On a Cahn-Hilliard model for phase separation with elastic misfit

Garcke, Harald

doi:10.1016/j.anihpc.2004.07.001

Garcke, Harald

Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 2, pp. 165-185.

Zbl

DOI : 10.1016/j.anihpc.2004.07.001

@article{AIHPC_2005__22_2_165_0,
     author = {Garcke, Harald},
     title = {On a {Cahn-Hilliard} model for phase separation with elastic misfit},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {165--185},
     publisher = {Elsevier},
     volume = {22},
     number = {2},
     year = {2005},
     doi = {10.1016/j.anihpc.2004.07.001},
     zbl = {1072.35081},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2004.07.001/}
}

TY  - JOUR
AU  - Garcke, Harald
TI  - On a Cahn-Hilliard model for phase separation with elastic misfit
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2005
SP  - 165
EP  - 185
VL  - 22
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2004.07.001/
DO  - 10.1016/j.anihpc.2004.07.001
LA  - en
ID  - AIHPC_2005__22_2_165_0
ER  -

%0 Journal Article
%A Garcke, Harald
%T On a Cahn-Hilliard model for phase separation with elastic misfit
%J Annales de l'I.H.P. Analyse non linéaire
%D 2005
%P 165-185
%V 22
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2004.07.001/
%R 10.1016/j.anihpc.2004.07.001
%G en
%F AIHPC_2005__22_2_165_0

Garcke, Harald. On a Cahn-Hilliard model for phase separation with elastic misfit. Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 2, pp. 165-185. doi : 10.1016/j.anihpc.2004.07.001. http://www.numdam.org/articles/10.1016/j.anihpc.2004.07.001/

Bibliographie
Cité par

[1] Barrett J.W., Blowey J.F., An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy, Numer. Math. 72 (1995) 257-287. | MR | Zbl

[2] Barrett J.W., Blowey J.F., An error bound for the finite element approximation of a model for phase separation of a multi-component alloy, IMA J. Numer. Anal. 16 (1996) 257-287. | MR | Zbl

[3] Bonetti E., Colli P., Dreyer W., Gilardi G., Schimperna G., Sprekels J., A solid-solid phase change model accounting for mechanical effects, Physica D 165 (2002) 48-65. | MR | Zbl

[4] Cahn J.W., Hilliard J.E., Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958) 258-267.

[5] Carrive M., Miranville A., Piétrus A., The Cahn-Hilliard equation for deformable elastic media, Adv. Math. Sci. Appl. 10 (2000) 539-569. | MR | Zbl

[6] De Fontaine D., An analysis of clustering and ordering in multicomponent solid solutions - I. Stability criteria, J. Phys. Chem. Solids 33 (1972) 287-310.

[7] C.M. Elliott, The Cahn-Hilliard model for the kinetics of phase transitions, in: J.F. Rodrigues (ed.), Mathematical Models for Phase Change Problems, Internat. Ser. Numer. Math., vol. 88, Birkhäuser, Basel, pp. 35-73. | MR | Zbl

[8] C.M. Elliott, S. Luckhaus, A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy, SFB256 University Bonn, Preprint 195, 1991.

[9] Eshelby J.D., Elastic inclusions and inhomogeneities, Prog. Solid Mech. 2 (1961) 89-140. | MR

[10] Fratzl P., Penrose O., Lebowitz J.L., Modelling of phase separation in alloys with coherent elastic misfit, J. Statist. Phys. 95 (5/6) (1999) 1429-1503. | MR | Zbl

[11] H. Garcke, On mathematical models for phase separation in elastically stressed solids, habilitation thesis, 2000.

[12] Garcke H., On Cahn-Hilliard sytems with elasticity, Proc. Roy. Soc. Edinburgh. Ser. A 133 (2003) 307-331. | MR | Zbl

[13] Garcke H., Rumpf M., Weikard U., The Cahn-Hilliard equation with elasticity: finite element approximation and qualitative studies, Interfaces and Free Boundaries 3 (2001) 101-118. | MR | Zbl

[14] Gehring F.W., The $L^{p}$ -integrability of the partial derivatives of a quasi conformal mapping, Acta Math. 130 (1973) 265-277. | MR | Zbl

[15] Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud., Princeton University Press, 1983. | MR | Zbl

[16] Giaquinta M., Modica G., Regularity results for some classes of higher order nonlinear elliptic systems, J. Reine Angew. Math. 311/312 (1979) 145-169. | MR | Zbl

[17] Hoyt J.J., The continuum theory of nucleation in multicomponent systems, Acta Metall. 38 (1990) 1405-1412.

[18] Khachaturyan A.G., Some questions concerning the theory of phase transitions in solids, Fiz. Tverd. Tela 8 (1966) 2709-2717, English translation in, Sov. Phys. Solid. State 8 (1966) 2163.

[19] Kirkaldy J.S., Young D.J., Diffusion in the Condensed State, The Institute of Metals, London, 1987.

[20] Larché F.C., Cahn J.W., Thermochemical equilibrium of multiphase solids under stress, Acta Metall. 26 (1978) 1579-1589.

[21] Larché F.C., Cahn J.W., The effect of self-stress on diffusion in solids, Acta Metall. 30 (1982) 1835-1845.

[22] Leo P.H., Lowengrub J.S., Jou H.J., A diffuse interface model for microstructural evolution in elastically stressed solids, Acta Mater. 46 (1998) 2113-2130.

[23] Novick-Cohen A., The Cahn-Hilliard equation: mathematical and modelling perspectives, Adv. Math. Sci. Appl. 8 (1998) 965-985. | MR | Zbl

[24] Onsager L., Reciprocal relations in irreversible processes I, Phys. Rev. 37 (1931) 405-426. | Zbl

[25] Onsager L., Reciprocal relations in irreversible processes II, Phys. Rev. 38 (1931) 2265-2279. | Zbl

[26] Onuki A., Ginzburg-Landau approach to elastic effects in the phase separation of solids, J. Phys. Soc. Jpn. 58 (1989) 3065-3068.

[27] J.D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, Verhaendel. Kronik. Akad. Weten. Amsterdam, vol. 1 (1893) (in Dutch).

[28] Zeidler E., Nonlinear Functional Analysis and its Applications IV, Springer, New York, 1988. | MR | Zbl

Cité par Sources :