A two-dimensional (2D) model of a granular medium is represented as a square lattice consisting of elastically interacting round particles possessing one rotational and two translational degrees of freedom. The differential equations describing propagation and interaction of waves of various types in such a medium have been derived in the long-wavelength approximation. Accounting for microrotations of the particles and moment interactions between them leads to the consideration of so-called microrotation waves (spin waves). In the absence of microrotations, the governing equations degenerate into 2D Lamé equations for anisotropic media. A one-to-one correspondence has been established between the microstructure parameters and effective elasticity constants of the second-order. Dependence of elasticity constants on the size of grains has been analyzed. The proposed model is compared in the continuous approximation to the equations of the 2D Cosserat continuum possessing macroscopic anisotropy.

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This article is devoted to the study of auxetic properties of Cosserat hexagonal lattices composed of finite‐sized particles with complex connections. The description of complex connections is given; their mathematical model is elaborated and the properties are studied. The introduction of complex connections enables one varying their structure and component parameters. Due to that there arise possibilities for both simulation of nonchiral lattices with symmetrical bonds and with a chiral microstructure and construction of lattices with desired properties. The discrete and micropolar equations of the lattice are obtained. As a result, the macroparameters are expressed in terms of the lattice microparameters. The dependence of the Poisson's ratio on the lattice microparameters is obtained. It allows finding and analyzing parameters, for which the lattice possesses auxetic properties. The importance of rotational degrees of freedom of particles and chirality for the appearance of auxetic properties is shown. For verification, the results of the calculation of the Poisson's ratio obtained on the basis of theoretically obtained relations are compared with the results of numerical simulation of the stretching of the lattice.

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