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

OBRABOTKAMETALLOV Vol. 20 No. 3 2018 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 3 2 21 Fig. 4. Displacements in the strain plane while it is rotated Fig. 5. Coordinate system for determining the intersection line of the roller surface and the strain plane Substituting the deformation plane equation (3) into the roller surface equation (2) and introducing the notation will give 2 2 ( ) 0 z B C + − = , (5) where 2 2 , r pr B A R R = + − (6) 2 4 , r C AR = (7) 2 2 2 (1 tan ) ( 2 ) tan . A y R R y = + a + S S − a (8) Assuming z = z 2 , from (5) we obtain the quadratic equation 2 2 2 ( ) 0. B B C z + z + − = (9) By solving this equation, we can define z as z . Thus, the x and z coordinates of each intersection line point of the roller surface and the strain plane rotated by an angle a can be determined from the given value of the y coordinate using expressions (3) and (9). To calculate the intersection line points coordinates in the coordinate system associated with the strain plane, the following expressions can be used: 0; cp x = (10) cos ; cp y y = a (11) . cp z z = (12) Result and Discussion To estimate the tool profile change during the strain plane rotation, the intersection lines points coordinates of the roller surface and the strain plane were calculated at R pr =7 mm, R r =8 mm, R p =20 mm, Σ R =35 mm for the inclination angle of the strain plane a 0°, 2°, 4° and 6° (Fig. 6, Table 1).