OBRABOTKAMETALLOV Vol. 25 No. 2 2023 technology Following the kinematically admissible scheme of the flow of a plastically compressible medium for layer A-B, taking into account the boundary conditions, the velocity field is represented in the form: τ ς = − α = + α = 1 2 1 2 [ ] sin , [ ] cos , [ ] 0 n V V V V V V V (9) where [ ] i V is a spike of the velocity vector of material particles moving through the severe deformation layer. The velocities V1 n, V 2 n (fig. 3b) are connected by the condition of mass conservation: − ϑ = = − ϑ 1 1 1 2 2 2 1 2 , ; 1 ñ ñ 1 n n V V V V (10) where r1, r2 is the density of the pressed material in parts II and I of the mold channel; ϑ1, ϑ2 is the porosity of the pressed material. In the case of a plastically compressible medium, the system of equations also includes the continuity condition, which in [32] is reintegrated along the trajectory of the representative element of the volume. It follows from the mass conservation condition (10) that the intersection of the plastically compressible medium layer A-B leads to a change in the relative porosity of the medium. Taking into account the mass conservation condition and the continuity condition, the density of the extruded material from the mold channel is determined as: − = + 1 2 2 1 1 2 2 ñ ñ exp V V V V . (11) Assuming that the relative density ρ of the compacted material is known from the analysis of the first stage of the ECAP process, the dissipation power of the severe deformation layer is calculated: ( ) → = + sξ 0 AB dh W AB TH S dh lim . (12) The intensity of the shear deformation rate H and the deformation rate of change in the volume ξ of the A-B layer are determined by the following relations: τ = + ξ = 1/2 2 2 1 4 [ ] [ ] [ ] ; 3 n n V H V V dh dh . The power balance equation (2), in which the value W | AB is calculated using equation (12), is applied to determine the energy-power parameters of the second stage of ECAP. Dividing the dissipative functions of the power balance equation by the values τs, V1, πD 2/4 equation (2) results in a dimensionless form: ( ) ( ) − r + χ = χ + + χ + + α − − χ − χ − − r 2 1 2 1 2 1 2 2/3 ln 1 4 1 2 4 cot( ) 1 3 1 3 1 (1 ) L L p p k k D D (13) τ τ = = τ τ 1 2 1 * 2 * , c c s c k k ;χ = 2 1 V V where 1 2 , k k are coefficients in the Siebel friction law; χ = r r 1 2 / is the parameter characterizing the compaction of the compressible medium in the A-B layer. Equation (13) is solved by the method of successive approximations. It is assumed that the density of the extruded blank is calculated for the previous stage of the studied process. The system of equations (1)–(13) makes it possible to unambiguously predict a set of technological parameters from cycle to cycle, which are necessary for the analysis and improvement of the ECAP process. A series of computational experiments were performed to determine the effect of the angle a on the extrusion pressure 1 p and the relative density of the extruded blank.
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