Abstract
The modelling of twins in crystals with strain gradient theories provides interesting problems both in thermodynamics and in the calculus of variations. Here, Dunn and Serrin's thermomechanical theory of interstitial working is used to obtain a variational principle that governs the equilibria of materials with non-convex Helmholtz free energy. In some geometries, this principle reduces to a novel calculus of variations problem; an example is described in which symmetry-related uniform equilibrium states can be connected by nonconstant extremals which realiselocal minima of the free energy. Certain implications of different definitions of local minima are also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D.E. Lieberman, Crystal geometry and mechanisms of phase transformations in crystalline solids, in ‘Phase Transformations’, American Society for Metals, Ohio, 1970.
R.H. Richman, The diversity of twinning in body-centered cubic structures, inDeformation Twinning, ed., R.E. Reed-Hill, J.P. Hirth and H.C. Rogers, Gordon and Breach Science Publishers Inc., New York, London (1964).
R.A. Toupin, Theories of elasticity with couple-stress.Arch. Rational Mech. Anal. 17 (1964) 85–112.
J.E. Dunn and J. Serrin, On the thermomechanics of interstial working, I.M.A. Preprint Series #24, (1983), to appear inArch. Rational Mech. Anal.
N. Chafee, Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions,J. Diff. Eqns. 18 (1975) 111–134.
J. Carr, M.E. Gurtin and M. Slemrod, One dimensional structured phase transformations under prescribed loads,J. Elasticity, 15 (1985) 133–142.
L.D. Landau, and E.M. Lifshitz,Statistical Physics, Vol. 5 of Course of Theoretical Physics, 2nd ed., Pergammon Press, London (1968).
L. Cesari, Optimization—Theory and Applications,Applications of Mathematics #17 Springer-Verlag, New York, (1983).
J.M. Ball and V.J. Mizel, Singular Minimizers for regular one-dimensional problems in the calculus of variations,Bulletin AMS 11 #1 (1984) 143–147.
R.G. Casten and C.J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions,J. Diff. Eqns. 27 (1978) 266–273.
H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations,Publ. R.I.M.S., Kyoto University 15 (1979) 401–454.
J. Carr, M.E. Gurtin and M. Slemrod, Structured phase transitions on a finite interval, to appear inArch. Rational Mech. Anal.
N.C. Owen, Thesis, Heriot-Watt University, (1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Maddocks, J.H., Parry, G.P. A model for twinning. J Elasticity 16, 113–133 (1986). https://doi.org/10.1007/BF00043580
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00043580