Structural Modeling of Metamaterials

The principles of the structural modeling method, the development of the theoretical foundations of which this monograph is devoted, are formulated in the first chapter. Moreover, the problem of the applicability of the classical mechanics laws to a theoretical description of media with micro- and nanostructure is discussed here.

Mechanical properties of a granular consolidated medium depend on the geometry of the microparticles, their location, and the forces of interaction between them. One of the main goals of mathematical modeling of such media is obtaining equations of motion and equations of state, which are capable to describe a discrete nature of a medium.

In this chapter, we consider a model of a granular medium as a rectangular lattice of rigid ellipse-shaped particles. Each particle of such a lattice possesses two translational and one rotational degrees of freedom. The space between the particles is a massless medium through which the force and coupled interactions are transmitted. In limiting cases, this model degenerates either into a chain of ellipse-shaped particles or into a square lattice of round particles. The main objectives of this chapter are to derive dynamic equations of a granular medium consisting of anisotropic particles and to identify the relationships between the physicomechanical properties of a granular material and the parameters of its microstructure. Using the results obtained in the chapter, it is possible to determine the elastic properties of an anisotropic nanocrystalline (granular) material with non-dense packing of particles by measuring the velocities of elastic waves propagating along different crystallographic directions [1].

Theoretical estimates [1] and experimental data [2‐5] show that rotational waves can exist in solids in the high-frequency field (> 10⁹ – 10¹¹ Hz), where it is rather difficult to carry out acoustic experiments with the technical viewpoint. The question arises: is it possible to obtain some information about the microstructure of a medium from acoustic measurements in the low-frequency range (10⁶ – 10⁷ Hz), when the rotational waves do not propagate in the medium? To this purpose, we will consider in this chapter the low-frequency approximation of Eqs. (2.8) and (3.6), in which the microrotations of the particles of the medium are not independent and are determined by the displacement field. Further, by comparing the obtained equations describing the propagation and interaction of longitudinal and transverse waves in a granular medium in the low-frequency approximation with the equations of the classical theory of elasticity, we will consider the problem of parametric identification of the developed models.

In Chaps. 2–4, the linear models of microstructured media have been considered, the particles of which have three degrees of freedom. In this chapter, the dynamic equations of a rectangular lattice of ellipse-shaped particles and a square lattice of round particles are generalized to the nonlinear case.

In the previous chapters, two-dimensional models of microstructured media were discussed, and the particles of which have three degrees of freedom. This chapter is devoted to the elaboration of a three-dimensional model of a crystalline medium consisting of spherical particles with six degrees of freedom. Such a medium is structurally similar to a fullerite crystal with a simple cubic lattice (see Sect. 1.​2). The main objectives of this chapter are to obtain dynamic equations of a crystalline medium consisting of spherical particles by the method of structural modeling and to establish the relationships between the coefficients of these equations and the microstructure parameters of the material at issue.

The main goal of the final chapter of the monograph is to study the features of the propagation of nonlinear elastic waves in metamaterials and constructions made of them.