A shell fracture framework based on a full discontinuous Galerkin formulation combined with an extrinsic cohesive law

The cohesive method can be combined easily with Finite Element method to simulate a fracture problem which can contains fracture initiation and propagation. Nevertheless, the insertion of cohesive elements is not straightforward. Indeed, the two classical approaches suffer from severe limitations. On one hand, in the intrinsic approach, as the cohesive element is inserted at the beginning, this element has to model the continuum stage of deformation before fracture. This is ensured by an initial slope in the cohesive law which leads to a stiffness modification and to an alteration of propagation of wave. On the other hand, the introduction of the cohesive element during the simulation in extrinsic approach requests a dynamic modification of mesh. This operation is very difficult to implement especially in the case of a parallel implementation which is almost mandatory due to the very important number of degrees of freedom inherent to a fine mesh used to track the crack path. A solution to these limitations, pioneered by J. Mergheim and R. Radovitzky is to recourse to a discontinuous Galerkin formulation. Indeed this one used discontinuous test functions and integration at the interface of elements to discretize a structure with discontinuous elements. The integration on the boundary of elements allows ensuring weakly the continuity of displacements in a stable and consistent manner. As interface elements are present they can be easily substituted by cohesive elements when a fracture criterion is reached. The interest of the method has been recently proved by R. Radovitzky etal. for 3D elements and by the authors for Euler-Bernoulli beams. An extension of the formulation to Kirchhoff-Love shell is presented here. A novel extrinsic cohesive law is developed to model a through the thickness fracture. In fact, as in thin bodies formulation the thickness is not “discretized” this operation is not straightforward. Indeed, as the fracture occurs only in tension, in a pure bending case the position of neutral axis has to be move to propagate the fracture. To avoid this complicated step, it is suggested to integrate on the thickness the cohesive law which is then applies on resultant efforts. The coupling between the openings in displacement and rotation is performed in a way which guarantees a proper release of energy for any loading. Furthermore, the combination between fracture modes I and II is realized as suggested by M. Ortiz etal. Some numerical quasi-static and dynamic benchmarks are simulated to show the interest and the good performance of the presented framework