Phase-Field Models of Fracture (19w5207)
The Banff International Research Station will host the "Phase-Field Models of Fracture" workshop in Banff from March 3, 2019 to March 8, 2019. Phase-field models of fracture were originally devised in the mathematics community as numerical approximation of the variational theories of brittle fracture, based on revisiting Griffith's theory from the 1920's with modern mathematical tools. The underlying idea of these models is to represent cracks --two-dimensional surfaces in three dimensions and curves in two dimensions-- using a smooth auxiliary "phase-field" diffused over a small regularization length. Part of their appeal is that their deep theoretical roots can be leveraged to address many of the most pressing challenges in fracture mechanics including crack path identification, crack nucleation prediction, and the need for efficient three-dimensional numerical simulations. As such, they have seen an explosive growth in the last few years, in particular in the mechanics and engineering communities. As the community of users and the range of applications have gotten larger, it has become increasingly difficult to ensure proper communication between key players from the mathematics and applications communities. This workshop brings together a broad multi-disciplinary group from academia and the industry at various stage of their career. The make-up of the organization committee is meant to reflect the diversity of viewpoints expected, with organizers from the applied mathematics, theoretical mechanics, and computational science communities, originating from three continents. The objectives of the workshop are manifold: It seeks to restore proper communication between the communities involved by informing each other of the key success and challenges. It will tackle a new class of inverse problems arising from optimal design of materials and structures or materials characterization, for instance. Finally, it will allow candid comparison of recently developed numerical approaches, and elaboration of benchmark problems. The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).
19
2019
180
12 hours 30 minutes
19 results
27:31
11Almi, StefanoIn 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
37:41
6Negri, MatteoWe 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
43:49
5Nishiura, YasumasaWe 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
40:33
4Lazzaroni, GiulianoIn 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
38:13
6Reinoso, JoseHeterogeneity 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
21:07
10Marazzato, FrédéricSince 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
34:17
21Iurlano, FlavianaIn 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
38:47
9Yoshioka, KeitaIn 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
52:43
8Crismale, VitoI 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
38:11
7Mesgarnejad, AtaPhase-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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
43:04
3Maso, Gianni DalWe 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 Γ-limit as ε tends to zero and show that it contains an energy term involving the crack opening.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
40:02
2Akagi, Goro2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
42:01
3Bleyer, JeremyIn 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
40:51
6Lopez-Pamies, OscarIn 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
31:56
7Lancioni, GiovanniIn 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
26:16
12Carrara, PietroA 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
42:57
12Larsen, ChristopherMathematicians have generally not emphasized the difference between Γ-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 Γ-convergence can fail to deliver them, with an emphasis on phase-field approximations and some surprising problems.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
51:20
29Gerasimov, TymofiyThe 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.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
59:23
19Babadjian, Jean-François2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery