Existence of solutions to a phase–field model of dynamic fracture with a crack–dependent dissipation

Caponi, Maicol

doi:10.1007/s00030-020-0617-z

Cited by 5 publications

(14 citation statements)

References 19 publications

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“…The generalization for the vectorial case is due to Iurlano [] under some additional regularity assumptions, and in full generality to []. Later, it was extended to the evolution situation, namely for a rate‐independent cohesive damage, in [], see also [] where inertial forces are incorporated in the description. Note however that plasticity was not involved in all these references.…”

Section: Setting Of the Problem And Statement Of The Main Resultsmentioning

confidence: 99%

Dynamic perfect plasticity and damage in viscoelastic solids

2019

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In this paper we analyze an isothermal and isotropic model for viscoelastic media combining linearized perfect plasticity (allowing for concentration of plastic strain and development of shear bands) and damage effects in a dynamic setting. The interplay between the viscoelastic rheology with inertia, elasto‐plasticity, and unidirectional rate‐dependent incomplete damage affecting both the elastic and viscous response, as well as the plastic yield stress, is rigorously characterized by showing existence of weak solutions to the constitutive and balance equations of the model. The analysis relies on the notions of plastic‐strain measures and bounded‐deformation displacements, on sophisticated time‐regularity estimates to establish a duality between acceleration and velocity of the elastic displacement, on the theory of rate‐independent processes for the energy conservation in the dynamical‐plastic part, and on the proof of the strong convergence of the elastic strains. Existence of suitably defined weak solutions (even conserving energy) is proved rather constructively by using a staggered two‐step time discretization scheme.

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Section: Setting Of the Problem And Statement Of The Main Resultsmentioning

confidence: 99%

Dynamic perfect plasticity and damage in viscoelastic solids

2019

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“…11 Such schemes are known in engineering literature under the adjective "monolithic" and the mentioned iterative solution is e.g. by the Newton-Raphson (or here equivalently SQP = sequential quadratic programming) method without any guaranteed convergence, however, or alternating-minimization algorithm (AMA) 12 . In general, such schemes even do not seem numerically stable because the a-priori estimates are not available.…”

Section: Implicit "Monolithic" Discretisation In Timementioning

confidence: 99%

“…The generalization for the vectorial case is in[19,20,33]. Later, it was extended for evolution situation, namely for a rate-independent damage, in[25], see also also[9,10,12,39,49] where also inertial forces are sometimes considered 7. In fact, as φ (1) = 0, the initiation of damage has zero threshold and is happening even on very low stress but then, if ε > 0 is very small, stops and high stress is needed to continue damaging.…”

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confidence: 99%

Models of Dynamic Damage and Phase-field Fracture, and their Various Time Discretisations

Roubíček

2019

CIM Series in Mathematical Sciences

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Several variants of models of damage in viscoelastic continua under small strains in the Kelvin-Voigt rheology are presented and analyzed by using the Galerkin method. The particular case, known as a phase-field fracture approximation of cracks, is discussed in detail. All these models are dynamic (i.e. involve inertia to model vibrations or waves possibly emitted during fast damage/fracture or induced by fast varying forcing) and consider viscosity which is also damageable. Then various options for time discretisation are devised. Eventually, extensions to more complex rheologies or a modification for large strains are briefly exposed, too.

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“…Such a shortcoming is overcome in [12] where the author considers a variation of (1.9), (1.10) that has no such damping term in (1.9). Rather than include a dissipative term in (1.9), a ratedependent term is included in the minimisation problem (1.10).…”

Section: Introductionmentioning

confidence: 99%

“…Rather than include a dissipative term in (1.9), a ratedependent term is included in the minimisation problem (1.10). The author of [12] considers the balance law (1.11)…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamic fracture problems with polynomial and strain-limiting constitutive relations

Patel

2021

Preprint

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We extend the framework of dynamic fracture problems with a phase-field approximation to the case of a nonlinear constitutive relation between the Cauchy stress tensor T, linearised strain ε(u) and strain rate ε(u t ). The relationship takes the form ε(u t ) + αε(u) = F (T) where F satisfies certain p-growth conditions. We take particular care to study the case p = 1 of a 'strain-limiting' solid, that is, one in which the strain is bounded a priori. We prove the existence of long-time, large-data weak solutions of a balance law coupled with a minimisation problem for the phase-field function and an energy-dissipation inequality, in any number d of spatial dimensions. In the case of Dirichlet boundary conditions, we also prove the satisfaction of an energy-dissipation equality.

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Existence of solutions to a phase–field model of dynamic fracture with a crack–dependent dissipation

Cited by 5 publications

References 19 publications

Dynamic perfect plasticity and damage in viscoelastic solids

Dynamic perfect plasticity and damage in viscoelastic solids

Models of Dynamic Damage and Phase-field Fracture, and their Various Time Discretisations

Nonlinear dynamic fracture problems with polynomial and strain-limiting constitutive relations

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