1 Non-Linear Piezoelectric Exact Geometry Solid-Shell Element Based on 9-Parameter Model Gennady M. Kulikov Department of Applied Mathematics & Mechanics 2 Shell Kinematics (Kulikov, 2001) Laminated shell with embedded piezoelectric layer (PZT) h =z z shell thickness; u ( 1, 2 ) displacement vectors of S-surfaces (I = , M, ) I + (1)(2) 3 Representation of Displacement Vectors Strain-Displacement Equations (Kulikov & Carrera, 2008) ij ( 1, 2 ) Green-Lagrange strains of S-surfaces (I = , M, ) I (3)(4)(5) 4 Strain Parameters (6) A , k Lame coefficients and principal curvatures of reference surface 5 Electric Potential Electric Field (7)(8)(9) ( 1, 2 ) electric potentials of outer surfaces of th piezoelectric layer (A = , ) A 6 Constitutive Equations (10)(11)(12) strain vector; ( ) stress vector; E ( ) electric field vector D ( ) electric displacement vector; A ( ), C ( ) elastic matrices D ( ) electric displacement vector; A ( ), C ( ) elastic matrices d ( ), e ( ) piezoelectric matrices; ( ), ( ) dielectric matrices d ( ), e ( ) piezoelectric matrices; ( ), ( ) dielectric matrices 7 Hu-Washizu Variational Equation Displacement-Independent Strains Stress Resultants (13)(14) 8 (15) Exact geometry piezoelectric solid-shell element based on 9-parameter model, where = ( d )/ normalized curvilinear coordinates; 2 element lengths 9 D uu, D u , D mechanical, piezoelectric and dielectric constitutive matrices p i surface tractions; q surface charge densities (A = , ) ( ) ( ) AA 10 Finite Element Formulation Displacement Interpolation Electric Potential Interpolation Assumed Natural Strain Method (16)(17)(18) 11 Assumed Electric Field Interpolation Displacement-Independent Strains Interpolation (19)(20) B r, B r, A r (U) constant inside the element nodal matrices u ( ) u ij = 0 except for ij = 1, 11 = 13 = 33 = 22 = 23 = 33 = r r 1 2 Stress Resultants Interpolation Finite Element Equations Matrix K T is evaluated using analytical integration Matrix K T is evaluated using analytical integration No matrix inversion is needed to derive matrix K T No matrix inversion is needed to derive matrix K T Use of extremely coarse meshes Use of extremely coarse meshes K T tangent stiffness matrix of order 36 36; U incremental displacement vector (21)(22) 13 1. Cantilever Plate with Segmented Actuators (geometrically linear solution) (geometrically linear solution) 14 Bending w 1 = u 3 (B)/b for [30/30/0] s plate M Twisting w 2 = (u 3 (C) u 3 (A))/b for [30/30/0] s plate MM 15 Deformed configuration of [0/45/-45] s plate at voltage =1576 V NI number of Newton iteration; RN = ||R || Euclidean norm of residual vector DN = ||U G U G || Euclidean norm of global displacement vectors DN = ||U G U G || Euclidean norm of global displacement vectors [NI+1] [NI] [NI] 2. Cantilever Plate with Segmented Actuators (geometrically non-linear solution) (geometrically non-linear solution) 16 3. Spiral Actuator (PZT-5H) r min = mm, r max = 15.2 mm, h = 0.2 mm L = 215 mm, b = 3.75 mm, =100 V L = 215 mm, b = 3.75 mm, =100 V r = r min + a 2, 2 [0, 8 ] 17 4. Shape Control of Pinched Hyperbolic Shell a b a b c d c d r = 7.5 cm, R = 15 cm, L = 20 cm h C = 0.04 cm, h PZT = 0.01 cm, F = 200 N [90/0/90] graphite/epoxy shell Shell configurations at: (a) F = 0, = 0; (b) F = 200 N, = 0 (c) F = 200 N, = 1000 V; (d) F = 200 N, = 1960 V (c) F = 200 N, = 1000 V; (d) F = 200 N, = 1960 V 18 Midsurface displacements of hyperbolic shell at points A, B, C and D versus: (a) force F for = 0, (b) voltage for F = 200 N a b a b 19 a b a b Midsurface displacements of hyperbolic shell at points belonging to: (a) hyperbola BD and (b) hyperbola AC 20 Thanks for your attention!