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.

One-dimensional Eikonal Non-Local (ENL) damage models: Influence of the integration rule for computing interaction distances and indirect loading control on damage localization

“…Damage evolves according to the exponential law A quasi-newton method is used to solve the global equilibrium problem. To follow the structural response in the snap-back phase, the external loading is controlled indirectly based on a simple path-following technique [8,34].…”

A Continuation Procedure for the Quasi-Static Analysis of Materially and Geometrically Nonlinear Structural Problems

“…In the present study, a proper tradeoff between quality and computational costs was found by taking η equal to 0.1. A new value of ω k , corresponding to the pointx(s i+1 ), can be computed and, according to Equations (18) and 20, a new estimate ofx * and ∆s * is also available. The procedure is repeated until convergence is met according to the criterion:…”

“…An example is found in Ref. [18], where path following techniques were discussed in the framework of Embedded Finite Element Method (E-FEM).…”

Damage-to-fracture transition through an Eikonal Non-Local (ENL) continuum damage formulation with embedded strong discontinuities

“…In order to trigger the localization, a stiffness defect is introduced at the middle of the bar through a weakened finite element. To describe the system's snap-back response in the load post-peak phase, the applied displacement at the end of the bar is indirectly controlled by imposing a constant rate of variation to the deformation of the weakened finite element [8]. The material is initially linear elastic.…”