OBRABOTKAMETALLOV Vol. 25 No. 2 2023 technology The power balance equation was applied to determine the power parameters of the second stage of the ECAP process: − =π τ + τ + 1 1 2 2 1 1 1 2 2 2 ( ) | , c c PV P V D LV L V W AB (2) where V1, V2 are flow velocities of plastically compressible mass from mold channels I and II; W | AB is a power dissipation in the severe deformation layer (layer thickness Δh→0). The physical equations of a representative element of the volume of a plastically compressible medium [28–30] have the form: ϑ s =s+ ξ − ξδ ( ) 1 2 3 ij ij ij T H (3) where σij, ξij are components of the stress tensor and deformation rate tensor; s ia an average normal stress; ξ is a volume strain rate; T is a shear stress intensity; H is a shear strain rate intensity; δij is a Kronecker symbol. The yield strengths in shear τ* s and isostatic compression * s p , depending on the relative porosity of the deformable medium, are given by the relations: τ = = τ − ϑ * 2/3 ( 1 ) s s T ; =−s=− τ ϑ * 2 ln 3 s s p , (4) where τs is a shear yield strength of titanium particles; ϑ is a relative porosity of the titanium sponge volume element. The dependencies τ τ τ = ϑ * / ( ) s s f and τ = ϑ * / ( ) s s p p f are shown in fig. 4. Consider the stage of the ECAP process in which briquette compression is carried out similar to the compression of a porous mass in a closed mold, using the results of [31]. For the first approximation, it is assumed that external friction can be neglected; the motion of plunger 2 is given; the pressure on the plunger is determined from the power balance equation (2); at the initial moment of pressing, the porosity of the briquette material is equal to ϑb. The boundary conditions in the cylindrical coordinate system (r, ϕ, z) have the following form: srz|r = R = 0, R = D/2; νr|r = 0 = νr|r = R = 0, νz|z = 0 = 0, νz|z = L1 = V1 = dl/dt. For these conditions, the kinematically permissible velocity field is nr = 0 , nz = V1 ∙ z/L1; components of the strain rate tensor: ξij = 0, except ξzz = –V1/L1; the rate of volume change in part I of the mold channel ξ = ξij. The degree of shear deformation Λ and the degree of volumetric deformation ε are of the form: Λ = ε = 1 1 2 , ln 3 b b L L L L ln . (5) The values of the relative porosity ϑ1 of the compressible medium in part I of the mold channel are the function of the movement dl of the working plunger: Fig. 4. Dependence of the yield strength of the compressible medium on porosity ϑ: for isostatic compression τ = ϑ * / ( ) s s p p f (1); for shear τ τ τ = ϑ * / ( ) s s f (2)
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