Micromorphic continuum model: granular materials to designed granular metamaterials

Abstract

This work investigates the mechanical response of granular-microstructured solids (natural and synthetic) in static and dynamic problems. Firstly, a non-classical micromorphic theory of degree n based upon granular micromechanics approach is developed to model the mechanical behavior of granular materials. This model is derived based on Hamilton’s principle, and provides variationally consistent boundary conditions. Moreover, less expensive models, namely micromorphic model of degree one, micropolar model, and second gradient model, are derived.Secondly, the micromorphic model of degree one is specialized to describe one-dimensional granular structures and the effect of different material parameters and higher order inertia on the wave propagation characteristics of such systems is parametrically studied. This model is able to describe dispersion, negative group velocity, and frequency band gaps. Moreover, the proposed model is further extended to investigate the effect of external electric field on tuning the dispersive behavior of dielectric granular materials in quasi-electro-statics. The model shows that an external electric field can potentially create, remove, or change location and width of the stop band in the granular medium. In addition, the micromorphic model of degree one is specialized to describe and analyze the wave propagation characteristics in axially moving materials with granular microstructure by employing an Eulerian frame of reference. The model predicts elastic wave dispersion asymmetries and the emergence and removal of stop bands for non-vanishing axial velocity. Thirdly, the micromorphic model of degree one is used to study the static behavior of one-dimensional materials with granular microstructure. The model predicts localization of deformation energy in the boundary layers for particular boundary conditions. The model is thereafter utilized to study the free vibration characteristics of one-dimensional granular-microstructured solids. The model predicts mode shapes similar to those of a classical rod, and natural frequencies different from those of a classical rod. The model also predicts length-scale effects such as stiffening of the material as the size of the structure shrinks. Finally, a micropolar model is developed to describe one-dimensional chiral granular (meta-) materials in a two-dimensional deformation plane. The proposed model is used to predict the behavior of chiral granular strings in tension. The domain of validity of the proposed model is thereafter investigated through parametric experimentation. To this end, particular chiral granular strings composed of 11 grains with varying geometrical parameters are considered. The granular strings are fabricated using 3D printing technology, and undergo tensile testing. The images taken from the experiment are analyzed using digital image correlation technique. The results are used to investigate the range of applicability of the model to predict the behavior of granular strings by comparing the predicted displacements and rotation fields by the model and the experimental results.