Modeling solutions with jumps for rate-independent systems  on
metric spaces

Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe

doi:10.3934/dcds.2009.25.585

Cited by 86 publications

(121 citation statements)

References 19 publications

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“…Here the labels (S ) and (E ) stand respectively for stationarity and energy balance while the prime symbol suggests the dependence on derivatives of the stored energy. Close to our definition of evolutions by critical points are those of [16,26,32,37] while close to our existence proof are those on minimizing movements for gradient flows, e.g. [1,2,15,33,35].…”

Section: Introductionsupporting

confidence: 53%

“…Let us turn to evolutions by critical points, assuming from now on that F is of class C 1 . In the literature there are several equivalent definitions, most of them formulated in terms of the trajectory t → v(t), as the BV solutions of [26,27]. Here we will adopt a graph parametrization of solutions, inspired by [16], which provides in turn a parametrized BV solution (see Sect.…”

Section: Introductionmentioning

confidence: 99%

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Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics

Negri

2014

ESAIM: COCV

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We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals F of class C 1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals Fn and its Γ-limit F we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.

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Section: Introductionsupporting

confidence: 53%

Section: Introductionmentioning

confidence: 99%

Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics

Negri

2014

ESAIM: COCV

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“…The more general concept of BV-solution has been recently introduced in [MRS07]. This notion works on general metric spaces, but in the context of the present work it coincides with the local energetic solution.…”

Section: Discussion and Comparison With Other Types Of Solutionsmentioning

confidence: 96%

“…However, if the rate-independent solutions are not continuous, then the corresponding solution with small viscosity develops very large rates that are governed by the viscosity. The aim is to understand the limits of viscous solutions when the viscosity is made smaller and smaller, see [EfM06,MRS07] for the general philosophy. For nontrivial PDE applications see also [DD * 07, MiZ07].…”

Section: Introductionmentioning

confidence: 99%

On the Inviscid Limit of a Model for Crack Propagation

Knees

Mielke

Zanini

2008

Math. Models Methods Appl. Sci.

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We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith's local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove the existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.

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“…Unfortunately, that approach soon grinds to a halt because a lack of Lipschitz estimate in time on the plastic strain seems to impede the proof of the lower inequality in the energy balance. The idea, borrowed from [26,27], but whose germ is in [18], is then to recover a Lipschitz estimate on the plastic strain through an adequate re-parameterization of time. Unfortunately, once again the analysis grinds to a halt because the re-parameterization involves both piecewise constant and piecewise affine interpolations of the incremental plastic strains.…”

mentioning

confidence: 99%

Quasi-static Evolution in Nonassociative Plasticity: The Cap Model

Babadjian¹,

Francfort²,

Mora³

2012

SIAM J. Math. Anal.

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Non-associative elasto-plasticity is the working model of plasticity for soil and rock mechanics. Yet, it is usually viewed as non-variational. In this work, we prove a contrario the existence of a variational evolution for such a model under a natural capping assumption on the hydrostatic stresses and a less natural mollification of the stress admissibility constraint. The obtained elasto-plastic evolution is expressed for times that are conveniently rescaled. Contents1. Description of the model 4 1.1. The original model revisited 4 1.2. The cap model 7 1.3. Abstract setting 8 1.4. Mathematical setting 12 2. The visco-plastic model 14 2.

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Modeling solutions with jumps for rate-independent systems on metric spaces

Cited by 86 publications

References 19 publications

Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics

Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics

On the Inviscid Limit of a Model for Crack Propagation

Quasi-static Evolution in Nonassociative Plasticity: The Cap Model

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