self-modulation of a quasi-harmonic wave
V.I. Erofeev, V.V. Kazhaev, I.S. Pavlov
At present, the method of structural modeling is frequently used for construction of mathematical models of generalized continua – media of granular, fibrous, and cellular structures. This approach is based on the method of description of a medium by means of discrete elements (inertial, elastic, viscous, etc.) and subsequent continualization. Structural models contain parameters characterizing the geometry of a material (the lattice period, size and shape of the particles), therefore they are very effective in studying of the influence of size effects on macroproperties of a material.
In this paper, a one-dimensional model of a granular medium has been elaborated using the method of structural modeling. This model represents a chain consisting of nondeformable homogeneous round particles (granules). The space between the particles is assumed to be an inertialess elastic medium, through which the force and moment interactions transmit. These interactions are simulated by elastic springs of three types (central, noncentral and diagonal) with the corresponding stiffness. Each particle is supposed to interact only with two nearest neighbors in the chain and to have three degrees of freedom: two translational and one rotational. Only small (in comparison with the chain period) deviations of the particles from the equilibrium states and small rotations of the particles around the axis passing through the mass center f the particle perpendicularly to the plane of the chain are considered. The nonlinear differential equations describing the propagation of longitudinal, transverse and rotational waves in a one-dimensional granular medium have been derived under these assumptions, as well as analytical dependences of the linear and nonlinear macroparameters of the medium on the particle size and the parameters of the interparticle interactions (the stiffness of springs of various types) have been found.
At low frequencies, when the rotational wave does not propagate, the obtained three-mode set of equations degenerates into the two-mode system. The "memory" of the microstructure of the medium remains in the form of relationships between the macroscopic characteristics of the medium and the micromodel parameters. Within the scope of the two-mode system, areas of modulation instability (self-modulation) of the shear strain wave in the presence of a static longitudinal deformation, according to the Lighthill criterion, have been found in the space of the microstructure parameters of the material. In the various ranges of wavelengths, the dependences of the height and width of the wave train generated as a result of self-modulation of a quasiharmonic wave on the microstructure parameters of the granular material are determined.