Дослідження кінематики пластичної течії металу за допомогою змінних Ейлера і Лагранжа

Abstract

In continuum mechanics, two equivalent points of view on the flow of a metal in the process of deformation - Euler and Lagrange - have found application. When attention is focused on a given point in space, where different particles of a continuous medium come in the process of deformation, this is the essence of Euler's point of view on the studied motion of a continuous medium. From the point of view of Lagrange, the motion is considered known if the speed, acceleration, temperature and other quantities are given as functions of the coordinates of a point in space and time. The essence of Lagrange's view of the studied motion lies in the fact that they focus on a specific particle of a continuous medium and investigate the history of its deformation and motion. The coordinates that individualize the points of the continuous medium are called Lagrange variables. Lagrange's view on the study of the motion of a continuous medium underlies physical laws, since it is associated with the motion of individual particles. At the initial moment, the coordinates of a certain point according to Euler and Lagrange coincide. The article discusses the possibility of using mixed Euler and Lagrange coordinates to determine the component of the strain rate tensor in plastic deformation processes. The expediency of the approach of Lagrange and Euler is clarified in specific problems. It should be borne in mind that in the Euler coordinates, the equilibrium equations have a simpler form (they are linear), and the boundary conditions are more complex due to the fact that it is impossible, without solving the problem, to determine the displacement components that are usually included in the boundary conditions. In Lagrange coordinates, the boundary conditions, as a rule, are written easier, and the equilibrium equation is more complicated. When processing metals by pressure, the metal undergoes significant deformation. However, it is convenient to consider each process at each fixed moment of its course, therefore, it is necessary to know the basic provisions and dependencies related to small deformations

В статті розглядається можливість застосування змішаних координат Ейлера та Лагранжа для визначення компонент тензора швидкостей деформацій в процесах пластичного деформування. Доцільність підходу Лагранжа та Ейлера з`ясовується в конкретних задачах. При цьому слід враховувати, що в координатах Ейлера рівняння рівноваги мають більш простий вигляд (вони лінійні), а граничні умови – складніший внаслідок того, що неможливо, не розв`язуючи задачі, визначити компоненти переміщення, які зазвичай входять до граничних умов. У координатах Лагранжа крайові умови, як правило, записуються простіше, а рівняння рівноваги – складніше.