Discontinuous velocity domain splitting in limit analysis

Abstract

We present a new limit analysis method, originated in Ionescu and Lachand-Robert (2005) and called discontinuous velocity domain splitting method (DVDS). DVDS is a mesh free method which focuses on the strain localization and completely neglect the bulk deformations. It considers the kinematic variational principle on a special class of virtual velocity fields to get an upper bound of the limit load. To construct this class of virtual velocity fields, the rigid-plastic body is splinted into simple connected sub-domains and on each such sub-domain a rigid motion is associated. The discontinuous collapse flow velocity field results in localized deformations only, located at the boundary of the sub-domains. In the numerical applications of the DVDS method, we introduce a numerical technique based on a level set description of the partition of the rigid-plastic body and on genetic minimization algorithms. For the anti-plane flow of a von Mises material, DVDS is exact in solving the limit analysis problem: the collapse flow is a rigid motion of a sub-domain. The associated deformation rate is localized on the smooth boundary of the moving sub-domain representing the fracture surface. In the case of in-plane deformation of pressure insensitive materials, the internal boundaries of the sub-domains are parts of circles or straight lines, tangent to the collapse velocity jumps. In this case, DVDS reduces to the block decomposition method, which was intensively used to get analytical upper bounds of the limit loads. When applied to the two notched tensile problem of a von Mises material, DVDS gives excellent results with a low computational cost. Furthermore, DVDS was applied to model collapse in pressure sensitive plastic materials. Illustrative examples for homogenous and heterogeneous Coulomb and Cam-Clay materials shows that DVDS gives excellent prediction of limit loads and on the collapse flow.