Discontinuity-scale path-following methods for the embedded discontinuity finite element modeling of failure in solids

Abstract

The initiation and propagation of cracks in solids often leads to unstable structural responses characterized by snap-backs. Path-following procedures allow finding a solution to the algebraic system of equations resulting from the numerical formulation of the considered problem. Accordingly, the boundary value problem is supplemented by a novel global unknown, namely, the loading factor, which should comply with a dedicated equation, the so-called path-following constraint equation. In this contribution, path-following methods are discussed within the framework of the Embedded Finite Element Method (E-FEM). Thanks to the enhanced kinematic description provided by the E-FEM, we show that it is possible to formulate constraint equations where the prescribed quantities are directly related to the dissipative process occurring at the strong discontinuity level. After introducing the augmented E-FEM formulation, three discontinuity-scale path-following constraints and their numerical implementation (using an operator-splitting method) are described. Simple quasi-static strain localization problems characterized by unstable structural responses exhibiting multiple snap-backs are numerically simulated. A comparison with several well-known constraint equations (commonly used in non-linear finite element computations) is finally established. This allows for illustrating the main features of the proposed methods as well as their efficiency in controlling highly unstable embedded discontinuity finite element simulations.