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

这项更新提案旨在继续进行周期性复合材料的非线性动态特性的研究。我们打算审议在评价以前的赠款期间出现的新问题，这些问题对分析和核查所取得的成果至关重要。目前，我们研究的是平稳非线性波的传播。然而，许多重要的非线性效应与复合固体的非稳态动态行为有关。该项目的主要目标如下：一维和二维非线性波在不同初始激励下的演化研究。研究是否可以发展稳定的静止模式，以及发展速度有多快。数值结果与分析结果的比较。二维非线性平面波在不同方向上传播的共振相互作用研究。由这种相互作用、过滤或放大二维孤立波而产生的。粘弹性复合材料中非线性波传播的研究。具有色散项和耗散项的宏观非线性波动方程的推导。非线性与耗散效应相互作用的研究。除了上述新任务外，我们计划完成前一个项目中几个问题的解决，即澄清非线性复合材料的离散模型与连续模型之间的联系；推导出的宏观动力学方程的验证及其适用范围的估计。在方法上，将使用基于正则和奇异摄动技术的渐近方法以及渐近均匀化方法。非平稳动力问题将采用伪谱数值方法进行研究。时域内的积分将通过有限差分方法进行。对于空间离散化，将应用傅里叶级数展开。我们打算从本质上改进级数的收敛性，并借助傅里叶-帕德帕尔近似法减少吉布斯-威尔布拉罕现象。

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.