Asymptotic Analysis and Synthesis in Mechanics of Solids and Nonlinear Dynamics

TL;DR

This paper discusses asymptotic methods and the construction of functions valid across entire parameter ranges to improve modeling in mechanics of solids and nonlinear dynamics, focusing on asymptotic analysis, summation, and interpolation techniques.

Contribution

It introduces methods for extending perturbation series applicability and constructing asymptotically equivalent functions for continuous models from discrete micro-structures.

Findings

Construction of functions valid for all parameter ranges.

Analysis of continualization procedures for non-local interactions.

Application of summation and interpolation in asymptotic analysis.

Abstract

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 $\ve\rightarrow 0$ implies an existence of the asymptotics for $\ve\rightarrow\infty$. Therefore, the idea to construct one function valid for the whole parameter interval for $\ve$ is very attractive. The construction of asymptotically equivalent functions possessing a known asymptotic behaviour for $\ve\rightarrow 0$ and $\ve\rightarrow \infty$ 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.