Carpio Rodríguez, Ana María and Chapman, S.J. and Velázquez, J.J. L.
(2001)
*Pile-up solutions for some systems of conservation laws modelling dislocation interaction in crystals.*
SIAM Journal on applied mathematics, 61
(6).
pp. 2168-2199.
ISSN 0036-1399

PDF
Restricted to Repository staff only until 2020. 302kB |

Official URL: http://epubs.siam.org/siap/resource/1/smjmap/v61/i6/p2168_s1?isAuthorized=no

## Abstract

Some continuum models for dislocation interactions in a simple crystal geometry are studied. The simplest models are mixed systems ofconservation laws which are shown to exhibit singularities and instabilities. These are then regularized, leading to parabolic free-boundary problems. In both cases, solutions describing the formation of structures such as dislocation pile-ups are discussed.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Dislocation dynamics, degenerate systems of conservation laws, free-boundary problems, pattern formation |

Subjects: | Sciences > Physics > Mathematical physics Sciences > Mathematics > Differential equations |

ID Code: | 15097 |

References: | E.C. Aifantis, On the problem of dislocation patterning and persistent slip bands, Solid Phenomena State, Vols. 3 and 4 (1988), pp. 397–406. D.G. Aronson, Nonnegative solutions of linear parabolic equations, Ann. Scuola. Norm. Sup. Pisa, 22 (1968), pp. 607–694. D.J. Bacon and D. Hull, Introduction to Dislocations, International series on Materials Science and Technology 37, Pergamon Press, Elmsford, NY, 1984. Z.S. Basinski, M.S. Duesbery, and R. Taylor, Influence of shear stress on screw dislocations in a model sodium lattice, Canad. J. Phys., 49 (1971), pp. 2160–2180. A. Carpio and S.J. Chapman, On the modelling of instabilities in dislocation interaction, Philos. Magazine B, 78 (1998), pp. 155–157. A. Carpio, S.J. Chapman, S.J. Howison, and J.R. Ockendon, Dynamics of line singularities, Philos. Trans. Roy. Soc. London Ser. A, 355 (1997), pp. 2013–2024. B. Devincre and L.P. Kubin, Simulations of forest interactions and strain hardening in fcc crystals, Model. Simul. Mater. Sci. Eng., 2 (1994), pp. 559–570. A. Fasano, M. Primicerio, and A.A. Lacey, New results on some classical free boundary problems, Quart. Appl. Math., 38 (1981), pp. 439–460. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964. W.G. Johnston, J.J. Gilman, Dislocation velocities, dislocation densities and plastic flow in lithium fluoride crystals, J. Appl. Phys., 30 (1958), p. 129. A.K. Head, S.D. Howison, J.R. Ockendon, and S.P. Tighe, An equilibrium theory of dislocation continua, SIAM Rev., 35 (1993), pp. 580–609. J.P. Hirth, and J. Lothe, Theory of Dislocations, John Wiley and Sons, New York, 1982. N.M. Kubin, Y. Estrin, and G. Canova, Dislocation patterns and plastic instabilities, in Patterns, Defects and Materials Instabilities, NATO Adv. Sci. Inst. Ser., D. Walgraef and N.M. Ghoniem, eds., Kluwer, Dordrecht, The Netherlands, 1990, pp. 277–301. P.S. Lomdahl and D.J. Srolovitz, Dislocation generation in the two dimensional Frenkel-Kontorova model at high stresses, Phys. Rev. Lett., 57 (1986), pp. 2702–2705. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, Berlin, 1983. M. Shearer and D. Schaeffer, The influence of material nonuniformity preceding shear-band formation in a model for granular flow, European J. Appl. Math., 8 (1998), pp. 457–483. V. Vitek, Effect of dislocation core structure on the plastic properties of metallic materials,in Dislocations and Properties ofReal Materials, The Institute ofMetals, London, 1985,pp. 30–50. G.B. Whitham, Linear and Nonlinear Waves, Wiley Science, New York, London, Sydney,1974. |

Deposited On: | 04 May 2012 11:44 |

Last Modified: | 06 Feb 2014 10:16 |

Repository Staff Only: item control page