# Nucleation, Kinetics and Admissibility Criteria for Propagating Phase Boundaries

## Abstract

This paper reviews our recent studies on the nucleation and kinetics of propagating phase boundaries in an elastic bar and relates them to various admissibility criteria. First, we discuss how the field equations and jump conditions of the quasi-static theory of such a bar must be supplemented with additional constitutive information pertaining to the initiation and evolution of phase boundaries. The kinetic relation relates the *driving traction* *f* at a phase boundary to the phase boundary velocity *ṡ*; thus *f = φ* (*ṡ*), where *φ* is a materially-determined function. The nucleation criterion specifies a critical value of *f* at an incipient phase boundary. We then incorporate inertial effects, and we find in the context of the Riemann problem that, as long as phase boundary velocities are subsonic, the theory again needs — and has room for — a nucleation criterion and a kinetic relation. Finally, we describe the sense in which each of three widely studied admissibility criteria for phase boundaries is equivalent to a specific kinetic relation of the form *f = φ* (*ṡ*) for a particular choice of *φ* A kinetic relation based on thermal activation theory is also discussed.

### Keywords

Combustion Entropy Argon Martensite Assure## Preview

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### References

- [1]R. Abeyaratne,
*Discontinuous deformation gradients in the finite twisting of an elastic tube*, Journal of Elasticity,**11**(1981), pp. 43–80.MathSciNetMATHCrossRefGoogle Scholar - [2]R. Abeyaratne and J.K. Knowles,
*On the dissipative response due to discontinuous strains in bars of unstable elastic material*, International Journal of Solids and Structures,**24**(1988), pp. 1021–1044.MathSciNetMATHCrossRefGoogle Scholar - [3]R. Abeyaratne_,
*Unstable elastic materials and the viscoelastic response of*bars*in tension*Journal of Applied Mechanics,**55**(1988), pp. 491–492.ADSCrossRefGoogle Scholar - [4]R. Abeyaratne,
*On the driving traction acting on a surface of strain discontinuity in a**continuum*Journal of the Mechanics and Physics of Solids,**38**(1990), pp. 345–360.MathSciNetADSMATHCrossRefGoogle Scholar - [5]R. Abeyaratne,
*Kinetic relations and the propagation of phase boundaries in solids*toappear in Archive for Rational Mechanics and Analysis.Google Scholar - [6]R. Abeyaratne_,
*Implications of viscosity and strain gradient effects for the kinetics of**propagating phase boundaries in solids*to appear in SIAM Journal on Applied Mathematics.Google Scholar - [7]R. Abeyaratne_,
*On the propagation of maximally dissipative phase boundaries in solids*to appear in Quarterly of Applied Mathematics.Google Scholar - [8]H.B. Callen,
*Thermodynamics and an Introduction to Thermostatistics*, second edition, Wiley, New York, 1985.MATHGoogle Scholar - [9]J.W. Christian,
*The Theory of Transformations in Metals and Alloys*, Part I, Pergamon, Oxford, 1975.Google Scholar - [10]B.D. Coleman and M.E. Gurtin,
*Thermodynamics with internal state variables*, Journal of Chemical Physics,**47**(1967), pp. 597–613.ADSCrossRefGoogle Scholar - [11]R. Courant and K.O. Friedrichs,
*Supersonic Flow and Shock Waves*, Interscience, New York, 1948.MATHGoogle Scholar - [12]C.M. Dafermos,
*Hyperbolic systems of conservation laws*, in Systems of nonlinear partial differential equations, (J.M. Ball, editor), pp. 25–70, D. Riedel, Dordrecht, 1983.Google Scholar - [13]J.E. Dunn and J. Serrin,
*On the thermomechanics of interstitial working*, Archive for Rational Mechanics and Analysis,**88**(1985), pp. 95–133.MathSciNetADSMATHCrossRefGoogle Scholar - [14]J.L. Ericksen,
*Equilibrium of bars*, Journal of Elasticity,**5**(1975), pp. 191–201.MathSciNetMATHCrossRefGoogle Scholar - [15]J.D. Eshelby,
*Continuum theory of lattice defects*, in Solid State Physics, (F. Seitz and D. Turnbull, editors), vol. 3, pp. 79–144, Academic Press, New York, 1956.Google Scholar - J.D. Eshelby,
*Energy relations and the energy-momentum tensor in continuum mechanics*, in Inelastic Behavior of Solids, (M.F. Kanninen et al., editors), pp. 77–115, McGraw-Hill, New York, 1970.Google Scholar - [17]H. Hattori, G.B. Olson and W.S. Owen,
*Mobility of martensite interfaces*, Metallurgical Transactions A, 16A (1985), pp. 1713–1722.ADSCrossRefGoogle Scholar - [18]H. Hattori,
*Mobility of the β*_{1}*— γ*_{1}*martensitic interface in Cu-Al-Ni:PartI. Experimental measurements*Metallurgical Transactions A,**16A**(1985), pp. 1723–1734.ADSCrossRefGoogle Scholar - [19]H. Hattori,
*The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion. Isothermal case*, Archive for Rational Mechanics and Analysis,**92**(1986), pp. 247–263.MathSciNetADSMATHCrossRefGoogle Scholar - [20]H. Hattori,
*The Riemann problem for a van der Waals fluid with entropy rate admissibilitycriterion. Nonisothermal case*Journal of Differential Equations,**65**(1986), pp. 158–174.MathSciNetMATHCrossRefGoogle Scholar - [21]W.D. Hayes,
*Gasdynamic Discontinuities*, Princeton University Press, Princeton, 1960.MATHGoogle Scholar - [22]R.D. James,
*The propagation of phase boundaries in elastic bars*, Archive for Rational Mechanics and Analysis,**73**(1980), pp. 125–158.MathSciNetADSMATHCrossRefGoogle Scholar - [23]J.K. Knowles,
*On the dissipation associated with equilibrium shocks in finite elasticity*, Journal of Elasticity,**9**(1979), pp. 131–158.MathSciNetMATHCrossRefGoogle Scholar - [24]D.J. Korteweg,
*Sur la forme que prennent les équations du mouvement des fluides si l’ontient compte des forces capillaires causées par des variation de densité*, Archives Néerlandaises des Science Exactes et Naturelle, Series II,**6**(1901), pp. 1–24.Google Scholar - [25]R.V. Krishnan and L.C. Brown,
*Pseudo-elasticity and the strain-memory effect in an Ag-45 at. pct. Cd alloy*, Metallurgical Transactions,**4**(1973), pp. 423–429.CrossRefGoogle Scholar - [26]T.P. Liu,
*Uniqueness of weak solutions of the Cauchy problem for general 2x2 conservation laws*, Journal of Differential Equations,**20**(1976), pp. 369–388.MathSciNetADSMATHCrossRefGoogle Scholar - [27]J. Lubliner,
*A maximum dissipation principle in generalized plasticity*, Acta Mechanica,**52**(1984), pp. 225–237.MathSciNetMATHCrossRefGoogle Scholar - [28]O.A. Oleinik,
*On the uniqueness of the generalized solution of the Cauchy problem for a nonlinear system of equations occurring in mechanics*, Uspekhi Matematicheskii Nauk (N.S.),**12**(1957), pp. 169–176 (in Russian).MathSciNetGoogle Scholar - [29]O.A. Oleinik,
*Uniqueness and stability of the generalized solution of the Cauchy problemfor a quasi-linear equation*Uspekhi Matematicheskii Nauk (N.S.), 14 (1959), pp. 165–170 (in Russian).MathSciNetGoogle Scholar - [30]R. Pego,
*Phase transitions in one dimensional nonlinear viscoelasticity: admissibility and stability*, Archive for Rational Mechanics and Analysis,**97**(1986), pp. 353–394.MathSciNetADSCrossRefGoogle Scholar - [31]T.J. Pence,
*On the encounter of an acoustic shear pulse with a phase boundary in an elastic material: energy and dissipation*to appear in Journal of Elasticity.Google Scholar - [32]T. Poston and I.N. Stewart,
*Catastrophe Theory and its Applications*, Pitman, London, 1978.MATHGoogle Scholar - [33]J.R. Rice,
*On the structure of stress-strain relations for time–dependent plastic deformation in metals*, Journal of Applied Mechanics,**37**(1970), pp. 728–737.ADSCrossRefGoogle Scholar - [34]J.R. Rice,
*Inelastic constitutive relations for solids: An internal variable theory and itsapplications to metal plasticity*Journal of the Mechanics and Physics of Solids,**19**(1971), pp. 433–455.ADSMATHCrossRefGoogle Scholar - [35]J.R. Rice,
*Continuum mechanics and thermodynamics of plasticity in relation to microscale**deformation mechanisms*, in Constitutive Equations in Plasticity, (A.S. Argon, editor), pp. 23–79, MIT Press, Cambridge, Massachusetts, 1975.Google Scholar - [36]M. Shearer,
*The Riemann problem for a class of conservation laws of mixed type*, Journal of Differential Equations**46**(1982), pp. 426–443.MathSciNetMATHCrossRefGoogle Scholar - [37]M. Shearer,
*Nonuniqueness of admissible solutions of Riemann initial value problems fora system of conservation laws of mixed type*Archive for Rational Mechanics and Analysis,**93**(1986), pp. 45–59.MathSciNetADSMATHCrossRefGoogle Scholar - [38]M. Shearer,
*Dynamic phase transitions in a van der Waals gas*Quarterly of Applied Mathematics,**46**(1988), pp. 631–636.MathSciNetMATHGoogle Scholar - [39]M. Slemrod,
*Admissibility criteria for propagating phase boundaries in a van der Waals fluid*, Archive for Rational Mechanics and Analysis,**81**(1983), pp. 301–315.MathSciNetADSMATHCrossRefGoogle Scholar - [40]M. Slemrod,
*Dynamics of first order phase transitions*, in Phase Transformations and Material Instabilities in Solids (M.E. Gurtin, ed.), pp. 163–203, Academic Press, New York, 1984.Google Scholar - [41]M. Slemrod,
*A limiting*“*viscosity*”*approach to the Riemann problem for materials exhibiting change of phase*Archive for Rational Mechanics and Analysis, 105 (1989), pp. 327–365.MathSciNetADSMATHCrossRefGoogle Scholar - [42]C. Truesdell,
*Rational Thermodynamics*, Springer-Verlag, New York, 1969.Google Scholar - [43]L. Truskinovsky,
*Equilibrium phase interfaces*, Soviet Physics Doklady,**27**(1982), pp. 551–553.ADSGoogle Scholar - [44]L. Truskinovsky,
*Structure of an isothermal phase discontinuity*Soviet Physics Doklady,**30**(1985), pp. 945–948.ADSGoogle Scholar - [45]L. Truskinovsky,
*Dynamics of non-equilibrium phase boundaries in a heat-conductingnon-linearly elastic medium*Journal of Applied Mathematics and Mechanics (PMM USSR),**51**(1987), pp. 777–784.MathSciNetADSCrossRefGoogle Scholar