A simple model for phase transitions: from the discrete to the continuum problem

Authors:
S. Pagano and R. Paroni

Journal:
Quart. Appl. Math. **61** (2003), 89-109

MSC:
Primary 74N15; Secondary 49J45, 74G65, 82C26

DOI:
https://doi.org/10.1090/qam/1955225

MathSciNet review:
MR1955225

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Abstract: In this paper we study a one-dimensional model simulating the shear in a two-dimensional body. We analyse the discrete system and we deduce the continuum limit of the lattice model as the lattice parameter goes to zero. Different energies are introduced and linked together.

**[1]**J. M. Ball and R. D. James,*Fine phase mixtures as minimisers of energy*, Arch. Rational Mech. Anal.**100**, 13-52 (1987) MR**906132****[2]**J. M. Ball and R. D. James,*Proposed experimental tests of a theory of fine microstructure and the two-well problem*, Phil. Tran. R. Soc. Lond. A**338**, 389-450 (1992)**[3]**A. Braides, G. Dal Maso, and A. Garroni,*Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case*, Arch. Rational Mech. Anal.,**146**, 23-58 (1999) MR**1682660****[4]**A. Braides and M. S. Gelli,*Continuum limits of discrete systems without convexity hypotheses*, Mathematics and Mechanics of Solids, 7, issue 1, 41-66 (2002) MR**1900933****[5]**A. Braides and M. S. Gelli,*Limits of discrete systems with long-range interactions*, Accepted by Journal of Convex Analysis MR**1970562****[6]**C. I. Christov, G. A. Maugin, and M. Velarde,*Well-posed Boussinesq paradigm with purely spatial higher-order derivatives*, Physical Review E,**54**, 3621-3638 (1996)**[7]**G. Dal Maso,*An introduction to*-*convergence*, Birkhäuser, Boston, 1993 MR**1201152****[8]**O. Iosefescu, C. Licht, and G. Michaille,*Variational limit of a one-dimensional discrete and statistically homogeneous system of material points*, Asymptotic Analysis,**28**, 309-329 (2001) MR**1878798****[9]**R. D. James,*Wiggly Energies*, Symposium in honour of J. L. Ericksen, Maryland, June 12-14, 1996**[10]**G. A. Maugin and S. Cadet,*Existence of solitary waves in martensitic alloys*, Int. J. of Engng. Sci.,**29**, 243-258 (1991) MR**1094233****[11]**G. A. Maugin,*Nonlinear waves in elastic crystals*, Oxford Mathematical Monographs, O.U.P., 1999 MR**1772390****[12]**J. Novak and E. K. H. Salje,*Simulated mesoscopic structures of a domain wall in a ferroelastic lattice*, The European Physical Journal B,**4**, 279-284 (1998)**[13]**J. Novak and E. K. H. Salje,*Surface structure of domain walls*, J. Phys.: Condens. Matter,**10**, 359-366 (1998)**[14]**R. Paroni,*From discrete to continuum: a Young measure approach*, accepted by Z. Angew. Math. Phys. MR**1967332****[15]**J. Pouget,*Dynamics of patterns in ferroelastic-martensitic transformations*. I.*Lattice model*, Physical Review B**43**, 3575-3581 (1991)**[16]**J. Pouget,*Dynamics of patterns in ferroelastic-martensitic transformations*. II.*Quasicontinuum model*, Physical Review B**43**, 3582-3592 (1991)**[17]**J. Pouget,*Nonlinear dynamics of a two-dimensional lattice model for ferroelastic materials*, Proceedings of the 8th International Symposium, Varna, Bulgaria, June 11-16, 1995**[18]**G. Puglisi and L. Truskinovsky,*Mechanics of a discrete chain with bi-stable elements*, J. Mech. Phys. Solids**48**, 1-27 (1999) MR**1727553****[19]**X. Ren and L. Truskinovsky,*Finite scale microstructures in nonlocal elasticity*, J. Elasticity, to appear MR**1833329****[20]**R. Rogers and L. Truskinovsky,*Discretization and hysteresis*, Physica B**233**, 370-375 (1997)**[21]**P. Rosenau,*Dynamics of dense lattices*, Physical Review B**36**, 5868-5876 (1987) MR**914756****[22]**L. Truskinovsky and G. Zanzotto,*Ericksen's bar revisited: energy wiggles*, J. Mech. Phys. Solids,**44**, 1371-1408 (1996) MR**1400578**

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Additional Information

DOI:
https://doi.org/10.1090/qam/1955225

Article copyright:
© Copyright 2003
American Mathematical Society