Submitted Abstract
In order to model the macroscopic behavior of crystalline materials, it is vital to find robust coarse-grained models. Thus, it is important to understand the microscopic effects that arise during inelastic deformation which is primarily caused by dislocation glide, the motion of twin boundaries and the motion of coherent phase boundaries. While a dislocation is a topological line defect, twin and phase boundaries are surface-like defects that separate the crystal in two regions of different orientation or structure. Most of the commonly used materials are precipitate hardened in order to increase their yield strength and hence the interaction of the above-mentioned defects and the precipitates has to be well understood. Caused by the motion of these defects, there are different effects, e.g., hysteresis, that emerge. In this project, we will try to identify the appropriate mathematical models that incorporate these emergent effects and discuss their properties. As the propagation of the defects can be described by an energy balance equation, we will investigate the right notion of solution, e.g., energy solutions or viscosity solutions for the induced partial differential equation and use them to describe the microscopic behavior of the material. To obtain a coarse-grained model, we will need to transition from the microscopic to the macroscopic layer. Classical homogenization methods cannot be used as they do not incorporate long range collective behavior that emerges from these systems. As a first step to derive these models, we will prove scaling laws and used them to explain how the motion of the defects is affected if the concentration of precipitates decreases. Our long-term objective is to rigorously derive models suitable for the macro-scale materials simulation and the design of improved materials.