1. 1 Non-Linear Piezoelectric Exact Geometry Solid-Shell Element Based on 9-Parameter Model Gennady M. Kulikov Department of Applied Mathematics & Mechanics

2. 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. 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. 4 Strain Parameters (6) A , k  – Lame coefficients and principal curvatures of reference surface

5. 5 Electric Potential Electric Field (7)(8)(9)   (  1,  2 ) – electric potentials of outer surfaces of th piezoelectric layer (A = ,  ) A

6. 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. 7 Hu-Washizu Variational Equation Displacement-Independent Strains Stress Resultants (13)(14)

8. 8 (15) Exact geometry piezoelectric solid-shell element based on 9-parameter model, where   = (    d  )/  – normalized curvilinear coordinates; 2  – element lengths

9. 9 D uu, D u , D  – mechanical, piezoelectric and dielectric constitutive matrices p i – surface tractions; q – surface charge densities (A = ,  ) ( ) ( ) AA

10. 10 Finite Element Formulation Displacement Interpolation Electric Potential Interpolation Assumed Natural Strain Method (16)(17)(18)

11. 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 = 1 00 01 01 01 10 10 10 r r 1 2

12. 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. 13 1. Cantilever Plate with Segmented Actuators (geometrically linear solution) (geometrically linear solution)

14. 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. 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. 16 3. Spiral Actuator (PZT-5H) r min = 1.875 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. 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. 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. 19 a b a b Midsurface displacements of hyperbolic shell at points belonging to: (a) hyperbola BD and (b) hyperbola AC

20. 20 Thanks for your attention!