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

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Assumptions Finite Element Equations Buckling Analysis Numerical Results Summary and Conclusions

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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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= i th element node displacement vector; five displacements per node: u, v, w, x, y = i th element node electromagnetic potential = i th element Gauss point transverse shear stress resultant vector; two per node: Q x and Q y

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= mechanical load vector = electrical load vector = temperature-stress load vector = pyroelectric load vector = nonlinear temperature-stress load vector

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= linear stiffness matrix for = linear coupling matrix between and = linear matrix for = linear stiffness matrix for = linear coupling matrix between and

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= 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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= geometric stiffness matrix = linear coefficient matrix = inplane stress magnification factor

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= 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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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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PVDFPZTGraphite-Epoxy E 1 = E 2 = E 3 =2 GPaE 1 = E 2 = E 3 = 60 GpaE 1 =138 GPa, E 2 = 8.28 GPa 12 = 13 = 23 = 0.333 12 = 0.33 G 12 = G 13 = G 23 = 0.75 GPaG 12 = G 13 = G 23 = 22.5 GPaG 12 = G 13 = G 23 = 6.9 GPa 1 = 2 = 3 = 1.2x10 -4 / 0 C 1 = 2 = 3 = 1x10 -6 / 0 C 1 = 0.18x10 -6 / 0 C 2 = 27x10 -6 / 0 C 11 = 22 = 33 = 1x10 -10 F/m 11 = 22 = 33 = 1.5x10 -8 F/m --- d 31 = d 32 = -d 24 = -d 15 23x10 -12 0 C/N d 31 = d 32 = -1.75x10 -8 0 C/N d 24 = d 15 = 6x10 -10 0 C/N --- p 3 = -2.5x10 -5 0 C/K/m 2 p 3 = 7.5x10 -4 0 C/K/m 2 ---

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a/h AnalyticalMF 1 UC 2 C3C3 51.457 1.502 101.8111.8131.869 151.8981.8991.958 201.9301.9321.992 251.9461.9472.008 301.9541.9562.016 351.9601.9612.022 401.9631.9642.025 601.9691.9702.031 801.9711.9722.034 1001.9721.9732.035 10001.9731.9752.037 1 MF FE Mixed Formulation 2 UC Uncoupled Piezoelectric Analysis 3 C Coupled Piezoelectric Analysis

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a/h MF 1 UC 2 C3C3 54.208-6.584 105.475-9.010 155.799-9.675 205.922-9.931 255.981-10.055 306.013-10.123 356.033-10.165 406.045-10.192 606.069-10.242 806.077-10.260 1006.081-10.268 10006.088-10.283 1 MF FE Mixed Formulation 2 UC Uncoupled Piezoelectric Analysis 3 C Coupled Piezoelectric Analysis

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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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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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Looking into different stacking sequences – symmetric and anti-symmetric stacking Looking into the effect of the piezoelectric thickness effect on buckling for the two cases above.

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

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