Abstract
We consider dynamics of phase boundaries in a bistable one-dimensional lattice with harmonic long-range interactions. Using Fourier transform and Wiener–Hopf technique, we construct traveling wave solutions that represent both subsonic phase boundaries (kinks) and intersonic ones (shocks). We derive the kinetic relation for kinks that provides a needed closure for the continuum theory. We show that the different structure of the roots of the dispersion relation in the case of shocks introduces an additional free parameter in these solutions, which thus do not require a kinetic relation on the macroscopic level. The case of ferromagnetic second-neighbor interactions is analyzed in detail. We show that the model parameters have a significant effect on the existence, structure, and stability of the traveling waves, as well as their behavior near the sonic limit.
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References
Abeyaratne R., Knowles J.K.: Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114, 119–154 (1991)
Atkinson W., Cabrera N.: Motion of a Frenkel-Kontorova dislocation in a one-dimensional crystal. Phys. Rev. A 138(3), 763–766 (1965)
Bhattacharya K.: Microstructure of Martensite—Why It Forms and How It Gives Rise to the Shape-Memory Effect. Oxford University Press, USA (2003)
Celli V., Flytzanis N.: Motion of a screw dislocation in a crystal. J. Appl. Phys. 41(11), 4443–4447 (1970)
Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Heidelberg (2000)
Ericksen J.L.: Equilibrium of bars. J. Elast. 5, 191–202 (1975)
Escobar J.C., Clifton R.J.: On pressure-shear plate impact for studying the kinetics of stress-induced phase transformations. J. Mater. Sci. Eng. A 170, 125–142 (1993)
Escobar J.C., Clifton R.J.: Pressure-shear impact-induced phase transitions in Cu-14.4Al-4.19Ni single crystals. SPIE 2427, 186–197 (1995)
Ishioka S.: Uniform motion of a screw dislocation in a lattice. J. Phys. Soc. Jpn. 30, 323–327 (1971)
Kresse O., Truskinovsky L.: Mobility of lattice defects: discrete and continuum approaches. J. Mech. Phys. Solids 51, 1305–1332 (2003)
Kresse O., Truskinovsky L.: Lattice friction for crystalline defects: from dislocations to cracks. J. Mech. Phys. Solids 52, 2521–2543 (2004)
LeFloch P.G.: Hyperbolic Systems of Conservation Laws. ETH Lecture Note Series. Birkhäuser, Basel (2002)
Marder M., Gross S.: Origin of crack tip instabilities. J. Mech. Phys. Solids 43(1), 1–48 (1995)
Noble B.: Methods Based on the Wiener–Hopf Technique for the Solution of Partial Differential Equations. 2nd edn. Chelsea Publishing Company, New York (1988)
Niemczura J., Ravi-Chandar K.: Dynamics of propagating phase boundaries in NiTi. J. Mech. Phys. Solids 54, 2136–2161 (2006)
Serre D.: Systems of Conservation Laws, Volume 1, 2. Cambridge University Press, Cambridge (1999)
Shaw J.A., Kyriakides S.: On the nucleation and propagation of phase transformation fronts in a NiTi alloy. Acta Mater. 45, 683–700 (1997)
Slemrod M.: Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Ration. Mech. Anal. 81, 301–315 (1983)
Slepyan L.I.: Antiplane problem of a crack in a lattice. Mech. Solids 16(5), 101–115 (1982)
Slepyan L.I.: Models and Phenomena in Fracture Mechanics. Springer, Berlin (2002)
Slepyan L.I., Troyankina L.V.: Fracture wave in a chain structure. J. Appl. Mech. Tech. Phys. 25(6), 921–927 (1984)
Slepyan L.I., Troyankina L.V.: Impact waves in a nonlinear chain. In: Gol’dstein, R.V. (eds) Plasticity and Fracture of Solids, pp. 175–186. Nauka, Moscow (1988) (in Russian)
Slepyan L.I., Cherkaev A., Cherkaev E.: Transition waves in bistable structures. II. Analytical solution: wave speed and energy dissipation. J. Mech. Phys. Solids 53, 407–436 (2005)
Titchmarsh E.C.: Theory of Functions. 2nd edn. Oxford University press, USA (1976)
Truskinovsky L.: Equilibrium interphase boundaries. Sov. Phys. Doklady 27, 306–331 (1982)
Truskinovsky L.: Dynamics of nonequilibrium phase boundaries in a heat conducting elastic medium. J. Appl. Math. Mech. 51, 777–784 (1987)
Truskinovsky L.: Kinks versus Shocks. In: Fosdick, R., Dunn, E., Slemrod, M. (eds) Shock Induced Transitions and Phase Structures in General Media, The IMA Volumes in Mathematics and Its Applications, vol. 52, pp. 185–229. Springer, Berlin (1993)
Truskinovsky L., Vainchtein A.: The origin of nucleation peak in transformational plasticity. J. Mech. Phys. Solids 52, 1421–1446 (2004)
Truskinovsky L., Vainchtein A.: Kinetics of martensitic phase transitions: Lattice model. SIAM J. Appl. Math. 66, 533–553 (2005)
Truskinovsky L., Vainchtein A.: Quasicontinuum modelling of short-wave instabilities in crystal lattices. Phil. Mag. 85(33–35), 4055–4065 (2005)
Truskinovsky L., Vainchtein A.: Dynamics of martensitic phase boundaries: discreteness, dissipation and inertia. Cont. Mech. Thermodyn. 20(2), 97–122 (2008)
Vainchtein A.: The role of spinodal region in the kinetics of lattice phase transitions. J. Mech. Phys. Solids 58(2), 227–240 (2010)
Vainchtein A., Van Vleck E.S.: Nucleation and propagation of phase mixtures in a bistable chain. Phys. Rev. B 79(14), 144123 (2009)
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Communicated by Prof. Marshall Slemrod.
An erratum to this article can be found at http://dx.doi.org/10.1007/s00161-012-0276-3
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Trofimov, E., Vainchtein, A. Shocks versus kinks in a discrete model of displacive phase transitions. Continuum Mech. Thermodyn. 22, 317–344 (2010). https://doi.org/10.1007/s00161-010-0148-7
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DOI: https://doi.org/10.1007/s00161-010-0148-7