A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity

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Micromorphic Approach for Gradient Elasticity, Viscoplasticity, and Damage (2009, background): “In Bayley et al. 2006 , 18 dislocation densities were treated as additional degrees-of-freedom.”
Crystal Plasticity Finite Element Methods: in Materials Science and Engineering (2010, background): “Grain boundary mechanics, Hall–Petch behavior, grain interaction, grain size effects, strain gradient effects, nonlocal formulations, interface mechanics, superplasticity Becker and Panchanadeeswaran (1995) Mika and Dawson (1998), Acharya and Beaudoin (2000), Meissonnier, Busso, and O’Dowd (2001) Ba”
On Strain Hardening Mechanism in Gradient Nanostructures (2021, background): “…to the long-range internal stress (Bayleyet al., 2006; Evers et al., 2004; Groma et al., 2003; Ye ﬁ mov et al., 2004), the internal stress resulted from single edge GND can be given as (Bayley et al., 2006) where R is the integral circular domain for GNDs to contribute to the back stress.”