New finite elements with embedded strong discontinuities in the finite deformation range

This paper develops new finite elements with embedded strong discontinuities for the modeling of the failure of solids in the finite deformation range. The cases of interest include cracks, shear bands, delamination in composites and similar failures modeled by a propagating discontinuity in the deformation mapping that incorporates a cohesive law in terms of the associated displacement jumps. The new elements accommodate the discontinuity in their interiors with a linear interpolation of the displacement jumps in contrast to the existing elements with constant jumps. A general methodology is proposed for the development of such finite elements, but the final focus is on quadrilateral elements for plane problems where a constant jump interpolation leads to a spurious transfer of stresses through the discontinuity (or stress locking) in certain common configurations. The key aspect of the new elements is the proper enhancement of the deformation gradient that can represent correctly the kinematics of linearly separating discontinuities, including a relative finite rotation across the discontinuity, and in a frame indifferent manner. The enhanced parameters modeling the displacement jumps are introduced independently for each element, allowing for their static condensation at the element level and, hence, to arrive at a global system of equations with the same number and connectivity of the original global degrees of freedom for the underlying finite element mesh. This situation leads to a particularly simple implementation of the proposed approach in an existing finite element code. The performance of the new elements is illustrated with several numerical simulations, including delamination in composite materials and propagating shear bands in a finitely deforming elastoplastic solid.