Propagation of nonlinear elastic waves and non-local dynamic behaviour of composite materials

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This renewal proposal aims to continue ongoing studies of nonlinear dynamic properties of periodic composite materials. We intend to consider new problems that have emerged during evaluation of the previous DFG grant and are crucially important for the analysis and verification of the obtained results. To the time being, we studied propagation of stationary nonlinear waves. However, a number of significant nonlinear effects are related to non-stationary dynamic behaviour of composite solids. The key objectives of the project are as follows:1. Study of evolution of 1D and 2D nonlinear waves from different initial excitations. Investigation whether and how fast stable stationary modes can be developed. Comparison of numerical and analytical results.2. Study of resonance interactions between 2D nonlinear plane waves propagating in different directions. Resulting from such interactions, filtering or amplification of 2D solitary waves.3. Study of nonlinear waves propagating in viscoelastic composites. Derivation of macroscopic nonlinear wave equations with dispersive and dissipative terms. Investigation of the interplay between the effects of nonlinearity and dissipation.Additionally to the new tasks listed above, we plan to complete the solution of several problems from the previous project, namely, clarification of the links between discrete and continuous models of nonlinear composite materials; verification of the derived macroscopic dynamical equations and estimation of their area of applicability. As to the methodology, asymptotic approaches based on regular and singular perturbation techniques and the asymptotic homogenization method will be used. Non-stationary dynamic problems will be studied by the pseudo-spectral numerical procedure. Integration in the time domain will be performed by finite-difference approaches. For the spatial discretization, the Fourier series expansions will be applied. We intend to improve essentially the convergence of the series and to reduce the Gibbs-Wilbraham phenomenon with the help of the method of Fourier-Padé approximants.