Diffusion-Driven Fracture

断裂力学通常与预测材料在给定力作用下裂纹的发展和扩展有关。然而，在某些情况下，实际的力在很大程度上取决于裂缝本身的几何形状。一个经典的例子是由蒸发引起的泥裂模式，它本身受到新裂纹表面发展的强烈影响。这类问题还包括水力压裂、铁和压电材料的断裂、金属的腐蚀应力断裂、玻璃和聚合物的环境应力断裂、锂离子电池电极断裂、热裂纹以及水泥中的离子扩散等。在这个项目中，研究者为这种扩散驱动的断裂问题构建严格的模型，在并行超级计算机上为这些模型开发计算机代码，并使用这些模型和代码来研究工业、工程和科学应用中感兴趣的具体问题。研究生参与项目工作。裂缝的变分相场模型最初是由研究者设计的，并在应用数学界得到进一步发展，在过去几年中得到了爆炸性的发展。它们的主要优势在于它们能够解决脆性材料准静态断裂中两个最关键的问题：裂纹路径识别和裂纹成核。虽然数学家通常关心的是在给定载荷下寻找固体的平衡位移和裂纹形态，但在许多情况下，载荷必须从次要问题中导出，并且可能受到断裂几何形状的影响。这些问题包括水力压裂、铁和压电材料的断裂、金属的腐蚀应力断裂、玻璃和聚合物的环境应力断裂、锂离子电池的电极断裂、热裂缝和水泥中的离子扩散，这些问题的研究大多是临时进行的。该项目的目标是为裂缝由扩散过程驱动的相场模型建立坚实的理论基础。研究者和他的学生实现了这些模型，并将其作为开源软件发布。他们使用这些数学和计算工具来研究二维和三维扩散载荷下的三维裂纹形核。研究生参与项目工作。

Fracture mechanics is often concerned with predicting the development and propagation of cracks in materials subject to given forces. In some situations, however, the actual forces depend strongly on the geometry of the cracks themselves. A classical example is that of mud-crack patterns induced by evaporation, which is itself strongly influenced by the development of new crack surfaces. Other problems of this class are commonly encountered in hydraulic fracturing, fracture in ferro- and piezoelectric materials, corrosion stress fracture of metals, environmental stress fracture of glass and polymers, fracture of electrodes in lithium-ion batteries, thermal cracks, and ion diffusion in cements. In this project, the investigator constructs rigorous models for such diffusion-driven fracture problems, develops computer codes for the models on parallel supercomputers, and uses the models and codes to study specific problems of interest in industrial, engineering, and scientific applications. Graduate students participate in the work of the project.Variational phase-field models of fracture, originally devised by the investigator and further developed in the applied mathematics community, have seen an explosive growth in the last few years. Their main strength is their ability to address two of the most critical issues in the quasi-static fracture of brittle materials: crack path identification and crack nucleation. While mathematicians are typically concerned to find the equilibrium displacement and crack configuration of a solid under a given load, in many situations the loadings must be derived from a secondary problem and may be affected by the fracture geometry. These include hydraulic fracturing, fracture in ferro- and piezoelectric materials, corrosion stress fracture of metals, environmental stress fracture of glass and polymers, fracture of electrodes in lithium-ion batteries, thermal cracks, and ion diffusion in cements, which have been studied in a largely ad hoc manner. The goal of this project is to establish a strong theoretical foundation for phase-field models where the fracture is driven by a diffusive process. The investigator and his students implement these models and release them as open source software. They use these mathematical and computational tools to study three-dimensional crack nucleation under diffuse loads in two and three dimensions. Graduate students participate in the work of the project.