Features of a New Method for Solving Practical Problems of Continuum Mechanics Under Conditions of Complex Loading

Abstract

This work, carried out within the framework of grant № AP23488953, funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan, considers an applied problem of continuum mechanics associated with solving a new closed problem of plasticity theory with respect to an asymmetric deformation focus during loading. The analytical solution of this problem is based on the method of argument of functions of a complex variable. The proposed approach is invariant to various areas of continuum mechanics, including not only the theory of plasticity, but also the theory of elasticity, the theory of dynamic processes. In particular, using the example of the rolling process, a closed planar problem of the theory of plasticity in an analytical form was posed and solved. The formulated system of equations includes: differential equations of equilibrium, the Huber-Mises plasticity condition, coupling equations, the equation of continuity of strain rates, the condition of constancy of volume, the equation of thermal conductivity, boundary conditions for stresses and strain rates. The solution uses approaches of limited nonlinearity, fundamental and trigonometric substitution. Under the conditions of new variables through the function argument, it was possible to simplify the intermediate result and determine not the solutions themselves, but the conditions of their existence under given boundary conditions. The solution of the asymmetric problem contributed to the identification of the effects of shape change in the processing center, which make it possible to control the processes of plastic deformation. The fundamental approach considered in this paper makes it possible to determine not only the stress-strain state of the plastic medium, but also the mathematical model of the deformed space and further possibilities for finding generalized solutions to problems of continuum mechanics.

Keywords

• Continuum mechanics
• Rolling
• Closed problem
• Asymmetric problem
• Plasticity theory

References

1. Chigirinsky, V., Kuzmin, S., & Lezhnev, S. (2025). Features of a new method for solving practical problems of continuum mechanics under conditions of complex loading. The Eurasia Proceedings of Science, Technology, Engineering and Mathematics (EPSTEM), 34, 8-14.