Probabilistic model of surface layer removal when grinding brittle non-metallic materials

OBRABOTKAMETALLOV Vol. 23 No. 2 2021 TECHNOLOGY For each revolution (pass), the change in the increment of the indicator  1 a is determined by the expression [16]:       1 ( , ) c z a y k b , (5) where c k – is the chip formation coef fi cient;  ( , ) z b y – is grain width at the level y ;  – is the number of abrasive grains that have passed through the section under consideration. When grinding brittle materials, the chip formation coef fi cient c k is 1, since there are no plastic deformation processes. To calculate the indicator  1 a , characterizing the change in the area of the depressions formed due to the mechanical cutting process, grains that cut off the material are taken into account (piercing grains are not considered in this case). Based on this, taking into account equation (5), the indicator  1 a can be calculated as follows:        1 ( , ) (1 ) z ck a y b P , (6) where ck P – is the probability of brittle chipping of the workpiece material. Through a single section of the surface, with a thickness of  u (Fig. 2), the tops of the abrasive grains  will pass over time  . The number of tops of abrasive grains can be calculated from the density of its distribution in the working layer of the tool ( ) f u along the coordinate u :          ( ) ( ) z k u n f u u V V , (7) where z n – is the number of grains per unit area of the working layer of the tool; k V – the peripheral speed of the tool (circle); u V – the peripheral speed of the workpiece. Fig. 2. Scheme for calculating the number of tops of abrasive grains, pass- ing through a unit surface area by the thickness of the tool per unit of time The distribution of cutting edges over the depth of the working surface of the tool was studied in [2, 15, 18]. In the analytical description of the distribution curves J. Cassen assumes that the number of cutting edges on the surface of the circle is proportional to the square of the distance inside the circle [19]. The probability density curve of the distribution of cutting edges is modeled by its straight-line dependence   ( ) f f u C u . According to the author, the simulation of the distribution curve by a straight- line dependence is valid for the section of the circle directly lying near the surface. To describe the distribution density of the tops of abrasive grains, O.Coyle suggested using a dependence of the form [17]: