On the Formulation and the Micromechanical Origin of Non-Classical Models of Diffusion

扩散过程在工程和自然科学的各种应用中是至关重要的。生物组织中的药物运输、电池中的充放电循环或合金中微结构的形成只是其中的几个例子。然而，经典的菲基式扩散模型往往无法解释这些现象的复杂性，因此，更复杂的非经典扩散模型被采用。鉴于这些模型的多样性，该项目的首要目标是澄清非经典扩散模型的微观力学起源。因此，在项目的第一个阶段，使用不同的计算方法，如有限元、自然单元和等几何分析，制定并实现了一类通用的第一类和第二类梯度型刚性扩散器和微形态型扩散器。在此基础上，进一步进行计算均匀化，从微观尺度的本构响应中确定未知的宏观尺度本构响应。由底层微观结构的特征长度决定，我们认为微观尺度的问题要么是固定的，要么是固定的，这对宏观尺度的解决方案产生了尺寸效应。为了更好地解释微观层面上的小长度尺度，我们建议在第二个项目阶段引入和制定微观尺度上的能量界面，以描述沿界面或跨界面的异常扩散或表面张力耦合变形时的现象。在许多工程应用中，扩散伴随着变形，因此它们的耦合特别重要，将在计算均匀化的框架内进行深入研究。此外，为了进一步阐明扩散过程的微观力学，我们计划采用离散模型并确定其对宏观解决方案的影响。我们相信，该项目的成果在工程和材料科学方面具有特别重要的意义，例如，对于新材料或连接工艺的开发。

Diffusion processes are of utmost importance for a variety of applications in engineering and natural sciences. Drug transport in biological tissue, charge and discharge cycles in batteries, or the formation of microscructures in alloys are just a few examples. There, however, the classical diffusion model of Fickian-type often fails in explaining the complex nature of these phenomena and thus, more sophisticated non-classical diffusion models are employed. In view of the diversity of these models, the overarching goal of the project is thereby the clarification of the micromechanical origin of non-classical diffusion models. Therefore, a generic class of first and second gradient-type rigid diffusors and micromorphic-type diffusors has been formulated and implemented in the first project phase using different computational methods, e.g. finite elements, natural elements, and isogeometric analysis. Based on that, computational homogenization is and will further be carried out to determine the unknown macroscale constitutive response from the constitutive response at the microscale. Dictated by the characteristic length of the underlying microstructure, we consider the microscale problem to be either stationary or instationary, which induces size effects to the macroscale solution. To better account for small length scales at the micro-level, we propose for the second project phase the introduction and formulation of energetic interfaces at the microscale to describe phenomena like anomalous diffusion along or across an interface or surface tension when coupled to deformation. In numerous engineering applications, diffusion accompanies deformation, for which reason their coupling is particularly important and will be intensively studied within the framework of computational homogenization. Moreover, to further elucidate the micromechanics of diffusion processes we plan to employ a discrete model and identify its impact on the macroscale solution. We believe that the outcome of the of this project is of particular importance in engineering and material sciences, e.g. for the development of new materials or joining processes.