Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

On the propagation of maximally dissipative phase boundaries in solids


Authors: Rohan Abeyaratne and James K. Knowles
Journal: Quart. Appl. Math. 50 (1992), 149-172
MSC: Primary 73B30
DOI: https://doi.org/10.1090/qam/1146630
MathSciNet review: MR1146630
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Abstract: This paper is concerned with the kinetics of propagating phase boundaries in a bar made of a special nonlinearly elastic material. First, it is shown that there is a kinetic law of the form $ f = \varphi \left( {\dot s} \right)$ relating the driving traction $ f$ at a phase boundary to the phase boundary velocity $ \dot s$ that corresponds to a notion of maximum dissipation analogous to the concept of maximum plastic work. Second, it is shown that a modified version of the entropy rate admissibility criterion can be described by a kinetic relation of the above form, but with a different $ \varphi $ . Both kinetic relations are applied to the Riemann problem for longitudinal waves in the bar.


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  • [1] R. Abeyaratne and J. K. Knowles, Kinetic relations and the propagation of phase boundaries in solids, Arch. Rational Mech. Anal. 114, 119-154 (1991) MR 1094433
  • [2] O. A. Oleinik, On the uniqueness of the generalized solution of the Cauchy problem for a nonlinear system of equations occurring in mechanics, Uspekhi Mat. Nauk (N.S.) 12, no. 6 (78), 169-176 (1957) (Russian) MR 0094543
  • [3] 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) MR 0393871
  • [4] C. M. Dafermos, Discontinuous thermokinetic processes, Appendix 4B of Rational Thermodynamics (C. Truesdell, ed.), Springer-Verlag, New York, 1984
  • [5] J. W. Christian, The Theory of Transformations in Metals and Alloys, Part I, Pergamon Press, Oxford, 1975
  • [6] R. Abeyaratne and J. K. Knowles, On the dissipative response due to discontinuous strains in bars of unstable elastic material, Internat. J. Solids and Structures 24, 1021-1044 (1988) MR 974437
  • [7] 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, 345-360 (1990) MR 1051343
  • [8] J. R. Rice, On the structure of stress-strain relations for time-dependent plastic deformation in metals, J. Appl. Mech. 37, 728-737 (1970)
  • [9] J. Lubliner, A maximum dissipation principle in generalized plasticity, Acta Mech. 52, 225-237 (1984) MR 765250
  • [10] 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) MR 822335
  • [11] M. Shearer, Dynamic phase transitions in a van der Waals gas, Quart. Appl. Math. 46, 631-636 (1988) MR 973380
  • [12] M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81, 301-315 (1983) MR 683192
  • [13] M. Slemrod, Dynamics of first order phase transitions, Phase Transformations and Material Instabilities in Solids (M. E. Gurtin, ed.), Academic Press, New York, 1984, pp. 163-203 MR 802225
  • [14] L. Truskinovsky, Equilibrium phase interfaces, Soviet Phys. Dokl. 27, 551-553 (1982)
  • [15] L. Truskinovsky, Structure of an isothermal phase discontinuity, Soviet Phys. Dokl. 30, 945-948 (1985)
  • [16] C. M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14, 202-212 (1973) MR 0328368
  • [17] C. M. Dafermos, Hyperbolic systems of conservation laws, Systems of Nonlinear Partial Differential Equations (J. M. Ball, ed.), Reidel, Dordrecht, 1983, pp. 25-70 MR 725517
  • [18] H. Hattori, The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion. Isothermal case, Arch. Rational Mech. Anal. 92, 247-263 (1986) MR 816624
  • [19] H. Hattori, The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion. Nonisothermal case, J. Differential Equations 65, 158-174 (1986) MR 861514
  • [20] R. D. James, The propagation of phase boundaries in elastic bars, Arch. Rational Mech. Anal. 73, 125-158 (1980) MR 556559
  • [21] T. J. Pence, On the encounter of an accoustic shear pulse with a phase boundary in an elastic material: energy and dissipation, J. Elasticity, to appear MR 1128426
  • [22] R. Abeyaratne and J. K. Knowles, Implications of viscosity and strain gradient effects for the kinetics of propagating phase boundaries in solids, Technical Report No. 11, ONR contract N00014-87-K-0117, April 1990; to appear in SIAM J. Appl. Math. MR 1127848
  • [23] D. A. Porter and K. E. Easterling, Phase Transformations in Metals and Alloys, van Nostrand-Reinhold, New York, 1981

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DOI: https://doi.org/10.1090/qam/1146630
Article copyright: © Copyright 1992 American Mathematical Society

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