Hyperbolic phase field modeling of brittle fracture: Part I—Theory and simulations

Earthquake fault zones are more complex, both geometrically and rheologically, than an idealized infinitely thin plane embedded in linear elastic material. To incorporate nonlinear material behaviour, natural complexities and multi-physics coupling within and outside of fault zones, here we present a first-order hyperbolic and thermodynamically compatible mathematical model for a continuum in a gravitational field which provides a unified description of nonlinear elasto-plasticity, material damage and of viscous Newtonian flows with phase transition between solid and liquid phases. The fault geometry and secondary cracks are described via a scalar function ξ ∈ [0, 1] that indicates the local level of material damage. The model also permits the representation of arbitrarily complex geometries via a diffuse interface approach based on the solid volume fraction function α ∈ [0, 1]. Neither of the two scalar fields ξ and α needs to be mesh-aligned, allowing thus faults and cracks with complex topology and the use of adaptive Cartesian meshes (AMR). The model shares common features with phase-field approaches, but substantially extends them. We show a wide range of numerical applications that are relevant for dynamic earthquake rupture in fault zones, including the co-seismic generation of secondary off-fault shear cracks, tensile rock fracture in the Brazilian disc test, as well as a natural convection problem in molten rock-like material. This article is part of the theme issue ‘Fracture dynamics of solid materials: from particles to the globe’.

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“…fracture dynamics and wave propagation. We note that a hyperbolic reformulation of phase-field models is possible as recently proposed in [ 55 ].…”

Section: Introductionmentioning

confidence: 99%

A unified first-order hyperbolic model for nonlinear dynamic rupture processes in diffuse fracture zones

Gabriel

Chiocchetti

et al. 2021

Phil. Trans. R. Soc. A.

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“…We have examined in detail two types of phase-field models, but we expect similar results from other phase-field models, e.g. [KMB18], because the key features that lead to these findings are similar in those other models. In turn, we expect that the augmentation proposed in this paper will prove useful in augmenting also those other models.…”

Section: Discussionmentioning

confidence: 71%

Phase-Field Modeling and Peridynamics for Defect Dynamics, and an Augmented Phase-Field Model with Viscous Stresses

Chua,

Agrawal,

Breitzman

et al. 2021

Preprint

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This work begins by applying peridynamics and phase-field modeling to predict 1-d interface motion with inertia in an elastic solid with a non-monotone stress-strain response. In classical nonlinear elasticity, it is known that subsonic interfaces require a kinetic law, in addition to momentum balance, to obtain unique solutions; in contrast, for supersonic interfaces, momentum balance alone is sufficient to provide unique solutions. This work finds that peridynamics agrees with this classical result, in that different choices of regularization parameters provide different kinetics for subsonic motion but the same kinetics for supersonic motion. In contrast, conventional phase-field models coupled to elastodynamics are unable to model, even qualitatively, the supersonic motion of interfaces. This work identifies the shortcomings in the physics of standard phase-field models to be: (1) the absence of higher-order stress to balance unphysical stress singularities, and (2) the ability of the model to access unphysical regions of the energy landscape.Based on these observations, this work proposes an augmented phase-field model to introduce the missing physics. The augmented model adds: (1) a viscous stress to the momentum balance, in addition to the dissipative phase-field evolution, to regularize singularities; and (2) an augmented driving force that models the physical mechanism that keeps the system out of unphysical regions of the energy landscape. When coupled to elastodynamics, the augmented model correctly describes both subsonic and supersonic interface motion. The augmented model has essentially the same computational expense as conventional phase-field models and requires only minor modifications of numerical methods, and is therefore proposed as a replacement to the conventional phase-field models.

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“…Constraining the foreground solid to the background fluid kinematics as in [7] gives the desired modularity together with strong coupling, however, the modeling of fracture and fragmentation in the immersed FSI simulations remains a challenge. While the foreground discretization such as PD can easily support discontinuous kinematic fields by locally breaking bonds between material points [11][12][13][14][15][16][17][18][19], the smooth background discretization of IGA [20][21][22][23][24][25][26][27][28] is not designed to excel in approximating discontinuous kinematics. Thus, constraining the foreground solution to its background counterpart results in an overly smooth foreground solution and, when coupled with continuum-damage (or phasefield [26,27]) approaches to model fracture and fragmentation, results in the size of damage zones that scales with that of the background mesh.…”

Section: Introductionmentioning

confidence: 99%

“…While the foreground discretization such as PD can easily support discontinuous kinematic fields by locally breaking bonds between material points [11][12][13][14][15][16][17][18][19], the smooth background discretization of IGA [20][21][22][23][24][25][26][27][28] is not designed to excel in approximating discontinuous kinematics. Thus, constraining the foreground solution to its background counterpart results in an overly smooth foreground solution and, when coupled with continuum-damage (or phasefield [26,27]) approaches to model fracture and fragmentation, results in the size of damage zones that scales with that of the background mesh. As a result, unless the background mesh is sufficiently fine, the damage bands appear to be artificially thick and often predict non-physical behavior, especially in the brittle fracture regime.…”

Section: Introductionmentioning

confidence: 99%

IGA-PD Penalty-Based Coupling for Immersed Air-Blast Fluid-Structure Interaction: A Simple and Effective Solution for Fracture and Fragmentation

Behzadinasab¹,

Hillman²,

Bazilevs³

2021

Preprint

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We present a novel formulation for the immersed coupling of Isogeometric Analysis (IGA) and Peridynamics (PD) for the simulation of fluid-structure interaction (FSI). We focus on air-blast FSI and address the computational challenges of immersed FSI methods in the simulation of fracture and fragmentation by developing a weakly volume-coupled FSI formulation by means of a simple penalty approach. We show the mathematical formulation and present several numerical examples of inelastic ductile and brittle solids that clearly demonstrate the power and robustness of the proposed methodology.

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Hyperbolic phase field modeling of brittle fracture: Part I—Theory and simulations

Cited by 45 publications

References 46 publications

A unified first-order hyperbolic model for nonlinear dynamic rupture processes in diffuse fracture zones

A unified first-order hyperbolic model for nonlinear dynamic rupture processes in diffuse fracture zones

Phase-Field Modeling and Peridynamics for Defect Dynamics, and an Augmented Phase-Field Model with Viscous Stresses

IGA-PD Penalty-Based Coupling for Immersed Air-Blast Fluid-Structure Interaction: A Simple and Effective Solution for Fracture and Fragmentation

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