Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model

TL;DR

This work addresses size effects in metamaterial beams by leveraging the relaxed micromorphic continuum to bridge micro- and macro-scale responses under bending. It develops a parameter-identification workflow that ties the macro elasticity to standard homogenization and introduces two approaches to bound the micro-elasticity, including non-affine boundary conditions, with a consistent coupling condition enabling full-scale operation between macro and micro ranges. Through calibration against fully resolved metamaterial beams and validation on additional loadings, the study demonstrates that the relaxed micromorphic model can reproduce size-dependent bending behavior with bounded stiffness and provides practical guidelines for boundary conditions and curvature scaling. The findings offer a tractable, scale-aware modeling framework for architected materials, enabling efficient prediction of size effects in metamaterial beams across multiple loading scenarios.

Abstract

In this paper we model the size-effects of metamaterial beams under bending with the aid of the relaxed micromorphic continuum. We analyze first the size-dependent bending stiffness of heterogeneous fully discretized metamaterial beams subjected to pure bending loads. Two equivalent loading schemes are introduced which lead to a constant moment along the beam length with no shear force. The relaxed micromorphic model is employed then to retrieve the size-effects. We present a procedure for the determination of the material parameters of the relaxed micromorphic model based on the fact that the model operates between two well-defined scales. These scales are given by linear elasticity with micro and macro elasticity tensors which bound the relaxed micromorphic continuum from above and below, respectively. The micro elasticity tensor is specified as the maximum possible stiffness that is exhibited by the assumed metamaterial while the macro elasticity tensor is given by standard periodic first-order homogenization. For the identification of the micro elasticity tensor, two different approaches are shown which rely on affine and non-affine Dirichlet boundary conditions of candidate unit cell variants with the possible stiffest response. The consistent coupling condition is shown to allow the model to act on the whole intended range between macro and micro elasticity tensors for both loading cases. We fit the relaxed micromorphic model against the fully resolved metamaterial solution by controlling the curvature magnitude after linking it with the specimen's size. The obtained parameters of the relaxed micromorphic model are tested for two additional loading scenarios.

Figures (23)

• Figure 1: Q2NQ2 Element. Black dots represent the displacement nodes. Red arrows and crosses indicate the edge and inner vectorial dofs, respectively, of the micro-distortion field used in Nédélec formulation.
• Figure 2: The beam models, compare Fig. \ref{['Figure:beam_metamaterials_BCs']}.
• Figure 3: Illustration shows the geometry of the specimens for $n=1,2,3,4,5$ with the assumed unit cell. The number of finite elements with degrees of freedom (dofs) are shown in parentheses.
• Figure 4: The boundary conditions of the fully resolved metamaterial shown exemplarily for $n=2$ ($H \times L = 2 \, l \times 24 \, l$).
• Figure 5: The normalized bending stiffness varying the beam size $H \times L = n\,l \times 12 \, n \, l$.