A novel continuum approach to gradient plasticity based on the complementing concepts of dislocation and disequilibrium densities

Abstract

A geometrically linear continuum mechanics framework is proposed for gradient plasticity combining 'strain gradients' and, with a novel approach, 'stress gradients'. Thereby the duality of kinematic and kinetic quantities is exploited in view of the 'div-grad-curl orthogonality' in continuum eld theories. On the one hand the non-integrability of the plastic distortion results in the well-established dislocation density-often denoted as the geometrically-necessary-dislocation (GND) density-that enters the energy storage function. On the other handas entirely novel concept introduced in this contribution-the non-equilibrium of the plastic stress results in the disequilibrium density that parameterizes the dual dissipation potential within the convex analysis setting of plasticity. Consequently both, the dislocation density as well as the disequilibrium density contribute in modelling the size-dependent hardening state of a material in a continuum mechanics setting. The novel approach is eventually elucidated in much detail for the specic case of single crystal plasticity.

A thermodynamically consistent theory of stress-gradient plasticity

“…The torque-twist curves for different values of the stress-gradient length scale and the interaction coefficient q are shown in Figure 3, and in magnified form in Figure 4. The results obtained using the disequilibrium density formulation introduced in [37] are also shown.…”

“…We consider the problem studied in [37], of a single crystal cylindrical specimen with face centred cubic unit cells subjected to simple torsion. The cylinder is assumed to be long, so that it suffices to assume periodic behaviour over a length L, and to consider a section of length L. The cylinder has radius R = 10 µm.…”

“…The material parameters for the micro wire torsion problem are taken from [37], except for the reference slip rate which is omitted in the current formulation since it only scales 1, while the length scale and the self/latent ratio q in (44) are varied and given in the result plots.…”

“…Table 2 gives the limiting torque, defined as T l = T (κ = 5mm −1 ) and q = 1.0. The Torque in nN m = 0.00 = 0.025 q = 0 = 0.025 q = 1.4 = 0.050 q = 0 = 0.050 q = 1.4 DE densities [10] Figure 3: Torque vs. twist for different values of and q compared with results from the disequilibrium density model introduced in [37]. Torque in nN m = 0.00 = 0.025 q = 0 = 0.025 q = 1.4 = 0.050 q = 0 = 0.050 q = 1.4 DE densities [10] Figure 4: Torque vs. twist for different values of and q as in Figure 3, magnified strengthening, that is, the increase in limiting torque with length scale, is clearly seen.…”

A Predictive Discrete-Continuum Multiscale Model of Plasticity With Quantified Uncertainty

“…Another class of models that address microscale plasticity, without limitation in length and time scales, is the strain gradient plasticity (SGP). After the initial introduction of the gradient theory of plasticity by Aifantis [3,2], there have been extensive developments of SGP models by Fleck and Hutchinson [40], and Nix and Gao [90], along with the theories developed within a thermodynamically consistent framework found in [43,44,49,48,55,52,51,129,7,87,125]. For a comprehensive review of the theoretical developments of SGP models and their applications, the interested readers are referred to Voyiadjis and Song [139], and the references cited therein.…”