Mathematical Analysis on Peridynamic Models

该奖项支持首席研究人员正在进行的研究计划，该计划对“非局部模型”进行了仔细的分析检查，这是一种相对较新的数学模型，已被证明对模拟某些具有挑战性的现象非常有效，例如固体力学中的断裂。例如，了解材料的行为、失效以及变形时的强度，对于正确使用材料以及设计对制造、材料工程和相关技术具有潜在影响的新材料至关重要。为此，过去已经提出了不同程度成功的模式。这项研究项目旨在发展基本的数学技术，为最近提出的连续介质力学的“周期动力学模型”提供适当的分析基础。这些发现也将适用于其他类似结构的非局部模型，并在社会科学和生物科学中应用。这些活动不仅将有助于成功和有效地进行建模、开发和实验验证，而且还将确保基于这些非局部理论的未来建模和仿真工作将更加定量和可靠。该研究项目将为研究生培养提供机会和支持。首席调查员将把该项目的成果融入课堂教学和其他教育工作中。该项目涉及一般非本地模型，特别是动态模型的理论和技术的发展。这些模型的特点是在单一的数学框架内用积分方程组代替微分方程组，有效地描述了连续和不连续的场。这些模型已被成功地应用于描述跳跃随机过程、异常扩散以及固体中裂纹的自发形成和扩展，仅举几个例子。然而，这些模型也给科学界带来了新的数学挑战。这项研究致力于探索一些分析问题，同时为未来对非局部和动态模型的分析奠定必要的数学基础。要解决的问题包括研究活动，这些活动促进了对具有实际意义的线性化周期动力学模型的分析的知识；非局部方程解作为作用力、初始数据和系数的函数的正则性；以及对基于周期动力学的非线性行为的理解。这些方法涉及到各种工具，这些工具导致将经典数学概念和技术扩展到非局部环境，包括微扰法、变分法和非线性泛函分析。此外，将制定的基本数学基础设施可能会影响有效和可靠的有限元方法和其他数值方案的发展，以通过动力学来解决复杂的工程问题。这项研究将使基于非局部和动态的建模与仿真在数学上更加一致，并有助于使此类建模与仿真在实际应用中更具定量化和预测性。

This award supports the ongoing research program of the Principal Investigator on the close analytical examination of "nonlocal models," a type of mathematical model of relatively recent vintage that has proved to be very effective for modeling certain challenging phenomena, such as fracture in solid mechanics. For example, understanding how materials behave, their failure as well as their strength when deformed, is crucial for their proper usage and also for the design of new materials with potential impact on manufacturing, materials engineering, and related technologies. For this purpose, models with varied levels of success have been proposed in the past. This research project aims to develop basic mathematical techniques that will deliver proper analytical footing for the recently proposed "peridynamic model" of continuum mechanics. The findings will also be applicable to other nonlocal models of similar structure, with applications in social and biological sciences. The activities not only will contribute to the success and effectiveness of attempts on modeling development and experimental validation but also will ensure that future modeling and simulation efforts based on these nonlocal theories will be more quantitative and reliable. The research project will provide opportunities and support for the training of graduate students. The Principal Investigator will integrate the findings of the project into classroom teaching and other educational endeavors.This project concerns the development of theory and techniques for nonlocal models in general and for the peridynamic model in particular. These models are characterized by their effective description of continuous as well as discontinuous fields within a single mathematical framework by using integral equations in lieu of differential equations. The models have been successfully applied to better describe jump stochastic processes, anomalous diffusion, and spontaneous formation and propagation of cracks in solids, to name a few applications. However, the models have also presented the scientific community with new mathematical challenges. This research is devoted to exploring some analytical issues while at the same time laying the necessary mathematical foundation for future analyses on nonlocal and peridynamic models. Issues to be addressed include research activities that advance knowledge on the analysis of linearized peridynamic models of practical interest; regularity properties of solutions of nonlocal equations as a function of applied force, initial data, and coefficients; and understanding of peridynamic-based nonlinear behavior. The approaches involve various tools that lead to extensions of classical mathematical concepts and techniques to the nonlocal setting, including perturbation methods, calculus of variations, and nonlinear functional analysis. Furthermore, the basic mathematical infrastructures that will be worked out are likely to impact the development of effective and reliable finite element methods and other numerical schemes to solve complex engineering problems via peridynamics. The research will make nonlocal and peridynamics-based modeling and simulation more mathematically consistent, and it will contribute to making such modeling and simulation more quantitative and predictive in practical applications.