A discontinuous Galerkin method for non-linear electro-thermo-mechanical problems: application to shape memory composite materials

A coupled electro-thermo-mechanical discontinuous Galerkin (DG) method is developed considering the non-linear interactions of electrical, thermal, and mechanical fields. In order to develop a stable discontinuous Galerkin formulation the governing equations are expressed in terms of energetically conjugated fields gradients and fluxes. Moreover, the DG method is formulated in finite deformations and finite fields variations. The multi-physics DG formulation is shown to satisfy the consistency condition, and the uniqueness and optimal convergence rate properties are derived under the assumption of small deformation. First the numerical properties are verified on a simple numerical example, and then the framework is applied to simulate the response of smart composite materials in which the shape memory effect of the matrix is triggered by the Joule effect.