Pinning and Relaxation of Dislocations in Continuum and Atomistic Models

晶格位错的形核和滑移是延性晶体金属在外载荷作用下塑性变形的最重要机制。因此，研究位错运动对于理解金属和合金的机械强度具有重要意义。在这里，我们建议使用离散原子论和连续统方法来研究位错运动的一些公开问题，这对两个建模领域都是有益的。在连续体方面，我们将研究非均质环境中基于线张力的位错的变分演化。位错与非均匀环境的相互作用导致位错线的粘滑行为。现在的数学兴趣在于推导有效的演化模型，这需要描述有效位错线张力。为此，我们还将考虑放松问题。我们将使用全原子或介观(即，Peierls-Nabarro-type)模型来构建不同长度尺度上能量最优的微结构。从Peierls-Nabarro模型得到的微结构能否在实验中观察到，这是一个悬而未决的问题。这种松弛现象还没有用现实的原子论模型来研究。因此，我们建议使用具有真实相互作用势的分子动力学模拟来研究从Peierls-Nabarro模型预测的位错结构是否能够稳定。为了能够在观察这种介观弛豫现象所需的长度尺度上进行原子模拟，我们将开发基于变分格林函数的新方法来为原子模拟提供精确的弹性边界条件。除了经典材料外，我们还将考虑高熵合金(HEAs)，这是一类通常由五种或五种以上元素组成的高浓度材料。HEAS因其潜在的卓越强度和硬度、耐磨性、耐腐蚀性和抗氧化性以及其他所需的性能而备受关注。从建模的角度来看，HEAS带来了新的挑战，因为这些材料中的位错被浸没在一个随机的空间波动环境中。我们将推导出新的数学工具以及原子模拟方法，以实现这种随机环境的随机同质化。

Nucleation and glide of lattice dislocations is the most important mechanism of plastic deformation of ductile crystalline metals under external load. Studying dislocation motion is therefore of fundamental importance for understanding the mechanical strength of metals and alloys. Here, we propose to study some open problems of dislocation motion using both discrete atomistic and continuum approaches, of mutual benefit to both fields of modeling. On the continuum side, we will study line-tension based variational evolution of dislocations in heterogeneous environments. Interaction of dislocations with a heterogeneous environment leads to a stick-slip-behavior of the dislocation line. The mathematical interest now lies in the derivation of effective evolution models, which requires a description of the effective dislocation line tension. To this end, we will also consider relaxation problems. We will use full atomistic or mesoscopic (i.e., Peierls-Nabarro-type) models to construct energetically optimal microstructures on various length scales. It is an open question whether the microstructures derived from Peierls-Nabarro-type models can be observed in experiments. Such relaxation phenomena have not been studied using realistic atomistic models. We thus propose to use molecular dynamics simulation with realistic interaction potentials to study whether dislocation structures predicted from Peierls-Nabarro models can be stabilized. In order to be able to perform atomistic simulations on the length scales necessary to observe such mesoscopic relaxation phenomena, we will develop novel variational Green's function-based methods to provide exact elastic boundary conditions for atomistic simulation. Aside from classical materials we will also consider High Entropy Alloys (HEAs), a class of materials composed of usually five or more elements in high concentration. HEAs are interesting due to their potentially exceptional strength and hardness, wear resistance, and corrosion and oxidation resistance, among other desirable properties. From a modeling point of view, HEAs pose new challenges, as dislocations in these materials are immersed in a random spatially fluctuating environment. We will derive both novel mathematical tools as well as atomistic simulation approaches for stochastic homogenization of this random environment.