Modeling, Analysis, and Computation in Nonlinear Elasticity

该项目致力于对高度变形的弹性结构和固体中的力学行为进行数学预测，包括软弹性系统和脆性材料。软结构自然存在于生物系统中，如皮肤和生物膜，也可制成薄膜和弹性体。在本项目的这一部分中，对它们的初始不稳定性(如起皱)和在加载和/或增长下形成临界图案的预测起着突出的作用。对于脆性固体(如低温下的陶瓷或钢)，我们将使用一种新的断裂方法，预测在增量加载下裂纹的萌生和形成。总体而言，该项目旨在提供新的连续统-力学模型和分析它们的新方法，从而导致对这些系统的力学行为的定量理解。这项工作具有一系列潜在的应用--从生物分子结构到工程机械和结构。该项目包括研究生的研究性培训机会。将分析一类弹性表面结构、软弹性固体和脆性固体的非线性模型。这项工作的主要目标是：(1)提供适当制定的基于力学的模型。(Ii)获得严格的数学结果，即建立存在定理--这是“确保对物理现象的数学描述是有意义的”的唯一真正的方法(R.库兰特)。(三)发现新现象。目标(一)和(二)相互通报和丰富；目标(三)由目标(一)和(二)促成。这项研究是高度跨学科的，需要来自多个领域的工具和视角，如非线性连续介质力学、椭圆型偏微分方程组、分叉理论、变分、数值方法、对称思想和生物物理学，同时提供它们之间的新联系。该项目将提供有关软弹性固体表面折痕形成、脆性材料断裂、弹性固体超临界增长和脂质双层结构或生物膜中图案形成的新结果和见解。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is focused on the mathematical prediction of mechanical behavior in highly deformable elastic structures and solids, including both soft elastic systems and brittle materials. Soft structures occur naturally in biological systems such as skin and bio-membranes and are also manufactured as thin films and elastomers. The prediction of their initial instability, such as wrinkling, and post-critical pattern formation under loading and/or growth play a prominent role in this part of the project. For brittle solids (such as ceramics or steels at low temperature), we will use a new approach to fracture, predicting the initiation and formation of cracks under incremental loading. Overall, the project aims to provide new classes of continuum-mechanical models and novel approaches to their analyses, leading to a quantitative understanding of the mechanical behavior of these systems. The work has a range of potential applications – from bio-molecular structures to engineering machines and structures. This project includes opportunities for the research training of graduate students. Classes of nonlinear models of elastic-surface structures, soft elastic solids, and brittle solids will be analyzed. The main goals of the work are: (i) To provide properly formulated mechanics-based models. (ii) To obtain rigorous mathematical results, viz., establish existence theorems – the only true way to “ensure that the mathematical description of a physical phenomenon is meaningful” (R. Courant). (iii) To detect new phenomena. Goals (i) and (ii) inform and enrich each other; goal (iii) is enabled by goals (i) and (ii). This research is highly interdisciplinary, requiring tools and perspectives from several fields, e.g., nonlinear continuum mechanics, elliptic PDE systems, bifurcation theory, calculus of variations, numerical methods, symmetry ideas and biophysics, while providing new links between them. The project will provide new results and insights pertaining to surface-creasing formation in soft elastic solids, fracture of brittle materials, post-critical growth in elastic solids and pattern formation in lipid-bilayer structures or bio-membranes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.