Damage processes in thermoviscoelastic materials with damage‐dependent thermal expansion coefficients

Heinemann, Christian; Rocca, Elisabetta

doi:10.1002/mma.3393

Cited by 5 publications

(7 citation statements)

References 19 publications

(48 reference statements)

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“…Instead, the coefficient b in the elastic energy density (1.4) can possibly vanish, and both b and the eigenstrain ε * are required to be sufficiently regular functions; -the thermal expansion coefficient ρ is assumed to be a positive constant. For more general behavior of ρ possibly depending on the damage parameter z the reader can refer to [24], while the fact that ρ is chosen to be independent of ϑ is justified by the fact that we assume to have a constant specific heat c v (equal to 1 in (1.3d) for simplicity): indeed they are related (by thermodynamical laws) by the relation ∂ ϑ c v = ϑ∂ ϑ ρ; -the initial data are taken in the energy space, except for the initial displacement and velocity which, jointly with the boundary Dirichlet datum for u, must enjoy the regularity needed in order to perform elliptic regularity estimates on the momentum balance (1.3e).…”

Section: Constitutive Relationsmentioning

confidence: 99%

“…Regarding the previous results on this type of problems in the literature, let us point out that, by now, several contributions on systems coupling rate-dependent damage and thermal processes (cf., e.g. [4,37,38,24]) as well as rate-dependent damage and phase separation (cf., e.g., [21,22]) are available in the literature. Up to our knowledge, this is one of the first contributions on the analysis of a model encompassing all of the three processes (temperature evolution, damage, phase separation) in a thermoviscoelastic material.…”

Section: Constitutive Relationsmentioning

confidence: 99%

“…We will use an iteration argument as in [38,24] (see also [2]) and sketch the proof for the case d = 3, since the calculations for d = 2 are completely analogous. We already know that µ k ∈ H 1 (Ω) satisfies the elliptic equation [24]), with the exception that one needs to take the Dirichlet data d k ∈ H 2 (Ω; R d ) into account. This is the very point where we need to assume that V = ωC for some ω > 0 (cf.…”

Section: 24mentioning

confidence: 99%

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A Temperature-Dependent Phase-Field Model for Phase Separation and Damage

Heinemann

Kraus

Rocca

et al. 2017

Arch Rational Mech Anal

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Abstract. In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature (cf., e.g., [21,22]), we encompass in the model thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More in particular, we prove the existence of "entropic weak solutions", resorting to a solvability concept first introduced in [10] in the framework of Fourier-Navier-Stokes systems and then recently employed in [9,38] for the study of PDE systems for phase transition and damage. Our global-in-time existence result is obtained by passing to the limit in a carefully devised time-discretization scheme.

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Section: Constitutive Relationsmentioning

confidence: 99%

Section: Constitutive Relationsmentioning

confidence: 99%

Section: 24mentioning

confidence: 99%

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A Temperature-Dependent Phase-Field Model for Phase Separation and Damage

Heinemann

Kraus

Rocca

et al. 2017

Arch Rational Mech Anal

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“…The case of a χ -dependent coefficient has been treated for similar PDE systems, e.g. in [5], where local-in-time results were obtained, and more recently in [26] where the existence of global-in-time weak solutions has been proved, but under the small perturbation assumption. Nonetheless, let us mention that, especially in case of phase transition phenomena, the choice of a constant ρ is quite reasonable (cf., e.g., [32] for further comments on this topic).…”

Section: Introductionmentioning

confidence: 99%

``Entropic” Solutions to a Thermodynamically Consistent PDE System for Phase Transitions and Damage

Rocca¹,

Rossi²

2015

SIAM J. Math. Anal.

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126

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In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L 1 . The whole system has a highly nonlinear character.We address the existence of a weak notion of solution, referred to as "entropic", where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics, as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data.We prove our results by passing to the limit in a time-discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its "entropic" formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.for all t ∈ (0, T ] and almost all s ∈ (0, t), with ξ a selection in the (convex analysis) subdifferential ∂ β( χ ) = ∂I [0,+∞) ( χ ) of I [0,+∞) . In [47, Prop. 2.14] (see also [24]), we prove that, under additional regularity properties any weak solution in fact fulfills (1.3) pointwise.Let us also mention that other approaches to the weak solvability of coupled PDE systems with an L 1 -righthand side are available in the literature: in particular, we refer here to [54] and [49]. In [54], the notion of renormalized solution has been used in order to prove a global-in-time existence result for a nonlinear system in thermoviscoelasticity. In [49] the focus is on rate-independent processes coupled with viscosity and inertia in the displacement equation, and with the temperature equation. There the internal variable equation is not of gradient-flow type as (1.3), but instead features a 1-positively homogeneous dissipation potential. For the resulting PDE system, a weak solution concept partially mutuated from the theory of rate-independent processes by A. Mielke (cf., e.g., [39]) is analyzed. An existence result is proved combining techniques for rate-independent evolution, with Boccardo-Gallouët type estimates of the temperature gradient in the heat equation with L 1 -right-hand side. Our existence results. The main results of this paper, Theorems 1 and 2, state the existence of entropic solutions for system (1.1-1.3), supplemented with the boundary conditions (1.4) (cf. Remark 2.12), in the irreversible (µ = 1) and reversible (µ = 0) cases.More precisely, in the case of unidirectional evolution for χ we can prove the existence of a global-intime entropic solution (i.e. satisfying the entropy (1.6) and the total energy (1.7) inequalities, the (pointwise) momentum balance (1.2), the one-sided variational inequality (1.9) and the energy (1.10) inequalities for...

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“…-Coupled thermoviscoelastic and isothermal damage models incorporating p-Laplacian operators are analyzed in [39] (see also [40] for the full heat equation including all dissipative terms and [20] for damage-dependent heat expansion coefficients). In those works homogeneous Dirichlet boundary conditions for the displacements are assumed.…”

Section: Introductionmentioning

confidence: 99%

A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media

Farshbaf-Shaker

Heinemann

2015

Math. Models Methods Appl. Sci.

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In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with non-homogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model. To this end, we prove existence of minimizers and study a family of "local" approximations via adapted cost functionals.

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Damage processes in thermoviscoelastic materials with damage‐dependent thermal expansion coefficients

Cited by 5 publications

References 19 publications

A Temperature-Dependent Phase-Field Model for Phase Separation and Damage

A Temperature-Dependent Phase-Field Model for Phase Separation and Damage

``Entropic” Solutions to a Thermodynamically Consistent PDE System for Phase Transitions and Damage

A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media

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