Enrichment of enhanced assumed strain approximations for representing strong discontinuities: addressing volumetric incompressibility and the discontinuous patch test

We present a geometrically non-linear assumed strain method that allows for the presence of arbitrary, intra- nite element discontinuities in the deformation map. Special attention is placed on the coarsemesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit. Given an underlying mesh and an arbitrary failure surface, we rst construct an enriched approximation for the deformation map with the non-linear analogue of the extended nite element method (X-FEM). With regard to the richer space of functions spanned by the gradient of the enriched approximation, we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain. In particular, in those elements intersected by the failure surface, we construct enhanced strain approximations which are orthogonal to piecewise-constant stress elds. Contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress elds. Importantly, the present formulation gives rise to a symmetric tangent sti ness matrix, even in localized elements. The present modi cation also allows for the satisfaction of a discontinuous patch test, wherein two di erent constant stress elds (on each side of the failure surface) lie in the solution space. We demonstrate how the proposed modi cations eliminate spurious stress oscillations along the failure surface, particularly for nearly incompressible material response. Additional numerical examples are provided to illustrate the e cacy of the modi ed method for problems in hyperelastic fracture mechanics