Intrinsic mixed-dimensional beam-shell-solid couplings in linear Cosserat continua via tangential differential calculus

TL;DR

This paper introduces a novel method for coupling mixed-dimensional continua using the linear Cosserat micropolar model and tangential differential calculus, enabling seamless integration of different dimensional domains without intermediate elements.

Contribution

It develops a mathematically enriched coupling approach for mixed-dimensional continua based on Cosserat models and tangential calculus, avoiding traditional interface methods.

Findings

Successfully couples 3D silicone-rubber with curved graphite shell

Demonstrates coupling of silver blocks with graphite reinforcements

Validates the approach with numerical examples involving complex geometries

Abstract

We present an approach to the coupling of mixed-dimensional continua by employing the mathematically enriched linear Cosserat micropolar model. The kinematical reduction of the model to lower dimensional domains leaves its fundamental degrees of freedom intact. Consequently, the degrees of freedom intrinsically agree even at the interface with a domain of a different dimensionality. Thus, this approach circumvents the need for intermediate finite elements or mortar methods. We introduce the derivations of all models of various dimensions using tangential differential calculus. The coupling itself is then achieved by defining a mixed-dimensional action functional with consistent Sobolev trace operators. Finally, we present numerical examples involving a three-dimensional silicone-rubber block reinforced with a curved graphite shell on its lower surface, a three-dimensional silver block reinforced with a graphite plate and beams, and lastly, intersecting silver shells reinforced with graphite beams.