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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 σ 0 ( 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 Young's 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.

A statistical Ritchie-Knott-Rice (RKR) [1 ] model for brittle fracture is considered for an FGM containing a slender notch. The FMG is modeled as linear elastic, with its strength described by two-parameter Weibull statistics. The Young's modulus is assumed to vary either linearly or sigmoidally. A compact tension (C(T)) fracture mechanics specimen is analyzed via the finite element method, considering the effect of modulus variation on the near-tip stress state. Results can be characterized by the stress intensity, K. For spatially constant Weibull parameters, the RKR model is used to predict the expected fracture toughness, K Φ , i.e., the K at which the first flaw failure occurs with probability Φ. For sufficiently high Weibull modulus, the failure occurs essentially at the notch tip. For sufficiently low Weibull modulus (m < 4), K Φ for an FGM is found to vary up to 25% from that of a homogeneous body.

The mixed-mode delaminating beam (MMDB) is a widely used test geometry designed to measure the fracture resistance of bimaterial interfaces under mixed-mode loading conditions. In the present work, linear-elastic finite element analyses are employed to determine the complex stress intensity factor, K , for an interfacial crack in this sample; results are found to confirm those of previous studies of the bilayer specimen. However, the numerical results further reveal that the region of K -dominance near the crack tip is very limited, extending merely ∼ 1/100 to 1/1000 of the sample height, about an order of magnitude smaller than for other common fracture-mechanics test samples. Analyses performed for this specimen geometry modified to include a thin sandwiched interlayer also indicate a very limited region of K -dominance, for example, extending ∼ 1/10 of the height of the sandwiched layer from the crack tip for very thin sandwiched layers. For the sandwiched geometry, two situations leading to a useful small scale yielding condition are described. Examples of the use of this specimen in the evaluation of the fracture toughness of ceramic joints are cited from the literature and comparisons are made between the size of the K-dominant region and the extent of crack tip plasticity. Based on these comparisons, the geometry-independent predictive power of linear-elastic fracture mechanics for common bimaterial couples and specimen dimensions using this geometry is challenged.