An Iterated Homogenization Method to Study Cavitation in Soft Solids

实验证据表明，具有足够大的三轴性的加载条件可以诱导弹性体（和其他软）固体内腔的突然出现。这种不稳定性的发生，通常被称为空化，可归因于预先存在的缺陷增长到有限尺寸。由于空化现象与材料的破坏起始密切相关，因此受到了材料界和力学界的广泛关注。空化也一直是数学界感兴趣的课题，因为它的建模促进了处理一类广泛的非凸变分问题的技术的发展。虽然近年来通过能量最小化方法建立存在性结果取得了相当大的进展，但关于在实际材料系统中定量预测空化发生的基本问题在很大程度上仍未解决。在这个项目中，首席研究员开发了一个新的研究空化的框架：(i)适用于实际感兴趣的大类非线性弹性固体，（ii）允许具有任意三轴性的3D一般加载条件，（iii）包含有关空化可能引发的潜在缺陷的初始形状，空间分布和机械特性的直接信息，并且（iv）同时在计算上易于处理。这是通过一种创新的迭代均匀化方法来实现的，该方法允许构建包含初始无限小空腔（或缺陷）随机分布的非线性弹性材料的机械响应的精确解。这些包括作为应用负载条件的函数的底层空腔尺寸变化的解决方案，由此可以确定空化的开始。尽管其具有普遍性，但对所提出的公式的分析可以归结为可处理的汉密尔顿-雅可比方程的研究，其中空腔的初始尺寸起时间的作用，施加的载荷起空间的作用。该项目提供了一种全新的方法，完全不同于现有的方法来研究固体中缺陷的影响。这是力学中的一个核心课题，对于理解和预测现实世界材料的失效具有重要意义。更一般地说，该项目旨在开发分析技术，将软非均质材料的宏观特性直接与它们的微观特性和潜在的微观结构联系起来，这是现代科学许多领域的核心问题。除了提出微观行为如何影响宏观行为的基本理解之外，这些技术为工程师和科学家提供了数学工具来表征和预测广泛的软复合材料的机械响应和失效，包括弹性体复合材料（例如，填充弹性体）和生物组织（例如，动脉壁）。

Lopez-PamiesDMS-1009503 Experimental evidence has shown that loading conditions withsufficiently large triaxialities can induce the sudden appearanceof internal cavities within elastomeric (and other soft) solids. The occurrence of such instabilities, commonly referred to ascavitation, can be attributed to the growth of pre-existingdefects into finite sizes. Because of its close connection withmaterial failure initiation, the phenomenon of cavitation hasreceived much attention from the materials and mechanicscommunities. Cavitation has also been a subject of interest inthe mathematical community because its modeling has prompted thedevelopment of techniques to deal with a broad class ofnon-convex variational problems. While in recent yearsconsiderable progress has been made via energy minimizationmethods to establish existence results, fundamental problemsregarding the quantitative prediction of the occurrence ofcavitation in real material systems remain largely unresolved. In this project, the principal investigator develops a novelframework to study cavitation that: (i) is applicable to largeclasses of nonlinear elastic solids of practical interest, (ii)allows for 3D general loading conditions with arbitrarytriaxiality, (iii) incorporates direct information on the initialshape, spatial distribution, and mechanical properties of theunderlying defects at which cavitation can initiate, and (iv) is,at the same time, computationally tractable. This isaccomplished by means of an innovative iterated homogenizationmethod that allows for the construction of exact solutions forthe mechanical response of nonlinear elastic materials containingrandom distributions of initially infinitesimal cavities (ordefects). These include solutions for the change in size of theunderlying cavities as a function of the applied loadingconditions, from which the onset of cavitation can be determined. In spite of its generality, the analysis of the proposedformulation reduces to the study of tractable Hamilton-Jacobiequations in which the initial size of the cavities plays therole of time and the applied load plays the role of space. This project makes available a fresh methodology radicallydifferent from existing approaches to investigate the influenceof defects in solids. This is a core topic in mechanics, ofgreat importance for understanding and predicting the failure ofreal-world materials. More generally, the project aims todevelop analytical techniques that link the macroscopicproperties of soft heterogeneous materials directly to theirmicroscopic properties and underlying microstructures, a centralissue in many fields of modern science. Beyond putting forwardfundamental understanding of how microscopic behavior influencesmacroscopic behavior, these techniques provide engineers andscientists with mathematical tools to characterize and predictthe mechanical response and failure of a broad spectrum of softcomposite materials, including elastomeric composites (e.g.,filled elastomers) and biological tissues (e.g., arterial walls).