An Iterated Homogenization Method to Study Cavitation in Soft Solids

Lopez-PamiesDMS-1009503 实验证据表明，具有足够大的三轴度的载荷条件可以引起弹性体（和其他软）固体内部突然出现内部空腔。这种不稳定性的发生，通常被称为空化，可以归因于预先存在的缺陷到有限尺寸的增长。 由于其与材料失效的发生有着密切的联系，空蚀现象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技术的发展，以处理广泛的一类非凸变分问题。 虽然近年来已经取得了相当大的进展，通过能量最小化方法建立存在的结果，基本问题，定量预测的发生ofcavitation在真实的材料系统仍然在很大程度上没有得到解决。在这个项目中，主要研究者开发了一个新颖的框架来研究空化：（i）适用于实际感兴趣的大类非线性弹性固体，（ii）允许具有任意三轴度的3D一般加载条件，（iii）包含关于初始形状、空间分布和空化可能引发的潜在缺陷的机械性质的直接信息，以及（iv）同时，计算上可处理的。 这是通过一个创新的迭代均匀化方法，允许为非线性弹性材料的力学响应包含初始无限小的空腔（或缺陷）的随机分布的建设的精确解。 这些包括解决方案的大小的变化作为一个功能的底层空腔所施加的loadingconditions，从其中可以确定空化的发病。尽管它的一般性，分析proposedformulation减少到研究易于处理的Hamilton-Jacobi方程，其中腔的初始尺寸扮演时间的角色和施加的负载扮演空间的角色。 该项目提供了一种新的方法，从根本上不同于现有的方法来调查固体中的缺陷的影响。 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，将软异质材料的宏观性质直接与其微观性质和潜在的微观结构联系起来，这是现代科学许多领域的中心问题。 除了提出微观行为如何影响宏观行为的基本理解之外，这些技术还为工程师和科学家提供了数学工具来表征和预测各种软复合材料的机械响应和失效，包括弹性体复合材料（例如，填充的弹性体）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).