Direct localized boundary-domain integro-differential formulations for physically nonlinear elasticity of inhomogeneous body

A static mixed boundary value problem (BVP) 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 BVP 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

Comparison of a Material Point Method and a Galerkin meshfree method for the simulation of cohesive-frictional materials

The simulation of large deformation problems, involving complex history-dependent constitutive laws, is of paramount importance in several engineering fields. Particular attention has to be paid to the choice of a suitable numerical technique such that reliable results can be obtained. In this paper, a Material Point Method (MPM) and a Galerkin Meshfree Method (GMM) are presented and verified against classical benchmarks in solid mechanics. The aim is to demonstrate the good behavior of the methods in the simulation of cohesive-frictional materials, both in static and dynamic regimes and in problems dealing with large deformations. The vast majority of MPM techniques in the literature are based on some sort of explicit time integration. The techniques proposed in the current work, on the contrary, are based on implicit approaches, which can also be easily adapted to the simulation of static cases. The two methods are presented so as to highlight the similarities to rather than the differences fromPeer ReviewedPostprint (published version