Scientists of the Institute of Problems in Mechanical Engineering of the Russian Academy of Sciences described analytically and numerically the changes in the final deformation in a material with specified properties. The results of the study were published in the scientific journal International Journal of Non-Linear Mechanics, Q1.
When scientists study complex, nonlinear processes such as wave propagation, a difficulty arises: there is no universal mathematical way to solve the equations which describe them.
Computer methods can be used to simulate these processes in order to understand their behavior better. But, when non-linearity is taken into account, new features appear which are difficult to be noticed with the help of computer calculations. Therefore, it is important to find out whether special mathematical approaches can be used to explain the certainty of numerical calculations and to identify a likely error in the numerical method. The work of scientists at IPMash RAS examines the problem using the example of bending waves in a special material – a metamaterial, for which the desired properties can be specified. The scientists simplified the complex equations of dynamic deformation processes and derived a new equation which describes nonlinear deformations in a material accurately. In addition, they obtained formulas which make it possible to predict the appearance of modulated waves depending on the initial and boundary conditions.
Alexey Porubov, the Head of the Laboratory of Micromechanics of Materials at IPMash RAS, said that, together with Nikolai Bessonov, the chief researcher of the Laboratory, they performed calculations by setting the periodic perturbations at the boundary of the metamaterial. It turned out that the results of these calculations coincide with those obtained analytically from simplified equations. This shows how important it is to conduct a theoretical analysis before performing complex calculations.
«To explain the behavior of changes inside the material, we used a model where the internal structure is represented by solid particles connected by springs. Such a model helps to understand the interaction of forces between atoms in a material. To describe the movements of particles, special equations were obtained which take into account the influence of neighboring particles through these springs. These originally difference equations were transformed into continual differential equations of motion.
“This work is part of a large series of articles on dynamic processes in metamaterials published in recent years in leading international journals from the first and second quartiles,» said Alexey Porubov.

It should be added that A. Porubov has been among the most cited scientists in Russia for several years. The other employees of the Laboratory, founded by the prominent Soviet and Russian scientist E.L. Aero, also demonstrate high publication activity.