Mathematical and computational aspects of solid mechanics

骨折是一种普遍存在的现象，具有重大的经济、技术和社会影响。它的预测性理解影响到许多学科，包括制造、材料设计、医疗保健以及气候变化影响的预测和缓解。拟议的固体力学研究计划的数学和计算方面的Tier-1加拿大研究教席专注于由提名者共同创建的裂缝的变分相场模型。这种方法已经证明了它能够预测沿复杂路径的裂纹形核和演化，而不需要经典模型所要求的任何先验假设。它适用于在广泛的材料和情况下对裂缝演化的定量预测。第一个目标是开发和实现新一代裂缝模拟算法。为了充分利用现有的高性能计算资源，并允许对大规模真实生活问题进行数值模拟，这些都是必需的。次要目标是研究反问题，即通过规定边界变形、引起材料性质的变化或施加电场、磁场或温度场来控制裂缝的扩展。这些问题直接应用于制造技术和具有极端断裂性能的材料的系统设计。第三个目标是将变分相场模型的适用范围扩展到活体组织的断裂。具体应用包括研究天然材料和人造材料(如植入物和假体)之间的相互作用，以及活体组织在老化和各种刺激下的断裂性能的长期演变。该计划的最终目标是缩小断裂力学知识和使用之间的差距。这将通过多学科合作和研究生教育，通过组织研究讲习班和专业发展课程来实现。此外，作为该项目的一部分开发的所有模型和算法都将在一个在开放源码许可下发布的强大且有良好文档记录的软件包中实施。

Fracture is a ubiquitous phenomenon with significant economic, technological, and societal impacts. Its predictive understanding affects many disciplines, including manufacturing, materials design, healthcare, and prediction and mitigation of the effects of climate change.The proposed Tier-1 Canada Research Chair on mathematical and computational aspects of solid mechanics research program focuses on variational phase-field models of fracture, co-created by the nominee. This approach has demonstrated its ability to predict crack nucleation and evolution along complex paths without any of the a priori assumptions required by classical models. It is applicable to the quantitative prediction of fracture evolution in a broad range of materials and situations.The first objective is the development and implementation of a new generation of algorithms for fracture simulation. These are needed in order to fully leverage existing high-performance computing resources, and allow the numerical simulation of large-scale real-life problems.The secondary objective is the study of inverse problems, i.e., controlling fracture propagation by prescribing boundary deformations, inducing changes in materials properties, or applying electrical, magnetic, or temperature fields. Such problems have direct applications to manufacturing techniques and to the systematic design of materials with extreme fracture properties.The third objective aims at extending the reach of variational phase-field models to the fracture of live tissue. Specific applications include the study of interactions between natural and made made materials such as implants and prostheses, as well as the long term evolution of fracture properties of live tissues subject to aging and diverse stimuli.The final objective of this program aims at reducing the gap between knowledge and its use in fracture mechanics. This will be achieved through multidisciplinary collaborations and graduate education, through the organization of research workshops and professional development courses. Furthermore, all models and algorithm developed as part of this project will be implemented in a robust and well-documented software package released under an open-source license.