 -2.5x10-5 0C/K/m2 p3 7.5x10-4 0C/K/m2 ---
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Nondimensionalized Thermal Buckling Loads ( )
for a Ten-Layer Symmetric Piezoelectric Composite
Laminate (PVDF/0/90/0/90)s
a/h Analytical MF1 MF1
a/h UC2 UC2 C3
5 1.457 1.457 1.502
10 1.811 1.813 1.869
15 1.898 1.899 1.958
20 1.930 1.932 1.992
25 1.946 1.947 2.008
30 1.954 1.956 2.016
35 1.960 1.961 2.022
40 1.963 1.964 2.025
60 1.969 1.970 2.031
80 1.971 1.972 2.034
100 1.972 1.973 2.035
1000 1.973 1.975 2.037
1MF ? FE Mixed Formulation 2UC ? Uncoupled
Piezoelectric Analysis 3C ? Coupled Piezoelectric
Analysis
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Nonlinear Thermal Buckling for a Ten-Layer
Symmetric Piezoelectric Composite Laminate
(PVDF/0/90/0/90)s for a/h 10
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Nonlinear Thermal Buckling for a Ten-Layer
Symmetric Piezoelectric Composite Laminate
(PVDF/0/90/0/90)s for a/h 40
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Nonlinear Thermal Buckling for a Ten-Layer
Symmetric Piezoelectric Composite Laminate
(PVDF/0/90/0/90)s for a/h 100
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Table 2. Nondimensionalized Thermal Buckling
Loads ( ) for a Ten-Layer Symmetric
Piezoelectric Composite Laminate (PZT/0/90/0/90)s
a/h MF1 MF1
a/h UC2 C3
5 4.208 -6.584
10 5.475 -9.010
15 5.799 -9.675
20 5.922 -9.931
25 5.981 -10.055
30 6.013 -10.123
35 6.033 -10.165
40 6.045 -10.192
60 6.069 -10.242
80 6.077 -10.260
100 6.081 -10.268
1000 6.088 -10.283
1MF ? FE Mixed Formulation 2UC ? Uncoupled
Piezoelectric Analysis 3C ? Coupled Piezoelectric
Analysis
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Summary and Conclusion

• 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.

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Summary and Conclusion

• 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.

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Mechanically Loaded Plate

• Six layer laminate (PZT5/0/90)s
• Simply supported
• a b 0.2m
• h 0.001 m

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Buckling Mode (1,1)
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Buckling Mode (2,1)
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Buckling Mode (3,1)
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Buckling Mode (4,1)
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Nonlinear Buckling Response for a Symmetric
Piezoelectric Composite Laminate (PZT/0/90)s
Subjected to a Uniaxial Line Load (Qref -1 kN/m)