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

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. Outline. Assumptions Finite Element Equations Buckling Analysis Numerical Results

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

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  1. 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

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

  3. Laminated Plate

  4. 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

  5. Finite Element Equations

  6. 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

  7. 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

  8. 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

  9. Nonlinear Matrices = nonlinear stiffness matrix consistent with the von Karman approximation = nonlinear coupling matrix between displace-ments and electromagnetic potentials in the piezoelectric laminae

  10. Hierarchic Lagrangian Nodes

  11. Stress Resultant Nodes (x)

  12. Buckling Analysis = linear coefficient matrix = geometric stiffness matrix = inplane stress magnification factor

  13. Nonlinear Analysis = residual force vector = nonlinear stiffness matrix consistent with a total Lagrangian formulation = linear and nonlinear force vectors

  14. Nonlinear Solution Schematic

  15. 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

  16. Material Property Data

  17. Nondimensionalized Thermal Buckling Loads ( ) for a Ten-Layer Symmetric Piezoelectric Composite Laminate (PVDF/0/90/0/90)s 1MF  FE Mixed Formulation 2UC  Uncoupled Piezoelectric Analysis 3C  Coupled Piezoelectric Analysis

  18. Nonlinear Thermal Buckling for a Ten-Layer Symmetric Piezoelectric Composite Laminate (PVDF/0/90/0/90)s for a/h = 10

  19. Nonlinear Thermal Buckling for a Ten-Layer Symmetric Piezoelectric Composite Laminate (PVDF/0/90/0/90)s for a/h = 40

  20. Nonlinear Thermal Buckling for a Ten-Layer Symmetric Piezoelectric Composite Laminate (PVDF/0/90/0/90)s for a/h = 100

  21. Table 2. Nondimensionalized Thermal Buckling Loads ( ) for a Ten-Layer Symmetric Piezoelectric Composite Laminate (PZT/0/90/0/90)s 1MF  FE Mixed Formulation 2UC  Uncoupled Piezoelectric Analysis 3C  Coupled Piezoelectric Analysis

  22. 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.

  23. 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.

  24. Mechanically Loaded Plate • Six layer laminate: (PZT5/0/90)s • Simply supported • a = b = 0.2m • h = 0.001 m

  25. Buckling Mode (1,1)

  26. Buckling Mode (2,1)

  27. Buckling Mode (3,1)

  28. Buckling Mode (4,1)

  29. Nonlinear Buckling Response for a Symmetric Piezoelectric Composite Laminate (PZT/0/90)s Subjected to a Uniaxial Line Load (Qref = -1 kN/m)

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