A crystal plasticity continuum theory with length scale dependent internal residual stress and free surface effect

The long range elastic interaction between dislocations is naturally accounted in
discrete dislocation plasticity through stress field of individual dislocation.
In addition, the dislocation boundaries elastic interaction is considered via
image stress superposition approach in the finite element framework. In current
study these interaction terms are considered in crystal plasticity framework
through length scale dependent internal residual stresses which are arise from
two sources: (I) long-range elastic interaction between geometrically necessary
dislocations (GNDs) in an infinite medium and (II) their interaction with
boundaries appearing as image effects.

The formulation is applied to the case of a long, thin specimen subjected to uniform curvature
which mimics the micro beam under pure bending. The analysis shows that even
under nominally uniform GND density distribution, internal stresses present due
to the finite spatial extent of the GND density field and free surfaces effects
which were not captured in the previous developed strain gradient theories. A
comparison with experimental results suggests that the length-scale for
internal stresses, described as a correlation length-scale, should increase
with decreasing specimen thickness (Motz et al, 2005). This observation is rationalized by
associating the internal length-scale with the average slip-plane spacing,
which may increase with decreasing specimen size due to paucity of dislocation
sources. Finally, we also discuss the length-scale dependent image stress in
terms of the Peach-Koehler force density proposed by Gurtin (2002). 

1) http://www.sciencedirect.com/science/article/pii/S0022509612001391 

2) http://www.sciencedirect.com/science/article/pii/S0022509610002371