Finite element implementation of virtual internal bond model for simulating crack behavior Ganesh Thiagarajan a , K. Jimmy Hsia b, * , Yonggang Huang c a Department of Civil Engineering, University of Missouri-Kansas City, Kansas City, MO 64110, USA b Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA c Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Received 10 January 2001; received in revised form 16 September 2002; accepted 21 January 2003 Abstract The virtual internal bond (VIB) model has been recently proposed to describe material deformation and failure under both static and dynamic loading. The model is based on the incorporation of a cohesive type law in a hyperelastic framework, and is capable of fracture simulation as a part of the constitutive formulation. However, with an implicit integration scheme, difficulties are often encountered in the finite element implementation of the VIB model due to possible negative eigenvalues of the stiffness matrix. This paper describes the implementation of an explicit integration scheme of the VIB model. Issues pertaining to the implementation, such as mesh size and shape dependence, loading rate dependence, crack initiation and growth characteristics, and solution time are examined. Both quasi-static and dynamic loading cases have been studied. The experimental validation of the VIB model has been done by calibrating the model parameters using the experimental data of Andrews and Kim [Mech. Mater. 29 (1988) 161]. The simulations using the VIB model are shown to agree well with the experimental observations. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Cohesive model; VIB model; Finite elements; Static and dynamic crack propagation; Explicit integration scheme 1. Introduction Numerical simulations of crack initiation, propagation and branching are a computationally intensive process. Traditional finite element packages simulate the crack propagation problem either by using sin- gularity elements or by using line spring elements with built-in fracture criteria. Recently, a popular method to do this task is the cohesive surface modeling of the fracture zone. Following the work of Barenblatt [2], Dugdale [3], and Willis [4], many researchers have addressed the issues pertaining to this approach. Nee- dleman [5] provided a framework for the separation process starting from initial debonding in the cohesive zone. Larsson [6] used this approach to simulate crack growth in brittle materials, while Xia and Shih [7] simulated fracture in ductile materials under static loading. Camacho and Ortiz [8] have used cohesive * Corresponding author. Tel.: +1-217-333-2321; fax: +1-217-244-5707. E-mail address: kj-hsia@uiuc.edu (K.J. Hsia). 0013-7944/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0013-7944(03)00102-4 Engineering Fracture Mechanics 71 (2004) 401–423 www.elsevier.com/locate/engfracmech