×

zbMATH — the first resource for mathematics

The role of spinodal region in the kinetics of lattice phase transitions. (English) Zbl 1193.82042
Summary: We consider a one-dimensional chain of phase-transforming springs with harmonic long-range interactions. The nearest-neighbor interactions are assumed to be trilinear, with a spinodal region separating two material phases. We derive the traveling wave solutions governing the motion of an isolated phase boundary through the chain and obtain the functional relation between the driving force and the velocity of a phase boundary which can be used as the closing kinetic relation for the classical continuum theory. We show that a sufficiently wide spinodal region substantially alters the phase boundary kinetics at low velocities and results in a richer solution structure, with phase boundaries emitting short-length lattice waves in both direction. Numerical simulations suggest that solutions of the Riemann problem for the discrete system converge to the obtained traveling waves near the phase boundary.

MSC:
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
70F45 The dynamics of infinite particle systems
65L99 Numerical methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Atkinson, W.; Cabrera, N., Motion of a frenkel – kontorova dislocation in a one-dimensional crystal, Physical review A, 138, 3, 763-766, (1965)
[2] Celli, V.; Flytzanis, N., Motion of a screw dislocation in a crystal, Journal of applied physics, 41, 11, 4443-4447, (1970)
[3] Chin, R.C.Y., Dispersion and Gibbs phenomenon associated with difference approximations to the initial boundary-value problems for hyperbolic equations, Journal of computational physics, 18, 233-247, (1975) · Zbl 0313.65067
[4] Flytzanis, N.; Crowley, S.; Celli, V., High velocity dislocation motion and interatomic force law, Journal of physics and chemistry of solids, 38, 539-552, (1977)
[5] Ishioka, S., Uniform motion of a screw dislocation in a lattice, Journal of physical society of Japan, 30, 323-327, (1971)
[6] Kresse, O.; Truskinovsky, L., Lattice friction for crystalline defects: from dislocations to cracks, Journal of the mechanics and physics of solids, 52, 2521-2543, (2004) · Zbl 1084.74005
[7] Marder, M.; Gross, S., Origin of crack tip instabilities, Journal of the mechanics and physics of solids, 43, 1-48, (1995) · Zbl 0878.73053
[8] Ngan, S.-C.; Truskinovsky, L., Thermo-elastic aspects of dynamic nucleation, Journal of the mechanics and physics of solids, 50, 1193-1229, (2002) · Zbl 1022.74034
[9] Purohit, P.K., 2002. Dynamics of phase transitions in strings, beams and atomic chains. Ph.D. Thesis, California Institute of Technology, Pasadena, California.
[10] Slemrod, M., Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Archive for rational mechanics and analysis, 81, 301-315, (1983) · Zbl 0505.76082
[11] Slepyan, L.I., Models in the theory of brittle fracture waves, Mechanics of solids, 12, 170-175, (1977)
[12] Slepyan, L.I., Dynamics of a crack in a lattice, Soviet physics doklady, 26, 5, 538-540, (1981) · Zbl 0497.73107
[13] Slepyan, L.I., Models and phenomena in fracture mechanics, (2002), Springer New York · Zbl 1047.74001
[14] Slepyan, L.I.; Cherkaev, A.; Cherkaev, E., Transition waves in bistable structures. II. analytical solution: wave speed and energy dissipation, Journal of the mechanics and physics of solids, 53, 407-436, (2005) · Zbl 1146.74336
[15] Slepyan, L.I.; Troyankina, L.V., Fracture wave in a chain structure, Journal of applied mechanics and technical physics, 25, 6, 921-927, (1984)
[16] Truskinovsky, L., Structure of an isothermal phase jump, Soviet physics doklady, 30, 945-948, (1985)
[17] Truskinovsky, L.; Vainchtein, A., Peierls – nabarro landscape for martensitic phase transitions, Physical review B, 67, 172103, (2003)
[18] Truskinovsky, L.; Vainchtein, A., The origin of nucleation peak in transformational plasticity, Journal of the mechanics and physics of solids, 52, 1421-1446, (2004) · Zbl 1079.74012
[19] Truskinovsky, L.; Vainchtein, A., Explicit kinetic relation from “first principles”, (), 43-50 · Zbl 1192.74287
[20] Truskinovsky, L.; Vainchtein, A., Kinetics of martensitic phase transitions: lattice model, SIAM journal on applied mathematics, 66, 533-553, (2005) · Zbl 1136.74362
[21] Truskinovsky, L.; Vainchtein, A., Quasicontinuum models of dynamic phase transitions, Continuum mechanics and thermodynamics, 18, 1-2, 1-21, (2006) · Zbl 1101.74049
[22] Truskinovsky, L.; Vainchtein, A., Dynamics of martensitic phase boundaries: discreteness, dissipation and inertia, Continuum mechanics and thermodynamics, 20, 2, 97-122, (2008) · Zbl 1160.74398
[23] Weiner, J.H.; Sanders, W.T., Peierls stress and creep in a linear chain, Physical review, 134, 4A, 1007-1015, (1964)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.