Yurchuk V. Scenarios of propagation of waves of different initial profiles in materials that deform nonlinearly elastically

In this dissertation, the propagation of plane and cylindrical waves in structural materials that deform elastically nonlinearly, and seismic waves in soil layers that deform viscoelastically, revealing nonlinearity during deformation is studied. The waves in materials are considered to be solitary waves of different initial profiles, except for the torsional waves, the profile of which is described by a harmonic function. The waves in the soil are considered seismic ones (in these waves, the frequencies are small). The nonlinear longitudinal and transverse plane waves and nonlinear cylindrical waves are studied, two types of which are selected – cylindrical radial and torsional. In the work, the main attention is paid to the influence of deformation nonlinearity on the evolution of waves of different initial profiles. At that, the new approaches to solving the nonlinear wave equations obtained for different types of waves and different media are developed. All waves, except for a seismic wave, are studied using one of the most developed models of nonlinear elasticity theory - the five-constant Murnaghan model. In this dissertation, the nonlinearity of the models is based on the consideration of quadratic and cubic nonlinearities, which can be treated as the element of novelty of the carried-out research. The dissertation compares the evolution of the initial wave profile at different levels of approximation (the first, first two and three approximations) of the solution of the nonlinear wave equation. An influence of the parameters specified in the nonlinear statement of the problem on the general picture of wave propagation (the kind of evolution of the initial profile) and other wave characteristics is studied. The scenarios of evolution are constructed for four types of waves (plane longitudinal, plane transverse - shear, cylindrical radial, and torsional). These waves are characterized by different initial profiles - harmonic cosine, single symmetric with one hump (Gauss function), single asymmetric with one hump (Whittaker functions), asymmetric without hump (McDonald and Friedlander functions). The approach applied to the group of waves made it possible to construct new scenarios that correspond to the physical sense of the analyzed problems. When studying seismic waves, the deformation of soil layers is described by a standard three-constant rheological model, the propagation of waves in soil layers is analyzed, and the method of determining the parameters of the rheological model from an experiment on the creep of a specific soil layer is applied. The determined rheological parameters are used to construct the scenarios of the change in the speed of two variants of the shear harmonic seismic wave and the wave amplitude for the three investigated soil types. In the dissertation, longitudinal harmonic wave, the nonlinear wave equations are analyzed by the methods of successive approximations and constraint on the displacement gradient to compare these approaches when constructing the solutions that take into account the first three approximations. In the analysis of cylindrical radial solitary waves, the method of restrition on displacement gradient is applied, and a nonlinear wave equation is obtained, for which a solution is found in the form of the first two and three approximations for their theoretical and numerical comparison. When studying a nonlinear elastic torsional wave, a generalized method of restriction on the (velocity) displacement is proposed for finding the first two approximations. Since the analyzed nonlinear wave equation is new, it demonstrates the applicability of the method for a wider class of wave problems. In the analysis of the harmonic seismic wave propagating in the soil layers, the direct method is used for the first time in the construction of the approximate solution of the wave. Numerical analysis of the evolution of the initial wave profile was performed using computer algebra system procedures. This anables to analyze numerically the influence of different orders of nonlinearities for the longitudinal, transverse, cylindrical, and torsional waves, as well as to determine the rheological parameters to construct the scenarios of evolution. Therefore, the obtained theoretical and numerical results made it possible to construct the scenarios of wave propagation in materials described by the nonlinear models of elastic and viscoelastic deformation, and a better understanding the influence of the number of approximations when solving a wave problem on the scenarios of propagation of different types of waves.