AEI-DFG: Nonlocal gradients in variational analysis and materials modelling: limits, kernels, boundaries

• Applicant: Professorin Dr. Carolin Kreisbeck
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541520348

Traditionally, many phenomena in nature, science and engineering are modelled with differential equations and local variational principles. Locality in this context means that the behaviour of an object depends only on its immediate neighbourhood. However, there are situations where the usual local modelling falls short because longer-range interactions and global effects have to be taken into account. This gives rise to nonlocal models, whose benefits are very prominent in many application areas, partly because they provide effective ways to bridge between different length scales and have shown to lead to refined predictions. Some of those areas are materials science, which is the focus of this proposal, as well as imaging and machine learning. This project will contribute to a better understanding of nonlocality by addressing a new relevant class of variational problems, where the functionals are integrals depending on nonlocal gradients. These problems differ from standard hyperelasticity, which involves usual gradients, and from peridynamics, where the nonlocality is not expressed with a gradient. Problems with nonlocal gradients are not yet well explored, despite their advantageous properties. In starting from functionals that involve a specific nonlocal gradient, the overall goal of our research programme is twofold: (A) We aim to provide the theoretical foundations to establish these functionals as suitable energies in a refined nonlocal model for hyper elasticity. Potential benefits refer to accuracy and the possibility of going beyond purely elastic behaviour by accounting for singular effects. Our justification of the new model comprises several facets, including relations with local and linear models, a discussion of suitable local boundary values, and non-interpenetration of matter. (B) We propose to develop a variational theory in a broad framework of nonlocal gradients, considering also different types of boundary conditions. This involves the introduction of new function spaces and requires a careful investigation of fundamental structural properties of nonlocal gradients that differ from those of their local counterparts. Despite impressive developments in the last years, many conceptual questions still remain open and new techniques need to be established to address relevant types of nonlocal structures. Apart from the foundations already laid by the applicants and others in recent works, we will borrow and adapt some tools from other classical areas of mathematical analysis.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partner: Professor Dr. Carlos Mora Corral