Crack growth in polyconvex materials

Knees, Dorothee; Zanini, Chiara; Mielke, Alexander

doi:10.1016/j.physd.2009.02.008

Cited by 43 publications

(55 citation statements)

References 23 publications

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“…Starting from the seminal paper [EM06], this technique has by now been thoroughly developed both for abstract rate-independent systems [MRS09, MRS12,MZ13], and in the applications to fracture [TZ09,Cag08,KMZ08,KZM10,LT11], and to plasticity [DDMM08,BFM12,DDS11,DDS12,FS13].…”

Section: Introductionmentioning

confidence: 99%

A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

Knees

Rossi

Zanini

2015

Nonlinear Analysis: Real World Applications

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This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps.Therefore we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling, and approximate the model by its viscous regularization. In fact, the analysis of the latter PDE system presents remarkable difficulties, due to its highly nonlinear character. We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with careful regularity estimates tailored to this specific system, relying on a q-Laplacian type gradient regularization of the damage variable. Hence for the viscous problem we conclude the existence of weak solutions, satisfying a suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis. The latter leads to the notion of (weak) parameterized solution to our rate-independent system, which encompasses the influence of viscosity in the description of the jump regime.

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Section: Introductionmentioning

confidence: 99%

A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

Knees

Rossi

Zanini

2015

Nonlinear Analysis: Real World Applications

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“…By (3.2), (6.4e), (6.4g),(6.4i), (6.5h), (6.6e), and (6.6g), using the Cauchy inequality, we get a positive constant D 0 such that 14) for every ∈ (0, 1). By Poincaré and Korn inequalities there exists a constant c such that…”

Section: Limit Of Dynamic Solutionsmentioning

confidence: 98%

“…The quasistatic evolution of rate independent systems has been often obtained as the limit case of a viscosity driven evolution (see [27], [20], [9], [6], [30], [22], [13], [14], [16], [17], [21], [23]). In this paper we present a case study on the approximation of a quasistatic evolution by dynamic evolutions, in a mechanical problem governed by partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes

Maso

Scala

2014

J Dyn Diff Equat

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“…We also refer to [77] for some developments in the framework of finite strain elasticity and rate-independent model. A general point of view is that we should consider a perturbed problem with respect to (5.1)-(5.5).…”

Section: Shape Derivatives Of Energy Functionalsmentioning

confidence: 99%