The performance of a Cosserat/micropolar solid as a numerical vehicle to represent dispersive media is explored. The study is conducted using the finite element method with emphasis on Hermiticity, positive definiteness, principle of virtual work and Bloch-Floquet boundary conditions. The periodic boundary conditions are given for both translational and rotational degrees of freedom and for the associated force-and couple-traction vectors. Results in terms of band structures for different material cells and mechanical parameters are provided.

In this work, we present a computational analysis of the planar wave propagation behavior of a one-dimensional periodic multi-stable cellular material. Wave propagation in these materials is interesting because they combine the ability of periodic cellular materials to exhibit stop and pass bands with the ability to dissipate energy through cell-level elastic instabilities. Here, we use Bloch periodic boundary conditions to compute the dispersion curves and introduce a new approach for computing wide band directionality plots. Also, we deconstruct the wave propagation behavior of this material to identify the contributions from its various structural elements by progressively building the unit cell, structural element by element, from a simple, homogeneous, isotropic primitive. Direct integration time domain analyses of a representative volume element at a few salient frequencies in the stop and pass bands are used to confirm the existence of partial band gaps in the response of the cellular material. Insights gained from the above analyses are then used to explore modifications of the unit cell that allow the user to tune the band gaps in the response of the material. We show that this material behaves like a locally resonant material that exhibits low frequency band gaps for small amplitude planar waves. Moreover, modulating the geometry or material of the central bar in the unit cell provides a path to adjust the position of the band gaps in the material response. Also, our results show that the material exhibits highly anisotropic wave propagation behavior that stems from the anisotropy in its mechanical structure. Notably, we found that unlike other multi-stable cellular materials reported in the literature, in the system studied in this work, the configurational changes in the unit cell corresponding to its different stable phases do not significantly alter the wave propagation behavior of the material.

In most of standard Finite Element (FE) codes it is not easy to calculate dispersion relations from periodic materials. Here we propose a new strategy to calculate such dispersion relations with available FE codes using user element subroutines. Typically, the Bloch boundary conditions are applied to the global assembled matrices of the structure through a transformation matrix or row-and-column operations. Such a process, is difficult to implement in standard FE codes since the user does not have access to the global matrices. In this work, we apply those Bloch boundary conditions directly at the elemental level. The proposed strategy can be easily implemented in any FE code. This strategy can be used either in real or complex algebra solvers. It is general enough to permit any spatial dimension and physical phenomena involving periodic structures. A detailed process of calculation and assembly of the elemental matrices is shown. We verify our method with available analytical solutions and external numerical results, using different material models and unit cell geometries.

The role played by the diffraction field on the problem of seismic site effects is studied. For that purpose we solve and analyze simple scattering problems under P and SV in-plane wave assumptions, using two well known direct boundary-element-based numerical methods. After establishing the difference between scattered and diffracted motions, and introducing the concept of artificious and physically based incoming fields, we obtain the amplitude of the Fourier spectra for the diffracted part of the response: this is achieved after establishing the connection between the spatial distribution of the transfer function over the studied simple topographies and the diffracted field. From the numerical simulations it is observed that this diffracted part of the response is responsible for the amplification of the surface ground motions due to the geometric effect. Furthermore, it is also found that the diffraction field sets in a fingerprint of the topographic effect in the total ground motions. These conclusions are further supported by observations in the time-domain in terms of snapshots of the propagation patterns over the complete computational model. In this sense the geometric singularities are clearly identified as sources of diffraction and for the considered range of dimensionless frequencies it is evident that larger amplifications are obtained for the geometries containing a larger number of diffraction sources thus resulting in a stronger topographic effect. The need for closed-form solutions of canonical problems to construct a robust analysis method based on the diffraction field is identified.

The problem of diffraction of cylindrical and plane SH waves by a finite crack is revisited. We construct an approximate solution by the addition of independent diffracted terms. We start with the derivation of the fundamental case of a semi-infinite crack obtained as a degenerate case of generalized wedge. This building block is then used to compute the diffraction of the main incident waves. The interaction between the opposite edges of the crack is then considered one term at a time until a desired tolerance is reached. We propose a recipe to determine the number of required interactions as a function of frequency. The solution derived with the superposition technique can be applied at low and high frequencies.