Asymptotic Analysis and Synthesis in Mechanics of Solids and Nonlinear Dynamics

In this lectures various methods which give a possibility to extend an area of applicability of perturbation series and hence to omit their local character are analysed. While applying asymptotic methods as a rule the following situation appears: the existence of asymptotics for $\text{ve}\to 0$ implies an existence of the asymptotics for $\text{ve}\to \mathrm{\infty }$. Therefore, the idea to construct one function valid for the whole parameter interval for $\text{ve}$ is very attractive. The construction of asymptotically equivalent functions possessing a known asymptotic behaviour for $\text{ve}\to 0$ and $\text{ve}\to \mathrm{\infty }$ will be discussed. Using summation and interpolation procedures we focus on continuous models derived from a discrete micro-structure. Various continualization procedures that take the non-local interaction between variables of the discrete media into account are analysed.