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January  2011, 15(1): 309-323. doi: 10.3934/dcdsb.2011.15.309

Traveling waves for models of phase transitions of solids driven by configurational forces

Shuichi Kawashima 1, and  Peicheng Zhu 2,

1. 

Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan
2. 

Basque Center for Applied Mathematics, Building 500, Bizkaia Technology Park, E-48160 Derio, Spain

Received  December 2009 Revised  April 2010 Published  October 2010

This article is concerned with the existence of traveling wave solutions, including standing waves, to some models based on configurational forces, describing respectively the diffusionless phase transitions of solid materials, e.g., Steel, and phase transitions due to interface motion by interface diffusion, e.g., Sintering. These models were proposed by Alber and Zhu in [3]. We consider both the order-parameter-conserved case and the non-conserved one, under suitable assumptions. Also we compare our results with the corresponding ones for the Allen-Cahn and the Cahn-Hilliard equations coupled with linear elasticity, which are models for diffusion-dominated phase transitions in elastic solids.
Keywords: Traveling waves, configurational forces., phase transitions, models.
Mathematics Subject Classification: Primary: 35D30; 35M13; Secondary: 74N2.
Citation: Shuichi Kawashima, Peicheng Zhu. Traveling waves for models of phase transitions of solids driven by configurational forces. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 309-323. doi: 10.3934/dcdsb.2011.15.309
References:
[1]

R. Abeyaratne and J. Knowles, On the driving traction acting on a surface of strain discontinuity in a continuum, J. Mech. Phys. Solids, 38 (1990), 345-360. doi: doi:10.1016/0022-5096(90)90003-M.

[2]

H. D. Alber, Evolving microstructure and homogenization, Continuum. Mech. Thermodyn., 12 (2000), 235-287. doi: doi:10.1007/s001610050137.

[3]

H. D. Alber and P. Zhu, Evolution of phase boundaries by configurational forces, Archive Rat. Mech. Anal., 185 (2007), 235-286. doi: doi:10.1007/s00205-007-0054-8.

[4]

H. D. Alber and P. Zhu, Solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces, SIAM J. Appl. Math., 66 (2006), 680-699. doi: doi:10.1137/050629951.

[5]

H. D. Alber and P. Zhu, Solutions to a model for interface motion by interface diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 923-955. doi: doi:10.1017/S0308210507000170.

[6]

H. D. Alber and P. Zhu, Interface motion by interface diffusion driven by bulk energy: Justification of a diffusive interface model, preprint, accepted for publication in Conti. Mech. Thermodyna, 2010.

[7]

S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Met., 27 (1979), 1084-1095. doi: doi:10.1016/0001-6160(79)90196-2.

[8]

D. Aronson and H. Weiberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: doi:10.1016/0001-8708(78)90130-5.

[9]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in "Perspectives in Nonlinear Partial Differential Equations," Contemp. Mathematics, 446, Amer. Math. Soc., Providence, RI, (2007), 101-123.

[10]

T. Blesgen and U. Weikard, Multi-component Allen-Cahn equation for elastically stressed solids, Electronic J. Diff. Eqs., 2005 (2005), 1-17.

[11]

J. Cahn, Free energy of a nonuniform system. II. Thermodynamic basis, J. Chem. Phys., 30 (1959), 1121-1124. doi: doi:10.1063/1.1730145.

[12]

J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1959), 258-267. doi: doi:10.1063/1.1744102.

[13]

J. Cahn and J. Hilliard, Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699. doi: doi:10.1063/1.1730447.

[14]

J. Cahn and J. Taylor, Surface motion by surface diffusion, Acta Metall. Mater., 42 (1994), 1045-1063. doi: doi:10.1016/0956-7151(94)90123-6.

[15]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Diff. Eq., 96 (1992), 116-141.

[16]

P. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Springer Verlag, 1979.

[17]

P. Fife and J. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rati. Mech. Anal., 65 (1977), 335-361. doi: doi:10.1007/BF00250432.

[18]

M. Gurtin, "Configurational Forces as Basic Concepts of Continuum Physics," Springer Verlag, New York, 2000.

[19]

H. Garcke, On the Cahn-Hilliard system with elasticity, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 307-331. doi: doi:10.1017/S0308210500002419.

[20]

E. Hornbogen and H. Warlimont, "Metallkunde," 4th edition, Springer-Verlag, 2001.

[21]

Y. Kanel, On the stabilization of solutions of the Cauchy problem for equations arising in the theory of combustion, Mat. Sbornik, 59 (1962), 245-288.

[22]

W. Mullins, Theory of thermal grooving, J. Appl. Phys., 28 (1957), 333-339. doi: doi:10.1063/1.1722742.

[23]

H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Diff. Eq., 213 (2005), 204-233.

[24]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Disc. Conti. Dyna. Syst., 15 (2006), 819-832. doi: doi:10.3934/dcds.2006.15.819.

[25]

R. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. R. Soc. Lond. Sect. A, 422 (1989), 261-278. doi: doi:10.1098/rspa.1989.0027.

[26]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer Verlag, New York, 1983.

[27]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: doi:10.1137/060661788.

[28]

J. Taylor and J. Cahn, Linking anisotropic sharp and diffuse surface motion laws via gradient flows, J. Stat. Phys., 77 (1994), 183-197. doi: doi:10.1007/BF02186838.

[29]

A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, Rhode Island, 1994.

show all references

References:
[1]

R. Abeyaratne and J. Knowles, On the driving traction acting on a surface of strain discontinuity in a continuum, J. Mech. Phys. Solids, 38 (1990), 345-360. doi: doi:10.1016/0022-5096(90)90003-M.

[2]

H. D. Alber, Evolving microstructure and homogenization, Continuum. Mech. Thermodyn., 12 (2000), 235-287. doi: doi:10.1007/s001610050137.

[3]

H. D. Alber and P. Zhu, Evolution of phase boundaries by configurational forces, Archive Rat. Mech. Anal., 185 (2007), 235-286. doi: doi:10.1007/s00205-007-0054-8.

[4]

H. D. Alber and P. Zhu, Solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces, SIAM J. Appl. Math., 66 (2006), 680-699. doi: doi:10.1137/050629951.

[5]

H. D. Alber and P. Zhu, Solutions to a model for interface motion by interface diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 923-955. doi: doi:10.1017/S0308210507000170.

[6]

H. D. Alber and P. Zhu, Interface motion by interface diffusion driven by bulk energy: Justification of a diffusive interface model, preprint, accepted for publication in Conti. Mech. Thermodyna, 2010.

[7]

S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Met., 27 (1979), 1084-1095. doi: doi:10.1016/0001-6160(79)90196-2.

[8]

D. Aronson and H. Weiberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: doi:10.1016/0001-8708(78)90130-5.

[9]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, in "Perspectives in Nonlinear Partial Differential Equations," Contemp. Mathematics, 446, Amer. Math. Soc., Providence, RI, (2007), 101-123.

[10]

T. Blesgen and U. Weikard, Multi-component Allen-Cahn equation for elastically stressed solids, Electronic J. Diff. Eqs., 2005 (2005), 1-17.

[11]

J. Cahn, Free energy of a nonuniform system. II. Thermodynamic basis, J. Chem. Phys., 30 (1959), 1121-1124. doi: doi:10.1063/1.1730145.

[12]

J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1959), 258-267. doi: doi:10.1063/1.1744102.

[13]

J. Cahn and J. Hilliard, Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699. doi: doi:10.1063/1.1730447.

[14]

J. Cahn and J. Taylor, Surface motion by surface diffusion, Acta Metall. Mater., 42 (1994), 1045-1063. doi: doi:10.1016/0956-7151(94)90123-6.

[15]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Diff. Eq., 96 (1992), 116-141.

[16]

P. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Springer Verlag, 1979.

[17]

P. Fife and J. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rati. Mech. Anal., 65 (1977), 335-361. doi: doi:10.1007/BF00250432.

[18]

M. Gurtin, "Configurational Forces as Basic Concepts of Continuum Physics," Springer Verlag, New York, 2000.

[19]

H. Garcke, On the Cahn-Hilliard system with elasticity, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 307-331. doi: doi:10.1017/S0308210500002419.

[20]

E. Hornbogen and H. Warlimont, "Metallkunde," 4th edition, Springer-Verlag, 2001.

[21]

Y. Kanel, On the stabilization of solutions of the Cauchy problem for equations arising in the theory of combustion, Mat. Sbornik, 59 (1962), 245-288.

[22]

W. Mullins, Theory of thermal grooving, J. Appl. Phys., 28 (1957), 333-339. doi: doi:10.1063/1.1722742.

[23]

H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Diff. Eq., 213 (2005), 204-233.

[24]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Disc. Conti. Dyna. Syst., 15 (2006), 819-832. doi: doi:10.3934/dcds.2006.15.819.

[25]

R. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. R. Soc. Lond. Sect. A, 422 (1989), 261-278. doi: doi:10.1098/rspa.1989.0027.

[26]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer Verlag, New York, 1983.

[27]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: doi:10.1137/060661788.

[28]

J. Taylor and J. Cahn, Linking anisotropic sharp and diffuse surface motion laws via gradient flows, J. Stat. Phys., 77 (1994), 183-197. doi: doi:10.1007/BF02186838.

[29]

A. Volpert, V. Volpert and V. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, Rhode Island, 1994.

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