New Hamiltonian Semi-analytical Approach for 3D Solution of Piezoelectric Smart Composites

Abstract

This chapter addresses the development of a new semi-analytical Lagrangian-Hamiltonian method for the three-dimensional solution of piezoelectric smart composite plates. It is based on the analytic state space symplectic Hamiltonian approach to fulfil the electromechanical multilayer interface continuity constraints and two-dimensional in-plane finite element (FE) numerical discretization to deal with arbitrary boundary conditions (BC) on the composite lateral edges. The originality of the proposed semi-analytical solution is that the latter feature (arbitrary BC handling) is reached through a mechanical displacements-electric potential primary variables based Lagrangian formalism, while the solution accuracy feature is reached through a primary and dual (transverse stresses and electric displacement) variables-based partial mixed Hamiltonian formalism. The transformation of the Lagrangian FE discretized formulation to a state space Hamiltonian one is made through the Legendre transformation. The proposed methodology is applied to the static actuation and sensing of piezoelectric hybrid laminated composite plates subjected to various BC. The obtained results comparison to reference ones of various benchmarks solutions, for non classical BC (cantilever), multilayer composite layups (angle-ply) and electromechanical loadings (uniform), from the open literature shows good computational convergence (coarse mesh), low cost (few FE degrees of freedom) and high accuracy (exact through-the-thickness) of the present new Hamiltonian semianalytical solutions. Thus, the provided tabulated numerical results can be used safely for benchmarking other closed-form, numerical or semi-analytical solutions.