Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-01T07:15:48.325Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  09 November 2012

CHRISTIAN HEINEMANN
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany emails: [email protected], [email protected]
CHRISTIANE KRAUS
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany emails: [email protected], [email protected]

Abstract

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.

Type
Papers
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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, 10851095.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), 487517.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), 409435.Google Scholar
[4]Bartkowiak, L. & Pawlow, I. (2005) The Cahn-Hilliard-Gurtin system coupled with elasticity. Control Cybern. 34, 10051043.Google Scholar
[5]Blesgen, T. & Weikard, U. (2005) Multi-component Allen-Cahn equation for elastically stressed solids. Electron. J. Differ. Equ. 89, 117.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, 124.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, 4865.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), 91116.Google Scholar
[9]Cahn, J. W. (1961) On spinodal decomposition. Acta Metal. 9, 795801.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), 877909.Google Scholar
[11]Carrive, M., Miranville, A. & Piétrus, A. (2000) The Cahn-Hilliard equation for deformable elastic continua. Adv. Math. Sci. Appl. 10 (2), 539569.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), 727748.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), 519541.Google Scholar
[14]Dreyer, W. & Mueller, W. H. (2000) A study of the coarsening in tin/lead solders. Int. J. Solids Struct. 37 (28), 38413871.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, 151167.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, 137.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), 3150.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), 10831103.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. 4377.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), 165185.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, 101118.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, 178192.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, 2033.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), 321359.Google Scholar
[28]Larché, F. C. & Cahn, J. W. (1982) The effect of self-stress on diffusion in solids. Acta Metal. 30, 18351845.Google Scholar
[29]Mielke, A. (2005) Evolution in rate-independent systems. Handbook Differ. Equ Evolutionary Equ. 2, 461559.CrossRefGoogle Scholar
[30]Mielke, A. & Roubícek, T. (2006) Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16, 177209.Google Scholar
[31]Mielke, A., Roubícek, T. & Zeman, J. (2010) Complete damage in elastic and viscoelastic media. Comput. Methods Appl. Mech. Eng. 199, 12421253.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, 88112.Google Scholar
[33]Nirenberg, L. (1959) On elliptic differential equations. Ann. Scuola Norm. Pisa (III) 13, 148.Google Scholar
[34]Simon, J. (1986) Compact sets in the space L p(0,T;B). Annali di Mat. Pura ed Appl. 146, 6596.Google Scholar
[35]Sobolev, S. L. (1938) On a theorem of functional analysis. Mat. Sbornik 46, 471497.Google Scholar