In the context of crystalline materials, an inhomogeneous plastic slip, which is closely related to a nonlocal mechanical behavior, is often associated with the pile-up of dislocations at ob- stacles, e.g., grain boundaries [1]. The interaction between dislocations and obstacles always involve internal length scales such as the distance between two dislocations and cause so-called size effects. These size effects are more pronounced the smaller a sample or the grains that it consists of. A prominent example, experimentally investigated for several metals [2], is the Hall-Petch effect [3, 4]. Gradient crystal plasticity theories are based on extended continua and can account for a nonlocal mechanical behavior. Thus, they are frequently considered to model size effects at a continuum scale while accounting for interactions at the grain scale. In contrast to the classical Cauchy-Boltzmann continuum, extended continua take into account additional degrees of freedom (DOFs) [5]. Several methods are known to derive the field equations for the additional DOFs [6]. In this presentation, the derivation is based on the invariance of an extended energy balance with respect to a change of observer [7]. The extended energy balance expands the classical energy balance by supplementary contributions that are associated with the plastic slip as an additional, scalar-valued DOF. Regarding single-slip, a thermodynami- cally consistent, nonlocal flow rule for the additional DOF is obtained by exploitation of the Coleman-Noll procedure [8]. For oligo- and polycrystals, grain boundaries, which are classi- cally considered as singular material surfaces, can be treated numerically efficient as a diffuse interface. A corresponding phase-field model is briefly discussed.

References

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[3] Hall, E. O. The deformation and ageing of mild steel: III discussion of results, Proc. Phys. Soc., London, Sect. B (1951) 64(9):747–753

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[7] Prahs, A., Böhlke, T. On invariance properties of an extended energy balance. Continuum. Mech. Thermodyn. (2019) 32:843–859

[8] Coleman, B. D., Noll, W. The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. Anal. (1963) 13(1):167–178