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。这项研究项目涉及一种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-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--在任意大的机械力作用下的断裂开始和扩展。众所周知，软有机固体的断裂方式与标准硬固体(如金属和陶瓷)非常不同。定义不同之处在于，软有机固体中的内部断裂是通过固有缺陷突然扩展到大的封闭空穴/裂缝(通常称为空化现象)而开始的。随着软材料在新技术中的使用不断增加，对有机固体何时以及如何破裂的基本和定量的了解对于它们的进步至关重要。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疗治疗是至关重要的，例如冲击波碎石术，或处理动脉瘤的治疗。本项目的中心是一个新的变分理论，用于有限变形的固体，它符合质量守恒原理(一个在相关公式中被忽略的高度非平凡的特征)，其中新产生的表面(通过骨折)不限于超曲面(如在经典的脆性骨折中)，但也可以是N维腔的边界，N是空间维度。该项目的主要目标是：(1)发展任意大的准静态变形下软有机固体中裂纹的起始和扩展的变分演化公式，以及(2)数值实现该公式，并将其预测与新出现的高时空分辨率的实验证据相对抗。目标(1)需要严格的存在性结果，而目标(2)需要构造适当的近似泛函(相场型)，并在有约束(特别是不可压缩)的非凸形变背景下实现其稳定和收敛的数值实现。

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).