Derivation of F=FeFp as the continuum limit of crystalline slip

Abstract

In this paper we provide a proof of the multiplicative kinematic description of crystal elastoplasticity in the setting of large deformations, i.e. F = F e F p , for a two dimensional single crystal. The proof starts by considering a general configuration at the mesoscopic scale, where the dislocations are discrete line defects (points in the two-dimensional description used here) and the displacement field can be considered continuous everywhere in the domain except at the slip surfaces, over which there is a displacement jump. At such scale, as previously shown by two of the authors, there exists unique physically-based definitions of the total deformation tensor F and the elastic and plastic tensors F e and F p that do not require the consideration of any non-realizable intermediate configuration and do not assume any a priori relation between them of the form F = F e F p . This mesoscopic description is then passed to the continuum limit via homogenization i.e., by increasing the number of slip surfaces to infinity and reducing the lattice parameter to zero. We show for two-dimensional deformations of initially perfect single crystals that the classical continuum formulation is recovered in the limit with F = F e F p , det F p = 1 and G = Curl F p the dislocation density tensor.

Homogenization of layered materials with rigid components in single-slip finite crystal plasticity

“…A substantial body of literature has emerged in materials science, engineering, and mathematics, see for instance [27,36] and the references therein, or more specifically, [23,29,40,44] for heterogeneous plastic materials, and [2,33,35] for fiber-reinforced materials, and [6,10] for high-contrast composites, to mention just a few references.…”

Homogenization of vector-valued partition problems and dislocation cell structures in the plane

“…The idea of approximating continuous maps by piecewise smooth maps appears also in the theory of structured deformations, proposed in [17] to study plasticity models (without a focus on dislocations) and analysed in [9,10], see also [18] for an overview and [37,38] for a related approach.…”

On plastic dislocation density tensor

“…G.I. Taylor [1] realized that plastic deformation could be explained in terms of the theory of dislocations, even since this view has become a consensus that mechanism of plastic deformation is the result of dislocation accumulation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][22][23][24]. Accordingly, some plastic dislocation density tensors have been proposed [2][3][4][5][6][7][8].…”

“…Reina et al [11][12][13][14] did a comprehensive and in depth studies on the Ortiz's definition T Ortiz = −F p × ∇. Berdichevsky [8] introduced a measure of the resultant closure failure leading to the dislocation density tensor T Berdichevsky = −F p−1 · (F p × ∇).…”