Constrained shearing of a thin crystalline strip: Application of a continuum dislocation-based model

Abstract

The problem of simple shearing of a constrained thin crystalline strip posed by Shu et al. ͓J. Mech. Phys. Solids 49, 1361 ͑2001͔͒ is reanalyzed using a nonlocal continuum dislocation-based model of plastic deformation that accounts for the flexibility of curved dislocations carrying the plastic deformation. Attention is paid to the inhomogeneous profiles of plastic slip and lattice rotation as well as to the size-dependent material response of the strip in single slip and symmetric double slip. Additional boundary conditions and an internal length scale with a clear physical interpretation appear quite naturally in the analysis.

Continuum theory of evolving dislocation fields

“…We note the boundary layer of the steeply changing orientation in the last loading step that has not been properly resolved. Already the earlier applications of the present model have shown that pronounced boundary layers characterized by steep gradients or even steps in plastic slip occur in the vicinity of impenetrable interfaces [22,23,25]. In accord with the above analysis, we observed upcoming spurious oscillations that we have suppressed by using extremely fine discretization of the space variable as well as extremely small time steps, which is computationally very expensive.…”

“…Unlike as in a local theory, the distribution and amount of plastic slip in the present model depends on the width H of the plastic channel, cf. Sedláček and Werner [25]. The results are shown in Fig.…”

“…At an impenetrable interface between an elastic and a plastically deforming materials, a surface dislocation density can be deposited, causing a step-like change in the plastic slip from zero in the elastic material to a finite value in the plastic region [47]. The surface dislocation density was discussed also in the framework of the present model [22,23,25]. It is convection-caused and presents a limit case of the boundary layer discussed above.…”

“…2 (left), the 'upper' and 'lower' parts of the loops can be considered as the single-valued fields. In this way, many problems of practical interest have been solved, among others formation of a dipolar dislocation structure [24], shearing [25] and bending [37] of a thin film, as well as deformation of a composite material in plane strain, where the present method has been coupled with finite elements [38]. Because of its simplicity, this pragmatic approach is adopted in the present paper as well.…”

“…An alternative approach to the density-based modelling of dislocations is to start with the continuum theory of dislocations that was developed originally to deal with the description of incompatibility and internal stresses resulting from a prescribed dislocation distribution [16]. In the present work, we further develop the non-local continuum theory of evolving dislocation fields based on the continuum theory of moving dislocations [16,17,18,19,20,21] that we have proposed earlier [22,23,24,25]. The length-scale dependency of this approach results from the line tension of the curved dislocations squeezing through small deforming volumes.…”