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

OBRABOTKAMETALLOV Vol. 23 No. 2 2021 TECHNOLOGY          1 0 lim 1 ( ) exp( ), n i i k l l P P M a a l (1) where: 0 a – an indicator characterizing the initial state of the surface of the workpiece in a given section before the start of the grinding process; a – an indicator characterizing the change in the area of the depressions formed by the sum of the pro fi les of abrasive grains passing through the considered section of the workpiece; ( ) P M – probability of material removal. To describe the regularities of material removal and tool surface wear in the contact zone, the concepts of the probability of removal ( ) P M and the probability of non-removal ( ) P M of the material are proposed in [17]. The fi rst indicator ( ) P M is determined by the probability of an event in which material at a point on the treated surface is removed. The second indicator ( ) P M is the probability of an event in which material is not removed from the treated surface. The sum of the probabilities, as the probabilities of opposite events, is equal to unity, and its values depend on the position of the point in the contact zone. For the processes of processing workpieces with abrasive tools, the probability of material removal is calculated from the dependence:         0 1 ( ... ... ) ( ) 1 k j a a a a P M e , (2) where 1 a , …, j a – are indicators characterizing the change in the areas of the depressions formed by the sum of the pro fi les of abrasive grains passing through the considered section of the workpiece after the corresponding contacts of the grains with the surface of the workpiece. In general, when fi nishing and fi ne grinding of holes in workpieces made of brittle non-metallic materials (glass, ceramics, quartz, ferrites, sitalls, etc.), as well as in workpieces with ceramic coatings, material removal is carried out due to a combination of micro-cutting-chipping and brittle volume destruction of the material. To obtain a mathematical model that allows calculating removal when grinding brittle non-metallic materials, consider the process of contacting the tool with the workpiece at the level y . As a result, of the impact of cutting and piercing grains on the surface of the workpiece, material is removed in the contact zone by micro-cutting and brittle chipping, which can be considered as a random event. It is characterized by the joint probability of material removal from the workpiece by the micro- cutting or shearing process. Thus, the probability of removal when grinding brittle non-metallic materials, is calculated by the formula:   1 2 ( ) ( ) ( ) P M P M P M , (3) where 1 ( ) P M – is the probability of an event in which the processed material is not removed due to the micro-cutting process; 2 ( ) P M – the probability of an event in which the processed material is not removed due to the brittle chipping process. Similarly to equation (1), dependence (3) can be described by the following expression:        0 1 2 ( ) 1 exp( ( , ) ( , )) P M a a y a y , (4) where 0 a – an indicator characterizing the initial state of the surface of the workpiece in a given section before the start of the grinding process;  1 ( , ) a y – an indicator characterizing the change in the area of the depressions formed due to the mechanical cutting process;  2 ( , ) a y – an indicator characterizing the change in the area of depressions formed due to the process of brittle chipping; y – distance from the outer surface of the workpiece to the current level;  – the moment in time of an ongoing event.