A viscoplastic strain gradient crystal plasticity theory based on the gradient of the equivalent plastic strain ∇γ eq is proposed. A grain boundary yield condition is introduced. The microstructural explanation of the Hall-Petch effect, accounting for notch-like stress concentrations at the grain boundary as a result of discrete slip bands, is reviewed. Periodic tensile test FEM simulation results illustrate the prediction of the numerical model. The requirements of modern products, like mechanical components for cars or airplanes, are demanding. Amongst other things they must be small, light-weight, robust and cheap. Therefore, a permanent product optimization is going on. This procedure demands high-performance materials and there is a significant industrial interest in material design and material development. The trend shows that material optimization as well as the design of new materials takes place. It is desired to develop materials by microstructural design. The preferred tool are simulations since these are often cheaper and faster than experiments. Additionally, the miniaturization trend requires micromechanical models for small systems. The problem is that these are often inadequate and/or computationally expensive material models. For example, they show weaknesses when size effects, interfaces or damage have to be modeled. Therefore, the focus of the contribution at hand is the modeling of metallic microstructures with interfaces (grain boundaries). This work is in accordance with [7]. The main new contribution constitutes details on the numerical solution based on a geometrical multigrid solver in combination with a locally refined mesh. This mesh has been obtained by deforming a perfectly cube shaped mesh.
Effects to be ModeledThe effects to be modeled are the effects which determine the macroscopic behavior. As far as elastic anisotropy is concerned, excellent continuum models exist. The same holds for crystal plasticity dominated by forest hardening. The models for size effects (e.g. dislocation pile-ups) are limited. Extended research is going on in this field like in the region of grain boundaries. Here, we focus on pile-ups and grain boundaries. An external loading of a given grain aggregate provokes dislocation motion inside of the grains. The grain boundaries (GBs) represent obstacles for dislocation motion. One main reason is the grain misorientation. As a consequence, dislocations pile up at the GBs. Initially, the GBs are often assumed impenetrable for dislocations. When more and more dislocations pile up there is a certain break through: dislocations enter and leave the GB (the grain boundary yields). This effect is considered later. At first, a short review of gradient plasticity model for impenetrable GBs is given.
Review of Gradient Plasticity Models for Impenetrable GBsFor simplicity a single slip setting at small strains is applied. As an example of a gradient plasticity model for impenetrable GBs the set of equations in Table 1 is considered. The equations are very similar to thos...