A complete three-dimensional continuum model of wing-crack growth in granular brittle solids

Failure of brittle materials containing embedded three dimensional pre-cracks and subjected to uniaxial compressive and tensile loading is considered here. The sliding crack (wing-crack) model of Ashby and Sammis (1990) is extended and further developed to formulate a 3D anisotropic continuum damage model.First, a frictional sliding condition of pre-cracks is formulated in three dimensions and a crack interaction function is proposed. To introduce inelastic strains due to cracking, crack opening displacements are derived from Castigliano's second theorem. Finally, a strain-stress relation is obtained from the Gibbs energy density equation.The model was implemented in Abaqus/Explicit finite element software. Material inhomogeneity was considered assuming that the pre-cracks are lognormally distributed between integration points.While testing the proposed model against experimental results of granular ice, the numerical simulations were in good agreement both under uniaxial compression and tension as a function of grain size and temperature-dependent kinetic friction. The model was able to predict qualitatively and quantitatively the brittle failure modes and strength both under compression and under tension. Due to the modelled inhomogeneity, the scatter in simulated strengths corresponded to that of the test results. Besides non-simultaneous and non-uniform damaging, the model revealed important phenomena observed during the experiments; e.g. under compression the sliding of the pre-cracks resembled "stick-slip" motion, and secondary cracks were observed to grow in a jerky manner. The effect of specimen end conditions on both the failure stress and failure mode was addressed in the simulations.