Well-posedness results for a model of damage in thermoviscoelastic materials

Bonetti, Elena; Bonfanti, Giovanna

doi:10.1016/j.anihpc.2007.05.009

Cited by 36 publications

(50 citation statements)

References 13 publications

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“…The papers [5,6,8,23] instead address models for damaging phenomena. In this case, the phase variable χ is related to the local proportion of damaged material.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials

Rocca

Rossi

2008

Journal of Differential Equations

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We address the analysis of a nonlinear and degenerating PDE system, proposed by M. Frémond for modelling phase transitions in viscoelastic materials subject to thermal effects. The system features an internal energy balance equation, governing the evolution of the absolute temperature ϑ, an evolution equation for the phase change parameter χ , and a stress-strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled.In this paper, we first prove a local (in time) well-posedness result for (a suitable initial-boundary value problem for) the above mentioned PDE system, in the (spatially) three-dimensional setting. Secondly, we restrict to the one-dimensional case, in which, for the same initial-boundary value problem, we indeed obtain a global well-posedness theorem.

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“…The papers [5,6,8,23] instead address models for damaging phenomena. In this case, the phase variable χ is related to the local proportion of damaged material.…”

Section: Introductionmentioning

confidence: 99%

“…Among others, we would like to cite the papers [3][4][5][6]8,16,23]. The analysis of a thermoviscoelastic system not subject to a phase transition has been tackled in [3,4], in which a linear viscoelastic equation for the displacement u and an internal energy balance equation for ϑ are considered.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials

Rocca

Rossi

2008

Journal of Differential Equations

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“…This allows them to estimate the term A 2 z and to gain enhanced spatial regularity for z, again by elliptic regularity. Refined estimates combined with regularity assumptions on the domain Ω are crucial also in [BB08], extending the analysis to a temperature-dependent model.…”

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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“…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%

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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Well-posedness results for a model of damage in thermoviscoelastic materials

Cited by 36 publications

References 13 publications

Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials

Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials

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

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

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