Revisiting brittle fracture as an energy minimization problem

Francfort, Gilles A.; Marigo, Jean‐Jacques

doi:10.1016/s0022-5096(98)00034-9

Cited by 2,597 publications

(1,652 citation statements)

References 13 publications

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“…On the contrary, in the presence of a point x of strong singularity the crack initiation is progressive: a crack departs from x at the initial time of loading and with zero velocity. These facts were proved by Francfort and Marigo in [20,Proposition 4.19], under the assumption that the path of the crack is given a priori by a finite number of fixed curves which can be parameterized by arc length. This is not the case in our larger class of admissible cracks.…”

Section: γ (T) = ∅ For Every T T Imentioning

confidence: 90%

“…As noticed by Francfort and Marigo in [20], if T is large enough, a crack will appear during the evolution, that is, Γ (s) = ∅ for some s ∈]0, T [. In fact if T is such that…”

Section: Crack Evolution {T → (U(t) γ (T))} Withmentioning

confidence: 94%

“…Although it is more general, our study is in part motivated by the variational model for quasistatic crack propagation proposed by Francfort and Marigo in [20]. The main features of this model are that the path of the increasing crack Γ (t) is not preassigned, the class of admissible cracks is given by all sets with finite length, and the growth is not assumed to be progressive, namely the length of the crack is not assumed to be continuous in time.…”

Section: (K − K(l(t)))l(t) = 0 That Is γ (L(t))mentioning

confidence: 99%

“…Inspired by the variational theory of quasistatic crack evolution proposed by Francfort and Marigo in [20], we replace Griffith's equilibrium condition with a static equilibrium condition and an energy balance. The static equilibrium condition is a unilateral minimality property which states that, during the crack evolution, the total energy is minimal among all configurations with larger cracks (so that discontinuities of the crack's length are allowed).…”

Section: Qualitative Properties Of Crack Initiationmentioning

confidence: 99%

“…We refer the reader to the paper by Francfort and Marigo [20] for further details on crack evolutions satisfying conditions (a), (b), and (c) (see also Mielke [27] for a connection with the theory of rate-independent processes). In order to treat the problem of crack initiation, we consider as in [20] the case in which f is p-homogeneous in the gradient, that is, for all x ∈ Ω, ξ ∈ R 2 and t > 0…”

Section: ) Energy Balance: the Total Energy E(u(t) γ (T)) Is Absolutmentioning

confidence: 99%

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Crack Initiation in Brittle Materials

Chambolle

Giacomini

Ponsiglione

2007

Arch Rational Mech Anal

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In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during a load process through a time-dependent boundary datum of the type t → tg(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation is brutal, that is, a big crack appears after a positive time t i > 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the problem is a local minimality result for the functionalwhere Ω ⊆ R 2 , k > 0 and f is a suitable Carathéodory function. We prove that if the uncracked configuration u of Ω relative to a boundary displacement ψ has at most uniformly weak singularities, then configurations (u Γ , Γ ) with H 1 (Γ ) small enough are such that E(u, ∅) < E(u Γ , Γ ).

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Section: γ (T) = ∅ For Every T T Imentioning

confidence: 90%

“…As noticed by Francfort and Marigo in [20], if T is large enough, a crack will appear during the evolution, that is, Γ (s) = ∅ for some s ∈]0, T [. In fact if T is such that…”

Section: Crack Evolution {T → (U(t) γ (T))} Withmentioning

confidence: 94%

Section: (K − K(l(t)))l(t) = 0 That Is γ (L(t))mentioning

confidence: 99%

Section: Qualitative Properties Of Crack Initiationmentioning

confidence: 99%

Section: ) Energy Balance: the Total Energy E(u(t) γ (T)) Is Absolutmentioning

confidence: 99%

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Crack Initiation in Brittle Materials

Chambolle

Giacomini

Ponsiglione

2007

Arch Rational Mech Anal

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Investigation of wing crack formation with a combined phase‐field and experimental approach

Lee

Reber

Hayman

et al. 2016

Geophysical Research Letters

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Fractures that propagate off of weak slip planes are known as wing cracks and often play important roles in both tectonic deformation and fluid flow across reservoir seals. Previous numerical models have produced the basic kinematics of wing crack openings but generally have not been able to capture fracture geometries seen in nature. Here we present both a phase‐field modeling approach and a physical experiment using gelatin for a wing crack formation. By treating the fracture surfaces as diffusive zones instead of as discontinuities, the phase‐field model does not require consideration of unpredictable rock properties or stress inhomogeneities around crack tips. It is shown by benchmarking the models with physical experiments that the numerical assumptions in the phase‐field approach do not affect the final model predictions of wing crack nucleation and growth. With this study, we demonstrate that it is feasible to implement the formation of wing cracks in large scale phase‐field reservoir models.

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Phase field model of fluid‐driven fracture in elastic media: Immersed‐fracture formulation and validation with analytical solutions

2017

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Propagation of fluid‐driven fractures plays an important role in natural and engineering processes, including transport of magma in the lithosphere, geologic sequestration of carbon dioxide, and oil and gas recovery from low‐permeability formations, among many others. The simulation of fracture propagation poses a computational challenge as a result of the complex physics of fracture and the need to capture disparate length scales. Phase field models represent fractures as a diffuse interface and enjoy the advantage that fracture nucleation, propagation, branching, or twisting can be simulated without ad hoc computational strategies like remeshing or local enrichment of the solution space. Here we propose a new quasi‐static phase field formulation for modeling fluid‐driven fracturing in elastic media at small strains. The approach fully couples the fluid flow in the fracture (described via the Reynolds lubrication approximation) and the deformation of the surrounding medium. The flow is solved on a lower dimensionality mesh immersed in the elastic medium. This approach leads to accurate coupling of both physics. We assessed the performance of the model extensively by comparing results for the evolution of fracture length, aperture, and fracture fluid pressure against analytical solutions under different fracture propagation regimes. The excellent performance of the numerical model in all regimes builds confidence in the applicability of phase field approaches to simulate fluid‐driven fracture.

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Revisiting brittle fracture as an energy minimization problem

Cited by 2,597 publications

References 13 publications

Crack Initiation in Brittle Materials

Crack Initiation in Brittle Materials

Investigation of wing crack formation with a combined phase‐field and experimental approach

Phase field model of fluid‐driven fracture in elastic media: Immersed‐fracture formulation and validation with analytical solutions

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