Mechanics of Fractal Materials

分形材料力学

• 批准号: 1030940
• 负责人: Martin Starzewski
• 金额: $ 20万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1030940&HistoricalAwards=false
• 关键词: Mechanics; Fractal; Materials

The objectives of this proposed research is to provide funding for (i) the analysis of formation of fractal patterns at elastic-inelastic transitions and (ii) the development of continuum poromechanics of fractal porous materials saturated with fluids. The first task will involve parallel computation, stochastic mathematics and physics of critical phenomena, so as to enable simulation of growth of fractal sets of plastic grains in 2D and 3D. Also, random field models will be developed in order to provide a theoretical framework for understanding elastic-inelastic transitions in spatially random solid-like and soil-like media, including functionally-graded materials. The second task will entail a generalization of continuum thermomechanics through dimensional regularization to materials with fractal geometries and arbitrary anisotropies, aiming at formulation of extremum and variational principles, governing equations and solution of initial-boundary value problems in static as well as dynamic settings. All the governing equations will explicitly depend on the fractal dimensions of mass distribution and bounding surfaces, so that a reduction to the well-known classical strong forms of partial differential equations can always be accomplished. While the subject matter of fractals in mechanics of materials offers several basic challenges, the pay-off will be significant for natural (geological, biological, etc.) as well as man-made materials. Thus, (i) the formation of fractal patterns at elastic-inelastic transitions in composites, solids, and soils will be better understood; and (ii) the highly complex, fractal-type media (polymer clusters, gels, fluid-saturated rock systems, percolating networks, neural and pulmonary systems, etc.) will become open to studies conventionally reserved for smooth materials. Overall, the results of this research will lead to a broadening of applicability of continuum mechanics and physics.

这项研究的目的是提供资金（一）在弹性-非弹性过渡的分形图案的形成分析和（ii）的分形多孔材料饱和流体的连续孔隙力学的发展。第一个任务将涉及并行计算，随机数学和物理学的关键现象，使模拟的分形集的塑料颗粒在2D和3D的增长。此外，随机场模型将被开发，以提供一个理论框架，用于理解弹性-非弹性过渡的空间随机固体和土壤介质，包括功能梯度材料。第二个任务将需要通过尺寸正则化的材料与分形几何形状和任意各向异性的连续热力学的推广，旨在制定极值和变分原理，控制方程和初边值问题的解决方案，在静态和动态设置。所有的控制方程将明确地依赖于质量分布和边界面的分形维数，使减少到众所周知的经典强形式的偏微分方程总是可以完成。虽然材料力学中的分形主题提供了几个基本挑战，但其回报将对自然（地质，生物等）以及人造材料。因此，（i）在复合材料，固体和土壤中的弹性-非弹性转变的分形图案的形成将得到更好的理解;和（ii）高度复杂的分形型介质（聚合物簇，凝胶，流体饱和岩石系统，膨胀网络，神经和肺系统等）。将对通常为光滑材料保留的研究开放。总体而言，本研究的结果将导致连续介质力学和物理的适用性的扩大。