×

zbMATH — the first resource for mathematics

Effect of nonlinearity on the steady motion of a twinning dislocation. (English) Zbl 1189.37082
Summary: We consider the steady motion of a twinning dislocation in a Frenkel-Kontorova lattice with a double-well substrate potential that has a non-degenerate spinodal region. Semi-analytical traveling wave solutions are constructed for the piecewise quadratic potential, and their stability and further effects of nonlinearity are investigated numerically. We show that the width of the spinodal region and the nonlinearity of the potential have a significant effect on the dislocation kinetics, resulting in stable steady motion in some low-velocity intervals and lower propagation stress. We also conjecture that a stable steady propagation must correspond to an increasing portion of the kinetic relation between the applied stress and dislocation velocity.

MSC:
37K60 Lattice dynamics; integrable lattice equations
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Müllner, P.; Chernenko, V.A.; Kostorz, G., Stress-induced twin rearrangement resulting in change of magnetization in a ni – mn – ga ferromagnetic martensite, Scr. mater., 49, 129-133, (2003)
[2] Bray, D.; Howe, J., High-resolution transmission electron microscopy investigation of the face-centered cubic/hexagonal close-packed martensite transformation in co-31.8 wt pct ni alloy: part I. plate interfaces and growth ledges, Metall. mater. trans. A, 27A, 3362-3370, (1996)
[3] Hirth, J.P., Ledges and dislocations in phase transformations, Metall. mater. trans. A, 25A, 1885-1894, (1994)
[4] Hirth, J.P.; Lothe, J., Theory of dislocations, (1982), John Wiley and Sons
[5] Abeyaratne, R.; Vedantam, S., A lattice-based model of the kinetics of twin boundary motion, J. mech. phys. solids, 51, 1675-1700, (2003) · Zbl 1048.74031
[6] Atkinson, W.; Cabrera, N., Motion of a Frenkel-Kontorova dislocation in a one-dimensional crystal, Phys. rev. A, 138, 3, 763-766, (1965)
[7] Carpio, A.; Bonilla, L.L., Discrete models of dislocations and their motion in cubic crystals, Phys. rev. B, 71, 13, 134105, (2005)
[8] Celli, V.; Flytzanis, N., Motion of a screw dislocation in a crystal, J. appl. phys., 41, 11, 4443-4447, (1970)
[9] Ishioka, S., Uniform motion of a screw dislocation in a lattice, J. phys. soc. Japan, 30, 323-327, (1971)
[10] Ishioka, S., Steady motion of a dislocation in a lattice, J. phys. soc. Japan, 34, 462-468, (1973)
[11] Kresse, O.; Truskinovsky, L., Mobility of lattice defects: discrete and continuum approaches, J. mech. phys. solids, 51, 1305-1332, (2003) · Zbl 1077.74512
[12] Flytzanis, N.; Crowley, S.; Celli, V., High velocity dislocation motion and interatomic force law, J. phys. chem. solids, 38, 539-552, (1977)
[13] Peyrard, M.; Kruskal, M.D., Kink dynamics in the highly discrete sine-Gordon system, Physica D, 14, 88-102, (1984)
[14] Boesch, R.; Willis, C.R.; El-Batanouny, M., Spontaneous emission of radiation from a discrete sine-Gordon kink, Phys. rev. B, 40, 2284-2296, (1989)
[15] Braun, O.M.; Kivshar, Y.S., The frenkel – kontorova model: concepts, methods and applications, () · Zbl 1140.82001
[16] Frenkel, J.; Kontorova, T., On the theory of plastic deformation and twinning, Proc. Z. sowj., 13, 1-10, (1938) · JFM 64.1422.02
[17] Carpio, A.; Bonilla, L.L., Oscillatory wave fronts in chains of coupled nonlinear oscillators, Phys. rev. E, 67, 056621, (2003)
[18] Vainchtein, A., The role of spinodal region in the kinetics of lattice phase transitions, J. mech. phys. solids, 58, 2, 227-240, (2010) · Zbl 1193.82042
[19] Slepyan, L.I., Models and phenomena in fracture mechanics, (2002), Springer-Verlag New York · Zbl 1047.74001
[20] Weiner, J.H.; Sanders, W.T., Peierls stress and creep in a linear chain, Phys. rev., 134, 4A, 1007-1015, (1964)
[21] O. Kresse, Lattice models of propagating defects, Ph.D. Thesis, University of Minnesota, Minneapolis, MN 2002.
[22] Carpio, A., Nonlinear stability of oscillatory wave fronts in chains of coupled oscillators, Phys. rev. E, 69, 046601, (2004)
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.