On the determination of missing boundary data for solids with nonlinear material behaviors, using displacement fields measured on a part of their boundaries

Abstract

The paper is devoted to the derivation of a numerical method for expanding available mechanical fields (stress vector and displacements) on a part of the boundary of a solid into its interior and up to unreachable parts of its boundary (with possibly internal surfaces). This expansion enables various identification or inverse problems to be solved in mechanics. The method is based on the solution of a nonlinear elliptic Cauchy problem because the mechanical behavior of the solid is considered as nonlinear (hyperelastic or elastoplastic medium). Advantage is taken of the assumption of convexity of the potentials used for modeling the constitutive equation, encompassing previous work by the authors for linear elastic solids, in order to derive an appropriate error functional. Two illustrations are given in order to evaluate the overall efficiency of the proposed method within the framework of small strains and isothermal transformation.