Three-dimensional explicit finite element formulation for shear localization with global tracking of embedded weak discontinuities

Abstract

Shear localization is a damage mechanism characterized by the concentration of plastic deformation within thin bands of materials, often observed as a precursor to failure in metals. In this paper, we present a three-dimensional explicit finite element formulation with embedded weak discontinuities for the treatment of shear localization under dynamic loading conditions. In this formulation, the onset of localization is detected via a material stability analysis suitable for rate-sensitive materials, and the continuity of propagating shear bands is ensured using a global tracking strategy involving a heat-conduction type boundary value problem, solved for a scalar level set function over the global domain of the problem. In addition, we propose a modified quadrature rule that is used to compute the contribution of an individual element to the global finite element arrays, taking into account the position of the embedded shear band within that element. The mechanical threshold stress (MTS) model, modified to account for dynamic recrystallization, is adopted as the constitutive model for rate- and temperature-sensitive metallic materials, and the constitutive equations are formulated in a corotational setting. The algorithmic implementation of the integrated finite element framework, including the global tracking strategy, is described in detail. Subsequently, a simple example problem involving shear band formation under uni-axial tension is presented, to illustrate the advantages of the proposed computational framework over conventional finite element techniques, in terms of convergence and mesh insensitivity of their numerical results. In addition, to demonstrate the capabilities of the proposed computational framework in more realistic problems, numerical results are compared to data from split-Hopkinson pressure bar dynamic experiments, conducted on two stainless steel samples with different geometric designs. In each problem presented, we study the effect of the mesh size on the numerical solution, including global results such as the load–displacement response, and local quantities such as material state variables at individual quadrature points. These numerical studies show that the geometry and total volume of the fully-developed embedded shear band are independent of the mesh size.