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Abstract
The mathematical analysis of plastic flow processes under uniform plane, axisymmetric and volumetric deformation is carried out. The analysis is based on the external shape change of the body, which determines the movement of material points. It is shown that the plastic flow of an isotropic rigid-plastic body under plane deformation obeys the hyperbolic law, and for axisymmetric and volumetric deformations – the inverse square law. Spatial-geometric expressions of these laws made it possible to reveal and explain in a new way the physical essence of plastic shear. It is proved that the stressed state of a body under uniform tension-compression deformation is complex and cannot be defined as “linear”. The normal stress, which coincides with the direction of the resulting deformation force, is not the main one, since in the areas perpendicular to this direction, the shear stresses are not equal to zero. Examples of solving technological problems are given: extrusion of cylindrical billets and wire drawing, rolling of a wide strip of rectangular profile. It is shown that the problems of determining the stress-strain state of isotropic rigid-plastic bodies along the known trajectories of movement of material points are statically definable.
Highlights
The mathematical analysis of plastic flow processes under uniform plane, axisymmetric and volumetric deformation is carried out
При однородной плоской деформации эта точка перемещается по траектории в виде равносторонней гиперболы (1) под действием результирующего сдвигающего напряжения, равного константе пластичности τ = k
Обозначения, принятые в статье σ, τ – нормальное и касательное напряжения; u – перемещение материальной точки; ε – степень деформации; h – высота образца; Δh – путь деформирования; k – константа пластичности
Summary
The mathematical analysis of plastic flow processes under uniform plane, axisymmetric and volumetric deformation is carried out. Рассмотрим случай идеальной однородной деформации образца – одноосное растяжение или сжатие 1. Пластическое течение металла при однородной деформации: а – плоская, b – осесимметричная деформация Fig. 1. Деформируемый образец сохраняет форму прямоугольного параллелепипеда при плоской или объемной деформации
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