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The article considers a mathematical model of an acoustic (mechanical) metamaterial, which is a chain of oscillators consisting of nonlinear elastic elements and masses, each of which contains an internal nonlinear oscillator. It is shown that, in the long-wavelength approximation, the resulting system of equations can be reduced to the Benjamin–Bon–Mahoney nonlinear evolutionary equation, in the framework of which interaction of three modulated quasi-harmonic waves (wave packets) is studied under the phase matching conditions. We investigate the formation of coupled three-frequency envelope solitons, i.e., wave packets that retain their amplitude–phase profiles as they propagate in the metamaterial due to the compensating effect of nonlinear effects. It is noted that in addition to solutions describing quasi-harmonic processes, the resulting evolutionary equation has an exact analytical solution in the form of a solitary stationary wave (soliton). Differences in the properties of this soliton and the classical Korteweg–de Vries soliton are revealed. [ABSTRACT FROM AUTHOR]

A self-consistent mathematical model is stated in linear and nonlinear formulations, which includes the equation of bending vibrations of a beam and the kinetic equation of damage accumulation in its material. The beam is assumed to be infinite. Such idealization is permissible if its boundaries are attached to optimal damping devices, i.e., the parameters of the boundary fixation are such that incident perturbations will not be reflected. This makes it possible to consider the beam model without taking into account the boundary conditions and regard vibrations propagating along the beam as traveling bending waves. As a result of analytical studies and numerical modeling, it is shown that material damage introduces frequency-dependent attenuation and significantly changes the character of the dispersion of the phase velocity of a bending elastic wave. In a classical Bernoulli–Euler beam, bending waves have one dispersive branch at any frequency, while, for a beam made of a damage-accumulating material, there are two pairs of dispersive branches in the entire frequency range, with one pair describing the wave propagation and the other, the wave attenuation. Within a geometrically nonlinear model of a damaged beam, the formation of intense bending waves of a stationary profile is studied. It is shown that such essentially nonsinusoidal waves can be both periodic and solitary (localized in space). Dependences have been determined relating the parameters of the waves (amplitude, width, and wave number) with the material damage. It was found that, with an increase in the material damage parameter, the amplitudes of the periodic and solitary waves as well as wave number of the periodic waves increase, while the width of the solitary wave decrease. [ABSTRACT FROM AUTHOR]

Within the framework of the classical Biot theory, the propagation of plane longitudinal waves in a porous liquidsaturated medium is considered with account for the nonlinear connection between deformations and displacements of solid phase. It is shown that the mathematical model accounting for the geometrical nonlinearity of the medium skeleton can be reduced to a system of evolution equations for the displacements of the skeleton of medium and of the liquid in pores. The system of evolution equations, in turn, depending on the presence of viscosity, is reduced to the equation of a simple wave or to the equation externally resembling the Burgers equation. The solution of the Riemann equation is obtained for a bell-shaped initial profile; the characteristic wave breaking is shown. In the second case, the solution is found in the form of a stationary shock wave having the profile of a nonsymmetric kink. The relationship between the amplitude and width of the shock wave front has been established. It is noted that the behavior of nonlinear waves in such media differs from the standard one typical of dissipative nondispersing media, in which the propagation of waves is described by the classical Burgers equation. [ABSTRACT FROM AUTHOR]

The propagation of plane longitudinal waves in an infinite medium with point defects has been investigated. The medium is assumed to be placed in a nonstationary nonuniform temperature field. A self-consistent problem considering both the influence of the acoustic wave on the generation and displacement of point defects and, conversely, the influence of point defects on the propagation of the acoustic wave has been considered. It has been shown that in the absence of heat diffusion, the corresponding set of equations reduces to a nonlinear evolutionary equation in particle displacements in the medium. This equation can be viewed as a formal generalization of the Korteweg–de Vries–Burgers equation. Using the finite difference method, an exact solution to the evolutionary equation in the form of a monotonically decreasing stationary shock wave has been found. It has been noted that defect-induced dissipation dominates over dispersion due to defect migration. [ABSTRACT FROM AUTHOR]