A trend in the last decade towards models in which nonlinear crack tip processes are represented explicitly, rather than being assigned to a point process at the crack tip (as in linear elastic fracture mechanics), is reviewed by a survey of the literature. A good compromise between computational efficiency and physical reality seems to be the cohesive zone formulation, which collapses the effect of the nonlinear crack process zone onto a surface of displacement discontinuity (generalized crack). Damage mechanisms that can be represented by cohesive models include delamination of plies, large splitting (shear) cracks within plies, multiple matrix cracking within plies, fiber rupture or microbuckling (kink band formation), friction acting between delaminated plies, process zones at crack tips representing crazing or other nonlinearity, and large scale bridging by through-thickness reinforcement or oblique crack-bridging fibers. The power of the technique is illustrated here for delamination and splitting cracks in laminates. A cohesive element is presented for simulating three-dimensional, modedependent process zones. An essential feature of the formulation is that the delamination crack shape can follow its natural evolution, according to the evolving mode conditions calculated within the simulation. But in numerical work, care must be taken that element sizes are defined consistently with the characteristic lengths of cohesive zones that are implied by the chosen cohesive laws. Qualitatively successful applications are reported to some practical problems in composite engineering, which cannot be adequately analyzed by conventional tools such as linear elastic fracture mechanics and the virtual crack closure technique. The simulations successfully reproduce experimentally measured crack shapes that have been reported in the literature over a decade ago, but have not been reproduced by prior models.

An augmented finite element method ("A-FEM") is presented that is a variant of the method of Hansbo and Hansbo (Comput Methods Appl Mech Eng, 193: 3523-3540, 2004), which can fully account for arbitrary discontinuities that traverse the interior of elements. Like the method of Hansbo and Hansbo, the A-FEM preserves elemental locality, because element augmentation is implemented within single elements and involves nodal information from the modified element only. The A-FEM offers the additional convenience that the augmentation is implemented via separable mathematical elements that employ standard finite element nodal interpolation only. Thus, the formulation is fully compatible with standard commercial finite element packages and can be incorporated as a user element without access to the source code. Because possible discontinuities include both elastic heterogeneity and cracks, the A-FEM is ideally suited to modeling damage evolution in structural or biological materials with complex morphology. Elements of a multi-scale approach to analyzing damage

The difficult challenge of simulating diffuse and complex fracture patterns in tough structural composites is at last beginning to yield to conceptual and computational advances in fracture modeling. Contributing successes include the refinement of cohesive models of fracture and the formulation of hybrid stress-strain and traction-displacement models that combine continuum (spatially averaged) and discrete damage representations in a single calculation. Emerging hierarchical formulations add the potential of tracing the damage mechanisms down through all scales to the atomic. As the models near the fidelity required for their use as virtual experiments, opportunities arise for reducing the number of costly tests needed to certify safety and extending the design space to include material configurations that are too complex to certify by purely empirical methods.

SUMMARYWe demonstrate that traditional cohesive zone (CZ) elements cannot be accurate when used in conjunction with solid elements with arbitrary intra-element cracking capability, because they cannot capture the load transfer between cohesive interfaces and the solid elements when crack bifurcation or coalescence occurs. An augmented cohesive zone (ACZ) element based on the augmented finite element method formulation is therefore proposed. The new element allows for arbitrary separation of the cohesive element in accordance with the crack configuration of the abutting solid elements, thus correctly maintaining the non-linear coupling between merging or bifurcating cracks. Numerical accuracy and effectiveness of the proposed ACZ element are demonstrated through several examples.

We review the development of virtual tests for high-temperature ceramic matrix composites with textile reinforcement. Success hinges on understanding the relationship between the microstructure of continuous-fiber composites, including its stochastic variability, and the evolution of damage events leading to failure. The virtual tests combine advanced experiments and theories to address physical, mathematical, and engineering aspects of material definition and failure prediction. Key new experiments include surface image correlation methods and synchrotron-based, micrometer-resolution 3D imaging, both executed at temperatures exceeding 1,500• C. Computational methods include new probabilistic algorithms for generating stochastic virtual specimens, as well as a new augmented finite element method that deals efficiently with arbitrary systems of crack initiation, bifurcation, and coalescence in heterogeneous materials. Conceptual advances include the use of topology to characterize stochastic microstructures. We discuss the challenge of predicting the probability of an extreme failure event in a computationally tractable manner while retaining the necessary physical detail. 17.1