Local existence for Fr�mond?s model of damage in elastic materials

Bonetti, Elena; Schimperna, Giulio

doi:10.1007/s00161-003-0152-2

Cited by 68 publications

(87 citation statements)

References 16 publications

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“…Hence, χ is forced to take values in [0,1], with the convention that χ = 0 when the body is completely damaged, and χ = 1 in the damage-free case. Hence, in [7], [8] the equation for macroscopic movements has a degenerating character related to the parameter χ, which is however different from the one in (1.3). For, in their case the coefficients of the elliptic operators in the stress-strain relation vanish only as χ ց 0, contrary to the twofold degeneracy of the equation (1.6 c), which we shall further comment later on.…”

Section: Introductionmentioning

confidence: 91%

“…The model is analyzed in [20] and in [21] pertains to nonlinear thermoviscoplasticity: in the one-dimensional (in space) case, the authors prove the global well-posedness of a PDE system, incorporating both hysteresis effects and modelling phase change, which however does not display a degenerating character. Degenerating phase parameters appear in models for damaging phenomena, see [6], [7], [8]. In this case, the phase variable χ is related to the local proportion of damaged material.…”

Section: Introductionmentioning

confidence: 99%

“…For, in their case the coefficients of the elliptic operators in the stress-strain relation vanish only as χ ց 0, contrary to the twofold degeneracy of the equation (1.6 c), which we shall further comment later on. Within this framework, in [7], [8] local in time well-posedness results are proved for the resulting PDE system.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

Rocca

Rossi

2008

Appl Math

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

Rocca

Rossi

2008

Appl Math

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“…[BS04]. Therein, the Laplace operator (i.e., q = 2) was considered and the choice f = ∂I [0,1] , with I [0,1] the indicator function of [0, 1], was allowed for.…”

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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“…Thus, it seems very hard to obtain global well-posedness results for the above system. Similar difficulties arise in the investigations carried out, for example, in [2,3,4,5].…”

Section: Introductionmentioning

confidence: 72%

Asymptotic analysis for vanishing acceleration in a thermoviscoelastic system

Bonetti¹,

Bonfanti²

2005

Abstract and Applied Analysis

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We have investigated a dynamic thermoviscoelastic system (2003), establishing existence and uniqueness results for a related initial and boundary values problem. The aim of the present paper is to study the asymptotic behavior of the solution to the above problem as the power of the acceleration forces goes to zero. In particular, well-posedness and regularity results for the limit (quasistatic) problem are recovered.

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Local existence for Fr�mond?s model of damage in elastic materials

Cited by 68 publications

References 16 publications

Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

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

Asymptotic analysis for vanishing acceleration in a thermoviscoelastic system

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