American Institute of Mathematical Sciences

Advanced
January  2009, 8(1): 55-81. doi: 10.3934/cpaa.2009.8.55

A variational model allowing both smooth and sharp phase boundaries in solids

John M. Ball 1, and  Carlos Mora-Corral 2,

1. 

Mathematical Institute, University of Oxford, 24--29 St Giles', Oxford OX1 3LB, United Kingdom
2. 

Mathematical Institute, University of Oxford, 24--29 St Giles, Oxford OX1 3LB, United Kingdom

Received  June 2008 Revised  September 2008 Published  October 2008

We present models for solid-solid phase transitions with surface energy that allow both smooth and sharp interfaces. The models involve the minimisation of an energy that consists of three terms: the elastic energy (a double-well potential), the smooth-interface surface energy and the sharp-interface surface energy. Existence of solutions is shown in arbitrary dimensions. The second part of the paper deals with the one-dimensional case. For the first 1D model (in which the sharp-interface energy is the same regardless of the size of the jump of the gradient), we study the regime of the parameters (one parameter represents the boundary conditions, one models the energy of the sharp interface, and the third one models the energy of the smooth interfaces) for which the minimiser presents smooth interfaces, sharp interfaces or no interfaces. We also prove that a suitable scaling of the functional $\Gamma$-converges to a pure sharp-interface model, as the parameters penalising the formation of interfaces go to zero. For the second 1D model (in which the sharp-interface energy depends on the size of the jump and can tend to zero as the jump tends to zero), we describe general properties of the minimisers, and show that their gradients have a finite number of discontinuity points.
Keywords: surface energy, smooth interfaces, Solid-solid phase transitions, sharp interfaces..
Mathematics Subject Classification: Primary: 74A50; Secondary: 49J45, 74B20, 74G25, 74N1.
Citation: John M. Ball, Carlos Mora-Corral. A variational model allowing both smooth and sharp phase boundaries in solids. Communications on Pure & Applied Analysis, 2009, 8 (1) : 55-81. doi: 10.3934/cpaa.2009.8.55
[1]

Markus Gahn. Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6511-6531. doi: 10.3934/dcdsb.2019151
[2]

Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997
[3]

Matteo Novaga, Enrico Valdinoci. The geometry of mesoscopic phase transition interfaces. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 777-798. doi: 10.3934/dcds.2007.19.777
[4]

Michael L. Frankel, Victor Roytburd. Dynamical structure of one-phase model of solid combustion. Conference Publications, 2005, 2005 (Special) : 287-296. doi: 10.3934/proc.2005.2005.287
[5]

Amin Boumenir. Determining the shape of a solid of revolution. Mathematical Control & Related Fields, 2019, 9 (3) : 509-515. doi: 10.3934/mcrf.2019023
[6]

Shin-Ichiro Ei, Kei Nishi, Yasumasa Nishiura, Takashi Teramoto. Annihilation of two interfaces in a hybrid system. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 857-869. doi: 10.3934/dcdss.2015.8.857
[7]

Mario Ohlberger, Ben Schweizer. Modelling of interfaces in unsaturated porous media. Conference Publications, 2007, 2007 (Special) : 794-803. doi: 10.3934/proc.2007.2007.794
[8]

N. D. Alikakos, P. W. Bates, J. W. Cahn, P. C. Fife, G. Fusco, G. B. Tanoglu. Analysis of a corner layer problem in anisotropic interfaces. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 237-255. doi: 10.3934/dcdsb.2006.6.237
[9]

Tommaso Lorenzi, Alexander Lorz, Benoît Perthame. On interfaces between cell populations with different mobilities. Kinetic & Related Models, 2017, 10 (1) : 299-311. doi: 10.3934/krm.2017012
[10]

Marius Mitrea. On Bojarski's index formula for nonsmooth interfaces. Electronic Research Announcements, 1999, 5: 40-46.

[11]

Luis Alvarez, Jesús Ildefonso Díaz. On the retention of the interfaces in some elliptic and parabolic nonlinear problems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 1-17. doi: 10.3934/dcds.2009.25.1
[12]

Manuel del Pino, Michal Kowalczyk, Juncheng Wei. The Jacobi-Toda system and foliated interfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 975-1006. doi: 10.3934/dcds.2010.28.975
[13]

Volker Elling. Compressible vortex sheets separating from solid boundaries. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6781-6797. doi: 10.3934/dcds.2016095
[14]

Henk W. Broer, Carles Simó, Renato Vitolo. Chaos and quasi-periodicity in diffeomorphisms of the solid torus. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 871-905. doi: 10.3934/dcdsb.2010.14.871
[15]

Jianjun Paul Tian, Kendall Stone, Thomas John Wallin. A simplified mathematical model of solid tumor regrowth with therapies. Conference Publications, 2009, 2009 (Special) : 771-779. doi: 10.3934/proc.2009.2009.771
[16]

Leda Bucciantini, Angiolo Farina, Antonio Fasano. Flows in porous media with erosion of the solid matrix. Networks & Heterogeneous Media, 2010, 5 (1) : 63-95. doi: 10.3934/nhm.2010.5.63
[17]

Johannes Elschner, George C. Hsiao, Andreas Rathsfeld. An inverse problem for fluid-solid interaction. Inverse Problems & Imaging, 2008, 2 (1) : 83-120. doi: 10.3934/ipi.2008.2.83
[18]

Peter Monk, Virginia Selgas. An inverse fluid--solid interaction problem. Inverse Problems & Imaging, 2009, 3 (2) : 173-198. doi: 10.3934/ipi.2009.3.173
[19]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83
[20]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences