Solution of nonlinear Cauchy problem for hyperelastic solids

Abstract

The problem of expanding given (measured) fields at the surface of a solid within the solid and up to inaccessible parts of its boundary is addressed for a nonlinear hyperelastic medium. The problem is formulated as a nonlinear Cauchy problem and is solved thanks to a technique consisting of splitting the unknown field into two solutions of well posed problems and minimizing a specially designed error in constitutive equation between the two fields, taking advantage of the convexity of the hyperelastic potential. The minimization involves as unknowns the boundary conditions fields on the inaccessible part of the boundary of the solid. Two illustrations are given, the first one with a twice-differentiable hyperelastic potential describing a material with nonlinear compressibility, the second one deals with a geomaterial with asymmetric elasticity in the tension and compression ranges, and involves an only one-differentiable potential.