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。所提出的工作考虑了非线性弹性静力学问题，其中存在材料特性、边界数据和加载条件等问题数据的不确定性。该方法基于模型验证的概念，通过随机微尺度特征控制建模误差，这些特征在微机械失效机制的开始和发展中起着重要作用。研究的目标是：1)研究几种多尺度建模技术，其中容易计算的均匀材料模型被随机微结构局部增强；2)对使用多尺度代理描述所引起的建模误差进行估计；和3)建立一种方法，可以解决问题中的两种不确定性：由已知概率分布的随机变量控制的不确定性和根据最坏情况提供的不确定性。从认识到在许多工程学科中遇到非均匀介质，如电气工程(异质导电介质)、生物力学和生物医学工程(软组织或骨骼中的非均质性)、土木工程(混凝土结构)或石油工程(地下渗流通过多孔介质)，拟议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)。提出了高中、本科和研究生层面的外展倡议。

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.