American Institute of Mathematical Sciences

Advanced
November  2006, 15(4): 1169-1191. doi: 10.3934/dcds.2006.15.1169

Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids

Irena Pawłow 1,

1. 

System Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland

Received  May 2005 Revised  December 2005 Published  May 2006

The goal of this paper is to derive again the generalized Cahn-Hilliard and Allen-Cahn models in deformable continua introduced previously by E. Fried and M. E. Gurtin on the basis of a microforce balance. We use a~different approach based on the second law in the form of the entropy principle according to I. Müller and I. S. Liu which leads to the evaluation of the entropy inequality with multipliers.
    Both approaches provide the same systems of field equations. In particular, our differential equation for the multiplier associated with the balance law for the order parameter turns out to be identical with the Fried-Gurtin microforce balance.
Keywords: Allen-Cahn models, Cahn-Hilliard, phase transitions, elastic solids..
Mathematics Subject Classification: Primary: 74A30, 35K25; Secondary: 35Q72, 35L2.
Citation: Irena Pawłow. Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1169-1191. doi: 10.3934/dcds.2006.15.1169
[1]

Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024
[2]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[3]

Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127
[4]

Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099
[5]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301
[6]

Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669
[7]

Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308
[8]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations and Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[9]

Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027
[10]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations and Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[11]

Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1873-1894. doi: 10.3934/cpaa.2021074
[12]

Irena Pawłow, Wojciech M. Zajączkowski. Regular weak solutions to 3-D Cahn-Hilliard system in elastic solids. Conference Publications, 2007, 2007 (Special) : 824-833. doi: 10.3934/proc.2007.2007.824
[13]

Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423
[14]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Phase transition and separation in compressible Cahn-Hilliard fluids. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73
[15]

Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741
[16]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703
[17]

Suting Wei, Jun Yang. Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2575-2616. doi: 10.3934/cpaa.2020113
[18]

Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure and Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303
[19]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145
[20]

Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (136)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy