Given a set of inelastic material models, a microstructure, a macroscopic structural geometry, and a set of boundary conditions, one can in principle always solve the governing equations to determine the system's mechanical response. However, for large systems this procedure can quickly become computationally overwhelming, especially in three-dimensions when the microstructure is locally complex. In such settings multi-scale modeling offers a route to a more efficient model by holding out the promise of a framework with fewer degrees of freedom, which at the same time faithfully represents, up to a certain scale, the behavior of the system. In this paper, we present a methodology that produces such models for inelastic systems upon the basis of a variational scheme. The essence of the scheme is the construction of a variational statement for the free energy as well as the dissipation potential for a coarse scale model in terms of the free energy and dissipation functions of the fine scale model. From the coarse scale energy and dissipation we can then generate coarse scale material models that are computationally far more efficient than either directly solving the fine scale model or by resorting to FE-square type modeling. Moreover, the coarse scale model preserves the essential mathematical structure of the fine scale model. An essential feature for such schemes is the proper definition of the coarse scale inelastic variables. By way of concrete examples, we illustrate the needed steps to generate successful models via application to problems in classical plasticity, included are comparisons to direct numerical simulations of the microstructure to illustrate the accuracy of the proposed methodology.

翻译：给定一组非弹性材料模型、微结构、宏观结构几何以及边界条件，原则上总可以通过求解控制方程来确定系统的力学响应。然而，对于大型系统，这一过程会迅速变得计算量巨大，尤其是在微结构局部复杂的三维问题中。在此类情况下，多尺度建模通过提供一种自由度更少的框架，同时在一定尺度上忠实地反映系统行为，为构建更高效的模型提供了途径。本文提出了一种基于变分方案的非弹性系统建模方法。该方案的核心在于，根据细尺度模型的自由能和耗散函数，为粗尺度模型构建关于自由能及耗散势的变分表述。通过粗尺度的自由能与耗散函数，我们可以生成粗尺度材料模型，其计算效率远高于直接求解细尺度模型或采用FE平方型建模。此外，粗尺度模型保留了细尺度模型的基本数学结构。此类方案的关键特征在于正确定义粗尺度非弹性变量。通过经典塑性问题中的具体实例，我们阐述了生成成功模型所需的步骤，并与微结构直接数值模拟结果进行了对比，以验证所提出方法的准确性。