OBRABOTKAMETALLOV technology Vol. 25 No. 4 2023 and temperature of 600...1,500 ºС) transform the metal into a state similar to an ideal plastic, and the shear zone is narrowed into a thin strip, rather than a wedge as is seen in other types of cutting. According to Korchak, the thickness m of the shear zone is 1…5 µm [1]. As a result, it is also assumed for free cutting conditions that in the parallelogram shear zone (fig. 1), the stress intensity σ and deformation rate intensity e are constant on average. The metal is sheared in the cutting zone if the abrasive grain is absolutely sharp, when the length of the cutting edge is equal to bj and the length of the blunting area along the rear face is equal to zero, i.e. lj = 0. The area of the shear zone can be described by the thickness m of the zone itself, the cut depth as, and the shear angle β1 (fig. 1). The energy of an absolutely sharp abrasive grain is expended on the plastic deformation of the metal of a volume ωj in the shear zone during the shear time ∆t, when the peak of the sharp grain of the wheel passes the distance hj (see fig. 1, the distance between points O and О1) at a speed of Vk. The volume of chips being removed (fig. 1 and 2, a) consists of a set of volumes ωj. Fig. 1. Calculation model of plastic deformation in the shear zone when cutting with a single grain: 1 – outer metal surface; 2 – cutting grain motion pattern; 3 – cutting grain We transform expression (1) with regard to the shift of the elementary metal volume ωj in the cutting zone made by the j-th grain of the wheel to calculate the energy Аj spent on the plastic deformation of the metal volume ωj (fig. 1): ω = σe ω = σeω ∫∫∫ . j A d . (3) Let us express the power Nj of metal deformation in the shear zone through the increment of the elementary metal volume ωj and the elementary shear time ∆t: ω = = σe = σeω , j j dA d N dt dt (4) given that ω = β1 , sin s j j a mb (5) where as is the thickness of the cutoff made with a sharp single grain (fig. 1 and 2, b), m; m is the thickness of the metal shear zone in the cutting zone, m; β1 is the slope angle of the reference shear plane, deg. (fig. 2, b); bj is the length of the cutting edge of a single abrasive grain, m; Nj is the power of deformation of the metal volume ωj in the shear zone, N‧m/s (W).
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