Statistical fracture modeling: crack path and fracture criteria with application to homogeneous and functionally graded materials

Analysis has been performed on fracture initiation near a crack in a brittle material with strength described by Weibull statistics. This nonlocal fracture model allows for a direct correlation between near crack-tip stresses and failure. Predictions are made for both the toughness and average fracture initiation angle of a crack under mixed-mode loading. This is pertinent for composites and is especially interesting for functionally graded materials (FGMs), where the stress and strength fields vary from the homogeneous form away from the crack tip. Both analytic and finite element analyses of FGMs reveal that gradients in Weibull scaling stress σ0(x,y) usually lead to a dramatic decrease of initiation fracture toughness; moreover, gradients normal to the crack result in a crack growing toward the weaker material. When comparing FGMs with gradients in Youngʹs modulus in the direction of the crack path, E(x), and the same stress-intensity factor K, the crack growing into the steeper negative gradient will be tougher, if m, the Weibull modulus, is low; with growth in the stiff direction, the effect is opposite. These effects offset the higher-stress intensity for cracks growing into more compliant material, and the crack-tip shielding when growing into a stiffer material based upon expectations for the applied load. Perpendicular gradients in modulus can cause a far-field mode I loading to produce mixed-mode loading of the crack tip and other asymmetric adjustments in the stress field; the gradient induces non-coplanar cracking that depends strongly on m. The distribution of damage near a crack tip will vary strongly with m. For high m materials, failure is dominated by the very near-tip parameters, and effects of gradients are minimized. With low m, distributed damage leading to toughening can be exaggerated in FGMs. Finally, consideration is given to the role of several higher-order terms in the stress field.