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Self-Strained Piezothermoelastic Composite Beam Analysis Using First-Order Shear Deformation Theory* G.E. Blandford, T.R. Tauchert and Y. Du University of Kentucky Lexington, KY ICCE/5 Las Vegas July 5-11, 1998 _________ *To appear: Composites Part B Engineering Journal 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 beam depth Displacement assumptions consistent with Mindlin theory Finite Element Equations Coupled Equations [K ]{a u } [K c ]{a } {FP } {Ft } {F } [K c ]T {a u } [K ]{a } {Fq } {P } Uncoupled Equations [K ]{a u } {FP } {Ft } {F } [K c ]{a } [K] = Stiffness matrix [K c ] = Coupling matrix [ K ] = Permittivity matrix {FP } = Nodal concentrated load vector {Ft } = Traction load vector {F } = Thermal load vector {Fq } = Electric charge load vector {P } = Pyroelectric load vector {a u } = Nodal displacement vector {a } = Nodal electric potential vector Sample Problems Material Property Data PVDF PZT Graphite-Epoxy k 1 = k2 = k3 = k0 k 1 = k2 = k3 = k0 k1 = 100k0, k2 = k3 = k0 E1 = E2 = E3 = E0 E1 = E2 = E3 = 30 E0 E1 = 90E0, E2 = E3 = 5E0 12 = 13 = 23 = 0 12 = 13 = 23 = 0 12 = 13 = 23 = 0 G12 = G13 = G23 = 0.375E0 G12 = G13 = G23 = 11.25E0 G12 = G13 = 4E0, G23 = 1.5E0 1 = 2 = 3 = 0 1 = 2 = 3 = 0.01 0 1= 0.00020, 2= 3 = 0.20 11 = 22 = 33 = 0 11 = 22 = 33 = 150 0 --- d31 = d32 = -d24 = -d15 = d0 d31 = d32 = -7d0, d24 = d15 = --- d33 = -1.2d0 24d0, d33 = 14d0 p 3 = p0 p3 = -30p0 --- Problem 1 Simply Supported Beam Orthotropic Layers 2 and 4 – Graphite Epoxy Piezoelectric Layers 1, 3, 5 – PVDF or PZT Spacially Varying Electro-Magnetic Potential Gradient 1( x ) 0 sin (x / L) w *mid plane Layup L/h Ref. 1 - U FEM - U FEM - C *xx max Ref. 1 - U FEM - U FEM - C Layer, Position PVDF/90/ Isotropic/ 90/PVDF 40 -0.3729 -0.3717 -0.3704 8.359 8.417 8.407 2, t (1, U & C) 5 -0.3870 -0.3695 -0.3682 8.377 8.417 8.407 2, t (1, U & C) PZT/90/ Isotropic/ 90/PZT 40 5.5460 5.4423 5.1769 -661.3 -661.3 -675.6 3, t (1, U & C) 5 5.4609 5.2687 5.0103 -602.2 -661.3 -676.0 3, t (1, U & C) PVDF/90/ Isotropic/ 0/PVDF 40 -0.2761 -0.2760 -0.2758 24.824 25.068 25.066 4, t (1, U & C) 5 -0.2787 -0.2747 -0.2744 24.204 25.068 25.066 4, t (1, U & C) PZT/90/ Isotropic/ 0/PZT 40 5.6510 5.5588 5.3181 -656.9 -658.6 -710.3 4, t (1, U & C) 5 5.3630 5.3963 5.1660 -571.8 -658.6 -711.0 3, t (1); 4, t (U&C) Spacially Varying Temperature Gradient 1 ( x ) 0 sin (x / L) w *mid plane Layup L/h Ref. 1 - U FEM - U FEM - C *xx max Ref. 1 - U FEM - U FEM - C Layer, Position PVDF/90/ Isotropic/ 90/PVDF 40 -0.0732 -0.0721 -0.0659 1.503 1.503 1.350 2, t (1, U & C) 5 -0.0739 -0.0721 -0.0659 1.480 1.503 1.352 2, t (1, U & C) PZT/90/ Isotropic/ 90/PZT 40 -0.00206 -0.00204 0.00957 -0.698 -0.699 1.790 2,t (1 & U); 1,b (C) 5 -0.00183 -0.00204 0.00915 -0.687 -0.699 1.768 2,t (1 & U); 1,b (C) PVDF/90/ Isotropic/ 0/PVDF 40 -0.0742 -0.0738 -0.0689 7.461 7.525 7.064 4, t (1, U & C) 5 -0.0675 -0.0738 -0.0689 6.150 7.525 7.070 4, t (1, U & C) PZT/90/ Isotropic/ 0/PZT 40 -0.00294 -0.00291 0.00978 -0.709 -0.711 1.858 2,t (1 & U); 1,b (C) 5 -0.00247 -0.00291 0.00938 -0.649 -0.711 1.834 2,t (1 & U); 1,b (C) Problem 2 Simply Supported Beam, L/h = 20 T = Total Thickness of PZT Layers 1 ( x ) 0 sin (x / L) CONCLUSIONS Five-Layer Hybrid Laminate Finite element displacement and lamina stress results compare favorably with the uncoupled analytical elasticity solutions of Tauchert for moderately thick laminates. Coupled/Uncoupled analyses nearly identical using PVDF laminae for the electric flux load case. Coupled analyses predict a reduction in displacement and stress magnitudes using PVDF laminae for the temperature load case. Coupled analyses showed that the hybrid laminates with PZT top and bottom surface laminae resulted in a slight reduction in displacements but slightly increased maximum lamina stress values for electric flux loading. Hybrid laminates with PZT laminae subjected to the temperature gradient loading produced self-strain deformations due to the pyroelectric effect that exceeded the temperature load deformations, which resulted in a change in the maximum stress location compared to the uncoupled analyses. Thick beams (length-to-depth ratios of five) analytical elasticity and FE displacement results did not agree, as should be expected since first-order shear deformation theory is not applicable. Uncoupled analytical and finite element stress results through the thickness of the beam do compare favorably provided the same through thickness temperature distribution is used. Three-Layer Hybrid Laminate Results show that the direct piezoelectric effect has an excess capacity to eliminate the bending displacements caused by the temperature gradient for this particular beam configuration. Excess capacity occurred for all PZT thickness to laminate thickness ratios, but particularly for T/h = 0.15 – 0.75 with a maximum at T/h = 0.40.