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
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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.
R. Abeyaratne, Discontinuous deformation gradients in the finite twisting of an elastic tube, Journal of Elasticity 11, 43–80 (1981)
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)
R. Abeyaratne and J. K. Knowles, Kinetic relations and the propagation of phase boundaries in solids, Arch. Rational Mech. Anal. 114 (1991)
R. Abeyaratne and J. K. Knowles, Implications of viscosity and strain gradient effects for kinetics of propagating phase boundaries in solids, SIAM (to appear)
R. Abeyaratne and J. K. Knowles, On the propagation of maximally dissipative phase boundaries in solids, Quart. Appl. Math. 50, 149–172 (1992)
J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal. 100, 13–53 (1987)
J. W. Christian, The theory of transformations in metals and alloys, Part I, Purgations, Oxford, 1975
C. M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14, 202–212 (1973)
C. M. Dafermos, Discontinuous thermokinetic process, Appendix 4B of Rational Thermodynamics, by C. Truesdell, Springer-Verlag, New York, 1984
J. L. Ericksen, Equilibrium of bars, Journal of Elasticity 5, 191–201 (1975)
H. Hattori, The Riemann problem for a van der Waals fluid with entropy admissibility criterion. Nonisothermal case, J. Differential Equations 65, 158–174 (1986)
R. D. James, The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73, 125–158 (1980)
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)
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)
M. Slemrod, Admissibility criterion for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81, 301–315 (1983)
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)
L. Truskinovsky, Equilibrium phase interface, Soviet Physics Doklady 27, 551–553 (1982)
L. Truskinovsky, Transition to “detonation” in dynamic phase changes, preprint, September, 1992
Y. Lin, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1993
R. Abeyaratne, Discontinuous deformation gradients in the finite twisting of an elastic tube, Journal of Elasticity 11, 43–80 (1981)
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)
R. Abeyaratne and J. K. Knowles, Kinetic relations and the propagation of phase boundaries in solids, Arch. Rational Mech. Anal. 114 (1991)
R. Abeyaratne and J. K. Knowles, Implications of viscosity and strain gradient effects for kinetics of propagating phase boundaries in solids, SIAM (to appear)
R. Abeyaratne and J. K. Knowles, On the propagation of maximally dissipative phase boundaries in solids, Quart. Appl. Math. 50, 149–172 (1992)
J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal. 100, 13–53 (1987)
J. W. Christian, The theory of transformations in metals and alloys, Part I, Purgations, Oxford, 1975
C. M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14, 202–212 (1973)
C. M. Dafermos, Discontinuous thermokinetic process, Appendix 4B of Rational Thermodynamics, by C. Truesdell, Springer-Verlag, New York, 1984
J. L. Ericksen, Equilibrium of bars, Journal of Elasticity 5, 191–201 (1975)
H. Hattori, The Riemann problem for a van der Waals fluid with entropy admissibility criterion. Nonisothermal case, J. Differential Equations 65, 158–174 (1986)
R. D. James, The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73, 125–158 (1980)
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)
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)
M. Slemrod, Admissibility criterion for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81, 301–315 (1983)
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)
L. Truskinovsky, Equilibrium phase interface, Soviet Physics Doklady 27, 551–553 (1982)
L. Truskinovsky, Transition to “detonation” in dynamic phase changes, preprint, September, 1992
Y. Lin, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1993
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© Copyright 1995
American Mathematical Society
