Multi-Scale Modeling of the Stochastic Micro-Mechanical Response of Fibrous Composite Materials Based on Local Estimation and Control of Modeling Errors

建立了纤维复合材料细观力学响应的多尺度建模方法。提出的工作考虑非线性弹性静力学问题，其中问题数据如材料特性，边界数据和加载条件存在不确定性。该方法基于模型验证的概念，通过控制随机微尺度特征的建模误差，这些特征有助于微力学失效机制的启动和发展。研究目标是：1)研究几种多尺度建模技术的变化，其中易于计算的均质材料模型局部增强了随机微观结构；2)利用多尺度替代描述对建模误差进行估计；3)建立一种方法，可以解决问题中的两种不确定性：一种是由已知概率分布的随机变量控制的，另一种是根据最坏情况提供的。在许多工程学科中都遇到了异质介质，如电气工程（异质导电介质）、生物力学和生物医学工程（软组织或骨骼中的异质）、土木工程（混凝土结构）或石油工程（通过多孔介质的地下流动），从这一点来看，所提出的工作的更广泛的社会效益是显而易见的。所有这些应用都不可避免地伴随着问题数据的不确定性，这些数据通常是通过实验测量获得的（因此，本质上是统计的）。提出了在高中、本科和研究生层面的推广倡议。

A multi-scale modeling methodology of the micro-mechanical response of fibrous composites is established. The proposed work considers nonlinear elastostatics problems in which there exists uncertainty in problem data such as material characteristics, boundary data, and loading conditions. The approach is based on the notion of model validation by controlling the modeling errors in terms of stochastic micro-scale features that are instrumental in the initiation and progression of micro-mechanical failure mechanisms. The research objectives are to: 1) investigate several variations of multi-scale modeling techniques in which easily computable homogenized material models are locally enhanced with the stochastic micro-structure; 2) develop estimates of the modeling errors, incurred by using the multi-scale surrogate descriptions; and 3) establish a methodology that can address two types of uncertainty in the problem: those governed by random variables with known probability distributions and those provided in terms of worst-case scenarios.The broader societal benefits of the proposed work are evident from the realization that heterogeneous media are encountered in many engineering disciplines such as: electrical engineering (heterogeneous conducting media), biomechanics and biomedical engineering (heterogeneities in soft tissue or bones), civil engineering (concrete structures), or petroleum engineering (subsurface flow through porous media). All those applications are inevitably accompanied with uncertainty in the problem data, generally obtained through experimental measurements (thus, inherently statistical in nature). Outreach initiatives on the high school, undergraduate, and graduate levels are proposed.