INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2015; 101:230–250 Published online 11 November 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4814 Combined continuum damage-embedded discontinuity model for explicit dynamic fracture analyses of quasi-brittle materials Timo Saksala 1, * ,† , Delphine Brancherie 2 , Isaac Harari 3 and Adnan Ibrahimbegovic 4 1 Department of Mechanical Engineering and Industrial Systems, Tampere Universityof Technology, P.O. Box 589, FIN-33101 Tampere, Finland 2 Laboratoire Roberval, Université de Compiègne, BP 20529–60205, Compiègne, France 3 Faculty of Engineering, Tel Aviv University, 69978 Ramat Aviv, Israel 4 Ecole Normale Supérieure de Cachan, LMT-Cachan, 61, avenue du président Wilson, 94235 Cachan, France SUMMARY In this paper, a novel constitutive model combining continuum damage with embedded discontinuity is developed for explicit dynamic analyses of quasi-brittle failure phenomena. The model is capable of describ- ing the rate-dependent behavior in dynamics and the three phases in failure of quasi-brittle materials. The first phase is always linear elastic, followed by the second phase corresponding to fracture-process zone creation, represented with rate-dependent continuum damage with isotropic hardening formulated by utiliz- ing consistency approach. The third and final phase, involving nonlinear softening, is formulated by using an embedded displacement discontinuity model with constant displacement jumps both in normal and tangen- tial directions. The proposed model is capable of describing the rate-dependent ductile to brittle transition typical of cohesive materials (e.g., rocks and ice). The model is implemented in the finite element set- ting by using the CST elements. The displacement jump vector is solved for implicitly at the local (finite element) level along with a viscoplastic return mapping algorithm, whereas the global equations of motion are solved with explicit time-stepping scheme. The model performance is illustrated by several numerical simulations, including both material point and structural tests. The final validation example concerns the dynamic Brazilian disc test on rock material under plane stress assumption. Copyright © 2014 John Wiley & Sons, Ltd. Received 11 October 2013; Revised 3 June 2014; Accepted 22 September 2014 KEY WORDS: dynamic fracture; finite elements; embedded discontinuities; strain-rate dependency; continuum damage model; explicit time integration 1. INTRODUCTION Finite element modeling of fracture has become an active area of research in the field of computa- tional mechanics due to its importance in failure analyses of structures and machine components. The main challenge in this field is the numerical simulation of a crack propagation that involves both strong (displacement) and weak (strain) discontinuities. An additional challenge in dynamics concerns accounting for the strain rate and inertia effects. This paper is seeking to address any such challenge in the context of modeling the crack propagation in quasi-brittle materials under transient dynamic loading. Many techniques have been developed for finite element modeling of crack propagation and other localization phenomena involving strain softening. Early approaches introduced a weak dis- continuity in the finite element description of softening continua via damage or plasticity models. This approach was, however, quickly found defective resulting in severe mesh dependencies as the *Correspondence to: Timo Saksala, Department of Mechanics and Design, Tampere University of Technology, P.O. Box 589, FIN-33101 Tampere, Finland. † E-mail: timo.saksala@tut.fi Copyright © 2014 John Wiley & Sons, Ltd.