Study of dynamic crack branching using intrinsic cohesive surfaces with variable initial elastic stiffness

As observed by Needleman and co-workers [1], it is still not clear to what extent the initially elastic cohesive surfaces are more appropriate than the initially rigid cohesive surfaces for a given application. In this work, an analysis where the initially elastic model is approached asymptotically to the initially rigid is proposed. The purpose of this analysis is to gain more insight into the numerical mechanisms that lead to different results using different approaches. It was found that convergence is only achieved when the initial cohesive stiffness is much higher than the bulk stiffness. When a relative low value of this cohesive stiffness is chosen, crack tip speed is delayed and branching is artificially increased. GM R&D report # 9650, Aug. 2003. Introduction Cohesive Models have been gaining significant importance in the modeling of crack propagation in recent years. The most commonly used technique to incorporate the cohesive zone model into a finite element analysis is the discrete representation of the crack which is accomplished by introducing cohesive surfaces along inter-element boundaries. Within this framework, these cohesive interfaces can be classified into two approaches: the intrinsic potential-based law used by Xu and Needleman [2], and the extrinsic linear law developed by Camacho and Ortiz [3]. The disti+nction between these two approaches is associated with the way the crack initiation and evolution is modeled. In the intrinsic approach, zero-thickness interface elements are embedded between volumetric elements from the beginning of the analysis. The tensile and shear traction in the interface elements are calculated from the constitutive cohesive law. The interface between two surfaces is intact until the interface traction reaches the maximum value. Once the maximum traction is reached, the interface starts failing and the traction reduces to zero up as the displacement increases up to a critical value according to the traction-displacement relationship. The propagation of a crack can thus be simulated as the consecutive failure of interface elements. These cohesive laws are often called “initially elastic” laws and they can have different shapes, such as exponential [2], trapezoidal [4] and bi-lineal [5,6]. In the extrinsic approach, interface elements are introduced in the mesh only after the corresponding interface is predicted to start failing. Beginning the calculation with a regular mesh, the stress acting along the interface between two volumetric elements is monitored at any time to evaluate where crack will initiate. Once the stress reaches a critical value, a zero-thickness interface element is inserted by duplicating the nodes. Unlike the initially elastic interface elements, the initial cohesive response is rigid and the initial cohesive traction is equal to the critical stress [3]. Then, the interface opens in accordance with a prescribed traction-separation relation called “initially rigid” cohesive law. Recently, Falk et al. [1] have demonstrated that the prediction of crack branching strongly depends on whether the initially rigid or initially elastic approach is used. The reason why the initially rigid approach showed absence of crack branching is unclear. The purpose of this work is to study this issue using a bi-lineal initially elastic cohesive law where the initial stiffness is varied from low to high values without affecting the maximum cohesive traction or the cohesive energy. In this way, the initially elastic model is approached asymptotically to the initially rigid and, therefore, a detailed analysis of the material behavior can be carried out. 2. Approach used by M. Falk, A. Needleman and J. Rice[1] In [1], Falk and co-workers compare the extrinsic approach proposed by Camacho and Ortiz [5] with the intrinsic approach proposed by Xu and Needleman [2]. As explained in [3], using the extrinsic approach, new fracture surfaces are created by splitting nodes according to a brittle fracture criterion. In quadratic triangular elements, mid-side nodes can only be split in one way, namely along the unique element boundary crossing that node. By contrast, interior corner nodes can potentially open up along multiple fracture paths, all of which need to be evaluated in turn. To this end, Camacho and Ortiz [3] begin the process by computing the traction t acting at the node across all potential fracture surfaces. If this traction satisfies the fracture criterion, the nodes involved in that surface are split in order to create two new surfaces. The details of the computation are given in [3]. As previously mentioned, this kind of interface element is called “initially rigid” cohesive element. For the intrinsic approach, potential cohesive surfaces are introduced along all boundaries in a finite element mesh where cracks may initiate and propagate. When these “initially elastic” cohesive surfaces are introduced along all finite element boundaries, the mechanical response clearly depends on the mesh spacing, as well as on physical parameters. In fact, the exponential formulation proposed by Needleman has the feature that the cohesive surfaces contribute to the linear elastic response of the body. For example, the normal traction of this law is given by the following expression: