INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2005; 62:1857–1872 Published online 2 February 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1256 Continuum–discontinuum modelling of shear bands Esteban Samaniego ‡,  and Ted Belytschko ∗, † Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Room A212, Evanston, IL 60208-3111, U.S.A. SUMMARY A methodology to model shear bands as strong discontinuities within a continuum mechanics context is presented. The loss of hyperbolicity of the IBVP is used as the criterion for switching from a classical continuum description of the constitutive behaviour to a traction–separation model acting at the discontinuity surface. The extended finite element method (XFEM) is employed for the spatial discretization of the governing equations. This enables the shear bands to be arbitrarily positioned within the mesh. Examples that study the shear band progression within a rate-independent material are presented. Copyright  2005 John Wiley & Sons, Ltd. KEY WORDS: finite element method; shear bands; loss of hyperbolicity; traction–separation laws; extended finite element method 1. INTRODUCTION A shear band can be described as a narrow region in a solid undergoing intense shearing. After a shear band is fully developed, relative sliding of the two sides of the band similar to mode II fracture can be observed. Shear bands can be regarded as specific instances of the more general phenomenon of strain localization and as material instabilities. When dealing with rate independent materials undergoing strain softening, the development of bands of localized deformation has classically been linked to the loss of hyperbolicity of the linearized initial boundary value problem (IBVP). In Reference [1], a closed form solution was obtained for a wave propagation problem in a one-dimensional bar. It was shown that, despite the loss of positivity of the tangent modulus, a solution could be obtained if the region ∗ Correspondence to: T. Belytschko, Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Room A212, Evanston, IL 60208-3111, U.S.A. † E-mail: t-belytschko@northwestern.edu ‡ Post Doctoral Research Fellow, Northwestern University.  E-mail: E-msamaniego@northwestern.edu Contract/grant sponsor: Office of Naval Research Received 3 September 2004 Revised 6 October 2004 Copyright  2005 John Wiley & Sons, Ltd. Accepted 18 October 2004