Two‐scale method for shear bands: thermal effects and variable bandwidth

Abstract

SUMMARYA method for the analysis of shear bands using local partition of unity is developed in the framework of the extended finite element method (XFEM). Enrichments are introduced for both the displacement field and the thermal field. The shear band width is determined by minimizing the plastic work. A coupled finite strain thermo-elastoplastic constitutive law is used. The enrichment is injected into the mesh when the material law becomes unstable. The criterion based on a complete stability analysis for materials in the finite strain regime including heat conduction, strain hardening, strain rate hardening and thermal softening is presented. A mixed continuous quadrilateral element is employed. The method is applied to the Nesterenko experiments, which exhibit multiple propagating shear bands and other problems.

Efficient adaptive integration of functions with sharp gradients and cusps in n‐dimensional parallelepipeds

“…The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in n-dimensional parallelepiped elements. With the above setting, the adaptive quadrature function can be called (see Figure 1): [X, W] = ndimensional_adaptive_integration(fn, d, [5,8], 1e-6); The MATLAB code for the quadrature construction follows. …”

The XFEM for high‐gradient solutions in convection‐dominated problems

“…This can Three different choices of enrichment functions are investigated. The first function is adapted from Areias and Belytschko [11] and depends on a parameter that specifies the width of the function and a parameter n that specifies the gradient of the function. By "width" of the regularized step function we refer to the region where the function varies monotonically between 0 (or −1) and 1.…”

“…Thereby, they incorporate the non-smooth behavior in the approximation space. Arieas and Belytschko [11] embedded a fine scale displacement field with a high strain gradient around a shear band. They used a tangent hyperbolic type function for the enrichment.…”

“…In [10,11,12], the regularized Heaviside functions depend on physical considerations. Only one enrichment function is used in [11,13].…”

“…Only one enrichment function is used in [11,13]. A common feature of most previous applications is that the change from 0 to 1 takes place within one element.…”