Direct Localized Boundary-Domain Integro-Differential Formulations for Physically Nonlinear Elasticity of Inhomogeneous Body

A static mixed boundary value problem of physically nonlinear elasticity for a continuously inhomogeneous body is considered. Using the two-operator Green-Betti formula and the fundamental solution of an auxiliary linear operator, a non-standard boundary-domain integro-differential formulation of the problem is presented, with respect to the displacements and their gradients. Using a cut-off function approach, the corresponding localized parametrix is constructed to reduce the nonlinear boundary value problem to a nonlinear localized boundary-domain integro-differential equation. Algorithms of mesh-based and mesh-less discretizations are presented resulting in sparsely populated systems of nonlinear algebraic equations.