Mechanics and Thermodynamics of Continua pp 145-158 | Cite as

# Weak Martensitic Transformations in Bravais Lattices

## Abstract

Nonlinear thermoelasticity theory is being used, with some success, to analyze phenomena associated with phase transitions in some crystals, involving a change in crystal symmetry, what are often called Martensitic transformations. Roughly, these are the crystals which are, or at least behave as if they were Bravais lattices. For such lattices, molecular theories of thermoelasticity imply that the Helmholtz free energy should be invariant under an infinite discrete group. However, workers often use constitutive equations which are invariant only under a finite subgroup, to analyze behavior near transitions, Physicists are likely to use polynomials of as low degree as is feasible, what is sometimes called “Landau Theory”, to treat second-order or “weak” first-order transitions. I don’t know how to give a precise meaning to the notion of a weak transition. By one rather pragmatic interpretation a transition is weak if behavior near the transition of interest can be analyzed, satisfactorily, with a free energy function which is invariant only under some finite subgroup. Using this idea, one can deduce some properties which transitions must have, to be considered weak. My purpose is to elaborate this, to try to get some better understanding of what limits the ranges of applicability of what are, really, two versions of thermoelasticity theory.

### Keywords

Quartz Zirconium UREa Austenite Martensite## Preview

Unable to display preview. Download preview PDF.

### References

- 1.James, R. D., The stability and metastability of quartz, in
*Metastability and Incompletely Posed Problems*, IMA Vol. 3 (ed. S. Antman, J. L. Ericksen, D. Kinderlehrer & I. Müller), Springer-Verlag, 1987, 147–176.Google Scholar - 2.Rivlin, R. S., Some thoughts on material stability,
*Proc. IUTAM Symp. Finite Elasticity*(ed. D. E. Carlson & R. T. Shield), Martinus Nijhoff, 1982, 105–122.Google Scholar - 3.Ericksen, J. L., Multi-valued strain energy functions for crystals,
*Int. J. Solids Structures***18**, 913–916, 1982.MathSciNetMATHCrossRefGoogle Scholar - 4.Zanzotto, G., Twinning in minerals and metals: remarks on the comparison of a thermoelastic theory with some available experimental results, to appear in
*Rend. dell’accad. Lincei*.Google Scholar - 5.Schwarzenberger, R. L. E., Classification of crystal lattices,
*Proc. Camb. Phil. Soc.***72**, 325–349, 1972.MathSciNetADSMATHCrossRefGoogle Scholar - 6.Ericksen, J. L., On the symmetry of deformable crystals,
*Arch. Rational Mech. Anal.***72**, 1–13, 1979.MathSciNetADSMATHCrossRefGoogle Scholar - 7.Ball, J. M., & James, R. D., Fine phase mixtures as minimizers of energy,
*Arch. Rational Mech. Anal.***100**, 13–52, 1987.MathSciNetADSMATHCrossRefGoogle Scholar - 8.Pitteri, M., Reconciliation of local and global symmetries of crystals,
*J. Elasticity***14**, 175–190, 1984.MathSciNetMATHCrossRefGoogle Scholar - 9.Coleman, B. D., & Noll, W., Material symmetry and thermostatic inequalities in finite elastic deformations,
*Arch. Rational Mech. Anal.***15**, 87–111, 1964.MathSciNetADSMATHCrossRefGoogle Scholar - 10.Fonseca, I., Variational methods for elastic crystals,
*Ph. D. Thesis*, University of Minnesota, 1985.Google Scholar - 11.Hartshorne, N. H., & Stuart, A.,
*Practical Optical Crystallography*, American Elsevier Publishing Co., 1964.Google Scholar - 12.Truesdell, C.,
*A First Course in Rational Continuum Mechanics*, Vol. I, Academic Press, 1977.Google Scholar - 13.Kinderlehrer, D., Remarks about equilibrium configurations of crystals, in
*Material Instabilities in Continuum Mechanics*(ed. J. M. Ball), Oxford University Press, 1988.Google Scholar - 14.Kinderlehrer, D., Phase transitions in crystals: the analysis of microstructure, to appear in
*Proc. Int. Colloquium in Free Boundary Problems: Theory and Applications*.Google Scholar