Mobility of lattice defects: Discrete and continuum approaches. (English) Zbl 1077.74512
Summary: We study a highly idealized model of a moving lattice defect allowing for an explicit, “first principles” computation of a functional relation between the macroscopic configurational force and the velocity of the defect. The discrete model is purely conservative and contains information only about elasticities of the constitutive elements. The apparent dissipation is due to the presence of microinstabilities and the nonlinearity-induced tunneling of the energy from long to short wavelengths. This type of “radiative damping” is believed to be generic and accounting for a considerable fraction of inelastic irreversibility associated with fracture, plasticity and phase transitions. The paper contains direct comparison of the exact lattice solution with various continuum and quasicontinuum approximations. Despite its simplicity, the model can be used directly for the description of dynamic phase transitions in thin films.
MSC:
| 74A60 | Micromechanical theories |
| 74N20 | Dynamics of phase boundaries in solids |
| 74J40 | Shocks and related discontinuities in solid mechanics |
References:
| [1] | Abeyaratne, R.; Vedantam, S., Propagation of a front by kink motion, (Argoul, P.; etal., IUTAM Symposium on Variations of Domains and Free Boundary Problems in Solid Mechanics (1999), Kluwer: Kluwer Dordrecht), 77-84 |
| [2] | Al’shitz, V. A.; Indenbom, V. L., Dynamic dragging of dislocations, Sov. Phys.-Usp., 18, 1, 1-20 (1975) |
| [3] | Atkinson, W.; Cabrera, N., Motion of a Frenkel-Kontorova dislocation in a one-dimensional crystal, Phys. Rev. A, 138, 3, A763-A766 (1965) |
| [4] | Boussinesq, M. J., Theorie des ondes etde remous qui se propagent le longd’un canal rectangulaire horizontal, en communiquant au liquide contenu dans cecanal des vitesses sensiblement parailles de la surface au fond, J. Math. Pures. Appl. (Ser. 2), 17, 55-108 (1872) · JFM 04.0493.04 |
| [5] | Braun, O.; Kivshar, Y., Nonlinear dynamics of the Frenkel-Kontorova model, Phys. Rep., 306, 1-108 (1998) |
| [6] | Cahn, J. W.; Mallet-Paret, J.; Van Vleck, E. S., Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59, 2, 455-493 (1999) · Zbl 0917.34052 |
| [7] | Celli, V.; Flytzanis, N., Motion of a screw dislocation in a crystal, J. Appl. Phys., 41, 11, 4443-4447 (1970) |
| [8] | Christov, C. I.; Maugin, G. A.; Velarde, M. E., Well-posed Boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev. E, 54, 3621-3638 (1996) |
| [9] | Frenkel, J.; Kontorova, T., On the theory of plastic deformation and twinning, Proc. Z. Sowj., 13, 1-10 (1938) · JFM 64.1422.02 |
| [10] | Hobart, R., Peierls stress dependence on dislocation width. Peierls-Barrier minima, J. Appl. Phys., 36, 6, 1944-1952 (1965) |
| [11] | Kunin, I. A., Elastic Media with Microstructure. 1. One-dimensional Models (1982), Springer: Springer New York · Zbl 0527.73002 |
| [12] | Marder, M.; Gross, S., Origin of crack tip instabilities, J. Mech. Phys. Solids, 43, 1, 1-48 (1995) · Zbl 0878.73053 |
| [13] | Ngan, S. C.; Truskinovsky, L., Thermal trapping and kinetics of martensitic phase boundaries, J. Mech. Phys. Solids, 47, 1, 141-172 (1999) · Zbl 0959.74052 |
| [14] | Ngan, S. C.; Truskinovsky, L., Thermoelastic aspects of nucleation in solids, J. Mech. Phys. Solids, 50, 1193-1229 (2002) · Zbl 1022.74034 |
| [15] | Pego, R. L.; Smereka, P.; Weinstein, M. I., Oscillatory instability of solitary waves in a continuum model of lattice vibrations, Nonlinearity, 8, 921-941 (1995) · Zbl 0858.35014 |
| [16] | Rosenau, P., Dynamics of nonlinear mass spring chains near the continuum limit, Phys. Rev. Lett. A, 118, 5, 222-227 (1986) |
| [17] | Slepyan, L. I., Dynamics of a crack in a lattice, Sov. Phys.-Dokl., 26, 5, 538-540 (1981) · Zbl 0497.73107 |
| [18] | Slepyan, L. I., The relation between the solutions of mixed dynamical problems for a continuous elastic medium and a lattice, Sov. Phys.-Dokl., 27, 9, 771-772 (1982) · Zbl 0541.73115 |
| [19] | Slepyan, L. I., Dynamic factor in impact, phase transition and fracture, J. Mech. Phys. Solids, 48, 927-960 (2000) · Zbl 0988.74050 |
| [20] | Slepyan, L. I., Feeding and dissipative waves in fracture and phase transition. 1. Some 1D structures and a square-cell lattice, J. Mech. Phys. Solids, 49, 469-511 (2001) · Zbl 1003.74007 |
| [21] | Slepyan, L. I., Feeding and dissipative waves in fracture and phase transition. 2. Phase-transition waves, J. Mech. Phys. Solids, 49, 513-550 (2001) · Zbl 1003.74007 |
| [22] | Slepyan, L. I., Feeding and dissipative waves in fracture and phase transition. 3. Triangular-cell lattice, J. Mech. Phys. Solids, 49, 2839-2875 (2001) · Zbl 1140.74535 |
| [23] | Slepyan, L. I.; Troyankina, L. V., Fracture wave in a chain structure, J. Appl. Mech. Techn. Phys., 25, 6, 921-927 (1984) |
| [24] | Theil, F.; Levitas, V. I., A study of a Hamiltonian model for martensitic phase transformations including micro-kinetic energy, Math. Mech. Solids, 5, 3, 337-368 (2000) · Zbl 1223.74032 |
| [25] | Wattis, J., Approximations to solitary waves on lattices2. Quasi-continuum methods for fast and slow waves, J. Phys. A, 26, 1193-1209 (1993) · Zbl 0774.35069 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
