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

OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 4 boundary into the tool body (see Fig. 4), and to calculate σ y J, N +1 , equation (15) is used for lines with N = 2 or more. To calculate σ y J , 1 in the line N = 1, equations (10) and (15) should be rearranged in fi nite difference form for point (J, 0). After their transformation, considering that the values of σ y and  xy on the boundary outside the contact zone are equal to zero, we obtain the equation: ,1 1,0 ,0 1,0 1 ( 2 ) . 2 yJ J J J     + - = - + (16) To fi nd the values of σ y at the edges of the grid lines, equation (15) should be rearranged in fi nite differ- ence form using a doubled step along the y -axis, that is, replacing h y =2  h x (Fig. 3b on the left), and solved with respect to member  y J, N +2 : ( ) ( ) , 2 1, , 1, , 1, 1, , 2 4 2 10 4 . y J N J N J N J N y J N y J N yJ N yJ N         + + - + - - = ⋅ - ⋅ + + ⋅ - ⋅ + - (17) After determining  y J, N +2 , we can fi nd  y J -1, N +1 and  y J +1, N +1 at the edges of grid line N +1 (Fig. 3, b on the right), by transforming equation (15) with respect to the center node ( J, N +1). For zone B (see Fig. 4), the calculation was performed in the same manner, by taking into account the change in direction, that is, instead of layers N of zone A , layers J were considered. The normal stress  x at each node is determined by the equation: . x y    = - (18) The tangential stress component  xy can be determined from the equilibrium equations (11) and (12), if they are transformed into a fi nite difference form: , 1 , 1 1, 1, xyJ N xyJ N xJ N xJ N     + - + - = + - (19) and in zone B : 1, 1, , 1 , 1 . xyJ N xyJ N yJ N yJ N     - + + - = - + (20) For the fi rst grid line of zone A , the equation becomes: 1,0 1,0 ,1 . 2 xJ xJ xyJ    + - - = (21) In this way, using equations (15) to (21), it is possible to separate the stress sums Θ obtained as a result of the interference pattern analysis; that is, we can calculate the stress components  x ,  y and  xy . To calculate the temperature fi eld, the plot m 1 obtained before the application of the load (i.e., for a cold tool), is subtracted from the fringe order plot m 3 obtained immediately after the interruption of the cutting process. Thus, a plot of the fringe order m t = ( m 3 − m 1 ) can be obtained for the section of interest of the heated tool. If there are temperature deformations, taking into account temperature stresses and formula (3), Hooke’s law for a plane stress state can be represented as: 0 ( ) ( ) , t z x y t è m T T t E       = = - + + - (22) where ( T t − T 0 ) is the temperature change from the initial temperature T 0 to the reached temperature T t at the moment of interest during tool operation, ( σ x + σ y ) is the sum of thermal stresses, and α is the CTE of the tool material [26]. Thermal stresses, according to various research results, are usually less than 40 % of the stresses caused by cutting forces during tool operation. If we take ( σ x + σ y ) = 0, then during rough turning of steel (e.g., at T t = 740 К for cemented tungsten carbide WC-8Co [grade VK8 ]) with  = 4.7  10 − 6 К − 1 , Е = 596 GPa,