A modification of the phase-field model for mixed mode crack propagation in rock-like materials

Xue, Zhang; Sloan, Scott W.; Vignes, Chet; Sheng, Daichao

doi:10.1016/j.cma.2017.04.028

Cited by 211 publications

(107 citation statements)

References 35 publications

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“…Remark While the smaller the length scale l 0 is, the closer the fracture representation is to true cracks computational limitation preclude extremely small length scales . It is recommended that one should have at least two elements across the smeared crack width, which means the element size must be at least twice smaller than the length scale .…”

Section: Fundamentals Of the Phase Field Methodsmentioning

confidence: 99%

“…While the smaller the length scale l 0 is, the closer the fracture representation is to true cracks computational limitation preclude extremely small length scales. 42,[64][65][66] It is recommended that one should have at least two elements across the smeared crack width, which means the element size must be at least twice smaller than the length scale. 64 Nonetheless, to avoid some of those computational limitations, l 0 can be related to the Young modulus, critical energy and the strength of the material and could directly be calibrated against experimental results, as suggested in Borden 62 and Zhang et al 64 Remark 2.…”

Section: Governing Equationsmentioning

confidence: 99%

“…However, for many materials, the energy release rate due to tension might be quite different from the one due to shear. Hence, although in the current work, we follow the assumption that G c corresponds to tension, one should consider G c in relation to the dominant energy release rate or incorporate both tension or shear components, as proposed in Zhang et al 65 Remark 3. Note that for inhomogeneous materials, the critical fracture G c should also be varying in the domain.…”

Section: Governing Equationsmentioning

confidence: 99%

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A novel phase field method for modeling the fracture of long bones

Shen

Waisman

Yosibash

et al. 2019

Numer Methods Biomed Eng

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A proximal humerus fracture is an injury to the shoulder joint that necessitates medical attention. While it is one of the most common fracture injuries impacting the elder community and those who suffer from traumatic falls or forceful collisions, there are almost no validated computational methods that can accurately model these fractures. This could be due to the complex, inhomogeneous bone microstructure, complex geometries, and the limitations of current fracture mechanics methods. In this paper, we develop a novel phase field method to investigate the proximal humerus fracture. To model the fracture in the inhomogeneous domain, we propose a power‐law relationship between bone mineral density and critical energy release rate. The method is validated by an in vitro experiment, in which a human humerus is constrained on both ends while subjected to compressive loads on its head, in the longitudinal direction, that lead to fracture at the anatomical neck. CT scans are employed to acquire the bone geometry and material parameters, from which detailed finite element meshes with inhomogeneous Young modulus distributions are generated. The numerical method, implemented in a high performance computing environment, is used to quantitatively predict the complex 3D brittle fracture of the bone and is shown to be in good agreement with experimental observations. Furthermore, our findings show that the damage is initiated in the trabecular bone‐head and propagates outward towards the bone cortex. We conclude that the proposed phase field method is a promising approach to model bone fracture.

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Section: Fundamentals Of the Phase Field Methodsmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

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A novel phase field method for modeling the fracture of long bones

Shen

Waisman

Yosibash

et al. 2019

Numer Methods Biomed Eng

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“…The nonvariational approach of Zhang et al proposes the modified evolution equation,

\begin{align} \frac{\partial g false(s false)}{\partial s} ([], \frac{ψ^{+, I}}{G_{c,I}} + \frac{ψ^{+, I I}}{G_{c,II}}) + \frac{\partial ψ^{ph}}{\partial s} prefix- △ ψ^{ph} = 0, \end{align}

in order to consider the observed differences in crack propagation at modes I and II crack deformation. The two deformation modes are identified from the spectral decomposition as:

\begin{align} ψ^{+, I} = \frac{1}{2} λ {⟨ ϵ_{1} + ϵ_{2} + ϵ_{3} ⟩}_{prefix\pm}^{2}, \end{align}

\begin{align} ψ^{+, I I} = μ ([], {⟨ ϵ_{1} ⟩}_{prefix\pm}^{2} + {⟨ ϵ_{2} ⟩}_{prefix\pm}^{2} + {⟨ ϵ_{3} ⟩}_{prefix\pm}^{2}), \end{align}

where G c,I and G c,II are the critical fracture energies, respectively.…”

Section: Crack Kinematics Of Phase‐field Formulations For Fracturementioning

confidence: 99%

The concept of representative crack elements for phase‐field fracture: Anisotropic elasticity and thermo‐elasticity

Storm

Supriatna

Kaliske

2019

Numerical Meth Engineering

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Summary The realistic representation of material degradation at a fully evolved crack is still one of the main challenges of the phase‐field method for fracture. An approach with realistic degradation behavior is only available for isotropic elasticity in the small deformation framework. In this paper, a variational framework is presented for the standard phase‐field formulation, which allows to derive the kinematically consistent material degradation. For this purpose, the concept of representative crack elements (RCE) is introduced to analyze the fully degraded material state. The realistic material degradation is further tested using the self‐consistency condition, where the behavior of the phase‐field model is compared to a discrete crack model. The framework is applied to isotropic elasticity, anisotropic elasticity and thermo‐elasticity, but not restricted to these constitutive formulations.

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“…Therefore, it is difficult to choose a single critical value G c for a crack under mixed‐mode loading. To overcome this limitation in the conventional phase‐field model, a modified phase‐field model based on the F ‐criterion was proposed by Zhang et al for simulating a mixed‐mode crack propagation. The modified phase‐field evolution equation based on the F ‐criterion is rearranged as

2 ((), 1 - d) \frac{H}{G_{normalc}} - \frac{d}{ℓ_{0}} + ℓ_{0} normalΔ d = 0,

where

\frac{H}{G_{normalc}} = \frac{{scriptH}_{normalI}}{G_{normalcI}} + \frac{{scriptH}_{normalII}}{G_{normalcII}} .

…”

Section: Phase‐field Fracture Modelmentioning

confidence: 99%

An adaptive material point method coupled with a phase‐field fracture model for brittle materials

Cheon

Kim

2019

Numerical Meth Engineering

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Summary In this study, a new adaptive method for crack propagation analysis is developed by using the material point method coupled with a phase‐field fracture model for brittle materials. A background grid of material particles is adaptively refined based on the amount of material damage to resolve the length scale in the phase‐field evolution equation. A division process of the material particles associated with the refined background cells is also performed to increase the resolution of solutions near the crack tip. The effectiveness and validity of the proposed method is assessed through several numerical examples for crack propagation in brittle materials.

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A modification of the phase-field model for mixed mode crack propagation in rock-like materials

Cited by 211 publications

References 35 publications

A novel phase field method for modeling the fracture of long bones

A novel phase field method for modeling the fracture of long bones

The concept of representative crack elements for phase‐field fracture: Anisotropic elasticity and thermo‐elasticity

An adaptive material point method coupled with a phase‐field fracture model for brittle materials

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