Convergence of alternate minimization schemes for phase-field fracture and damage

Knees, Dorothee; Negri, Matteo

doi:10.1142/s0218202517500312

Cited by 37 publications

(73 citation statements)

References 35 publications

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“…This is satisfied for ϕ from (28a) with ε = 0 or from (32). The mentioned potential of the boundary-value problem (35)- (36) is…”

Section: Implicit "Monolithic" Discretisation In Timementioning

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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“…This is satisfied for ϕ from (28a) with ε = 0 or from (32). The mentioned potential of the boundary-value problem (35)- (36) is…”

Section: Implicit "Monolithic" Discretisation In Timementioning

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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“…This viewpoint on the Ambrosio-Tortorelli model has also been taken in the rate-independent setting e.g. in [KN17], where the alternate minimization scheme (23) has been further iterated in each time-step leading to parametrized BV -evolutions of the rate-independent problem, and in [Neg16], where also a viscous regularization has been taken into account.…”

Section: Wwwgamm-mitteilungenorgmentioning

confidence: 99%

Phase‐Field Fracture at Finite Strains Based on Modified Invariants: A Note on its Analysis and Simulations

2018

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Key words Phase-field model for fracture, Ambrosio-Tortorelli, viscous evolution, finite strains, modified invariants MSC (2010) 35K85,74B20,74B20,74R10,74R20We present recent results on the mathematical analysis of phase-field model for anisotropic fracture at finite strains. It is based on a modification of the Ambrosio-Tortorelli model, formulated for polyconvex energy densities in terms of the modified invariants of the right Cauchy-Green strain tensor and augmented by a viscous dissipation for the phase-field variable. A suitable notion of solution is introduced and used for the simulation of a conchoidal fracture in a brittle specimen.

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“…The total cost of a transition ϑ : E → X at a jump time t is therefore 17) and the corresponding cost c for a jump from u(t−) to u(t+) passing through the value u(t) is given by 18) where the infimum is attained whenever there is at least one admissible transition with finite cost. Notice that the cost c is always bigger than the corresponding value computed by the dissipation distance d, i.e.…”

Section: Viscous Corrections Of the Incremental Minimization Schemementioning

confidence: 99%

Viscous Corrections of the Time Incremental Minimization Scheme and Visco-Energetic Solutions to Rate-Independent Evolution Problems

Minotti

Savaré

2017

Arch Rational Mech Anal

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We propose the new notion of Visco-Energetic solutions to rate-independent systems (X, E, d) driven by a time dependent energy E and a dissipation quasi-distance d in a general metric-topological space X.As for the classic Energetic approach, solutions can be obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation quasi-distance d is incremented by a viscous correction δ (e.g. proportional to the square of the distance d), which penalizes far distance jumps by inducing a localized version of the stability condition.We prove a general convergence result and a typical characterization by Stability and Energy Balance in a setting comparable to the standard energetic one, thus capable to cover a wide range of applications. The new refined Energy Balance condition compensates the localized stability and provides a careful description of the jump behavior: at every jump the solution follows an optimal transition, which resembles in a suitable variational sense the discrete scheme that has been implemented for the whole construction.

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Convergence of alternate minimization schemes for phase-field fracture and damage

Cited by 37 publications

References 35 publications

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

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

Phase‐Field Fracture at Finite Strains Based on Modified Invariants: A Note on its Analysis and Simulations

Viscous Corrections of the Time Incremental Minimization Scheme and Visco-Energetic Solutions to Rate-Independent Evolution Problems

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