A Hamilton-Jacobi equation as a refined model for plasticity, based on the continuous theory of dislocations

We discuss the continuous theory of dislocations and show that this theory leads to a Hamilton- Jacobi equation as a refined model for plasticity. The Hamilton-Jacobi equation differs from the standard constitutive equations in plasticity by a gradient term describing the dislocation density. Therefore, for materials with a dislocation density, which is everywhere positive and which does not vary too much, this gradient term can be replaced by a constant, which yields the ordinary plasticity equation. The Hamilton-Jacobi equation should therefore be used as a model for plasticity for materials with low dislocation density or for thin metallic filme, where the mobility of dislocations is restricted.

We do not present existence and uniqueness results for the refined plasticity equations. To advance the theory, in a next step such results should be proved and other mathematical problems posed by the derivation of the model equations should be investigated. One of these problems is to determine the exact form of the singularites of the solution of a linear boundary value problem of elasticty modelling closed dislocation loops.