Numerical modeling of rate-dependent debonding processes in composites Alberto Corigliano * , Stefano Mariani, Anna Pandolﬁ Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Abstract In this paper we discuss analytical and numerical results concerning rate-dependent dynamic crack processes in homogeneous and heterogeneous materials. Rate-dependency is introduced in the model through a newly formulated interface law. Three sig- niﬁcant examples are presented: (i) the mode-I dynamic decohesion between two linear elastic half-planes under a tensile stress pulse; (ii) the uniaxial spall test and (iii) the debonding of a thin layer from a rigid substrate. The results show that rate-dependent cohesive laws can play an important role in dynamic crack processes. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Composite materials; Delamination and fracture; Interface models; Dynamic fracture; Rate-dependency 1. Introduction In recent years, dynamic crack propagation processes have raised the interest of the scientiﬁc literature (see e.g. [1–10]). As pointed out in [3], several phenomena related to the dynamic crack propagation are not completely clear and are in need of further investigations. Among these, we may mention here: (i) the maximum crack propagation speed in pure and mixed mode conditions [4,9,10]; (ii) impact induced damage phenomena in het- erogeneous materials [11], especially structural com- posites [6]; (iii) the inﬂuence of temperature and rate eﬀects in the propagation of dynamic cracks [8]. The main purpose of this paper is to give a contri- bution to the study of the latter issue, in particular to the investigation of the role of rate-dependency in dynamic crack propagation processes, either in homogeneous and heterogeneous materials. Various possible strategies are suitable for modeling dynamic crack processes. Here we make use of interface (or cohesive crack) models, which have become more and more popular in numerical simulations of debond- ing and fracture [12–15]. A simpliﬁed rate-dependent interface law is here formulated and used to simulate the nucleation and propagation of cracks subject to dy- namic pulse loading. The formulation of a rate-dependent interface law allows to partially answer a remarkable question related to the modeling of the crack process zone: should the process zone be described through a rate-dependent law? Such a problem becomes particularly important for impacts in composite materials, when dynamic delam- ination is involved. In these cases, rate eﬀects in the process zone could interact with the observed rate-de- pendency of the matrix materials (typically polymers) giving rise to complex phenomena. The outline of the paper is as follows. In Section 2 a rate-dependent interface constitutive law is proposed and discussed. Section 3 presents the semi-analytical solution of a mode-I dynamic debonding process of an interface surrounded by an unbounded linear elastic domain. Numerical simulations of the uniaxial spall test and of the debonding of a thin layer from a rigid sub- strate are discussed in Sections 4.1 and 4.2, respectively. 2. Interface constitutive model We consider a three-dimensional solid X split into two parts by a locus of potential decohesion (or inter- face) C d , whose surfaces are denoted by C þ d and C  d , respectively, see Fig. 1. We assume that all dissipative * Corresponding author. Tel.: +39-02-23994244; fax: +39-02- 23994220. E-mail addresses: alberto.corigliano@polimi.it (A. Corigliano), stefano.mariani@polimi.it (S. Mariani), anna.pandolﬁ@polimi.it (A. Pandolﬁ). 0263-8223/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00030-8 Composite Structures 61 (2003) 39–50 www.elsevier.com/locate/compstruct