An incremental Lagrangian formulation to the analysis of piezoelectric bodies subjected to geometric non-linearities

An incremental formulation to model geometric non-linearities in piezoelectric solids is presented. First, Lagrangian and Eulerian measures for the electric field and the electric displacement are discussed together with traditional mechanical measures. Using the Lagrangian energetic conjugated measures, Hamilton’s principle is used to derive a consistent variational incremental description for the finite movement of a piezoelectric body. Those equations are approximated using the finite element method, and the resulting equations are solved using the Newton–Raphson method and arc-length constraints. Examples with non-linear behaviour such as stress stiffening, large rotations and limit points are presented