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

OBRABOTKAMETALLOV Vol. 23 No. 2 2021 TECHNOLOGY After substituting the values of gamma functions at particular values   1.3 ,  0.7 x m and   2 into expression (25), we get:                              2 2 0 2 3 2 3 3 6 2 ( ) (1 ) 2 2 ( , ) 2 (3) z c z k u z f e L u u k V V n P z a y z t y dz D V H à à à                  4 2 1.3 2 13 2 ( ) (3.3) (1.7) 5 (5) z c z k u z f x e L u u f k V V n z t y r dz D à V H t à à . (26) After substituting the values of the gamma functions, we fi nally get:                     2 3 5 0 2 3 2 3 2 ( ) (1 )( ) 2 8 ( , ) 5 15 3 8 c z k u z f y y y u u k V V n P t y z z a y z z L L L V H         4 1.3 2 0.05 2 ( ) ( ) ( ) c z k u z f x u u f x k V V n t y r V H t r                 9 7 5 3 2 3 2 4 6 4 8 5 20 3 9 7 y y y y y z z z z z L L L L L . ( 27) Conclusions The developed mathematical models make it possible to trace the effect on material removal of the overlap of single sections on each other when grinding holes in ceramic materials. The proposed dependences show the regularity of stock removal within the arc of contact of the grinding wheel with the workpiece. The considered features of the change in the probability of material removal upon contact of the treated surface with an abrasive tool and the proposed analytical dependences are valid for a wide range of grinding modes, wheel characteristics and a number of other technological factors [20, 22]. The obtained expressions make it possible to fi nd the amount of material removal also for schemes of end, fl at and circular external grinding, for which it is necessary to know the amount of removal increment due to brittle fracture during the development of micro-cracks in the surface layer. One of the ways to determine the magnitude of this increment is to simulate the crack formation process using a computer. References 1. Malkin S., Guo C. Grinding technology: theory and applications of machining with abrasives. New York, Industrial Press, 2008. 372 р . ISBN 978-0-8311-3247-7. 2. Hou Z.B., Komanduri R. On the mechanics of the grinding process. Pt. 1. Stochastic nature of the grinding process. International Journal of Machine Tools and Manufacture, 2003, vol. 43, pp. 1579–1593. DOI: 10.1016/ S0890-6955(03)00186-X. 3. Lajmert P., Sikora V., Ostrowski D. A dynamic model of cylindrical plunge grinding process for chatter phenomena investigation. MATEC Web of Conferences , 2018, vol. 148, pp. 09004–09008. DOI: 10.1051/ matecconf/20181480900. 4. Leonesio M., Parenti P., Cassinari A., Bianchi G. , Monn M. A time-domain surface grinding model for dynamic simulation. Procedia CIRP , 2012, vol. 4, pp. 166–171. DOI: 10.1016/j.procir.2012.10.030. 5. Sidorov D., Sazonov S., Revenko D. Building a dynamic model of the internal cylindrical grinding process. Procedia Engineering , 2016, vol. 150, pp. 400–405. DOI: 10.1016/j.proeng.2016.06.739. 6. Zhang N., Kirpitchenko I., Liu D.K. Dynamic model of the grinding process. Journal of Sound and Vibration , 2005, vol. 280, pp. 425–432. DOI: 10.1016/j.jsv.2003.12.006.