DERIVATION AND STUDY OF DYNAMICAL MODELS OF DISLOCATION DENSITIES

In this paper, starting from the microscopic dynamics of isolated dislocations, we explain how to derive formally mean field models for the dynamics of dislocation densities. Essentially these models are tranport equations, coupled with the equations of elasticity. Rigorous results of existence of solutions are presented for some of these models and the main ideas of the proofs are given.

[1]  M. Jazar,et al.  Dynamics of Dislocation Densities in a Bounded Channel. Part II: Existence of Weak Solutions to a Singular Hamilton–Jacobi/Parabolic Strongly Coupled System , 2009, 0903.1435.

[2]  M. Jazar,et al.  Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system , 2009, 0903.0964.

[3]  R. Monneau,et al.  Time-homogenization of a first order system arising in the modelling of the dynamics of dislocation densities , 2009 .

[4]  R. Monneau,et al.  Global Existence for a System of Non-Linear and Non-Local Transport Equations Describing the Dynamics of Dislocation Densities , 2009, 0901.0219.

[5]  A. El Hajj,et al.  A convergent scheme for a non-local coupled system modelling dislocations densities dynamics , 2007, Math. Comput..

[6]  Ahmad El Hajj,et al.  Well-Posedness Theory for a Nonconservative Burgers-Type System Arising in Dislocation Dynamics , 2007, SIAM J. Math. Anal..

[7]  R. Monneau.  A transport formulation of moving fronts and application to dislocation dynamics , 2007 .

[8]  H. Ibrahim.  Existence and Uniqueness for a Nonlinear Parabolic/Hamilton-Jacobi Coupled System Describing the Dynamics of Dislocation Densities , 2007, math/0703783.

[9]  Michael Zaiser,et al.  A three-dimensional continuum theory of dislocation systems: kinematics and mean-field formulation , 2007 .

[10]  Michael Zaiser,et al.  Spatial Correlations and Higher-Order Gradient Terms in a Continuum Description of Dislocation Dynamics , 2003 .

[11]  H. Kozono,et al.  Limiting Case of the Sobolev Inequality in BMO,¶with Application to the Euler Equations , 2000 .

[12]  István Groma,et al.  Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation , 1999 .

[13]  J. Bogdanoff,et al.  On the Theory of Dislocations , 1950 .

[14]  H. Ishii,et al.  Viscosity solutions for monotone systems of second-order elliptic PDES , 1991 .