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Local well-posedness for Frémond’s model of complete damage in elastic solids

Part of: Parabolic equations and systems Equations and systems of special type Equations of mathematical physics and other areas of application Fracture and damage

Published online by Cambridge University Press:  01 March 2021

GORO AKAGI Open the ORCID record for GORO AKAGI [Opens in a new window]  and
GIULIO SCHIMPERNA Open the ORCID record for GIULIO SCHIMPERNA [Opens in a new window]
GORO AKAGI
Affiliation:
Mathematical Institute and Graduate School of Science, Tohoku University 6-3 Aoba, Aramaki, Aoba-ku, Sendai 980-8578 Japan e-mail: goro.akagi@tohoku.ac.jp
GIULIO SCHIMPERNA
Affiliation:
Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, I-27100 Pavia, Italy e-mail: giusch04@unipv.it
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Abstract

We consider a model for the evolution of damage in elastic materials originally proposed by Michel Frémond. For the corresponding PDE system, we prove existence and uniqueness of a local in time strong solution. The main novelty of our result stands in the fact that, differently from previous contributions, we assume no occurrence of any type of regularising terms.

Keywords

Degenerate elliptic–parabolic systemdamage phenomenalocal existencea priori estimates

MSC classification

Primary: 35K55: Nonlinear parabolic equations 35Q74: PDEs in connection with mechanics of deformable solids
Secondary: 35K86: Nonlinear parabolic unilateral problems and nonlinear parabolic variational inequalities 35M33: Initial-boundary value problems for systems of mixed type 74R05: Brittle damage

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 2 , April 2022 , pp. 309 - 327
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Arai, T. (1979) On the existence of the solution for \[\partial \varphi (u\prime (t)) + \partial \psi (u(t)) \ni f(t)\]. J. Fac. Sci. Univ. Tokyo Sec. IA Math. 26, 75–96.Google Scholar
Barbu, V. (1976) Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leiden,CrossRefGoogle Scholar
Bonetti, E. Bonfanti, G. & Rossi, R. (2014) Analysis of a temperature-dependent model for adhesive contact with friction. Phys. D 285, 42–62.CrossRefGoogle Scholar
Bonetti, E. Freddi, L. & Segatti, A. (2017) An existence result for a model of complete damage in elastic materials with reversible evolution. Contin. Mech. Thermodyn. 29, 31–50.CrossRefGoogle Scholar
Bonetti, E. & Schimperna, G. (2004) Local existence for Fremond’s model of damage in elastic materials. Contin. Mech. Thermodyn. 16, 319–335.CrossRefGoogle Scholar
Bonetti, E. Schimperna, G. & Segatti, A. (2005) On a doubly non linear model for the evolution of damaging in viscoelastic materials. J. Differ. Equ. 218, 91–116.CrossRefGoogle Scholar
Bouchitté, G., Mielke, A. & Roubček, T. (2009) A complete-damage problem at small strains. Z. Angew. Math. Phys. 60, 205–236.CrossRefGoogle Scholar
Brézis, H. (1973) Opérateurs Maximaux Monotones et Sémi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Mathematics Studies, Vol. 5, North-Holland, Amsterdam.Google Scholar
Fiaschi, A. Knees, D. & Stefanelli, U. (2012) Young-measure quasi-static damage evolution. Arch. Ration. Mech. Anal. 203, 415–453.CrossRefGoogle Scholar
Frémond, M. (2002) Non-smooth Thermomechanics, Springer, Berlin.CrossRefGoogle Scholar
Frémond, M. (2012) Phase Change in Mechanics, Springer-Verlag, Berlin, Heidelberg.CrossRefGoogle Scholar
Frémond, M. Kuttler, K. L. Nedjar, B. & Shillor, M. (1998) One-dimensional models of damage. Adv. Math. Sci. Appl. 8, 541–570.Google Scholar
Frémond, M. Kuttler, K. L. & Shillor, M. (1999) Existence and uniqueness of solutions for a dynamic one-dimensional damage model. J. Math. Anal. Appl. 229, 271–294.CrossRefGoogle Scholar
Frémond, M. & Nedjar, B. (1993) Damage and principle of virtual power. Comptes Rendus de l’Academie des Sciences, Serie II 317, 857–864.Google Scholar
Frémond, M. & Nedjar, B. (1996) Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct. 33, 1083–1103.CrossRefGoogle Scholar
Gasiński, L. & Ochal, A. (2015) Dynamic thermoviscoelastic problem with friction and damage. Nonlinear Anal. Real World Appl. 21, 63–75.CrossRefGoogle Scholar
Heinemann, C. & Kraus, C. (2015) Complete damage in linear elastic materials: modeling, weak formulation and existence results. Calc. Var. Partial Differ. Equ. 54, 217–250.CrossRefGoogle Scholar
Heinemann, C. & Kraus, C. (2015) Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects. Discrete Contin. Dyn. Syst. 35, 2565–2590.CrossRefGoogle Scholar
Heinemann, C. Kraus, C. Rocca, E. & Rossi, R. (2017) A temperature-dependent phase-field model for phase separation and damage. Arch. Ration. Mech. Anal. 225, 177–247.CrossRefGoogle Scholar
Knees, D., Rossi, R. & Zanini, C. (2013) A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23, 565–616.CrossRefGoogle Scholar
Lemaitre, J. (1992) A Course on Damage Mechanics, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Mielke, A. (2011) Complete-damage evolution based on energies and stresses. Discrete Contin. Dyn. Syst. Ser. S 4, 423–439.Google Scholar
Mielke, A. & Roubíček, T. (2006) Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16, 177–209.CrossRefGoogle Scholar
Nedjar, B. (2002) A theoretical and computational setting for a geometrically nonlinear gradient damage modelling framework. Comput. Mech. 30, 65–80.CrossRefGoogle Scholar
Nedjar, B. (2016) On a concept of directional damage gradient in transversely isotropic materials. Int. J. Solids Struct. 88–89, 56–67.CrossRefGoogle Scholar
Nirenberg, L. (1958) On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13, 115–162.Google Scholar
Simon, J. (1987) Compact sets in the space \[{L^p}(0,T;B)\]. Ann. Mat. Pura Appl. (4) 146, 65–96.CrossRefGoogle Scholar
Schimperna, G. & Pawłow, I. (2013) On a class of Cahn-Hilliard models with nonlinear diffusion. SIAM J. Math. Anal. 45, 31–63.CrossRefGoogle Scholar