Nonlocal continuum crystal plasticity with internal residual stresses

Coupling continuum dislocation transport with crystal plasticity for application to shock loading conditions

“…Gerken and Dawson (2008) directly address the kinematic significance of a configuration of GNDs through the three term multiplicative decomposition of the deformation gradient introduced previously by Hartley (2003), Clayton and McDowell (2003), and Clayton et al (2004). Within a small strain geometrical setting, Aghababaei et al (2011) use a kinematic decomposition similar to Acharya (2001) and compute the external and long-range internal stress fields by obtaining a Green's function solution to the Beltrami stress function adapted to the incompatible part of elastic distortion.…”

A crystal plasticity analysis of length-scale dependent internal stresses with image effects

“…To our knowledge most length-scale dependent continuum crystal plasticity frameworks with internal stresses do not explicitly discuss image stress fields arising from the long-range elastic interactions between the GNDs and free surfaces. (Aghababaei et al, 2011;Bayley et al, 2006;Evers et al, 2004;Gerken and Dawson, 2008;Gurtin, 2002;Kuroda and Tvergaard, 2008). Recently, Vinogradov and Willis (2008) and Cherednichenko (2010) derived a continuum crystal plasticity framework incorporating image stress fields using statistical mechanics based approach (Groma, 1997).…”

“…The approach expands on our previous work (Aghababaei et al, 2011) by introducing a generalized stress function that now incorporates appropriate boundary corrections through image fields. As a model system, we analyze a thin specimen experiencing uniform curvature under the action of an external bending moment.…”

“…The stress function corresponding to (i) is obtained via appropriate infinite medium Green's function (Aghababaei et al, 2011;Kröner, 1959) and the resulting stress fields are akin to the Volterra solution as applied to finite spatial extent of the GND density in an infinite medium. The solution to (ii) is incorporated by writing an available stress function for a single dislocation in a finite medium derived using complex Fourier transform approach (Fotuhi and Fariborz, 2008), equivalently in terms of the GND density.…”

“…We also compare our results with experiments and DD simulations and propose a likely origin of variable internal length-scale for internal stresses. In the following section, we first provide key equations pertaining to our previous work based on stress functions (Aghababaei et al, 2011) and then extend the approach by including additional stress functions providing image fields.…”