A kinematic model for continuous distributions of dislocations

On the dynamics of continuous distributions of dislocations

“…For this reason, each exact R3-valued two-form dto describes a dislocation density on M, cf. [28,Theorem 4.1]. Summarizing we have Theorem 3.1.…”

“…Thus, as an appropriate framework for describing elastic materials with dislocations, we consider so-called generalized configurations 7 = dj + /3 G Z(A/;R3) whose integrable component lrThis can be made precise in mathematical terms, cf. [28]. In fact, one can show that any possible dislocation density can be described in terms of generalized configurations 7 G V(M;R3), cf.…”

“…Following [28], we consider a material whose interior structure is given by some triple of vector fields Xi, X2, X3 G T(TM) such that 7ex(X,) = e,, l = 1,2,3,…”

“…A may play the role of Hooke's law (e.g., see [28]) and ||<ij7||2 is a measure for the volume of the configuration 7. Assuming that A o dj-y is an II2 i exact one-form, the equations of motion (32) given in Theorem 7.1 reduce to pdth = S(A 0 dh) for the elastic components and <j<9t2/?7 = /(||dj7||2)/?7 for the plastic components.…”

“…Following [28], we define a generalized configuration space for a material with dislocations V(M;R3) as a submanifold of fi1(M;R3) such that the gradient part dj7 € Q1 (M; R3) of each generalized configuration 7 G V(M; R3) stems from an embedding j7 6 E(AI; ]R3). In classical terms, dj7 is precisely the deformation gradient of an actual configuration j1 of the system, whereas e?7 corresponds to the dislocation density associated with this configuration.…”