Ahmed Elmarakbi
Faculty of Applied Sciences, University of Sunderland
United Kingdom
1. Introduction
This chapter provides an overview of the delamination growth in composite materials, cohesive interface models and finite element techniques used to simulate the interface elements. For completeness, the development and implementation of a new constitutive formula that stabilize the simulations and overcome numerical instabilities will be discussed in this chapter.
Delamination is a mode of failure of laminated composite materials when subjected to transverse loads. It can cause a significant reduction in the compressive load-carrying capacity of a structure. Cohesive elements are widely used, in both forms of continuous interface elements and point cohesive elements, (Cui & Wisnom, 1993; De Moura et al., 1997; Reddy et al., 1997; Petrossian & Wisnom, 1998; Shahwan & Waas, 1997; Chen et al., 1999; Camanho et al., 2001) at the interface between solid finite elements to predict and to understand the damage behaviour in the interfaces of different layers in composite laminates. Many models have been introduced including: perfectly plastic, linear softening, progressive softening, and regressive softening (Camanho & Davila, 2004). Several rate- dependent models have also been introduced (Glennie, 1971; Xu et al., 1991; Tvergaard & Hutchinson, 1996; Costanzo & Walton, 1997; Kubair et al., 2003). A rate-dependent cohesive zone model was first introduced by Glennie (Glennie, 1971), where the traction in the cohesive zone is a function of the crack opening displacement time derivative. Xu et al. (Xu et al., 1991) extended this model by adding a linearly decaying damage law. In each model the viscosity parameter (n ) is used to vary the degree of rate dependence. Kubair et al. (Kubair et al., 2003) thoroughly summarized the evolution of these rate-dependant models and provided the solution to the mode III steady-state crack growth problem as well as spontaneous propagation conditions.
A main advantage of the use of cohesive elements is the capability to predict both onset and propagation of delamination without previous knowledge of the crack location and propagation direction. However, when using cohesive elements to simulate interface damage propagations, such as delamination propagation, there are two main problems. The first one is the numerical instability problem as pointed out by Mi et al. (Mi et al., 1998), Goncalves et al. (Goncalves et al., 2000), Gao and Bower (Gao & Bower, 2004) and Hu et al.
(Hu et a!., 2007). This problem is caused by a well-known elastic snap-back instability, which occurs just after the stress reaches the peak strength of the interface. Especially for those interfaces with high strength and high initial stiffness, this problem becomes more obvious when using comparatively coarse meshes (Hu et al., 2007). Traditionally, this problem can be controlled using some direct techniques. For instance, a very fine mesh can alleviate this numerical instability, however, which leads to very high computational cost. Also, very low interface strength and the initial interface stiffness in the whole cohesive area can partially remove this convergence problem, which, however, lead to the lower slope of loading history in the loading stage before the happening of damages. Furthermore, various generally oriented methodologies can be used to remove this numerical instability, e. g. Riks method (Riks, 1979) which can follow the equilibrium path after instability. Also, Gustafson and Waas (Gustafson & Waas, 2008) have used a discrete cohesive zone method finite element to evaluate traction law efficiency and robustness in predicting decohesion in a finite element model. They provided a sinusoidal traction law which found to be robust and efficient due to the elimination of the stiffness discontinuities associated with the generalized trapezoidal traction law.
Recently, the artificial damping method with additional energy dissipations has been proposed by Gao and Bower (Gao & Bower, 2004). Also, Hu el al. proposed a kind of move – limit method (Hu et al., 2007) to remove the numerical instability using cohesive model for delamination propagation. In this technique, the move-limit in the cohesive zone provided by artificial rigid walls is built up to restrict the displacement increments of nodes in the cohesive zone of laminates after delaminations occurred. Therefore, similar to the artificial damping method (Gao & Bower, 2004), the move-limit method introduces the artificial external work to stabilize the computational process. As shown later, although these methods (Gao & Bower, 2004; Hu et al., 2007) can remove the numerical instability when using comparatively coarse meshes, the second problem occurs, which is the error of peak load in the load-displacement curve. The numerical peak load is usually higher than the real one as observed by Goncalves et al. (Goncalves et al., 2000) and Hu et al. (Hu et al., 2007). Similar work has also been conducted by De Xie and Waas (De Xie & Waas, 2006). They have implemented discrete cohesive zone model (DCZM) using the finite element (FE) method to simulate fracture initiation and subsequent growth when material non-linear effects are significant. In their work, they used the nodal forces of the rod elements to remove the mesh size effect, dealt with a 2D study and did not consider viscosity parameter. However, in the presented Chapter, the author used the interface stiffness and strength in a continuum element, tackled a full 3D study and considered the viscosity parameter in their model.
With the previous background in mind, the objective of this Chapter is to propose a new cohesive model named as adaptive cohesive model (ACM), for stably and accurately simulating delamination propagations in composite laminates under transverse loads. In this model, a pre-softening zone is proposed ahead of the existing softening zone. In this pre-softening zone, with the increase of effective relative displacements at the integration points of cohesive elements on interfaces, the initial stiffnesses and interface strengths at these points are reduced gradually. However, the onset displacement for starting the real softening process is not changed in this model. The critical energy release rate or fracture toughness of materials for determining the final displacement of complete decohesion is kept constant. Also, the traction based model includes a cohesive zone viscosity parameter (n ) to vary the degree of rate dependence and to adjust the peak or maximum traction.
In this Chapter, this cohesive model is formulated and implemented in LS-DYNA (Livermore Software Technology Corporation, 2005) as a user defined materials (UMAT). LS-DYNA is one of the explicit FE codes most widely used by the automobile and aerospace industries. It has a large library of material options; however, continuous cohesive elements are not available within the code. The formulation of this model is fully three dimensional and can simulate mixed-mode delamination. However, the objective of this study is to develop new adaptive cohesive elements able to capture delamination onset and growth under quasi-static and dynamic Mode-I loading conditions. The capabilities of the proposed elements are proven by comparing the numerical simulations and the experimental results of DCB in Mode-I.
