Study of the stress-strain and temperature fields in cutting tools using laser interferometry

OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. No. 4 2021 ô ó 0, xy x Y x y ¶ ¶ + + = ¶ ¶ (10) where X and Y are the components of the internal forces acting on the solid (tool). Because only external forces act on the tool, the internal forces are constant or zero; that is, they can be taken as X = Y = 0. Taking this into account, and differentiating equation (9) with respect to x and equation (10) with respect to y , we obtain: 2 2 2 0, y xy x y x   ¶ ¶ + = ¶ ¶ ¶ (11) 2 2 2 0 . y xy x y y   ¶ ¶ + = ¶ ¶ ¶ (12) Subtracting equations (11) and (12) we obtain: 2 2 2 2 0 . y x y x   ¶ ¶ + = ¶ ¶ (13) Transforming the last equation by making the substitution σ x = ( Θ  σ y ), we obtain 2 2 2 2 2 2 . y y x y x    ¶ ¶ ¶ + = ¶ ¶ ¶ (14) If equation (14) is represented in fi nite difference form (the node positions illustrated in Fig. 3, a ), we obtain 1, , 1 1, , 1 , 1, , 1, 4 2 . yJ N yJ N yJ N yJ N yJ N J N J N J N         + + - - + - + + + - = - + (15) For extrapolation (during line-by-line separation of the stress sums), the obtained equation (15) should be transformed with respect to members σ y J, N +1 , σ y J +1 , N , σ y J -1 , N , and σ y J, N -1 ; for harmonization equation should be transformed with respect to member σ y J, N . In zone A , the calculation is performed from the Fig. 4. Scheme to grid orientation relative to the cutting tool: C – length of the tool in contact with the chip; C 1 – length of the tool in contact with the workpiece; β – lip angle