A large-deformation gradient theory for elastic–plastic materials: Strain softening and regularization of shear bands Lallit Anand ⇑ , Ozgur Aslan, Shawn A. Chester Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA article info Article history: Received 5 August 2011 Received in final revised form 26 September 2011 Available online 12 October 2011 Keywords: Plasticity Strain softening Shear bands Strain gradients Finite elements abstract We present a large deformation gradient theory for rate-independent, isotropic elastic– plastic materials in which in addition to the standard equivalent tensile plastic strain   p , a variable e p is introduced for the purpose of regularization of numerical simulations of shear band formation under strain softening conditions. Specifically, in contrast to tradi- tional gradient theories which are based on   p and r   p , here we develop a theory which depends on   p ; e p , and the gradient re p , with the latter chosen to represent a measure of the inhomogeneity of the microscale plasticity. We have numerically implemented a two-dimensional plane strain version of our theory in a commercial finite element program by writing a user-element subroutine. Representative examples which demonstrate the ability of the theory and its numerical implementation to satisfactorily model large-defor- mation strain-softening response accompanied by intense localized shear bands — with no pathological mesh-dependence — are provided. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction As is well-known, strain softening in plastically deforming solids invariably leads to the formation of one or more shear bands. In a shear band the deformation is localized within a confined region of finite thickness — a thickness which is set by the underlying microstructure of the material. Continuum mechanics based analysis of localization of plastic deformation into shear bands has received considerable attention over the past four decades. For materials which may be adequately modeled as rate-independent and considered to be deforming quasi-statically and isothermally, a mathematical method is available for analyzing the onset of shear band formation. Specifically, the onset of localization in such an analysis is viewed as a material instability, and critical conditions are sought at which the rate-independent elastic–plastic constitutive equations first allow a bifurcation from a homogeneous deformation into a shear band mode. The results of such an analysis show that a necessary condition for the existence of shear bands is that the velocity equations of continuing equilibrium suf- fer a loss of ellipticity, 1 and this occurs when the rate of strain-hardening reaches a critically low value. Further, the boundaries of the emergent shear bands correspond to the associated characteristic lines (cf., e.g., Hill and Hutchinson, 1975; Rudnicki and Rice, 1975; Rice, 1977; Anand and Spitzig, 1980, 1982). The loss of ellipticity of the underlying field equations for strain soft- ening materials gives rise to ill-posed boundary value problems, and finite element simulations result in a pathological mesh- dependency of the calculated results (cf., e.g., Benallal et al., 1989). 0749-6419/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2011.10.002 ⇑ Corresponding author. Tel.: +1 617 253 1635. E-mail address: anand@mit.edu (L. Anand). 1 Classical elastic–plastic constitutive equations may be expressed in a form in which a suitable stress rate is related to the velocity gradient. The equation obtained from substituting such a rate-constitutive equation into the first-order rate equation for continuing equilibrium is called the velocity equation of continuing equilibrium. International Journal of Plasticity 30–31 (2012) 116–143 Contents lists available at SciVerse ScienceDirect International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas