New finite elements with embedded strong discontinuities to model failure of three-dimensional continua
This work addresses the developments of new finite elements with embedded strong discontinuities for the modeling of three-dimensional solids at failure in the infinitesimal small-strain or finite deformation regimes.
Cracks and shear bands involving the localized dissipative mechanism in a relatively narrow zone provide typical examples to be modeled by this strong discontinuity approach. The narrowness of such a localized region compared to the size of the overall mechanical problem then reveals the multi-scale character of the physical phenomena, thus allowing the given problem to be split into the typical global continua involving only smooth displacement fields and a small-scale problem to represent localized solutions.
The direct consequence of the multi-scale approach is on the element-based enhancement for the associated singular field in the discrete setting,
leading to the very same number of global degrees of freedom and mesh connectivity as the original problem without the discontinuity. This procedure is then achieved by the direct identification of the discrete kinematics associated to the sought separation modes of the individual finite elements. In particular, we focus on the direct enhancement of infinitesimal strains (for the infinitesimal case) or deformation gradients (for the finite deformation case) rather than attempting to find the associated displacement field in terms of discontinuous shape functions, also allowing the proposed formulations to be more generally applicable to the strain-based high performance finite elements.
Given the complex kinematics arising from discontinuities in three dimensions, the new finite elements consider full linear interpolations of the displacement jumps on both the normal and tangential components to the discontinuities in the interiors of the respective finite elements. The incorporation of the high order separation modes then allows a complete vanishing of stress locking, namely, over-stiff responses of the approximated solutions due to the poor resolutions of discrete kinematics associated to the discontinuities. A total of nine enhanced parameters for each element are required to represent the linear displacement jumps, but being condensed out at the element level in virtue of the proposed discrete multi-scale framework.
To illustrate the improved performance of the new three-dimensional finite elements with embedded strong discontinuities, several representative numerical examples such as a series of basic single element tests and a set of benchmark problems are implemented. The elements involving only piecewise constant displacement jumps are also considered there for comparison purposes, showing by design the overall improvement on the new elements in terms of the locking free properties and sharper resolution of the discontinuities that propagate in arbitrary directions.