A quadratic finite element for the relaxed micromorphic model

Abstract

In this work we discuss the relaxed micromorphic model and implementation details for a full three-dimensional formulation entailing a quadratic Lagrangian-Nédélec finite element and appropriate boundary conditions in the discrete setting.

The relaxed micromorphic model is a generalized continuum theory with the capacity to capture more complex kinematical behaviour than in the classical Cauchy continua. Such behaviour is commonly found in materials with a pronounced micro-structure such as porous media and metamaterials. The theory introduces the microdistortion field, encompassing nine additional degrees of freedom for each material point in the continuum, effectively turning each material point into a deformable micro-body. The uncommon discrete formulation stems from the employment of the Curl operator in the energy functional of the relaxed micromorphic model, thus requiring H(curl)-conforming finite elements for well-posedness to be inherited in the discrete setting. The model further introduces the so called consistent coupling condition, which requires some technical considerations in order to be upheld correctly.

This work demonstrates the finite element formulation, culminating with a numerical example.