Schedule for: 19w5207 - Phase-Field Models of Fracture

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The irreversibility constraint, the non-convexity of governing energy functional and the intrinsically small length-scale are the main sources of algorithmic and numerical challenges for phase-field models of fracture. The talk aims at summarizing the main ideas, results and challenges that we proposed and encountered in addressing the above issues in the past few years. We highlight - various solution strategies for the discretized coupled problem, such as partitioned (staggered) and frontal (monolithic) schemes, with a particular focus on their robustness and efficiency, - various options of incorporating the crack irreversibility constraint, with special focus on our newly proposed penalization approach with a practical and accurate bound for the penalty constant, - a posteriori estimation analysis for the discretization error and the induced adaptive mesh refinements, with a specified hierarchy of the “adapt” and “solve” processes. With intensive benchmarking, the implications of the above on simulation results are illustrated and discussed. This is a joint work with L. De Lorenzis

Mathematicians have generally not emphasized the difference between \(\Gamma\)-convergence and the convergence necessary for "approximate" dynamic solutions to converge to the correct limiting dynamics. I will discuss what properties limiting dynamic fracture models should have, and how \(\Gamma\)-convergence can fail to deliver them, with an emphasis on phase-field approximations and some surprising problems.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A novel variational framework to model the fatigue behavior of brittle materials based on a phase-field approach to fracture is presented. The standard regularized free energy functional is modified introducing a fatigue degradation function that effectively reduces the fracture toughness as a proper history variable accumulates. This macroscopic approach allows to reproduce the main known features of fatigue crack growth in brittle materials. Numerical experiments show that the Wöhler curve, the crack growth rate curve and the Paris law are naturally recovered, while the approximate Palmgren-Miner criterion and the monotonic loading condition are obtained as special cases.

In this talk, a variational model is proposed for the description of ductile failure in composite materials consisting of short strengthening fibers embedded in brittle matrices. The composite is schematized as a mixture of two phases coupled by elastic bonds: a brittle phase and a plastic phase account for matrix and fibers contributions, respectively. Balance and evolution equations are variationally deduced, and the role played by three different internal lengths is discussed. Finally, results of numerical simulations are shown.

I will review phase field methods that couple stress and composition, and then illustrate how fracture can effect and be effected by phase separation.

In this talk, I will present a macroscopic phase-field theory seemingly capable of explaining, describing, and predicting all of the classical and recent experimental observations on the internal fracture of rubber: from the nucleation of cavities/cracks, to their growth to micro-cracks, to their continued growth to macro-cracks, to the remarkable healing of some of the cracks. Following the outline of the theory, I will present its numerical implementation as well as comparisons with the classical poker-chip experiments of Gent and Lindley (1959) and recent experiments due to Ravi-Chandar.

In this talk, I will present a particular class of generalized continua called multiphase models which consist of different media possessing their own kinematics and in interaction with each other. This setting is particularly suited to fiber-reinforced media and enables to model in a macroscopic fashion phenomena like bridged cracks or fiber debonding. A variational phase-field combined with a debonding damage law will be proposed for simulating matrix cracks bridged by intact fibers.

In this talk, we shall discuss energy-dissipation phenomena and smoothing effect of solutions for an Allen-Cahn equation with nondecreasing constraint, which is inspired by a study of Damage Mechanics and corresponds to unidirectional evolution of damaging phenomena. More precisely, we shall treat the Cauchy-Dirichlet problem for the equation \[ u_t = \Big(\Delta u - W'(u) \Big)_+, \] where \(W(\cdot)\) is a double-well potential and \((\cdot)_+\) is the positive-part function. Hence solutions are constrained to be nondecreasing. Such a constraint prevents emergence of the energy-dissipation and smoothing effect, which are completely realized for classical Allen-Cahn equation. As a result, one can prove non-existence of global attractor in any $L^p$-spaces (and hence, in any Sobolev spaces). On the other hand, this equation still involves a gradient structure, and hence, energy-dissipation and smoothing effect emerge in an incomplete way. The main purpose of this talk is to explain how to extract such an incomplete emergence of energy-dissipation and smoothing effect for evolution equations with nondecreasing constraint from a functional analytic point of view. This talk is based on a joint work with M. Efendiev (München).

The objective of this work is two-fold. First, layered materials comprising alternating elastic brittle (resp., compliant) and tough (resp., rigid) phases are considered as a reference system. Depending on the layer angle, two regimes of propagation are observed: the crack can either be arrested at the interfaces or propagate along them. Computational results show that, when the material exhibits toughness heterogeneity, the effective toughness is equal to the largest pointwise value irrespective of the layer angle. Thus, smooth changes in the crack path do not to induce any toughening effect. On the other hand, when the material has contrasts in the elastic moduli, the effective toughness varies as a function of the layer angle. As a consequence, anisotropic effects arise. Second, reference is made to elastic-plastic homogeneous materials. Crack nucleation and propagation are investigated, by respectively considering V-notched specimens and large pre-cracked domains. In the first study case, the crack is observed to initiate at the notch tip and to suddenly jump into the intact material once nucleation occurs. The same jump is observed for the plastic process zone ahead of the tip. In the second study case, the crack propagates discontinuously through the material, leaving behind a discontinuous plastic wake.

We study the asymptotic behavior of a variational model for damaged elasto- plastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter epsilon, which forces damage concentration on regions of codimension one. We determine the \(\Gamma􏰀\)-limit as \(\varepsilon\) tends to zero and show that it contains an energy term involving the crack opening.

Phase-field models have shown great promise and flexibility for quantitative analysis of fracture. In this talk, I show our recent results where we extended these models to fracture of anisotropic materials and fatigue crack growth. By combining experiments using a biomimetic composite and phase-field modeling, in the first part of the talk, I show how one needs to take account of the process-zone size (and T-stress) to interpret the different kinking behavior in different sample geometries. In the second part of the talk, I show how the celebrated Paris power law emerges from a simple model based on the degradation of materials at the crack tip.

I will describe a class of numerical methods, known as threshold dynamics, that were originally derived from the Allen-Cahn equation for simulating two-phase motion by mean curvature of an interface, and were meant to be a much faster alternative. Indeed, what results is an unconditionally stable scheme, with very low per time step cost, that no longer contains a small parameter describing the interfacial thickness as in standard phase field models. Unfortunately, the original derivation does not generalize to more elaborate settings, such as multiphase and possibly anisotropic interfacial motions. A different perspective, namely a new variational formulation of threshold dynamics that forgets the original connection with phase field, allows vast extensions. I will discuss the current state of this class of algorithms.

I will present recent works about the minimisation of the Griffith energy for brittle fracture in elastic materials, under Dirichlet boundary conditions. Together with Antonin Chambolle (CMAP, École Polytechnique) we have proven the existence of minimisers and a phase-field approximation à la Ambrosio-Tortorelli for this energy.

In this talk, we will first go through the construction of a variational phase-field based coupled hydro-mechanical model in poor-elastic media. We will then revisit the problem of a single hydraulic fracture propagating in an infinite impermeable medium in order to justify our coupling strategy. Finally, we will discuss how a phase-field description of a system of cracks can be leveraged to model flow in a fractured porous medium.

Ambrosio-Tortorelli's functional (AT) is a well known relaxation of the classical Mumford-Shah model (MS). AT involves both a reconstruction function \(u\) and an approximation of the set of discontinuities \(v\) in its formulation. AT has the nice property to \(\Gamma\)-converge toward MS while being much simpler to solve. However its numerical approximation suffers from a technical difficulty: it is difficult to make the set of discontinuities thin at any digitisation scale, thus making the numerical result poor around discontinuities. We propose a discrete calculus model of AT, whose formulation authorises thin discontinuities at the scale of interest. We will recall the main aspects of discrete calculus, and present our discrete AT model. We will then show that this formulation is versatile enough to address several problems of image and geometry processing: image restoration, segmentation and inpainting, digital surface normal field regularisation, or geometric mesh denoising, inpainting or segmentation.

In this paper we propose a notion of irreversibility for the evolution of cracks in presence of cohesive forces, which allows for different responses in the loading and unloading processes, motivated by a variational approximation with damage models. We investigate its applicability to the construction of a quasistatic evolution in a simple one-dimensional model. This is a joint work with M. Bonacini and S. Conti.

I will talk about several applications of gradient flow-type phase field crack growth model to complex fracture problems related to thermal stress, hydrogen embrittlement and viscoelasticity. If time permits, I also mention some experimental results to which the gradient flow model may apply. This is a joint work with Takeshi Takaishi (Musashino Univ.) and Masato Kimura (Kanazawa Univ.)

Since their first use by Hoover et al (1974) in models for crystalline materials and Cundall & Strack (1979) in geotechnical problems, Discrete Elements Methods (DEM) have found a large field of applications in granular materials, soil and rock mechanics by allowing to compute materials' strain and cracking in a unified framework. This talk will present a possible formalization of DEM leading to a general discretization method for PDEs and allowing to write convergence proofs. Also several strategies for computing cracking under dynamical loading with DEM will be presented and their methodology compared to phase-field methods.

Heterogeneity is present in most of natural and engineering systems. In this contribution, I present the recent developments of a combined phase field method for bulk fracture and interface-like cracks. This methodology allows triggering the competition between crack penetration and deflection at an interface, recalling fundamental results from Linear Elastic Fracture Mechanics (LEFM). Following the fundamental developments, I revisit the numerical implementation as well as its application to different systems such as shell-like structures, composite materials, dynamics among others.

In this talk we discuss the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e., a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that we lack controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulation, we improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement. Joint work with Vito Crismale and Gianluca Orlando.

We apply the PCA and topological data analysis to the polymer materials like epoxy resin in order to classify its microstructure depending on the process. Based on this classification, we study the toughness problem via phase field approach.

We present two gradient flow evolutions, both obtained with alternate schemes for separately-quadratic phase-field energies. The first, in the plane strain setting, features a monotonicity constraint (in time) and a multi-step scheme, for better numerical results. The second, in higher dimension, features instead a penalty method. In this case, strong compactness of the phase-field variable allows to characterize evolutions in terms of curves of maximal slope with respect to the penalty-metric.

In a two dimensional setting, we present a result of convergence of an alternate minimization scheme applied to a phase field model of fracture with non-interpenetration. Our analysis is based on the study of suitable gradient flows of the phase field energy, which connect all the states of the algorithm. The limit evolutions are described in terms of parametrized \(BV\)-solutions. This is a joint work with M. Negri.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.

Glass is widely used for industrial applications such as glass housewares, automotive glass, and smartphone cover glass. Due to its low ductility, glass products tend to fail catastrophically and tougher glass is highly desired to add more value to industrial products. In AGC we have been developing a computer aided approach to develop glass products with higher resistance against crack propagation. In this talk we present our attempts at toughening glass based on computer simulation.