INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2008; 73:869–894 Published online 29 August 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2156 Multiscale aggregating discontinuities: A method for circumventing loss of material stability Ted Belytschko 1, ∗, †, ‡ , Stefan Loehnert 2, § and Jeong-Hoon Song 1, ¶ 1 Theoretical and Applied Mechanics, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, U.S.A. 2 Institute of Mechanics and Computational Mechanics, Leibniz University Hannover, Appelstr. 9A, 30167 Hannover, Germany SUMMARY New methods for the analysis of failure by multiscale methods that invoke unit cells to obtain the subscale response are described. These methods, called multiscale aggregating discontinuities, are based on the concept of ‘perforated’ unit cells, which exclude subdomains that are unstable, i.e. exhibit loss of material stability. Using this concept, it is possible to compute an equivalent discontinuity at the coarser scale, including both the direction of the discontinuity and the magnitude of the jump. These variables are then passed to the coarse-scale model along with the stress in the unit cell. The discontinuity is injected at the coarser scale by the extended finite element method. Analysis of the procedure shows that the method is consistent in power and yields a bulk stress–strain response that is stable. Applications of this procedure to crack growth in heterogeneous materials are given. Copyright 2007 John Wiley & Sons, Ltd. Received 25 June 2007; Accepted 25 June 2007 KEY WORDS: multiscale; fracture; discontinuity; finite elements; extended finite element method 1. INTRODUCTION Multiscale methods based on unit cells, often called FE 2 methods [1, 2], are becoming of increasing importance because of their ability to compute the effective properties of complex microstructures on the fly. For example, the modeling of the response of composite material structures is streamlined ∗ Correspondence to: Ted Belytschko, Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, U.S.A. † E-mail: tedbelytschko@northwestern.edu ‡ Walter P. Murphy and McCormick Professor. § Research Associate. ¶ Graduate Research Assistant. Contract/grant sponsor: Army Office of Scientific Research; contract/grant number: W911-NF-05-1-0049 Contract/grant sponsor: Office of Naval Research; contract/grant number: N00014-06-1-0380 Copyright 2007 John Wiley & Sons, Ltd.