Brittle Fracture of Dissipative Solids

几个世纪以来，由于骨折的普遍性和对由无机和活体材料组成的结构(如桥梁、飞机、骨骼和韧带)的力学性能的高风险影响，骨折引起了人类、研究人员和门外汉的关注。可以说，正是在这段漫长而丰富的历史的过去25年里，在寻求完整的裂缝数学公式方面取得了最大的进展。这是由于一个关键的想法，即将断裂现象塑造为能量之间的竞争：使结构变形所需的能量和在结构中产生裂缝所需的能量。关键的是，这一进展仅限于弹性固体中脆性断裂的基本情况，即材料以两种方式之一对机械力做出响应：它们要么弹性变形，要么产生新的表面，即它们断裂。然而，虽然在一定的限制条件下，某些材料可以安全地理想化为脆性弹性固体，就像室温下的玻璃一样，但所有材料在变形时都会耗散能量，主要是通过粘性或塑性变形，或两者兼而有之，例如橡胶和铝。在这种背景下，基于研究人员最近发现的一种普遍的能量竞争，该项目旨在开发一种严格的数学公式来描述耗散固体中的断裂。该项目将为研究生提供跨学科的研究培训机会。该项目有三个主要目标：1)发展一种在等温准静态机械载荷作用下的一大类耗散固体中脆性断裂演化的数学适定的时间离散公式；2)发展时间离散公式的相场正则化，并建立其收敛到精确极限；以及3)数值实施所发展的相场公式，并通过对不同类型固体的代表性实验来验证其预测。从根本上讲，该项目试图为任何类型的固体在准静态机械载荷下的断裂提供一个数学上合适的通用公式的第一步。换句话说，提供第一步，以确定所谓的脆性格里菲斯断裂是对任何类型固体的断裂的普遍描述。从应用的角度来看，将开发一种易于处理的计算工具，能够描述、解释和预测由大的预先存在的裂纹形成的裂缝的形核，以及在任意准静态载荷下由一大类耗散固体组成的结构中的裂缝的扩展。这样一个通用的量化工具将提供对由断裂主导的广泛现象的特殊洞察。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Because of its pervasiveness and high-stakes impact on the mechanical performance of structures made of inorganic and live matter alike, such as bridges, airplanes, bones, and ligaments, fractures have attracted the attention of humans, researchers and laymen alike, for centuries. Arguably, it is in the past 25 years of this long and rich history that most progress has been made in the quest for a complete mathematical formulation of fractures. This has been made possible by a pivotal idea, to wit, the casting of the phenomenon of fracture as a competition between energies: the energy required to deform the structure and the energy required to create a crack in the structure. Critically, this progress has been restricted to the elementary case of brittle fracture in elastic solids, that is, materials that respond in one of two ways to mechanical forces: they either deform elastically or create new surface, i.e., they fracture. Yet, while within certain restricted conditions some materials may be safely idealized as brittle elastic solids, as in the case of glass at room temperature, all materials dissipate energy when they deform, primarily by viscous or plastic deformation, or both, as for rubber and aluminum. In this context, based on a universal energy competition recently discovered by the investigator, this project aims to develop a rigorous mathematical formulation to describe fracture in dissipative solids at large. The project will provide interdisciplinary research training opportunities for graduate students. The project has three main objectives: 1) to develop a mathematically well-posed time-discrete formulation of brittle fracture evolution in a large class of dissipative solids subjected to isothermal quasistatic mechanical loading; 2) to develop a phase-field regularization of the time-discrete formulation and establish its convergence to the sharp limit; and 3) to numerically implement the developed phase-field formulation and validate its predictions against representative experiments on different types of solids. From a fundamental standpoint, the project seeks to provide a first step towards a mathematically well-posed universal formulation of fractures in any type of solid subjected to quasistatic mechanical loading. In other words, to provide a first step in establishing that the so-called brittle Griffith fracture is a universal description of fracture in any type of solid. From an applications standpoint, a tractable computational tool will be developed with the capability to describe, explain, and predict the nucleation of a fracture from large pre-existing cracks, as well as the propagation of fractures in structures made of a large class of dissipative solids under arbitrary quasistatic loading. Such a general quantitative tool would provide exceptional insight into a broad spectrum of phenomena dominated by fracture.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.