Abstract
We determine the detailed qualitative behavior of radially symmetric equilibrium states having coexistent phases for general classes of aeotropic nonlinearly thermoelastic materials. We treat both structured and non-structured interfaces. The aeolotropy is responsible for many novel effects. We show that linearly elastic materials cannot sustain coexistent radially symmetric phases unless the interfaces are structured. Our analysis is largely elementary, being based on a combination of geometric constructions with phase-plane methods. A few results, however, depend on our development of appropriate versions of the theory of asymptotically autonomous ordinary differential equations.
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References
R. Abeyaratne and J. K. Knowles, On the driving traction acting on a surface of strain discontinuity in a continuum.J. Mech. Phys. Solids 38 (1990) 345–360.
R. Abeyaratne and J. K. Knowles, Kinetic relations and the propagation of phase boundaries in solids.Arch. Rational Mech. Anal. 114 (1991) 119–154.
S. S. Antman and P. V. Negrón-Marrero, The remarkable nature of radially symmetric equilibrium states of aeolotropic nonlinearly elastic bodies.J. Elasticity 18 (1987) 131–164.
S. S. Antman and M. M. Shvartsman, The shrink-fit problem for aeolotropic nonlinearly elastic bodies.J. Elasticity 37 (1995) 157–166.
J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity.Phil. Trans. Roy. Soc. Lond. A 306 (1982) 557–611.
M. E. Gurtin, The dynamics of solid-solid phase transitions 1. Coherent interfaces.Arch. Rational Mech. Anal. 123 (1993) 305–335.
M. E. Gurtin, The nature of configurational forces.Arch. Rational Mech. Anal. 131 (1995) 67–100.
M. E. Gurtin and I. Murdoch, A continuum theory of elastic material surfaces.Arch. Rational Mech. Anal. 57 (1976) 291–323.
J. K. Hale,Ordinary Differential Equations. Wiley-Interscience (1969).
H. B. Keller,Numerical Solution of Two-Point Boundary Value Problems. S.I.A.M. (1976).
M. A. Krasnosel'skiy, A. I. Perov, A. I. Povolotskiy and P. P. Zabreiko,Plane Vector Fields. Academic Press (1966).
L. Markus, Asymptotically autonomous differential systems. In S. Lefschetz (ed.),Contributions to the Theory of Nonlinear Oscillations, Vol. III (1956) pp. 17–29.
P. V. Negrón-Marrero and S. S. Antman, Singular global bifurcation problems for buckling of anisotropic plates.Proc. Roy. Soc. Lond. A 427 (1990) 95–137.
M. M. Shvartsman, ‘Coherent Equilibrium Phases in Aeolotropic Nonlinear Thermoelasticity’. Dissertation, University of Maryland (1994).
A. K. Sinha,Ferrous Physical Metallurgy, Butterworths (1989).
J. Smoller,Shock Waves and Reaction-Diffusion Equations. Springer (1983).
L. Truskinovsky, Transition to detonation in dynamic phase change.Arch. Rational Mech. Anal. 125 (1994) 375–397.
J. L. Walker,Structure of Ingots and Castings, Liquid Metals and Solidification. American Society for Metals (1958).
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This paper is dedicated to Mort Gurtin on the occasion of his sixtieth birthday
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Shvartsman, M.M., Antman, S.S. Coexistent phases in nonlinear thermoelasticity: Radially symmetric equilibrium states of aeolotropic bodies. J Elasticity 41, 107–136 (1995). https://doi.org/10.1007/BF00042510
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DOI: https://doi.org/10.1007/BF00042510