Consistency and recovery in electroelasticity. Part II: Equilibrium and mixed finite elements

Developing finite elements for electroelastic analysis requires a special care as the coupling in the discrete equations depends on the matching between the approximations assumed for mechanical and electrical variables. To provide a formal basis to this intuitive remark, a notion of consistency is established and a rigorous consistency analysis is presented for the standard finite element model based on assumed displacement and electric potential. In this way, specific analytical requirements are obtained which serve as a guide to select the interpolation functions for primary variables. Moreover, the analysis shows that violating consistency can be reflected as spurious outcomes upsetting the local distributions of secondary variables. Indeed, this undesirable effect is shown to be typical of the standard approach if stress and electric flux density are computed via the constitutive equations. To cure the trouble, an alternative recovery procedure is devised based on the consistency analysis. The procedure is variationally correct and reconstitutes in a consistent manner the distributions of stress and electric flux density.