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Existence results for diffuse interface models describing phase separation and damage*

Published online by Cambridge University Press:  09 November 2012

CHRISTIAN HEINEMANN  and
CHRISTIANE KRAUS
CHRISTIAN HEINEMANN
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany emails: christian.heinemann@wias-berlin.de, christiane.kraus@wias-berlin.de
CHRISTIANE KRAUS
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany emails: christian.heinemann@wias-berlin.de, christiane.kraus@wias-berlin.de
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Abstract

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In this paper, we analytically investigate multi-component Cahn–Hilliard and Allen–Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form ∫Ω½Γ∇c : ∇c + ½|∇z|2+Wch(c)+Wel(e,c,z)dx with a polynomial or logarithmic chemical energy density Wch, an inhomogeneous elastic energy density Wel and a quadratic structure of the gradient of damage variable z. For the corresponding elastic Cahn–Hilliard and Allen–Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for strain e.

Keywords

Cahn–Hilliard systemsAllen–Cahn systemsPhase separationDamageElliptic-parabolic systemsEnergetic solutionWeak solutionDoubly nonlinear differential inclusionsExistence resultsRate-dependent systemsLogarithmic-free energy
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 2 , April 2013 , pp. 179 - 211
Copyright
Copyright © Cambridge University Press 2012

Footnotes

*

This project is partially supported by the DFG project A 3 ‘Modeling and sharp interface limits of local and non-local generalized Navier–Stokes–Korteweg Systems.’

References

[1]Allen, S. M. & Cahn, J. W. (1979) A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metal. 27, 1085–1095.CrossRefGoogle Scholar
[2]Barrett, J. W. & Blowey, J. F. (1999) Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility. Math. Comput. 68 (226), 487–517.Google Scholar
[3]Bartels, S. & Müller, R. (2011) Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential. Numer. Math. 119 (3), 409–435.Google Scholar
[4]Bartkowiak, L. & Pawlow, I. (2005) The Cahn-Hilliard-Gurtin system coupled with elasticity. Control Cybern. 34, 1005–1043.Google Scholar
[5]Blesgen, T. & Weikard, U. (2005) Multi-component Allen-Cahn equation for elastically stressed solids. Electron. J. Differ. Equ. 89, 1–17.Google Scholar
[6]Boldrini, J. L. & da Silva, P. N. (2004) A generalized solution to a Cahn-Hilliard/Allen-Cahn system. Electron. J. Differ. Equ. 126, 1–24.Google Scholar
[7]Bonetti, E., Colli, P., Dreyer, W., Gilardi, G., Schimperna, G. & Sprekels, J. (2002) On a model for phase separation in binary alloys driven by mechanical effects. Physica D 165, 48–65.Google Scholar
[8]Bonetti, E., Schimperna, G. & Segatti, A. (2005) On a doubly nonlinear model for the evolution of damaging in viscoelastic materials. J. Diff. Equ. 218 (1), 91–116.Google Scholar
[9]Cahn, J. W. (1961) On spinodal decomposition. Acta Metal. 9, 795–801.Google Scholar
[10]Cahn, J. W. & Novick-Cohen, A. (1994) Evolution equations for phase separation and ordering in binary alloys. J. Stat. Phys. 76 (3–4), 877–909.Google Scholar
[11]Carrive, M., Miranville, A. & Piétrus, A. (2000) The Cahn-Hilliard equation for deformable elastic continua. Adv. Math. Sci. Appl. 10 (2), 539–569.Google Scholar
[12]Cherfils, P. & Pierre, M. (2008) Non-global existence for an Allen-Cahn-Gurtin equation with logarithmic free energy. J. Evol. Equ. 8 (4), 727–748.CrossRefGoogle Scholar
[13]Colli, P., Gilardi, G., Podio-Guidugli, P. & Sprekels, J. (2010) Existence and uniqueness of a global-in-time solution to a phase segregation problem of the Allen-Cahn type. Math. Models Methods Appl. Sci. 20 (4), 519–541.Google Scholar
[14]Dreyer, W. & Mueller, W. H. (2000) A study of the coarsening in tin/lead solders. Int. J. Solids Struct. 37 (28), 3841–3871.CrossRefGoogle Scholar
[15]Efendiev, M. A. & Mielke, A. (2006) On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13, 151–167.Google Scholar
[16]Elliott, C. M. & Luckhaus, S. (1991) A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy. IMA Preprint Series 887, 1–37.Google Scholar
[17]Fernández, J. R. & Kuttler, K. L. (2009) An existence and uniqueness result for an elasto-piezoelectric problem with damage. Math. Mod. Meth. Appl. Sci. 19 (1), 31–50.Google Scholar
[18]Fremond, M. (2002) Non-Smooth Thermomechanics, Springer, Berlin, Germany.Google Scholar
[19]Frémond, M. & Nedjar, B. (1996) Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct. 33 (8), 1083–1103.CrossRefGoogle Scholar
[20]Garcke, H. (2000) On Mathematical Models for Phase Separation in Elastically Stressed Solids, Habilitation thesis, University of Bonn, Bonn, Germany.Google Scholar
[21]Garcke, H. (2005) Mechanical effects in the Cahn-Hilliard model: A review on mathematical results. In: Miranville, A. (editor), Mathematical Methods and Models in Phase Transitions, Nova Science, New York, pp. 43–77.Google Scholar
[22]Garcke, H. (2005) On a Cahn-Hilliard model for phase separation with elastic misfit. Annales de l'Institut Henri Poincare (C) Non-Linear Anal. 22 (2), 165–185.Google Scholar
[23]Garcke, H., Rumpf, M. & Weikard, U. (2001) The Cahn-Hilliard equation with elasticity: Finite element approximation and qualitative studies. Interfaces Free Bound. 3, 101–118.CrossRefGoogle Scholar
[24]Giaquinta, M. (1983) Multiple Integrals in the Calcula of Variations and Nonlinear Elliptic Systems, Annals of Mathematical Studies, Princeton University Press, Princeton, NJ.Google Scholar
[25]Gurtin, M. E. (1996) Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92, 178–192.Google Scholar
[26]Harris, P. G., Chaggar, K. S. & Whitmore, M. A. (1991) The effect of ageing on the microstructure of 60:40 tin–lead solders. Solder. Surf. Mount Technol. Improved Phys. Underst. Intermittent Failure Continuous 3, 20–33.CrossRefGoogle Scholar
[27]Heinemann, C. & Kraus, C. (2011) Existence of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2), 321–359.Google Scholar
[28]Larché, F. C. & Cahn, J. W. (1982) The effect of self-stress on diffusion in solids. Acta Metal. 30, 1835–1845.Google Scholar
[29]Mielke, A. (2005) Evolution in rate-independent systems. Handbook Differ. Equ Evolutionary Equ. 2, 461–559.CrossRefGoogle Scholar
[30]Mielke, A. & Roubícek, T. (2006) Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16, 177–209.Google Scholar
[31]Mielke, A., Roubícek, T. & Zeman, J. (2010) Complete damage in elastic and viscoelastic media. Comput. Methods Appl. Mech. Eng. 199, 1242–1253.Google Scholar
[32]Mielke, A. & Thomas, M. (2010) Damage of nonlinearly elastic materials at small strain – Existence and regularity results. ZAMM Z. Angew. Math. Mech. 90, 88–112.Google Scholar
[33]Nirenberg, L. (1959) On elliptic differential equations. Ann. Scuola Norm. Pisa (III) 13, 1–48.Google Scholar
[34]Simon, J. (1986) Compact sets in the space L p(0,T;B). Annali di Mat. Pura ed Appl. 146, 65–96.Google Scholar
[35]Sobolev, S. L. (1938) On a theorem of functional analysis. Mat. Sbornik 46, 471–497.Google Scholar