A strong discontinuity approach on multiple levels to model solids at failure

This paper presents a modification of the well established strong discontinuity approach to model failure phenomena in solids by extending it to multiple levels. This is achieved by the resolution of the overall problem to be solved into a main boundary value problem and identified sub-domains based on the concepts of domain decomposition. The initiation of those sub-domains is based on the detection of failure onset within finite elements of the main boundary value problem which takes place at the process zone in front of the propagating cracks. Those sub-domains are subsequently adaptively discretized during run-time and comprise the so called sub-boundary value problem to be solved simultaneously with the main boundary value problem. To model failure, only the sub-elements of those sub-boundary value problems are treated by the strong discontinuity approach which, depending on their state of stress, may develop strong discontinuities to be understood as jumps in the displacement field to model cracks and shear bands. Due to its resolution into many sub-elements, the single finite element of the main boundary value problem can therefore simulate a single propagating strong discontinuity arising in quasi-static problems as well as the propagation of multiple propagating strong discontinuities arising for simulations of crack branching in brittle materials undergoing dynamic failure. Whereas the advantages of the strong discontinuity approach in the form of its efficiency by statically condensing out the degrees of freedom related to the failure zone as well as its applicability to use standard displacement based, mixed, and enhanced formulations for the underlying finite element are kept, new challenges arise due to its proposed modification. Firstly, the solutions of the different sub-boundary value problems must be transferred to the main boundary value problem, which is achieved in this work based on concepts of domain decomposition. Secondly, since multiple strong discontinuities might propagate over the boundaries of the sub-boundary value problem, the applied boundary conditions must take into account the appearance of possible jumps in the displacement fields arising from the solution of the sub-boundary value problem itself. It is shown that for single propagating cracks arising in problems of quasi-static failure only minor differences are obtained through the proposed modification. For the simulation of solids undergoing dynamic fracture the modification allows though to predict the onset of crack branching without the need for any artificial crack branching criterion. A close agreement with experiments of the simulation results in terms of micro- and macro branching in addition to studying certain key parameters like critical velocity, dynamic stress intensity factor, and the strain energy release rate at branching is found.