where is the undeformed domain, denotes the interior ``cohesive'' boundary along which the cohesive tractions act, and corresponds to the part of the exterior boundary along which the external tractions are applied. and denote the acceleration and displacement fields, respectively. is the second Piola-Kirchoff stress tensor and , the Lagrangian strain tensor, which is related to the displacement field through

(2.7) |

Nonlinear kinematics is used in this study to account for the possible large rotations present in the structure due to the fracture process. The expression (2.6) of the principle of virtual work is fairly conventional, except for the presence of the fourth term, which corresponds to the virtual work done by cohesive traction for a virtual separation .

The resulting semi-discrete finite element formulation can be expressed in the following matrix form:

where is the lumped mass matrix, is the vector containing the nodal accelerations, and , and respectively denote the internal, cohesive and external force vectors.

The time stepping scheme is based on the classical explicit
second-order central difference scheme (Belytschko *et al.,* 1976):

where is the time step and denotes the nodal displacement vector at time .

The expression of the internal, cohesive and external force vectors can be found in Baylor (1997).While a variety of constitutive models can be used to characterize the response of the volumetric elements, we use, in this study, a simple linear isotropic relation between the second Piola-Kirchoff stresses and the Lagrangian strains :

(2.12) |

where and are the Lame's constants.