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The Cahn-Hilliard-Gurtin system coupled with elasticity

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The paper concerns the existence of weak solutions to the Cahn-Hilliard-Gurtin system coupled with nonstationary elasticity. The system describes phase separation process in elastically stressed material. It generalizes the Cahu-Hilliard equation by admitting a more general structure and by coupling diffusive and elastic effects. The system is studied with the help of a singularly perturbed problem which has the form of a well-known phase field model coupled with elasticity. The established existence results are restricted to the homogeneous problem with gradient, energy tensor and elasticity tensor independent of the order parameter.
Rocznik
Strony
1005--1043
Opis fizyczny
Bibliogr. 36 poz.
Twórcy
  • Extramural doctoral course Systems Research Institute, Polish Academy of Sciences Newelska 6, 01-447 Warsaw, Poland
autor
  • Systems Research Institute, Polish Academy of Sciences Newelska 6, 01-447 Warsaw, Poland, Institute of Mathematics and Operations Research Military University of Technology S. Kaliskiego 2, 00-908 Warsaw, Poland
Bibliografia
  • Bonetti, E., Colli, P., Dreyer, W., Giliardi, G., Schimperna, G. and Sprekels, J. (2002) On a model for phase separation in binary alloys driven by mechanical effects. Physica D 165, 48-65.
  • Bonetti, E., Dreyer, W. and Schimperna, G. (2003) Global solutions to a generalized Cahn-Hilliard equation with viscosity. Advances in Differential Equations 8, 231-256.
  • Carrive, M., Miranville, A. and Pietrus, A. (2000) The Cahn-Hilliard equation for deformable elastic continua. Advances in Mathematical Sciences and Applications 10 (2), 539-569.
  • Carrive, M., Miranville, A., Pietrus and A., Rakotoson, J.M. (1999) The Cahn-Hilliard equation for an isotropic deformable continuum. Applied Mathematics Letters 12, 23-28.
  • Dreyer, W. and Muller, W.H. (2000) A study of the coarsening in tin/lead solders. International Journal of Solids and Structures 37, 3841-3871.
  • Dreyer, W. and Muller, W.H. (2001) Modelling diffusional coarsening in eutetic tin/lead solders: a quantitative approach. International Journal of Solids and Structures 38, 1433-1458.
  • Duvaut, G. and Lions, J.L. (1972) Le Inequations en Mecanique et en Physique. Dunod, Paris.
  • Fratzl, P., Penrose, O. and Lebowitz, J.L. (1994) Dynamic solid-solid transitions with phase characterized by an order parameter. Physica D 72, 287-308.
  • Fratzl, P., Penrose, O. and Lebowitz, J.L. (1999) Modelling of phase separation in alloys with coherent elastic misfit. Journal of Statistical Physics 95 (5/6), 1429-1503.
  • Fried, E. and Gurtin, M. E. (1999) Coherent solid-state phase transitions with atomic diffusion: A thermodynamical treatment. Journal of Statistical Physics 95 (5/6), 1361-1427.
  • Garcke, H. (2000) On mathematical models for phase separation in elastically stressed solids. Habilitation Thesis, University of Bonn.
  • Garcke, H. (2003) On Cahn-Hilliard systems with elasticity. Proc. Roy. Soc. Edinburgh 133A, 307-331.
  • Garcke, H. (2005) On a Cahn-Hilliard model for phase separation with elastic misfit. Inst. H. Poincare Anal. Non Lineaire 22, 165-185.
  • Garcke, H., Rumpf, M. and Weikard, U., (2001) The Cahn-Hilliard equation with elasticity - finite element approximation and qualitative studies. Interfaces and Free Boundaries 3, 101-118.
  • Gurtin, M.E. (1996) Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92, 178-192.
  • Kroner, D. and Zajączkowski, W. M. (1996) Measure-valued solutions of the Euler equations for ideal compressible polytropic fluids. Mathematical Methods in the Applied Sciences 19, 235-252.
  • Larche, F.C. and Cahn, J.W. (1982) The effect of self-stress on diffusion in solids. Acta Metall. 30, 1835-1845.
  • Larche, F.C. and Cahn, J.W. (1985) The interactions of composition and stress in crystalline solids. Acta Metall. 33 (3), 331-357.
  • Larche, F.C. and Cahn, J.W. (1992) Phase changes in a thin plate with non-local self-stress effects. Acta Metall. 40 (5), 947-955.
  • Laurenc¸ot, Ph. (1994) Degenerate Cahn-Hilliard equation as limit of the phase-field equation with non-constant thermal conductivity. In: Niezgódka, M., Strzelecki, P., eds., Free Boundary Problems, Theory and Applications. Pitman Research Notes in Mathematics Series 363, Longman, 135-144.
  • Leo, P.H., Lowengrub, J.S. and Jou, H.J. (1998) A diffuse interface model for microstructural evolution in elastically stressed solids. Acta Mater. 46 (6), 2113-2130.
  • Lions, J.L. (1969) Quelques methodes de resolution des problemes aux limites non lineaires. Dunod, Paris.
  • Miranville, A., Pietrus, A. and Rakotoson, J.M. (1998) Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance. Asymptotic Analysis 16, 315-345.
  • Miranville, A. (1999) A model of Cahn-Hilliard equation based on a microforce balance. C.R. Acad. Sci. Paris 328 S´erie I, 1247-1252.
  • Miranville, A. (2000) Some generalizations of the Cahn-Hilliard equation. Asymptotic Analysis 22, 235-259.
  • Miranville, A. (2001a) Long-time behavior of some models of Cahn-Hilliard equations in deformable continua. Nonlinear Analysis: Real World Applications 2, 273-304.
  • Miranville, A. (2001b) Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions. Phys. D 158 (1-4), 233-257.
  • Miranville, A. (2003) Generalized Cahn-Hilliard equations based on a microforce balance. Journal of Applied Mathematics (4), 165-185.
  • Miranville, A. and Schimperna, G. (2005) Nonisothermal phase separation based on microforce balance. Discrete Cont. Dyn. Systems, Series A (to appear).
  • Muller, I. (1985) Thermodynamics. Pitman, London.
  • Neustupa, J. (1993) Measure-valued solutions of the Euler and Navier-Stokes equations for compressible barotropic fluids. Math. Nachr. 163, 217-227.
  • Onuki, A. (1989) Ginzburg-Landau approach to elastic effects in the phase separation of solids. Journal of the Physics Society of Japan 58 (9), 3065-3068.
  • Pawłow, I. (2005) The thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids. Discrete Cont. Dyn. Systems Series A (to appear).
  • Pawłow, I. and Zajączkowski, W.M. (2006) Measure-valued solutions of heterogeneous Cahn-Hilliard system in elastic solids. Research Report IBS PAN.
  • Simon, J. (1987) Compact sets in the space Lp(0, T ;B). Annali di Mathematica Pura et Applicata 146, 65-97.
  • Stoth, B.E. (1995) The Cahn-Hilliard equation as a degenerate limit of the phase-field equations. Quarterly of Applied Mathematics, LIII (4), 695-700.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BAT5-0010-0013
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