Brittle Fracture of Dissipative Solids

由于骨折的普遍性和对无机物和有生命物质构成的结构（如桥梁、飞机、骨头和韧带）的机械性能的重大影响，几个世纪以来，它一直吸引着人类、研究人员和外行的注意。可以说，在这段漫长而丰富的历史中，在寻求完整的裂缝数学公式方面取得了最大进展。这是由于一个关键的想法，即铸造断裂现象作为能量之间的竞争：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脆性断裂的基本情况，即材料以两种方式之一对机械力作出反应：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，一些材料可以被安全地理想化为脆性弹性固体，如室温下的玻璃，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，如橡胶和铝。在这种情况下，基于研究者最近发现的普遍能量竞争，该项目旨在开发一个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训练机会。该项目有三个主要目标：1)在等温准静态机械载荷下，开发一种数学上良好定态的时间离散脆性断裂演化公式；2)建立时间离散公式的相场正则化，并确定其收敛到锐极限；3)在数值上实现所开发的相场公式，并通过不同类型固体的代表性实验验证其预测。从基本角度来看，该项目旨在为任何类型的固体在准静态机械载荷下的裂缝提供数学上良好的通用公式的第一步。换句话说，为建立所谓的脆性格里菲斯断裂是任何类型固体断裂的通用描述提供了第一步。从应用的角度来看，一种易于处理的计算工具将被开发出来，具有描述、解释和预测大型预先存在的裂缝的裂缝成核的能力，以及在任意准静态载荷下由大量耗散固体构成的结构中裂缝的扩展。这种通用的定量工具将提供对以断裂为主导的广泛现象的独特见解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.