Tool profile stationarity while simulating surface plastic deformation by rolling as a process of flat periodically reproducible deformation

OBRABOTKAMETALLOV MATERIAL SCIENCE Том 20 № 3 2018 EQUIPMEN . INSTRUM TS Vol. 3 No. 2 2021 Fig. 3. Strain plane positions when the piece is rotated Fig. 2. Material particle displacement while moving along the flow line in the deformation zone The roller profile changes when the strain plane is rotated, and in each time it is defined as the intersec - tion line of the roller surface and the strain plane. At the time corresponding to the position of the strain plane 3 in Fig. 3, the tool profile is an arc of a circle. At the time corresponding to the positions of the strain plane 1 and 2, the intersection line is a fourth-order curve, the points coordinates of which are determined by the solution of a system of equations. One of it describes the equation of the roller surface, and the sec - ond – the strain plane. The surface of the roller under consideration is a torus. If choosing a coordinate system in which the z axis coin- cides with the axis of generating circle rotation (Fig. 5), the equation of the surface will have the following form ( ) 2 2 2 2 2 2 2 2 2 4 ( ) 0, r pr r x y z R R R x y + + + − − + = (2) where R r is the rotation radius of the generating circle; R pr is the roller profile radius (and the generating circle radius). The part axis is parallel to the z axis and lies in the yoz plane, the strain plane passes through the part axis and is located at a certain angle a relative to the yoz plane. Then the strain plane equation for the x coordinate will have the following form tan ( ), x R y = a S − (3) where S R is the distance from the axis of roller generating circle rotation to the part axis, determined by the sum of all radii , r pr p R R R R S = + + (4) where R p is the part radius.