Finite Element Formulations

Abstract

In this chapter, resultant strain and stress are introduced, such that the volume integration can be treat as surface integration. In order to describe the unrestricted finite rotations in thin-walled smart structures, five mechanical nodal DOFs are defined to represent the six kinematic parameters in strain-displacement relations by using Euler angles. Furthermore, an eight-node elements with five mechanical nodal DOFs and additionally integrated with one electrical DOF using full integration or uniformly reduced integration scheme are developed for both composite and smart structures. Implementing both linear constitutive equations and electroelastic nonlinear constitutive equations, one obtains nonlinear FE models by Hamilton’s principle and the principle of virtual work, in which various geometrically nonlinear phenomena discussed in Chap. 3 are considered. In the last part of this chapter, several numerical algorithms are developed for solving the nonlinear equilibrium equations and the equations of motion.