Phase-field approaches to fracture, initially designed as regularization of the Griffith model of brittle fracture, are now commonly viewed as gradient-damage models whose regularization length becomes a material property driving crack nucleation. One weakness of this approach is that the strength surface cannot be arbitrary: its shape is dictated by the elastic energy, and its magnitude by the regularization length. We focus on the antiplane version of the model introduced by Bourdin, Marigo, Maurini and Zolesi (arXiv:2506.22558), which handles crack propagation along unknown paths and nucleation governed by an arbitrary convex strength surface by degrading the strength instead of the stiffness. It can be interpreted as a regularization of softening plasticity in which localization bands obey an equivalent cohesive law set by the strength domain and the toughness, while the role of the regularization length, when small compared to the elasto-cohesive length, is purely numerical. Strength, stiffness, and toughness thus become independent material data, and limit analysis, perfect plasticity, cohesive fracture, and brittle fracture merge into a single variational framework. We derive closed-form solutions for a simple shear problem, propose a numerical scheme combining alternate minimization and conic programming, and numerically verify the equivalent cohesive law, its independence of the regularization, and the size effect governed by the elasto-cohesive length. A "surfing" simulation highlights the structure of the propagating crack while a re-entrant V-notch is used to show how the model bridges small-scale yielding, cohesive fracture, and brittle fracture without a priori hypotheses.

We study crack stability near grain boundaries using a phase-field fracture model. Using Mode-I crack opening boundary conditions, we estimate singularity exponents at the crack tips near bi-material interfaces. We compute and analyze trends in the singularity exponents by sampling the five macroscopic degrees of freedom in the grain boundary crystallography for different strengths of the cubic elastic anisotropy. Using the insights from the computed exponents, we perform time-dependent simulations of crack propagation through grain boundaries. We find that cracks can be pinned or attracted to grain boundaries due to changes in elastic constants across the boundary. Time-dependent simulations show that the crack pinning or attraction at grain boundaries is consistent with the computed singularity exponent of the stress field of a crack at a grain boundary. There is a long-range interaction of a crack with a grain boundary with a distance that is a function of the degree of elastic anisotropy. Increasing elastic anisotropy leads to a larger variation in the singularity exponents of the stress field. We conclude that it is possible to engineer the microstructure to improve fracture toughness by selecting grain boundaries that offer more resistance to crack propagation.

We investigate the systematic design of compliant morphing structures composed of materials reacting to an external stimulus. We add a perimeter penalty term to ensure existence of solutions. We propose a phase-field approximation of this sharp interface problem, prove its convergence as the regularization length approaches 0 and present an efficient numerical implementation. We illustrate the strengths of our approach through a series of numerical examples.

The role of grain size in determining fracture toughness in metals is incompletely understood with apparently contradictory experimental observations. We study this grain-size dependence computationally by building a model that combines the phase-field formulation of fracture mechanics with dislocation density-based crystal plasticity. We apply the model to cleavage fracture of body-centered cubic materials in plane strain conditions, and find non-monotonic grain-size dependence of plastic-brittle transgranular fracture. We find two mechanisms at play. The first is the nucleation of failure due to cross-slip in critically located grains within transgranular band of localized deformation, and this follows the classical Hall-Petch law that predicts a higher failure stress for smaller grains. The second is the resistance to the propagation of a mode I crack, where grain boundaries can potentially pin a crack, and this follows an inverse Hall-Petch law with higher toughness for larger grains. The result of the competition between the two mechanisms gives rise to non-monotonic behavior and reconciles the apparently contradictory experimental observations.

We propose a variational phase-field model of fracture capable of accounting for arbitrary closed convex strength domains. Unlike traditional models based on Ambrosio and Tortorelli regularization, the phase-field variable does not affect the material stiffness. Instead, our elastic energy exhibits linear growth outside a strength domain, which shrinks to 0 as the phase-field variable goes to 1. We characterize this model through a fundamental problem on a cube subject to boundary loads. We show that the solution of this problem is a transverse cohesive crack, provided that the applied load and the direction of the displacement jumps satisfy a compatibility criterion, which we formulate in terms of Mohr's circles for isotropic strength domains. This allows us to derive a hierarchy of strength criteria for which fracture is never possible, sometimes possible or always possible, depending on the direction of the stress tensor. We discuss the properties of the model and postulate a “sharp-interface” limit in the form of a cohesive law that can be explicitly derived from the form of the phase-field model. We give several examples of phase-field models and their cohesive limits. The proposed framework unifies within a single consistent variational theory key concepts developed over the centuries to predict or prevent material failure: Griffith and cohesive crack models, damage models, plasticity, strength criteria, and limit analysis.

Variational phase-field models of fracture are widely used to simulate nucleation and propagation of cracks in brittle materials. They are based on the approximation of the solutions of a free-discontinuity fracture energy by two smooth function: a displacement and a damage field. Their numerical implementation is typically based on the discretization of both fields by nodal ℙ^1 Lagrange finite elements. In this article, we propose a nonconforming approximation by discontinuous elements for the displacement and nonconforming elements, whose gradient is more isotropic, for the damage. The handling of the nonconformity is derived from that of heterogeneous diffusion problems. We illustrate the robustness and versatility of the proposed method through series of examples.

With recent advances in both responsive materials and fabrication techniques it is now possible to construct integrated functional structures, composed of both structural and active materials. We investigate the robust design of such structures through topology optimization. By applying a typical interpolation scheme and filtering technique, we prove existence of an optimal design to a class of objective functions which depend on the compliances of the stimulated and unstimulated states. In particular, we consider the actuation work and the blocking load as objectives, both of which may be written in terms of compliances. We study numerical results for the design of a 2D rectangular lifting actuator for both of these objectives, and discuss some intuition behind the features of the converged designs. We formulate the optimal design of these integrated responsive structures with the introduction of voids or holes in the domain, and show that our existence result holds in this setting. We again consider the design of the 2D lifting actuator now with voids. Finally, we investigate the optimal design of an integrated 3D torsional actuator for maximum blocking torque.

Experimental studies suggest that the fracture toughness of rocks increases with the confining pressure. Among many methods to quantify this dependency, a so-called burst experiment (Abou-Sayed, 1978) may be the most widely applied in practice. Its thick wall cylinder geometry leads to a stress state resembling the subsurface condition of a pressurized wellbore with bi-wing fractures. The fracture toughness of a sample, under a given confinement pressure, can be recovered from the critical pressure upon which the bi-wing cracks propagate. Traditionally, this critical pressure is thought to correspond to a sudden drop in injection pressure. However, as the standard configuration was deliberately designed to obtain stable fracture growth at the onset, propagation can take place well before this drop in pressure, and one may overestimate the fracture toughness from measured pressures. Here, we study crack stability in the burst experiment and propose modifications to the experimental design which promotes unstable fracture growth and makes the critical pressure less ambiguous to interpret. We found that experiments with the original, stable design can lead to inconsistent measurement of fracture toughness under confining pressure, while results from unstable configurations are more consistent. Our claim on the stability was also supported by the recorded acoustic emissions from both stable and unstable experiments.

Fracking, or hydraulic fracturing, is a ubiquitous technique for generating fracture networks in rocks for enhanced geothermal systems or hydrocarbon extraction from shales. For decades, models, numerical simulation tools, and practical guidelines have been based on the assumption that this process generates networks of self-similar parallel cracks. Yet, some field and laboratory observations show asymmetric crack growth, and material heterogeneity is routinely attributed for it. Here, we show that simultaneous growth of multiple parallel cracks is impossible and that a single crack typically propagates asymmetrically in toughness dominated hydraulic fracturing, in which viscous dissipation of the fluid is negligible. In other words, loss of symmetry is a fundamental feature of hydraulic fracturing in a toughness dominated regime and not necessary the result of material heterogeneities. Our findings challenge the assumptions of symmetrical growth of hydraulic fractures commonly made in practice, and point to yet another instability other than material heterogeneity.

We present a new approach to optimal design with state constraints based on active set optimization theory and implement it using a phase-field model. Our primary focus is on compliance minimization subject to inner and outer obstacles. We compare our approach to a classical penalization method and study the influence of initial guess, penalization parameters, and discretization.

Toughness describes the ability of a material to resist fracture or crack propagation. It is demonstrated here that fracture toughness of a material can be asymmetric, i.e., the resistance of a medium to a crack propagating from right to left can be significantly different from that to a crack propagating from left to right. Such asymmetry is unknown in natural materials, but we show that it can be built into artificial materials through the proper control of microstructure. This paves the way for control of crack paths and direction, where fracture-when unavoidable-can be guided through predesigned paths to minimize loss of critical components.

Twenty years in since their introduction, it is now plain that the regularized formulations dubbed as phase-field of the variational theory of brittle fracture of Francfort and Marigo (1998) provide a powerful macroscopic theory to describe and predict the propagation of cracks in linear elastic brittle materials under arbitrary quasistatic loading conditions. Over the past ten years, the ability of the phase-field approach to also possibly describe and predict crack nucleation has been under intense investigation. The first of two objectives of this paper is to establish that the existing phase-field approach to fracture at large - irrespectively of its particular version - cannot possibly model crack nucleation. This is so because it lacks one essential ingredient: the strength of the material. The second objective is to amend the phase-field theory in a manner such that it can model crack nucleation, be it from large pre-existing cracks, small pre-existing cracks, smooth and non-smooth boundary points, or within the bulk of structures subjected to arbitrary quasistatic loadings, while keeping undisturbed the ability of the standard phase-field formulation to model crack propagation. The central idea is to implicitly account for the presence of the inherent microscopic defects in the material - whose defining macroscopic manifestation is precisely the strength of the material - through the addition of an external driving force in the equation governing the evolution of the phase field. To illustrate the descriptive and predictive capabilities of the proposed theory, the last part of this paper presents sample simulations of experiments spanning the full range of fracture nucleation settings. (C) 2020 The Author(s). Published by Elsevier Ltd.

The problem of toughening heterogeneous materials with a stiff matrix and compliant inclusions is investigated through numerical simulations and experiments. Specifically, the problem of optimizing a combination of effective toughness and effective elastic modulus in the context of a square array of compliant inclusions in a stiff matrix is explored. Crack propagation in the heterogeneous material is simulated using a variational phase-field approach. It is found that the crack can meander between or get attracted to and trapped in the inclusions. Composite specimens with a stiff matrix and compliant circular inclusions were 3D printed, and their fracture toughness was measured using a specially designed loading fixture. The experimental results show agreement with the numerical predictions by demonstrating the attraction and trapping of cracks in the inclusions. This study demonstrates the potential for significant enhancement of toughness through elastic compliance contrast between the matrix and the inclusion without notably compromising the effective elastic modulus of the composite material.

This continues the study of the effective toughness of layered materials started in Hossain et al. (2014) and Hsueh et al. (2018), with a focus on anisotropy. We use the phase-field model and the surfing boundary condition to propagate a crack macroscopically at various angles to the layers. We study two idealized situations, the first where the elastic modulus is uniform while the toughness alternates and a second where the toughness is uniform and the elastic modulus alternates. We find that in the first case of toughness heterogeneity the effective toughness displays 'anomalous isotropy' in that it is independent of the propagation direction and equal to that of the tougher material except when the crack propagation is parallel to the layers. In the second case of elastic heterogeneity, we find the behavior more anisotropic and consistent with the toughening effects of stress fluctuation and need for crack renucleation at the compliant-to-stiff interface. In both cases, the effective toughness is not convex in the sense of interfacial energy or Wulff shape reflecting the fact that crack propagation follows a critical path. Further, in both cases the crack path is not straight and consistent with a maximal dissipation principle. Finally, the effective toughness depends on the contrast and pinning, rather than on the extent of crack fluctuation. (C) 2019 Elsevier Ltd. All rights reserved.

Crack initiation and propagation in elastic–perfectly plastic bodies is studied in a phase-field or variational gradient damage formulation. A rate-independent formulation that naturally couples elasticity, perfect plasticity and fracture is presented, and used to study crack initiation in notched specimens and crack propagation using a surfing boundary condition. Both plane strain and plane stress are addressed. It is shown that in plane strain, a plastic zone blunts the notch or crack tip which in turn inhibits crack nucleation and propagation. Sufficient load causes the crack to nucleate or unpin, but the crack does so with a finite jump. Therefore the propagation is intermittent or jerky leaving behind a rough surface. In plane stress, failure proceeds with an intense shear zone ahead of the notch or crack tip and the fracture process is not complete.

Rigorous coupling of fracture–porous medium fluid flow and topologically complex fracture propagation is of great scientific interest in geotechnical and biomechanical applications. In this paper, we derive a unified fracture–porous medium hydraulic fracturing model, leveraging the inherent ability of the variational phase-field approach to fracture to handle multiple cracks interacting and evolving along complex yet, critically, unspecified paths. The fundamental principle driving the crack evolution is an energetic criterion derived from Griffith’s theory. The originality of this approach is that the crack path itself is derived from energy minimization instead of additional branching criterion. The numerical implementation is based on a regularization approach similar to a phase-field model, where the cracks location is represented by a smooth function defined on a fixed mesh. The derived model shows how the smooth fracture field can be used to model fluid flow in a fractured porous medium. We verify the proposed approach in a simple idealized scenario where closed form solutions exist in the literature. We then demonstrate the new method’s capabilities in more realistic situations where multiple fractures turn, interact, and in some cases, merge with other fractures.

The contents of this chapter are also presented in "Fracture Diodes: Directional Asymmetry of Fracture Toughness" by N.R. Brodnik, S. Brach, C.M. Long, K. Bhattacharya, B. Bourdin, K.T. Faber, and G. Ravichandran, which is currently in preparation. N.R. Brodnik led the experimental portion of the work, and was assisted by C.M. Long. S. Brach led the numerical simulations. Bhattacharya and Bourdin supervised the numerical simulations while Faber and Ravichandran supervised the experimental work. All authors were involved in discussing all aspects of the work, and preparing the manuscript.

Phase-field models, sometimes referred to as gradient damage or smeared crack models, are widely used methods for the numerical simulation of crack propagation in brittle materials. Theoretical results and numerical evidences show that they can predict the propagation of a pre-existing crack according to Griffith' criterion. For a one-dimensional problem, it has been shown that they can predict nucleation upon a critical stress, provided that the regularization parameter be identified with the material's internal or characteristic length. In this article, we draw on numerical simulations to study crack nucleation in commonly encountered geometries for which closed-form solutions are not available. We use U- and V-notches to show that the nucleation load varies smoothly from that predicted by a strength criterion to that of a toughness criterion when the strength of the stress concentration or singularity varies. We present validation and verification numerical simulations for both types of geometries. We consider the problem of an elliptic cavity in an infinite or elongated domain to show that variational phase field models properly account for structural and material size effects. Our main claim, supported by validation and verification in a broad range of materials and geometries, is that crack nucleation can be accurately predicted by minimization of a nonlinear energy in variational phase field models, and does not require the introduction of ad-hoc criteria. (C) 2017 Elsevier Ltd. All rights reserved.

It has been established that contrast in the elastic properties can lead to enhancement of fracture toughness in heterogeneous materials. Focussing on layered materials as a model system, we show that this enhancement is a result of two distinct phenomena – first, fluctuations in stress leading to regions where the stress intensity at the crack is considerably smaller than that of the macroscopically applied value; and second, the lack of stress intensity when a crack is at a compliant to stiff interface thereby requiring renucleation. Using theoretical, computational and experimental methods, we study two geometries – a layered material and a layered material with a narrow channel – to separate the two phenomena. The stress fluctuation is present in both, but renucleation is present only in the layered medium. We provide quantitative estimates for the enhanced toughness.