Theoretical and Computational Foundations of Multi-Scale Analysis of Inelastic Solid Materials

该项目的关键方面是在多个尺度上对固体材料的微观到宏观转变和微观结构发展的新观点的发展。微观结构可以先验地定义为非均质材料的代表性体积元，也可以在均质材料中由于材料失稳现象在变形过程的某一阶段产生。最近发展的均匀化增量变分公式为非弹性材料的均匀化分析提供了一个全新的观点，其中细尺度波动场被定义为适当定义的均匀化泛函的能量最小值。方法基础是应用数学概念处理非凸能量最小化问题及其凸化问题。将这些概念应用到工程问题中，需要发展一种基于变分的非弹性固体稳定性一般理论，并构建新的计算工具，以便有效处理非线性固体力学中的分层多尺度均匀化过程。我们专注于有限弹塑性和损伤力学的本构模型在金属和岩土材料方面的应用。该项目的总体结果将是更好地理解固体中材料不稳定现象的数学基础，物理机制和数值。

The key aspect of the project is the development of new perspectives towards the formulation of micro-to-macro transitions and micro-structure developments in solid materials on multiple scales. The micro-structures may a priori be defined as representative volume elements of heterogeneous materials or may in homogeneous materials be generated at a certain stage of the deformation process as a result of a material instability phenomenon. A fundamentally new viewpoint to the homogenization analysis of inelastic materials is provided by recently developed incremental variational formulations of homogenization where fine-scale fluctuation fields are defined as energy minimizers of suitably defined homogenization functionals. Methodical basis is the application of mathematical concepts for the treatment of non-convex energy minimization problems and their convexifications. Specification of these concpts to engineering problems needs the development of a variational-based generic theory of stability for inelastic solids and the construction of new computational tools for an effective handling of hierarchical multi-scale homogenization procedures in nonlinear solid mechanics. We focus on constitutive models of finite elastoplasticity and damage mechanics with regard to applications to metals and geomaterials. The overall result of the project will be a better understanding of the mathematical basis, the physical mechanisms and the numerical of material instability phenomena in solids.