Forschungsprojekt ::

Variierende Horizonte: Mathematische Fundierung neuer Materialmodelle mit nichtlokalen Energien

Projektbeschreibung

The mathematical models that are traditionally used to describe many phenomena in nature and technology are local. However, they often fall short of capturing complex global phenomena or discontinuity effects. For this reason, modeling approaches involving nonlocality have attained increased attention in the last years. The need for a rigorous analysis of nonlocal models has brought up a myriad of challenging open problems in the mathematics community.
In this research project, which is a first step towards establishing a strong long-term research relationship between the applicant’s group at KU and the Spanish experts at UAM and UCLM, the focus lies on novel material models in elasticity theory. In response to the recent discovery that the widely used models in peridynamics are not suitable for recovering standard energy densities in hyperelasticity, Bellido, Cueto & Mora-Corral [BCMC22] took a new path by choosing energies of integral type with densities that depend on nonlocal gradients.
The cutting-edge research question we will address is that of how to establish a rigorous connections between the classical local model of hyperelasticity and the new one from [BCM22] through a localization procedure, that is, a limit passage for vanishing horizon. If successful, this analysis will confirm [BCM22] as a consistent modeling alternative, with the benefit of allowing any discontinuities. From the viewpoint of methodology, new to attack these problems comprises elements of asymptotic variational analysis, properties of fractional function spaces, and Fourier techniques.
This collaboration is intended to lead to a high-impact publication in a peer-reviewed journal on applied analysis, such as Nonlinear Analysis or SIMA. In the long run, it will serve as a basis for a sustainable mutual exchange of ideas, knowledge, and staff between the groups. It further gives the perspective for other joint activities, such as applying for international research grants.