Collaborative Research: Fracture in Soft Organic Solids --- The Variational View

该奖项支持在软材料失效的数学和数值建模和分析方面的研究合作。 本研究项目涉及数学理论的推导和数值实现，该理论能够描述，解释和预测软有机固体中断裂的起始和传播-即由长碳基大分子网络组成的固体，如弹性体，凝胶和生物组织-当受到任意大的机械力时。 已知软有机固体以与标准硬固体（例如金属和陶瓷）非常不同的方式断裂。 区别在于，软有机固体中的内部断裂通过固有缺陷突然生长成大的封闭空腔/裂纹（通常称为空化的现象）而开始。 随着软材料在新技术中的使用越来越多，对有机固体何时以及如何断裂的基本和定量理解对于其发展至关重要。 同样，这种基本的和定量的理解对于推进涉及软生物组织的医学治疗，如冲击波碎石术，本项目主要研究与质量守恒原理相一致的可变形固体断裂的新变分理论（在文献中被相关公式忽略的高度非平凡的特征），并且其中新创建的表面（通过断裂）不限于是超曲面（如在经典脆性断裂中），而是也可以是N维空腔的边界，N是空间维度。 该项目的主要目标是：（1）发展一个公式的变分演化的启动和扩展的断裂在软有机固体在任意大的准静态变形，和（2）实施这一配方数值和面对其预测与新兴的实验证据的高时空分辨率。 目标（1）需要严格的存在性结果，而目标（2）需要适当的近似泛函（相场型）的建设和他们的稳定和收敛的数值实现在非凸的有限变形与约束（特别是不可压缩性）。

This award supports a research collaboration on mathematical and numerical modeling and analysis of failure in soft materials. This research project concerns the derivation and numerical implementation of a mathematical theory capable of describing, explaining, and predicting the initiation and propagation of fracture in soft organic solids---namely, solids made up of networks of long carbon-based macromolecules such as elastomers, gels, and biological tissues---when subjected to arbitrarily large mechanical forces. Soft organic solids are known to fracture in a very different manner than standard hard solids (such as metals and ceramics). The defining difference is that internal fracture in soft organic solids initiates through the sudden growth of inherent defects into large enclosed cavities/cracks (a phenomenon popularly referred to as cavitation). With the ever-increasing use of soft materials in new technologies, a fundamental and quantitative understanding of when and how organic solids fracture is of utmost importance for their advancement. Likewise, such a fundamental and quantitative understanding is critical in advancing medical treatments involving soft biological tissues, such as shock-wave lithotripsy, or treatments dealing with aneurysms.This project centers on a novel variational theory of fracture for finitely deformable solids that is consistent with the principle of conservation of mass (a highly non-trivial feature that has been overlooked in the literature by related formulations) and wherein the newly created surfaces (by fracture) are not restricted to be hypersurfaces (as in classical brittle fracture) but can also be the boundaries of N-dimensional cavities, N being the spatial dimension. The main objectives of the project are: (1) to develop a formulation in terms of variational evolutions for the initiation and propagation of fracture in soft organic solids under arbitrarily large quasi-static deformations, and (2) to implement this formulation numerically and confront its predictions with emerging experimental evidence of high spatio-temporal resolution. Objective (1) entails rigorous existence results, while objective (2) entails the construction of appropriate approximate functionals (of the phase-field type) and their stable and convergent numerical implementation in the non-convex context of finite deformations with constraints (in particular, incompressibility).