Potential-based interactions of beams and shells

Various mechanical interactions between material bodies shape the world around us. We can describe these interactions with phenomena called forces, which often stem from an interaction potential between material bodies. Two well-known examples are the gravitational potential/force that exists between masses, and the electrostatic potential/force that exists between bodies with an electric charge. Other notable examples are the van der Waals and steric forces that are responsible for interactions between bodies at micro and nano levels. The aim of mechanics is to describe the motion of material bodies. Since experimental and analytical methods are often limited in scope, computational (numerical) methods have become the main tool for analyzing mechanical interactions. The numerical simulation of interactions between bodies is a well-established scientific method that can efficiently compute accurate results for many problems. There are several well-established computational formulations for the simulation of deformable continuous bodies, such as beam and shell models. However, an accurate and efficient method for the simulation of interactions at micro and nano levels is not yet developed. For example, the behavior of biological materials is defined by the interactions between molecular assemblies that are driven by various potentials/forces. The main idea of standard existing approaches is to model interaction forces between each pair of molecules, making these methods either inefficient or lacking in detail. Motivated by the fact that many bodies at micro and nano levels resemble shapes of beams (fibers) and shells (membranes), the goal of this research is to bridge the gap in existing approaches, and to apply the methods of continuum mechanics to the potential-based interactions between molecular assemblies. The computational mechanics aims to develop efficient reduced models without sacrificing accuracy. Some well-known examples are obtained by assuming that beams cross sections and shells normal fibers are rigid. Following this line of thought, this research aims to develop improved beam and shell models along with specific interaction laws. The main assumption is that the potential-driven interactions between beams and shells can be accurately modeled with forces and moments that act at the beams axis and the shells middle surface. For this, it is necessary to simplify the interaction laws by coarse-graining and homogenizing the interaction potential. The governing equations will be solved using modern numerical techniques that are suitable for modeling smooth geometries. The derived formulation will allow efficient and accurate simulations of the potential-based interactions between various deformable bodies that resemble the shapes of beams and shells.