The micromorphic constitutive parameters and dispersion behaviors in different granular crystals

Study on dispersion and wave velocity in 2D elliptic granular crystals by a micromechanics-based micromorphic model

“…Traditional continuous medium models 28 can quantitatively describe dispersion behaviors, but they are not enough to accurately simulate the effect of microstructure. The micromechanics-based continuum models 27,31,33 include advantages of discrete and continuum models in describing dispersion and the effect of microstructure. And among these models, the micromechanics-based micromorphic model can provide more macroscopic variables and deformation modes with respect to microstructural information in the process of homogenization for granular assembly, which is seen as a complete model.…”

“…Then, in the micromorphic theory, the most important hypothesis, the decomposition of motion, needs to be introduced. In our previous studies, 22,31 the microscopic actual motion (displacement and rotation) of a particle is a sum of an average of the microscopic actual motions of particles and a fluctuation related to the average motion. The average displacement u ¯ i and the average rotation ω ¯ 3 should be independent of X coordinate system, therefore, u ¯ i = u ¯ i ( x , t ) and ω ¯ 3 = ω ¯ 3 ( x , t ) .…”

“…This matches the conclusions in previous studies. 19,31 And if the rotation of material point is not considered, that is, the micropolar medium degenerates into a classical one, the non-dispersive acoustic branches are resulted from the DOFs u ¯ 1 and u ¯ 2 . To summarize, the optical mode of wave comes from two aspects: the rotation of material point and the microscopic deformation of material point.…”

“…21,34 Following this method, Xiu and Chu 22,35 proposed a first-order micromechanics-based micromorphic model to predict dispersions and frequency band gaps of granular materials. 22 Furthermore, Xiu et al 31 derived macroscopic micromorphic constitutive modulus tensors for granular crystals with different specific 3D microstructures, instead of granular materials based on a hypothesis of isotropic contact density distribution in above models. 22,35 In this way, more microscopic information is considered by specific particle arrangements and sizes, void ratios, and coordination numbers.…”

“…Comparing with classical and Cosserat continuum theories, the micromorphic continuum theory has the capacity of providing complete deformation modes of microstructures and more macroscopic measures relating to microscopic information. 31,32 Subsequently, some micromechanics-based micromorphic models [20][21][22]31 are developed and applied to describe wave propagations and dispersions in granular materials. Misra and Poorsolhjouy 33 first proposed a micromorphic model based on a micromechanical approach and applied the model to analyze dispersions in elastic granular media 20,33 and 1D granular crystal.…”