Abstract—The stage in which a central punch is introduced to form a depression during the plastic deformation of large disks or hard-to-deform disks is mathematically described.
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In this paper, we solve the problem formulated in [1]. In other words, we determine the basic deformational and mechanical parameters of sectional stamping, by theoretical means. That provides the basis for the design of a technological process.
The general methods employed were developed and verified in [2–5].
First, on the basis of Fig. 8, we consider the formation of a depression when a central punch is introduced. Note that, except in isolated cases, we use dimensionless geometric parameters referred to the radius r of the central punch. The plastic-deformation region is divided into zones 1 and 2. A rigid zone without plastic deformation lies below zones 1 and 2. We will use a cylindrical coordinate system (ρ, θ, z).
To simplify the analysis, we assume a rigid–plastic material. That corresponds to hot deformation, which is used in disk production. When cold deformation is employed, we take account of the mean yield point σs over the plastic-deformation region, which may expediently be determined by the method outlined in [6, 7].
In what follows, we use the relative stress (referred to σs). The contact-friction force is determined from the Siebel law: τco = μβσs, where μ is the frictional coefficient (with respect to σs); and β is the Lode coefficient.
In zone 2, the kinematically possible velocities are specified as follows.
The kinematically possible axial velocity is
It satisfies the following boundary conditions: \({{{v}}_{z}} = - {{{v}}_{0}}\) when z = 0 and z = –h1.
The incompressibility condition [4]
may be written in the form
or
From the boundary condition that vρ = 0 when ρ = 0, it follows that f1(z) = 0. Hence, we may write
We see that, in general, the radial velocity depends on the coordinate z. In other words, the barrel shape of the deformed workpiece may be taken into account by this means.
From Eq. (1), we may write the axial velocity in the specific form
That satisfies the boundary conditions already stated.
Then, from Eq. (6), we obtain an expression for the radial velocity
by means of Eqs. (7) and (8), we obtain
Taking account of Eq. (10), the effective strain rate takes the form [8]
to obtain the energy equation of plastic deformation [10]. In Eq. (12), Wq is the power of plastic deformation; Wσ is the power of the internal forces with effective stress σi in the plastic-deformation region V
WΔ is the power developed by the maximum tangential stress at all the surfaces SΔ corresponding to velocity discontinuity Δv
and Wτ is the power of all the contact-friction forces τco at slip velocity vco over the contact surface Sco
Thanks to the rigid–plastic model of the material and the Siebel frictional law, taking account of the plasticity condition σi = σs, dimensionless equations for the power (referred to the yield point) may be employed. Correspondingly, if we take account of the Lode coefficient β, we may write Eqs. (13)–(15) in the form
Then, in accordance with Eqs. (16) and (11), the power of the internal forces in zone 2 takes the form
It is evident from Eq. (10) that ξρ = ζθ in zone 2, and hence β = 1. Therefore, we may write the power of the contact friction at the punch surface in accordance with Eqs. (18) and (8) in the form
Since large disks are stamped when hot, as a rule, and the efficiency of this process is low even in the presence of lubricant, we adopt the value of the frictional coefficient corresponding to hot deformation: μ = 0.5 [11, 12]. Then
Note that, in cold stamping of a disk, the error due to the assumption that μ = 0.5 does not lead to marked discrepancy between the calculated and actual force in the equipment, since it is compensated by the familiar decrease in this force in the mechanics of a deformable solid because the conventional theories disregard the mismatch of the tangential stresses [13].
This is evident from the calculation of similar processes in [4]. In those calculations, the discrepancy with the experimental value (always a greater margin of deforming force in the calculation) is δ = 0.4 and 1.9% with increase in the frictional coefficient from μ1 = 0.1 to 0.5 (examples 4.11 and 4.12). Likewise, δ = 2.5 and 5.1% in examples 4.14 and 4.15; δ = 1.7 and 5.2% in examples 4.17 and 4.18; and δ = 0.3 and 6.0% in examples 6.12 and 6.13.
Taking account of Eqs. (17) and (18), we may write the power of the tangential surface at the discontinuity surfaces between zones 2 and the lower rigid zone in the form
In zone 1, the kinematically possible axial velocity takes the form
Obviously, vz1 = 0 when ρ = R. When ρ = 1, by contrast, vz1 = vz2 in Eq. (7); and vz1 = –v0 when z = 0. Thus, at the surface of the depression, vz1 varies from zero at the boundary with the lateral surface of the workpiece to the punch velocity at the contact point of the punch. That describes the formation of the depression.
Substituting Eq. (23) into the incompressibility condition in Eq. (3), we obtain
We find the constant C from the condition that vρ1 = vρ2 when ρ = 1. Then we may write
Substituting Eqs. (23) and (25) into Eq. (9), we find the strain rates
Substituting Eq. (26) into Eq. (11), we find the effective strain rate
It takes the following form when ρ = 1
By contrast, when ρ = R
It is simple to establish that the value in Eq. (29) is significantly less than that in Eq. (28). For example, when R = 2
Hence, for the sake of simplicity, we assume that the effective strain rate is constant in zone 1 and is determined by the mean version of Eq. (28)
Since the size of the depression is not known in advance, the indentation is ignored, and we determine the power of the internal forces in volume
This volume is slightly greater than that in Eq. (16) but is consistent with the upper bound. Hence, we obtain
Since β = 1 when ρ = 1 and β = 1.155 when ρ = R, we take the mean value at the horizontal discontinuity surface between zone 1 and the lower rigid zone: β = 1.1. Then, from Eq. (17), taking account of Eq. (25), we find the power of the tangential stress
In accordance with the power balance in Eq. (12), we sum Eqs. (19), (21), (22), (34), and (35) and regard the result as the deformational power
Hence we find the unit force qdef forming the depression
To determine the height of the plastic-deformation region, we use the condition of minimum unit deforming force
Since Eq. (38) is not solved algebraically when Eq. (37) is taken into account, we need to use the familiar numerical method of successive approximation in that case. In that method, selecting the calculation step, we gradually increase the depth hde of the depression from zero. Then, when R0 = 1, we find that hde = 0.889 and h1 = 1.075; and, when R0 = 2, we find that hde = 0.362 and h1 = 2.612. Note that the value hde = 0.362 when R0 = 2 is consistent with the experimental value hde = 0.36 (discrepancy δ = 0.5%) [12, Fig. 80c, p. 98].
Table 1 presents the calculation results.
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Translated by Bernard Gilbert
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Vorontsov, A.L. Production of Large Disks from Cylindrical Blanks by Plastic Deformation. 2. Introduction of Central Punch to Create a Depression. Russ. Engin. Res. 39, 567–570 (2019). https://doi.org/10.3103/S1068798X19070268
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DOI: https://doi.org/10.3103/S1068798X19070268