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Kinetic relations and the propagation of phase boundaries in solids

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Abstract

This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.

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References

  • Abeyaratne, R. (1981) Discontinuous deformation gradients in the finite twisting of an elastic tube, Journal of Elasticity 11, 43–80.

    Google Scholar 

  • Abeyaratne, R. (1983) An admissibility condition for equilibrium shocks in finite elasticity, Journal of Elasticity 13, 175–184.

    Google Scholar 

  • Abeyaratne, R. & Knowles, J. K. (1987a) Non-elliptic elastic materials and the modeling of dissipative mechanical behavior: an example, Journal of Elasticity 18, 227–278.

    Google Scholar 

  • Abeyaratne, R. & Knowles, J. K. (1987b) Non-elliptic elastic materials and the modeling of elastic-plastic behavior for finite deformation, Journal of the Mechanics and Physics of Solids 35, 343–365.

    Google Scholar 

  • Abeyaratne, R. & Knowles, J. K. (1988a) On the dissipative response due to discontinuous strains in bars of unstable elastic material, International Journal of Solids and Structures 24, 1021–1044.

    Google Scholar 

  • Abeyaratne, R. & Knowles, J. K. (1988b) Unstable elastic materials and the viscoelastic response of bars in tension, Journal of Applied Mechanics 55, 491–492.

    Google Scholar 

  • Abeyaratne, R. & Knowles, J. K. (1990) On the driving traction acting on a surface of discontinuity in a continuum, Journal of the Mechanics and Physics of Solids 38, 345–360.

    Google Scholar 

  • Bunshak, R. F. & Mehl, R. F. (1952) Time duration of the formation of a martensite plate and the influence of metallurgical variables, Journal of Metals 4, 1042–1042.

    Google Scholar 

  • Christian, J. W. (1975) The Theory of Transformations in Metals and Alloys, Part I, Pergamon, Oxford.

    Google Scholar 

  • Dafermos, C. M. (1973) The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, Journal of Differential Equations 14, 202–212.

    Google Scholar 

  • Dafermos, C. M. (1983) Hyperbolic systems of conservation laws, in Systems of nonlinear partial differential equations (J. M. Ball, editor), 25–70, D. Riedel, Dordrecht.

    Google Scholar 

  • Dafermos, C. M. (1989) Admissible wave fans in nonlinear hyperbolic systems, Archive for Rational Mechanics and Analysis 106, 243–260.

    Google Scholar 

  • Ericksen, J. L. (1975) Equilibrium of bars, Journal of Elasticity 5, 191–201.

    Google Scholar 

  • Eshelby, J. D. (1956) Continuum theory of lattice defects, in Solid State Physics, (F. Seitz & D. Turnbull, editors), Vol. 3, 79–144, Academic Press, New York.

    Google Scholar 

  • Eshelby, J. D. (1970) Energy relations and the energy-momentum tensor in continuum mechanics, in Inelastic Behavior of Solids, M. F. Kanninen et al, editors, 77–115, McGraw-Hill, New York.

    Google Scholar 

  • Fine, M. E. (1964) Introduction to Phase Transformations in Condensed Systems, Macmillan, New York.

    Google Scholar 

  • Grujicic, M., Olson, G. B. & Owen, W. S. (1985a) Mobility of martensite interfaces, Metallurgical Transactions A, 16 A, 1713–1722.

    Google Scholar 

  • Grujicic, M., Olson, G. B. & Owen, W. S. (1985b) Mobility of the β 1-γ′1martensitic interface in Cu-Al-Ni: Part I. Experimental measurements, Metallurgical Transactions A, 16 A, 1723–1734.

    Google Scholar 

  • Gurtin, M. E. (1983) Two-phase deformations of elastic solids, Archive for Rational Mechanics and Analysis 84, 1–29.

    Google Scholar 

  • Hattori, H. (1986a) The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion. Isothermal case, Archive for Rational Mechanics and Analysis 92, 247–263.

    Google Scholar 

  • Hattori, H. (1986b) The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion. Nonisothermal case, Journal of Differential Equations 65, 158–174.

    Google Scholar 

  • James, R. D. (1979) Co-existent phases in the one-dimensional theory of elastic bars, Archive for Rational Mechanics and Analysis 72, 99–140.

    Google Scholar 

  • James, R. D. (1980) The propagation of phase boundaries in elastic bars, Archive for Rational Mechanical and Analysis 73, 125–158.

    Google Scholar 

  • James, R. D. (1981) Finite deformation by mechanical twinning, Archive for Rational Mechanics and Analysis 77, 143–176.

    Google Scholar 

  • James, R. D. (1986) Displacive phase transformations in solids, Journal of the Mechanics and Physics of Solids 34, 359–394.

    Google Scholar 

  • Knowles, J. K. (1979) On the dissipation associated with equilibrium shocks in finite elasticity, Journal of Elasticity 9, 131–158.

    Google Scholar 

  • Krishnan, R. V. & Brown, L. C. (1973) Pseudo-elasticity and the strain-memory effect in an Ag-45 at. pct. Cd alloy, Metallurgical Transactions 4, 423–429.

    Google Scholar 

  • Lax, P. D. (1973) Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Regional Conference Series in Applied Mathematics, No. 11, SIAM, Philadelphia.

    Google Scholar 

  • Oleinik, O. A. (1957) On the uniqueness of the generalized solutions of the Cauchy problem for a nonlinear system of equations occurring in mechanics, Uspekhi Matematicheskii Nauk (N.S.) 12, no. 6(78), 169–176 (in Russian).

    Google Scholar 

  • Pence, T. J. (1986) On the emergence and propagation of a phase boundary in an elastic bar with a suddenly applied end-load, Journal of Elasticity 16, 3–42.

    Google Scholar 

  • Porter, D. A. & Easterling, K. E. (1981) Phase Transformations in Metals and Alloys, van Nostrand Reinhold, New York.

    Google Scholar 

  • Ralls, K. M., Courtney, T. H. & Wulff, J. (1976) Introduction to Materials Science and Engineering, John Wiley & Sons, New York.

    Google Scholar 

  • Rice, J. R. (1970) On the structure of stress-strain relations for time-dependent plastic deformation in metals, Journal of Applied Mechanics 37, 728–737.

    Google Scholar 

  • Rice, J. R. (1971) Inelastic constitutive relations for solids: An internal variable theory and its application to metal plasticity, Journal of the Mechanics and Physics of Solids 19, 433–455.

    Google Scholar 

  • Rice, J. M. (1975) Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms, in Constitutive Equations in Plasticity (A. S. Argon, editor), 23–79, MIT Press, Cambridge, Massachusetts.

    Google Scholar 

  • Shearer, M. (1982) The Riemann problem for a class of conservation laws of mixed type, Journal of Differential Equations 46, 426–443.

    Google Scholar 

  • Shearer, M. (1986) Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type, Archive for Rational Mechanics and Analysis 93, 45–59.

    Google Scholar 

  • Shearer, M. (1988) Dynamic phase transitions in a van der Waals gas, Quarterly of Applied Mathematics 46, 631–636.

    Google Scholar 

  • Šilhavý, M. (1988) On the admissibility of shocks and propagating phase boundaries in a van der Waals fluid, in Material Instabilities in Continuum Mechanics and Related Mathematical Problems (J. M. Ball, editor), 481–493, Clarendon Press, Oxford.

    Google Scholar 

  • Silling, S. A. (1988) Consequences of the Maxwell relation for anti-plane shear deformations of an elastic solid, Journal of Elasticity 19, 241–284.

    Google Scholar 

  • Silling, S. A. (1989) Phase changes induced by deformation in isothermal elastic crystals, Journal of the Mechanics and Physics of Solids 37, 293–316.

    Google Scholar 

  • Slemrod, M. (1983) Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Archive for Rational Mechanics and Analysis 81, 301–315.

    Google Scholar 

  • Slemrod, M. (1989) A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of phase, Archive for Rational Mechanics and Analysis 105, 327–365.

    Google Scholar 

  • Truskinovsky, L. M. (1985) Structure of an isothermal phase discontinuity, Soviet Physics Doklady 30, 945–948.

    Google Scholar 

  • Truskinovsky, L. M. (1987) Dynamics of non-equilibrium phase boundaries in a heat-conducting non-linearly elastic medium, Journal of Applied Mathematics and Mechanics (PMM) 51, 777–784.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Massachusetts Institute of Technology, 02139, Cambridge, Massachusetts

    Rohan Abeyaratne & James K. Knowles

  2. Division of Engineering and Applied Science, California Institute of Technology, 91125, Pasadena, California

    Rohan Abeyaratne & James K. Knowles

Authors
  1. Rohan Abeyaratne
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  2. James K. Knowles
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Communicated by C. Dafermos

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Abeyaratne, R., Knowles, J.K. Kinetic relations and the propagation of phase boundaries in solids. Arch. Rational Mech. Anal. 114, 119–154 (1991). https://doi.org/10.1007/BF00375400

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  • DOI: https://doi.org/10.1007/BF00375400

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