Determining Cosserat constants of 2D cellular solids from beam models

Abstract

We present results of a two-scale model of disordered cellular materials where we describe the microstructure in an idealized manner using a beam network model and then make a transition to a Cosserat-type continuum model describing the same material on the macroscopic scale. In such scale transitions, normally either bottom-up homogenization approaches or top-down reverse modelling strategies are used in order to match the macro-scale Cosserat continuum to the micro-scale beam network. Here we use a different approach that is based on an energetically consistent continuization scheme that uses data from the beam network model in order to determine continuous stress and strain variables in a set of control volumes defined on the scale of the individual microstructure elements (cells) in such a manner that they form a continuous tessellation of the material domain. Stresses and strains are determined independently in all control volumes, and constitutive parameters are obtained from the ensemble of control volume data using a least-square error criterion. We show that this approach yields material parameters that are for regular honeycomb structures in close agreement with analytical results. For strongly disordered cellular structures, the thus parametrized Cosserat continuum produces results that reproduce the behavior of the micro-scale beam models both in view of the observed strain patterns and in view of the macroscopic response, including its size dependence.

Size and disorder effects in elasticity of cellular structures: From discrete models to continuum representations

“…Diebels and Steeb, 2002;Tekoglu and Onck, 2005;Mora and Waas, 2007;Liebenstein et al, 2014) our approach allows to directly take the information of the microstructure into account. We showed this in an accompanying paper by Liebenstein and Zaiser (2017) where we identified constitutive parameters of a Cosserat continuum.…”

Topology optimization of periodic beam lattices using Cosserat elasticity

“…The only difficulty lies in determining the Cosserat constants for arbitrary beam structures. However, this can be done using various homogenization methods [25,32,40].…”

“…Two main approaches exist: (i) the micro-scale is represented by an inhomogeneous Cauchy continuum [25,22,24,23], and (ii) the lattices are modeled with either Euler-Bernoulli [48,54,49] or Timoshenko-Ehrenfest beams [40]. The important difference is that in the latter case, the rotational degrees of freedom are already present at the microscale.…”

A fully second‐order homogenization formulation for the multi‐scale modeling of heterogeneous materials

“…The conservation of the virtual power is ensured in the scale transition by employing the principle of multi-scale virtual power, an extension of the Hill-Mandel Principle, where the minimal constraints (31) and (32) are enforced through the Lagrange multipliers L and M, respectively:…”

“…The physical meaning of the associated material constants, like the length scale parameter, and its determination based on either experimental or numerical considerations, has been the subject of research and discussion. [27][28][29] Generalized continua models have been employed in a wide range of applications, like functionally graded materials, 30 cellular materials, 31 fiber reinforced composite materials, [32][33][34] modeling of damage and ductile fracture, [35][36][37] crystalline plasticity, [38][39][40] multi-physics problems like magneto-elasticity. 41 Dynamic effects have also been introduced, 42,43 allowing to predict dispersive wave propagation, with application to phononic crystals for example.…”