Ingenieria

Sadhana, Vol. 25, Part 6, December 2000, pp. 561–587. # Printed in India ¯ ¯

A cohesive finite element formulation for modelling fracture and delamination in solids
S ROY CHOWDHURY and R NARASIMHAN
Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India e-mail: narasi@mecheng.iisc.ernet.in Abstract. In recent years, cohesive zone models have been employedto simulate fracture and delamination in solids. This paper presents in detail the formulation for incorporating cohesive zone models within the framework of a large deformation finite element procedure. A special Ritz-finite element technique is employed to control nodal instabilities that may arise when the cohesive elements experience material softening and lose their stress carrying capacity.A few simple problems are presented to validate the implementation of the cohesive element formulation and to demonstrate the robustness of the Ritz solution method. Finally, quasi-static crack growth along the interface in an adhesively bonded system is simulated employing the cohesive zone model. The crack growth resistance curves obtained from the simulations show trends similar to thoseobserved in experimental studies. Keywords. Cohesive zone models; fracture; delamination; finite elements.

1.

Introduction

During the span of the last two decades, considerable research work in the field of fracture mechanics has been devoted to the analysis of stationary cracks in various structural systems. While such an analysis is useful to predict the behaviour of a system till the pointof initiation of crack propagation, its applicability reduces during the propagation phase. In many cases, such as homogeneous ductile materials, layered structures, composites etc., initiation of crack growth does not indicate catastrophic failure. A stable crack extension phase, which is associated with a steady increase in the external load or crack driving force, precedes catastrophic failurein such systems. In other words, in these cases, there is reserve strength in the system even after crack initiation has occurred and it is expended during propagation. Thus, design considerations call for detailed analyses and understanding of the crack growth phase to profitably make use of this reserve strength. Several investigators have contributed in providing an understanding of themechanics and the practical implications of stable crack growth by using both analytical and numerical techniques. However, primarily due to the difficulty involved in the treatment of the governing equations, analytical solutions have mostly been limited to idealised cases, 561

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such as, elastic-perfectly plastic homogeneous materials (Chitaley & McClintock1971; Drugan et al 1982) or materials with linear hardening (Amazigo & Hutchinson 1977). The early finite element studies (Rice & Sorensen 1978; Malluck & King 1980; Sham 1983; Narasimhan et al 1987) have extensively used the nodal release procedure to simulate quasi-static crack growth as well as dynamic crack propagation. In this method, crack extension is assumed to take place when a fracturecriterion, based on a critical stress or deformation measure near the crack tip, is satisfied. In the finite element procedure, the boundary condition at the crack tip node is then replaced by the point load acting on it. This point load is subsequently reduced to zero in several increments, at the end of which a traction-free element surface emerges, and the crack advances by one element length.Despite widespread uses, a clear defect in the nodal release technique is the lack of a material length scale. This implies that it is not possible to incorporate both the requirements of critical stress and fracture energy for simulating crack growth or interface debonding. Further, the absence of a material length scale leads to strong dependence of the finite element results on the size of the...