**Investigation of Christensen's two-parameter failure criterion for
brittle materials**

KJ Hanley and C O'Sullivan and X Huang, Geomechanics from Micro to Macro, Vols I and II, 129-134 (2015).

A number of recent experimental and numerical studies have highlighted the need to consider sand particle crushing in geomechanics. Changes in the particle size distribution affect sample permeability and the transmission of forces within a sample, particularly when large stresses are attained and hence the amount of crushing is substantial. The approaches that have been used to consider crushing in Discrete Element Modelling (DEM) can be divided into two broad categories: (i) crushable agglomerates are created by joining unbreakable fundamental particles with bonds possessing finite strengths or (ii) the fundamental particles themselves are deemed to fail when some pre-allocated measure of strength is reached. For the latter, the failure criterion for brittle materials proposed by Christensen (2000), a modification of the Von Mises criterion, has a justifiable theoretical basis and appears to be an improvement on the seemingly arbitrary failure criteria which have been used in some DEM crushing models. Suitable parameters for the Christensen model can be estimated from experimental single-particle crushing data, albeit with a large uncertainty. In this study, the sensitivity of the Christensen failure criterion to the input parameters was investigated by systematically varying the invariants of a particle's stress tensor until failure occurred. Failure surfaces were obtained for regular packings of monosized spheres (following Russell, Muir Wood, & Kikumoto (2009a, 2009b)) and compared with the equivalent failure surfaces for a maximum force failure criterion. The large disparity between the failure criteria, confirmed by large-scale DEM simulations, indicates that adopting the Christensen failure criterion using the per-particle stress tensor commonly calculated in DEM leads to significant underpredictions of the extent of crushing.

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