Thermodynamically consistent continuum dislocation dynamics

Abstract

Dislocation based modeling of plasticity is one of the central challenges at the crossover of materials science and continuum mechanics. Developing a continuum theory of dislocations requires the solution of two long standing problems: (i) to represent dislocation kinematics in terms of a reasonable number of variables and (ii) to derive averaged descriptions of the dislocation dynamics (i.e. material laws) in terms of these variables. The kinematic problem (i) was recently solved through the introduction of continuum dislocation dynamics (CDD), which provides kinematically consistent evolution equations of dislocation alignment tensors, presuming a given average dislocation velocity (Hochrainer (2015), Philos. Mag. 95 (12), 1321-1367).In the current paper we demonstrate how a free energy formulation may be used to solve the dynamic closure problem (ii) in CDD. We do so exemplarily for the lowest order CDD variant for curved dislocations in a single slip situation. In this case, a thermodynamically consistent average dislocation velocity is found to comprise five mesoscopic shear stress contributions. For a postulated free energy expression we identify among these stress contributions a back-stress term and a line-tension term, both of which have already been postulated for CDD. A new stress contribution occurs which is missing in earlier CDD models including the statistical continuum theory of straight parallel edge dislocations (Groma et al. (2003), Acta Mater. 51, 1271-1281). Furthermore, two entirely new stress contributions arise from the curvature of dislocations.

Dislocation patterning in a two-dimensional continuum theory of dislocations

“…One can easily see that the trivial homogeneous solution ρ = ρ 0 , κ = 0, and τ mf = τ 0 satisfies Eqs. (37), (38), and (51), where ρ 0 and τ 0 are constants representing the initial dislocation density and the external shear stress, respectively. The stability of the trivial solution can be analyzed by applying the standard method of linear stability analysis.…”

“…Recently this work was complemented by the derivation of matching energy functionals based upon averaging the elastic energy functionals of the corresponding discrete dislocation systems [36]. In parallel, it was demonstrated how such energy functionals can be used to derive closed-form dislocation dynamics equations which are consistent not only with thermodynamics, but also with the constraints imposed by the ways in which dislocations move in 3D [37].…”

On the Derivation of Boundary Conditions for Continuum Dislocation Dynamics

“…with a non-negative mobility function M (π α , ρ α ) as introduced in [15]. With the above constitutive relations, we find that the microscopic force balance yields…”

“…Such a constitutive theory has recently been derived [15] with methods from irreversible thermodynamics, provided that there is an energy density given in terms of the CDD density variables. A suitable energy density was recently derived from a local density approximation by Zaiser [16].…”

“…A suitable energy density was recently derived from a local density approximation by Zaiser [16]. However, the constitutive model in [15] excluded the discussion of boundary conditions. The derivation of thermodynamically consistent boundary conditions for CDD is the original objective of the current paper.…”