A linear failure index for the Christensen criterion

The Christensen failure criterion is a stress-based two-parameter failure criterion with two subcriteria for isotropic non-porous materials ranging from ductile materials like mild steel over plastics and cast iron to fully brittle materials like concrete or rocks. Despite reported experimental validation and wide applicability, the Christensen failure criterion is not widely used due to its complexity (combination of two subcriteria) and the lack of available implementations, e.g. in finite element software. In this work we propose a linear failure index for the Christensen criterion, akin to failure indices for fiber-reinforced plastic. The failure index is derived by projecting the stress state in spherical coordinates onto the failure surface, enabling a closed-form solution applicable also to other multi-parameter failure criteria. Additionally, the ductility number defined in the Christensen theory, describing the failure mode is implemented in the given framework. The methodology is further implemented in a new python package Christensen_FailureIndex as well as for the commercial finite element software Simulia ABAQUS (via a UMAT subroutine and also via a postprocessing Python GUI plugin) allowing for versatile use of the criterion. To assess efficiency and discuss its utility, the derived failure index is applied to a three point bending test with a notched specimen for a mildly brittle plastic to discuss critical load and failure location. The code for all implementations is publicly available on https://github.com/cld-rostock/Christensen_FailureIndex . • A linear failure index for the Christensen criterion is derived under consideration of proportional loading. • The ductility number Nd indicating the type of failure is derived for arbitrary stress states. • Both f Chr and Nd are provided as code, including FEA software plugins. • A complex loading case demonstrates the efficiency of the implementation and the practical relevance of the Christensen criterion. • The proposed methodology is generally applicable and can be extended to other multi-criteria failure theories.