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The aim of this paper is to define a new, simple and efficient method of solution of elasticity problems for 3D bodies containing cracks slightly perturbed out of their original plane or surface. This method extends that proposed by Rice (1985, 1989), from a re-formulation of Bueckner (1987)'s 3D weight function theory, for the treatment of coplanar crack perturbations. First, a general formula is derived for the variation of total energy of a cracked elastic body, resulting from some small but otherwise arbitrary geometric perturbation of the embedded crack(s). This formula, which involves integrals over both the front and the surface of the crack, is derived from an expression of deLorenzi (1982) and Destuynder et al. (1983) in the form of a volumic integral, originally limited to purely tangential crack perturbations but duly extended here to arbitrary perturbations having both tangential and normal components. It is then used to derive a general expression of the variation of displacement arising from a general perturbation of the crack(s), anywhere in the cracked body. The reasoning here basically follows the same lines as in the works of Rice (1985, 1989), but for the presence of an additional normal component of the crack perturbation. The possible use of the formalism developed to treat problems of out-of-plane, or out-of-surface perturbations of cracks is finally briefly evoked; the straightforwardness of the new method proposed is hoped to permit future applications to non-coplanar crack problems too complex to be accessible by more conventional methods. [ABSTRACT FROM AUTHOR]

When a crack interacts with material heterogeneities, its front distorts and adopts complex tortuous configurations that are reminiscent of the energy barriers encountered during crack propagation. As such, the study of crack front deformations is key to rationalize the effective failure properties of micro-structured solids and interfaces. Yet, the impact of a localized dissipation in a finite region behind the crack front, called the process zone , has often been overlooked. In this work, we derive the equation ruling 3D coplanar crack propagation in heterogeneous cohesive materials where the opening of the crack is resisted by some traction in its wake. We show that the presence of a process zone results in two competing effects on the deformation of crack fronts: (i) it makes the front more compliant to small-wavelength perturbations, and (ii) it smooths out local fluctuations of strength and process zone size, from which emerge heterogeneities of fracture energy. Their respective influence on front deformations is shown to strongly impact the stability of perturbed crack fronts, as well as their stationary shapes when interacting with arrays of tough obstacles. Overall, our theory provides a unified framework to predict the variety of front profiles observed in experiments, even when the small-scale yielding hypothesis of linear elastic fracture mechanics breaks down. [ABSTRACT FROM AUTHOR]

This paper addresses the question of the homogenization of fracture properties for three-dimensional disordered brittle solids. The effective toughness, identified as the minimum elastic energy release rate required to ensure crack growth, is predicted from a semi-analytical framework inspired by both micromechanics and statistical physics, that encompasses the decisive influences of both the material disorder and the mechanisms of interaction between a crack and heterogeneities. Theoretical predictions are compared to numerical values of the effective toughness that are computed with the fracture-mechanics-based semi-analytical method of Lebihain et al. (2020). Based on a perturbative approach of Linear Elastic Fracture Mechanics, this method allows for the efficient computation of crack propagation under tensile Mode I loading in composite materials containing several millions of inclusions, where the crack interacts with them through two mechanisms : crossing, wherein the crack penetrates the inclusion, and by-pass, wherein the crack wanders out-of-plane and follows the inclusion/matrix interface. We show that our homogenization procedure provides an accurate prediction of the homogenized fracture properties for a broad range of microstructural parameters such as the inclusion toughness, density or shape. This original theoretical framework constitutes a powerful mean to connect the microstructural parameters of materials to their crack growth resistance, beyond the particular cases considered in the simulations performed. As a result, it provides new strategies for the rational design of optimized brittle composites with tailored fracture properties. [ABSTRACT FROM AUTHOR]

A powerful semi-analytical method is developed to investigate the impact of tough inclusions on crack paths and the resulting effective fracture properties. Using a perturbative approach of fracture mechanics, this method allows to account for out-of-plane excursions of cracks resulting from the discontinuity of toughness at the inclusion/matrix interface, by applying a generalized maximum-energy-release-rate criterion. For a sufficiently large toughness mismatch between the matrix and the inclusions, the crack by-passes the obstacles, thus reducing the inclusion-induced reinforcement of the material. For spherical inclusions, obstacles of fracture energy larger than 3.85 times that of the matrix become ineffective to further reinforce the material, because of their systematic by-pass. The role played by the shape of the inclusions is also investigated; inclusions of non-spherical shape may reduce the reinforcement through three-dimensional collective effects emerging from the coupling between the in-plane and out-of-plane components of the crack front perturbation. Finally, we show that inclusion by-pass not only limits crack bowing but can also prevent crack bridging, for inclusions short enough in the direction orthogonal to the mean fracture plane. Thus this study provides a quantitative picture of the interaction mechanisms between a crack front and tough inclusions, which paves the way to the future microstructural design of brittle solids with improved resistance to failure. [ABSTRACT FROM AUTHOR]