Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A Riemann problem for an elastic bar that changes phase

Author: Yier Lin
Journal: Quart. Appl. Math. 53 (1995), 575-600
MSC: Primary 73D35; Secondary 35L67, 35Q72, 73B05, 73C50, 73K05
DOI: https://doi.org/10.1090/qam/1343469
MathSciNet review: MR1343469
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the dynamics of an elastic bar that can undergo reversible stress-induced phase transformations. We consider a Riemann problem in which the initial strains belong to a single metastable phase and prove uniqueness of solution that satisfies a nucleation criterion and a kinetic law at all subsonic and sonic phase boundaries. This paper generalizes the results of [3]; the authors of [3] considered a piecewise-linear material for which no wave fans exist, shock waves always travel at the acoustic speed, and shock waves are dissipation-free. The material model of the present paper does not suffer from these degeneracies.

References [Enhancements On Off] (What's this?)

  • [1] R. Abeyaratne, Discontinuous deformation gradients in the finite twisting of an elastic tube, Journal of Elasticity 11, 43-80 (1981)
  • [2] R. Abeyaratne and J. K. Knowles, On dissipative response due to discontinuous strains in bars of unstable elastic material, Internat. J. Solids Structures 24, 1021-1024 (1988)
  • [3] R. Abeyaratne and J. K. Knowles, Kinetic relations and the propagation of phase boundaries in solids, Arch. Rational Mech. Anal. 114 (1991)
  • [4] R. Abeyaratne and J. K. Knowles, Implications of viscosity and strain gradient effects for kinetics of propagating phase boundaries in solids, SIAM (to appear)
  • [5] R. Abeyaratne and J. K. Knowles, On the propagation of maximally dissipative phase boundaries in solids, Quart. Appl. Math. 50, 149-172 (1992)
  • [6] J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal. 100, 13-53 (1987)
  • [7] J. W. Christian, The theory of transformations in metals and alloys, Part I, Purgations, Oxford, 1975
  • [8] C. M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14, 202-212 (1973)
  • [9] C. M. Dafermos, Discontinuous thermokinetic process, Appendix 4B of Rational Thermodynamics, by C. Truesdell, Springer-Verlag, New York, 1984
  • [10] J. L. Ericksen, Equilibrium of bars, Journal of Elasticity 5, 191-201 (1975)
  • [11] H. Hattori, The Riemann problem for a van der Waals fluid with entropy admissibility criterion. Nonisothermal case, J. Differential Equations 65, 158-174 (1986)
  • [12] R. D. James, The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73, 125-158 (1980)
  • [13] T. P. Liu, Uniqueness of weak solutions of the Cauchy problem for general $ 2 \times 2$ conservation laws, J. Differential Equations 20, 369-388 (1976)
  • [14] O. A. Oleinik, On the uniqueness of the generalized solution of the Cauchy problem for nonlinear system of equations occurring in mechanics, Uspekhi Matematicheskii Nauk (N.S.) 12 6 (78), 169-176 (1957) (Russian)
  • [15] M. Slemrod, Admissibility criterion for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81, 301-315 (1983)
  • [16] M. Shearer, Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type, Arch. Rational. Mech. Anal. 93, 45-69 (1986)
  • [17] L. Truskinovsky, Equilibrium phase interface, Soviet Physics Doklady 27, 551-553 (1982)
  • [18] L. Truskinovsky, Transition to ``detonation'' in dynamic phase changes, preprint, September, 1992
  • [19] Y. Lin, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1993

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73D35, 35L67, 35Q72, 73B05, 73C50, 73K05

Retrieve articles in all journals with MSC: 73D35, 35L67, 35Q72, 73B05, 73C50, 73K05

Additional Information

DOI: https://doi.org/10.1090/qam/1343469
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society