Modeling, Analysis, and Computation in Nonlinear Elasticity

该项目的重点是在高度可变形的弹性结构和固体，包括软弹性系统和脆性材料的力学行为的数学预测。 软结构自然存在于生物系统中，例如皮肤和生物膜，并且也被制造为薄膜和弹性体。 它们的初始不稳定性的预测，例如在加载和/或增长下的扭曲和临界后的模式形成在该项目的这一部分中起着重要作用。 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，预测增量载荷下裂纹的萌生和形成。 总的来说，该项目旨在提供新的连续力学模型和新的分析方法，从而定量了解这些系统的力学行为。 这项工作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培训机会。 将分析弹性表面结构、软弹性固体和脆性固体的非线性模型。 工作的主要目标是：（一）提供适当制定的力学模型。 (ii)为了得到严格的数学结果，即，建立存在定理--唯一真正的方法“确保对物理现象的数学描述是有意义的”（R。Courant）。 (iii)去发现新的现象。 目标（i）和（ii）相互启发和丰富;目标（iii）通过目标（i）和（ii）得以实现。 这项研究是高度跨学科的，需要来自多个领域的工具和观点，例如，非线性连续介质力学，椭圆偏微分方程系统，分叉理论，变分法，数值方法，对称性思想和生物物理学，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、弹性固体临界后生长和脂质双层结构或生物膜图案形成的新成果和见解。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.