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Upper bound analysis of axial metal flow inhomogeneity in radial forging process
International Journal of Mechanical Sciences 93 (2015) 102–110
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Upper bound analysis of axial metal flow inhomogeneity in radial forging process Yunjian Wu, Xianghuai Dong n, Qiong Yu National Die & Mold CAD Engineering Research Center, Shanghai Jiao Tong University, 1954 Hua Shan Road, Shanghai 200030, China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 October 2014 Received in revised form 24 December 2014 Accepted 19 January 2015 Available online 28 January 2015
An axisymmetric upper bound method model is proposed to analyze the inhomogeneity of axial metal flow in the radial forging process from the viewpoint of end profile in the meridian plane of the workpiece. The velocity fields are newly derived using stream function approach so that they can automatically satisfy the volume constancy as well as the velocity boundary conditions on the contact surface between the hammers and the workpiece. The assumed stream function has considered the inhomogeneous deformation mode. As a result, compared to the parallel velocity fields proposed in previous studies, reasonable predictions of radial load can be achieved even though the value of radial reduction is relatively low. Besides, the axial flow inhomogeneity along the radial direction can be described by this model because the axial velocity is no longer independent of the radial position. This model is verified by comparing the predicted forging load with published experimental data. Finally, the influences of axial feed and radial reduction on the end profile in the meridian plane of the workpiece are investigated. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Radial forging Inhomogeneous deformation Stream function Concave–convex depth
1. Introduction In industrial metal forming processes, redundant or unwanted deformation such as bulged profiles of upset forging occurs commonly. The material undergoes more strain than required for the area reduction and consequently strain hardens more and is less ductile [1]. To obtain the optimum conditions for the minimization of increased forming energy or the improvement of inhomogeneous strain distribution due to the redundant work, investigations on various processes such as upset forging [2,3], bar or wire drawing [4,5], flow forming [6], are conducted. On the other hand, by affecting the kinetics of recrystallization and the microstructure during subsequent annealing, such redundant deformation can also influence the mechanical properties of materials [7–9]. Deformation in radial forging results from a large number of shortstroke and high speed pressing operations by usually four hammer dies, arranged radially around the workpiece, as illustrated in Fig. 1. The workpiece is enclosed by forging dies and is therefore only able to flow axially, thereby avoiding or reducing greatly the redundant work spending on lateral spread. However, the frictional effects between the hammers and the workpiece still lead to redundant deformation characterized by an inhomogeneous distribution of axial metal flow in the radial direction. The surface layers are sheared relative to the
n
Corresponding author. Tel./fax: þ 86 21 62813435. E-mail address: [email protected] (X. Dong).
http://dx.doi.org/10.1016/j.ijmecsci.2015.01.012 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
center so that the plane sections will no longer remain plane [1]. As a result, concave or sometimes convex profiles occur in the meridian plane of the workpiece, which usually have to be cut off in industrial production. Therefore, it is necessary to study the inhomogeneity of axial metal flow to avoid excessive concave or convex profiles in the meridian plane. Radial forging process is generally studied by two approaches: theoretical analyses such as the slab method and upper bound method, and numerical analyses represented by the finite element method. It is known that slab analysis is based on making a force balance on a differentially thick slab of material, thus it is employed to predict the forging load or pressure distribution. For example, Lahoti et al. [10,11] developed an analytical model which takes into account the strain, strain rate and temperature effects on the material for radial forging of rods and tubes. Ghaei et al. [12,13] studied the effects of hammer geometry on the radial pressure and consequently drew conclusions for forging dies design. In essence, slab analysis cannot solve displacement field, so it is incapable of solving inhomogeneous metal flow problems. Upper bound analyses for the radial forging process are usually conducted by assuming a kinematically admissible velocity field and then minimizing the deformation power with regards to the location of the neutral plane. In this way, Ghaei et al. [14] and Sanjari et al. [15] investigated the influence of process parameters on forging load in the case of radial forging of rods and tubes. Wu et al. [16] studied the axial metal flow, especially the forward flow in the cold radial forging process of rods. However, the aforementioned velocity fields
Y. Wu et al. / International Journal of Mechanical Sciences 93 (2015) 102–110
103
Fig. 1. Schematic representation of the radial forging process.
did not consider the redundant deformation mode. They are called the parallel velocity fields since the axial velocity was assumed to be completely independent of the radial position. Above all, an available theoretical model for the study of axial metal flow inhomogeneity in radial forging process has not been proposed. Comparing with the solutions of theoretical analyses, the finite element method can be more practical and more accurate, especially for the prediction of strain fields. Domblesky and Shivpuri [17] developed a finite element model to study the plastic strain distribution along the radial direction for multiple-pass radial forging. For the prediction of plastic strain and temperature along the axial and hoop directions in the deformed workpiece, Chen et al. [18] presented a 3D nonlinear thermo-mechanical coupled model. Furthermore, utilizing the finite element method verified by the microhardness test, Sanjari et al. [19,20] studied the effects of process conditions on heterogeneity of the strain field, which was defined as an inhomogeneity factor. In summary, numerical method is available for the study of axial metal flow inhomogeneity, but it has not yet been discussed from the viewpoint of end profile in the meridian plane in the literature. In the present work, an axisymmetric upper bound method model is proposed to simulate the axial metal flow from the viewpoint of end profile in the meridian plane in the radial forging process. The velocity fields are derived from a specific stream function considering inhomogeneous deformation mode. Consequently, it can automatically satisfy the volume constancy as well as the velocity boundary conditions on the contact surface between the hammers and the workpiece. This model is verified by comparing the predicted forging load with published experimental data. Then the influences of axial feed and radial reduction on the axial metal flow inhomogeneity are investigated.
2. Mathematic modeling In the following analysis, the upper bound method model is based on the assumptions that (a) the deformation mode is simplified to be axisymmetric [10,14,15]; (b) the material is rigid-plastic and strainhardened; (c) shear frictional law is applied on the contact surface; and (d) the front-pull and back-push forces are neglected. Compared to the previous studies [10,21], it is worth noting that the end surface of workpiece no longer remains planar because the distribution of axial metal flow along the radial direction is actually inhomogeneous. The concave or convex depth of end surface is defined as the difference between the axial distance of particles on the outside surface and that at the centerline of the workpiece, as marked with d in Fig. 2(a). A positive value of d means that the end profile of the workpiece is concave. A negative value indicates a convex surface. The concave or convex depth reflects the degree of
Fig. 2. Schematic diagram of the upper bound method model: (a) the pre-forging state and (b) the steady state of the radial forging process.
deformation inhomogeneity: the larger the absolute depth, the more inhomogeneous the deformation. In the present model, a redundant deformation mode has been considered, so the concave or convex depth can be solved and it is adopted to analyze the axial metal flow inhomogeneity in the radial forging process. In general, the analytical models are built at the steady state of the radial forging process. However, in order to solve the end profile in the meridian plane, it needs to be modeled since the first stroke of the hammers. Fig. 2 shows the schematic diagram of this model in meridian plane. At the beginning of each pass, the workpiece is fed axially; therefore the contact surface between the workpiece and the pre-forging part of hammers increases gradually until the workpiece is fed to touch the sizing part of hammers. This process usually contains first several or dozens of strokes and it is called the pre-forging state of radial forging in this paper, as seen in Fig. 2(a). Then, the hammers are in full contact with the workpiece at the end of each subsequent
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stroke. Accordingly it is called the steady state of radial forging, as seen in Fig. 2(b). At the steady state, the plastic zone of workpiece contains two distinct regions: the sinking zone II and the forging zone III. Zone I and zone IV are generally considered as rigid zones since they are assumed to be long enough. In this model, zone I is formed at the beginning of the steady state and expands gradually due to the axial feed. Therefore, it is taken as a plastic zone within a certain range of forging length. At the pre-forging state, in contrast, there is only the forging zone III and a rigid zone IV. 2.1. Velocity fields In Fig. 2, extending the hammer contour to intersect with the centerline of the workpiece at point O, a cylindrical coordinate system is defined with point O as origin and with radial, circumferential, axial directions as r, θ, and z axis, respectively. The contour plotted in dashed line illustrates the deformation state at the beginning of present stroke, namely the end of previous stroke. The solid line illustrates the current position of the hammer and workpiece. 2.1.1. The forging zone A kinematically admissible velocity field must satisfy the volume constancy and the velocity boundary condition. Besides, it is generally considered that there is a neutral plane located in the forging zone and deforming metal flows axially away from it to both sides [10]. In the following analysis, not only the preceding conditions are satisfied, but also a redundant deformation mode which has never been considered is presented in present velocity field. For incompressible materials, the velocity fields derived from a stream function which is a powerful method in fluid mechanic can automatically satisfy the volume constancy [22–24]. The axial velocity u_ z .and radial velocity u_ r in cylindrical coordinates can be expressed by a stream function ψ as follows: 1 ∂ψ ðz; r Þ r ∂r 1 ∂ψ ðz; r Þ u_ r ðz; r Þ ¼ r ∂z u_ z ðz; r Þ ¼
ð1Þ
Obviously, they satisfy the continuity condition in cylindrical coordinates expressed as ∂ðr u_ z ðz; r ÞÞ ∂ðr u_ r ðz; r ÞÞ þ ¼0 ∂z ∂r
where φz ðz; r Þ and φr ðz; r Þ are the first derivative of φðz; r Þ with respect to the coordinates z and r, respectively. The position of the neutral plane is defined by u_ z ¼ 0, so it is found that the neutral plane is affected by both parameters x1, x2 and expression φðz; r Þ, and parameter x1 plays a leading role. Namely, parameter x1 mainly determines the approximate location of the neutral plane. On the other hand, due to the introduction of expression φðz; r Þ, the axial velocity is a function of both the axial and the radial coordinates while axial velocities proposed in other studies are generally with respect to only the axial coordinate. Accordingly, redundant deformation characterized by a concave or convex contour occurred on the end cross sections of the workpiece can be analyzed. Furthermore, φðz; r Þ is associated with the end profile in the meridian plane and the coefficient x2 reflects the extent of concave or convex profile. When x2 is equal to zero, this velocity field correspondingly degenerates to a parallel velocity field proposed in [16]. Substituting Eq. (5) into Eq. (3), the expressions of φðz; r Þ can be obtained as zφz ðz; z tan αÞ þ r φr ðz; z tan αÞ ¼ 0
φðz; z tan αÞ ¼ 0
ð6Þ
The general solution of the first-order partial differential equation (Eq. (6)) is φðz; rÞ ¼ η zr
ηð tan αÞ ¼ 0
ð7Þ
where ηðxÞ is an arbitrary function satisfying the condition ηð tan αÞ ¼ 0. Considering the general use of exponential function in the simulation of bulged profile [25–27], φðz; r Þ is defined as ! r cot α 2 φðz; r Þ ¼ exp 1 1 ð8Þ z
ð2Þ
According to the boundary conditions on the contact surface and that at the centerline, as illustrated in Fig. 2(b), the following equations are obtained. u_ z ðz; z tan αÞ sin α u_ r ðz; z tan αÞ cos α ¼ u_ cos α u_ r ðz; 0Þ ¼ 0
Referring to Eqs. (1) and (4), the forms of the axial velocity and the radial velocity can be obtained. 1 x2 x2 u_ z ðz; r Þ ¼ u_ cot α 1 21 1 þ x2 φðz; r Þ þ x2 u_ cot α 1 21 r φr ðz; r Þ 2 z z 1 x2 x2 u_ r ðz; r Þ ¼ u_ cot α 31 r 1 þ x2 φðz; r Þ x2 u_ cot α 1 21 r φz ðz; r Þ 2 z z ð5Þ
ð3Þ
where u_ z ðz; z tan αÞ and u_ r ðz; z tan αÞ are the axial and radial velocities on the contact surface between the hammer and the workpiece respectively; u_ r ðz; 0Þ is the radial velocity at the centerline of the workpiece; u_ is the instantaneous radial velocity and α is the inlet angle of hammer, respectively. In order to model the metal flow accurately, deforming characteristics of the metal flow should be considered as many as possible. For example, the existence of the neutral plane discussed by many studies and the inhomogeneity of axial metal flow proposed in this analysis are both included in the stream function given as x2 1 ψ ðz; rÞ ¼ u_ cot α 1 21 r2 1 þx2 φðz; rÞ ð4Þ 2 z where x1 and x2 are undetermined parameters and they will be finally determined by the minimization of dissipated power in each solving procedure. φðz; r Þ is an expression to be determined later.
2.1.2. The sizing zone Compared with the pre-forging state, analysis of velocity field in the sizing zone also needs to be performed at the steady state of the radial forging process. In Fig. 2(b), the radial velocity in sizing zone II is assumed to change linearly from u_ on the outside surface to zero at the centerline of the workpiece [16]. It is expressed as r u_ IIr ðz; r Þ ¼ u_ R
ð9Þ
where R is the current radius of sizing zone. According to the geometrical relations in Fig. 2, it can be derived as R ¼ R1 þ Lf tan α h
ð10Þ
where Lf is the axial feed and h is the current reduction in the radial direction. The velocity boundary condition on the boundary between zone II and zone III (hereinafter referred to as boundary II–III) is u_ IIz ðR cot α; r Þ ¼ u_ z ðR cot α; r Þ
ð11Þ
and u_ z are the axial velocities in the sizing zone II and where that in the forging zone III, respectively. Then the velocity field in the sizing zone can be obtained by solving the simultaneous Eqs. (2), (9) and (11). u_ IIz
Y. Wu et al. / International Journal of Mechanical Sciences 93 (2015) 102–110
2.1.3. Zone I Because the material in zone I has been forged to the required dimensions in current pass, there is no deformation along the radial direction. Besides, according to the velocity boundary condition on boundary I–II, the velocity field in this zone can be written as u_ Ir ¼ 0 u_ Iz
¼ u_ IIz R u_ Ir
cot α Lf ; r
ð12Þ
u_ Iz
and are the radial velocity and axial velocity in zone I, where respectively. Eq. (12) indicates that the metal in this zone shears only along the axial direction and it degenerates to a rigid zone if the axial velocity on boundary I–II is equal to a constant. 2.2. Dissipated powers At the pre-forging and the steady states, the required total power is divided into three components: the plastic power, the shear power and the frictional power. Because the compositions of the deformation zones at these two states are different, slight differences between the two states for the calculation of each kind of dissipative power exist as well. Although the following expressions are derived for the model at the steady state, as seen in Fig. 2(b), they can also be applied to the pre-forging state when some corrections are made. Referring to the strain–displacement relations, effective plastic strain rates can be derived from the velocity fields, and finally plastic power can be computed as Z Z Z _ p¼ W σ I ε_ I dV þ σ II ε_ II dV þ σ III ε_ III dV ð13Þ VI
V II
V III
where σ and ε_ are the average effective stress and the effective plastic strain rate respectively. The subscripts “I, II, III” denote zones I, II and III, respectively. The total shear power results from the velocity discontinuities existed on the boundaries between every two connected zones. It can be derived as follows: Z Z Z _t¼ W K I;II Δu_ I II dΓ þ K II;III Δu_ II III dΓ þ K III;IV Δu_ III IV dΓ Γ I II
Γ II III
Γ I IV
ð14Þ where K is the average yield shear stress and Δu_ is the velocity discontinuity. They are written as pffiffiffi K ¼ σ= 3 Δu_ I II ¼ u_ IIr R cot α Lf ; r Δu_ II III ¼ u_ r ðR cot α; rÞ u_ IIr ðR cot α; r Þ Δu_ III IV ¼ u_ r ðR0 cot α; r Þ ð15Þ where the subscripts “I–II” denote boundaries I–II, etc. The frictional power dissipated on the contact surface between the hammer and the workpiece is computed by Z Z _f¼ τII Δu_ II dΓ þ τIII Δu_ III dΓ ð16Þ W Γ f II
Γ f III
where τ is the shear stress on the frictional surface. According to the assumption of shear frictional law, τ is expressed as Eq. (17). Δu_ is the velocity discontinuity written as Eq. (18).
τ ¼ mK
ð17Þ
where m is the friction factor.
Δu_ II ¼ u_ IIz ðz; RÞ
Δu_ III ¼ u_ z ðz; z tan αÞ cos α þ u_ r ðz; z tan αÞ sin α þ u_ sin α
ð18Þ
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According to the upper bound theorem, an inequality for the radial load is established, as shown in Eq. (19). The total power obtained by adding all the preceding powers is expressed as a function of the undetermined parameters x1 and x2 . _ t þW _ ¼ f ðx1 ; x2 Þ _ p þW F u_ r W f
ð19Þ
For the pre-forging state, the dissipated powers associated with zone I and the sizing zone II should be equal to zero because it does not exist actually. Hence, when the first two terms on the right of Eqs. (13) and (14), the first term on the right of Eq. (16) are corrected to be zero, the equations are available for the pre-forging state as well. 2.3. Solving procedure Based on the upper bound theorem, the radial load can be solved by minimizing the power with respect to x1 and x2 in the current step. Taking these results as the initial condition of next step and assuming that physical quantities such as the velocity and the flow stress are constant, the whole process at both the preforging and the steady states can be solved step by step. The average effective stress in next step σ i þ 1 is updated by ð20Þ σ i þ 1 ¼ σ εai þ 1 ; ε_ ai ; T i where the subscript denotes the step number. σ εai ; ε_ ai ; T i is the mapping relationship between the average flow stress and other parameters. It can be a fitting model such as the Swift equation, the Voce equation, the Johnson–Cook law and their improved models; or it is just the interpolation of the original experimental data. T is the average temperature of the workpiece. εa and ε_ a are the average effective plastic strain and strain rate, respectively. They are defined as R ε_ dV ε_ ai ¼ R i dV εai þ 1 ¼ εai þ ε_ ai Δt ð21Þ where Δt is the time increment between two steps. In this model, the integration domain of the plastic zone is regular, so the integral of the plastic power is implemented directly on the whole domain without any discretization. As seen in Eq. (21), the average effective plastic strain rate is correspondingly calculated on the whole domain and it finally results in an average flow stress. As a result, the computational efficiency is consequently improved, but the flow stress and the yield shear stress are simplified to distribute homogeneously on the whole domain. If both sides of Eq. (19) are divided by σ , it is obvious that on the left side of the equation, the radial load is affected by the flow stress; on the right side of the equation, the three terms are independent of the flow stress. Therefore, the flow stress model affects slightly the velocity field of the workpiece. It is the limitation of both the current model and the traditional upper bound method. To balance the efficiency and the accuracy of the theoretical model, numerical methods on the integral of plastic powers will be improved in future work. Finally, the contour of end profile can be predicted by tracing the position of particles. For a certain particle, the final position in absolute coordinate system is obtained from the vector sum of displacement vectors in all previous strokes. Take a particle on the outside surface of the workpiece for example, when it moves form position P0 at the beginning of present stroke (namely the end of previous stroke) to position P at the end of present stroke (namely the beginning of next stroke), as shown in Fig. 2(a), the displace! ment vector u P 0 P is expressed as ! !0 u P P ¼ u_ P 0 ðzP0 ; r P 0 ÞΔt
ð22Þ
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Y. Wu et al. / International Journal of Mechanical Sciences 93 (2015) 102–110
! where u_ P 0 is the velocity vector of the particle at position P0 at the beginning of present stroke. Δt is the time increment between two strokes. In order to take advantages of the remarkable properties of symbolic mathematics and optimization algorithms, the entire procedure is coded in MATLAB. Similar to the authors' previous study [16], the results are not sensitive to the number of steps. In addition, when the radial reduction of hammer in each step is less than about 3.3% of the whole stroke, the fractional error caused by the number of steps is less than 1%.
3. Results and discussions 3.1. Radial load The predicted radial load was compared with the experimental measurement of cold swaging of AISI 1015 steel rods by Uhlig [28], which was also used by other researchers to verify their predictions [10,14,29,30]. In Uhlig's work, the stress–strain relationship of the material is assumed as σ ¼ σ 0 εn . The basic experiment parameters are listed in Table 1 and the results are shown in Table 2. It is found that the fraction errors between the predicted results and the experimental data are lower than 5%. Hence the availability of the velocity fields is verified. It is worth noting that the prediction in experimental case no. 5 is reasonable, while this prediction is not available in the authors' previous model [16]. For a parallel velocity field, the axial velocity is assumed to be uniform across a section and the radial velocity distributes linearly from the outside surface to the centerline of the workpiece. Accordingly, if the radial reduction is small, plastic deformation concentrates near the outside surface of the workpiece, so that deviations of velocity distributions between the assumed velocity field and the experiment increases. The present velocity field considering the inhomogeneous deformation mode is closer to the reality, so reasonable prediction can be obtained even though the radial reduction is relatively small. In order to confirm this viewpoint, contour plots of the stream function, respectively corresponding to the inhomogeneous velocity field in present paper and the formerly adopted homogeneous one, are compared under forging conditions of experimental case no. 5 in Table 2. According to the stream function expressed as Eq. (4), every position in forging zone corresponds to a specific value of stream function ψ and they are plotted with different colors depending on these values. Besides, as discussed earlier, a homogeneous deformation mode can be obtained by setting the undetermined parameter x2 to be zero. In this way, contour plots of stream functions are drawn in Fig. 3. Lines of ψ ¼ constant are streamlines and show the plastic flow patterns, since the instantaneous velocity vector of a particle is along the tangent direction of the streamline passing through it [31]. At the beginning of the stroke, comparing with Fig. 3(a) and (b), it is obvious that the deformation zone predicted by an inhomogeneous deformation mode is much closer to the surface. In contrast, the deformation predicted by a homogeneous deformation mode is distributed almost along the entire radial direction on the longitudinal section of the workpiece. As a result, the left boundary of the forging zone in Fig. 3(a) presents to be a concave contour while that in (b) remains a vertical line. At the end of the stroke, Fig. 3(c) and (d) shows a similar distribution of deformation. It should be noted that Table 1 Process parameters of Uhlig's experiment. Material
K (MPa)
n
α (deg)
m
Lf (mm)
AISI 1015
618.14
0.1184
4.3
0.15
0.37
Table 2 Comparison of predicted radial loads with the experimental results. No.
1 2 3 4 5
Diameter (mm)
Radial load (kN)
Workpiece
Product
Experiment
Analytical model
15.97 15.97 15.03 13.99 13.17
13.18 13.25 13.11 13.03 13.01
172 167 124 74 54
166.05 163.05 120.43 75.41 52.97
Error (%)
3.46 2.37 2.88 1.91 1.9
the boundaries in this figure represent only the profiles of the forging zone in the current step; the final profiles of the end surface are the accumulation of deformations in all steps. In summary, during the forging process of a stroke, the deformation predicted by present model penetrates gradually to the centerline of the workpiece, while the deformation predicted by the previous parallel velocity field is always distributed along the entire radial direction. This difference finally results in the different predictions of radial load for experimental case no. 5. 3.2. The neutral plane Generally, the velocity fields are solved by minimizing the total power with respect to an undetermined parameter in theoretical analyses of the radial forging process. This parameter physically represents the position of the neutral plane. The material which is exactly on this neutral plane is only deformed radially and does not flow axially. In the present model, the neutral plane can also be obtained according to two parameters defined in the stream function Eq. (4). Expression of axial velocity can be derived from the stream function, and then solutions of the equation u_ z ¼ 0 illustrate exactly the position of the neutral plane. It is marked with dot dash line in Fig. 4. On the other hand, as shown in Fig. 4, contour plots of the stream function are drawn under certain forging conditions. Since the velocity vector of a particle is always along the tangent of the streamline passing through it, and the axial velocities of particles on the neutral plane are equal to zero, so the neutral plane is characterized by vertical streamline, and its position can be roughly estimated. In Fig. 4(a), the profile of “the neutral plane” is actually not a planar cross section of the workpiece. It is ascribed to the fact that the assumed velocity field has considered the inhomogeneous deformation mode. The axial velocity is no longer evenly distributed along the radial direction, especially at the pre-forging state. With the increase of the radial reduction rate from 2.1% in Fig. 4(a) to 4.7% in (b), “the neutral plane” tends to be a plane. Therefore, it is reasonable for Lahoti [10,11] and other researchers [12,14,15] to assume that the position on which the material is only deformed radially and does not flow axially, is exactly a planar section which is called the neutral plane. It is also found in Fig. 4 that (a) the streamlines are distributed on both sides of the neutral plane, and the farther from the neutral plane or the centerline of the workpiece, the higher the density of the streamlines; (b) the streamlines are concentrated near to the outside surface when the current radius is relatively large. The former indicates that the deforming material flows axially away from the neutral plane to both sides simultaneously; the velocity increases with an increasing distance to both the position of the neutral plane and the centerline of the workpiece, because the velocity is inversely proportional to the spacing of the streamlines. The later shows that as the radial reduction increases, the plastic deformation penetrates gradually to the centerline of the workpiece. In summary, the above corollaries are consistent with the conclusions of previous studies [10,16] and empirical knowledge; therefore the present model can be validated qualitatively as well.
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107
Fig. 3. Contour plots of the stream functions which have considered (a) inhomogeneous, (b) homogeneous deformation modes at the beginning of the stroke and (c) inhomogeneous, (d) homogeneous deformation modes at the end of the stroke. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
3.3. Concave–convex depth The end surface of the workpiece keeps planar if the metal flows homogeneously. Actually, the distribution of axial metal flow along the radial direction is inhomogeneous and consequently concave or sometimes convex profiles occur on the end surface of the workpiece, as indicated in Fig. 2. The concave or convex depth of end surface reflects the degree of deformation inhomogeneity: the larger the depth, the more inhomogeneous the deformation. According to the displacements of particles at different radial positions, as expressed by Eq. (22), the concave or convex depth of end surface can be obtained d ¼ z Q zP
ð23Þ
where zQ and zP are the axial coordinates of position at the centerline and that on the outside surface of the workpiece, respectively. In the following analysis, the concave or convex depth is analyzed under various process parameters listed in Table 3. The properties of material, geometry of hammer and friction factor are the same as those adopted in Section 3.1. 3.3.1. History plots of the end surface profile Fig. 5(a) shows the history plots of end surface profiles when the workpiece is forged from an initial radius R0 ¼7.985 mm to the final radius R1 ¼6.5 mm with a constant axial feed Lf ¼0.5 mm in a single pass. Each curve in this figure corresponds to one stroke shown in Fig. 2. It is assumed that deformation in radial forging results from the accumulation of successive short-strokes. The plastic deformation is distributed unevenly along the radial direction because a short stroke tends to make the plastic deformation concentrate mainly near the outside surface rather than penetrate to the centerline of the workpiece. The axial velocity of material is decreased with a decreasing distance to the centerline of workpiece. As a result, the end surface presents to be a concave profile. As the hammer presses radially, the concave depth increases gradually. If the curves are rotated anticlockwise 90º, they coincide with the end surface profile illustrated in the schematic diagram at the top left corner of Fig. 5(a). History plot of the concave–convex depth during this process, namely the depth of each curve in Fig. 5(a), is drawn in (b). In this
figure, the abscissa axis, forging length, means the length of the deformed workpiece. The curve on both sides of the dashed line represents the history plot of concave–convex depth at the preforging state and that at the steady state, respectively. At the preforging state, due to the accumulation of deformation inhomogeneity during the successive strokes, the concave–convex depth of end surface becomes increased. On the other hand, when the radial reduction increases, the plastic deformation zone expands gradually to the centerline of the workpiece and consequently the difference between the velocity on the outside surface and that at the centerline of the workpiece decreases. Finally, the depth increasing rate will decline and it is proved in Fig. 5(b). It is observed in Fig. 5(b) that at the steady state the concave–convex depth tends to be a stable value. To show detailed variation of concave–convex depth at the steady state, the enlarged history plot of this state is redrawn in Fig. 5(c). As mentioned in Section 2, the calculation at the steady state is performed within a certain range of forging length; if the forging length, and then the length of zone I defined in Fig. 2(b), is long enough, the plastic deformation zone could not extend to the end surface. In the following analyses, for the sake of convenience, the maximal forging length at the steady state is assumed to be equal to the sizing part length of the hammer, namely L in Table 3. As a result, the concave– convex depth decreases gradually to a stable value. It indicates that zone I will change from a plastic zone to a rigid zone within a forging length of L. Accordingly, it is reasonable for other researchers to assume that zone I is a rigid zone at the steady state.
3.3.2. Effects of the radial reduction In order to study the effects of radial reduction on the concave– convex depth, calculation instances are performed, in which the workpiece is forged form an initial radius R0 ¼7.985 mm to the final radius range of R1 ¼6.0–7.5 mm with a constant axial feed Lf ¼0.5 mm, as listed in Table 3. In Fig. 6, the relative radial reduction is defined as a dimensionless ratio of the radial reduction to the initial radius R0 of the workpiece. The curve marked with open circle and that marked with open triangle illustrate the net increments of concave–convex
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Fig. 4. Streamlines and the neutral plane of the workpiece forged from an initial radius R0 ¼ 7.985 mm to current radii (a) R¼ 7.816 mm and (b) R ¼7.609 mm.
Table 3 Process parameters in present study. R0 (mm)
R1 (mm)
Lf (mm)
L (mm)a
7.985
6–7.5
0.1–1.1
18
a
L is the length of the sizing part of hammer, as illustrated in Fig. 2.
depths at the pre-forging state and that at the steady state, respectively. At the pre-forging state, the workpiece is fed axially and forged by the forging part of the hammer, as seen in Fig. 2(a). During the
Fig. 5. History plots of (a) the end surface profiles, (b) the concave–convex depth d and (c) the enlarged curve of concave–convex depth d at the steady state when the workpiece is forged from an initial radius R0 ¼ 7.985 mm to the final radius R1 ¼ 6.5 mm with a constant axial feed Lf ¼ 0.5 mm.
descent of the hammer, the real radial reduction of the workpiece is lower than the required radial reduction and actually depends on the axial feed per stroke. Consequently, for the forging processes with different radial reductions, although the final radial reductions are different with each other, the real radial reductions of the workpiece
Y. Wu et al. / International Journal of Mechanical Sciences 93 (2015) 102–110
Fig. 6. The effects of relative radial reduction (R0 R1)/R0 on the concave–convex depths d at different states (R0 ¼7.985 mm, R1 ¼ 6.0–7.5 mm, Lf ¼ 0.5 mm).
are identical since the workpieces are forged with the same axial feed. In other words, the radial reduction affects only the number of total strokes rather than the process condition of each stroke. Therefore, with the increase of relative radial reduction, the concave–convex depth at the pre-forging state becomes increased due to the accumulation of strokes. At the steady state, however, the different radial reductions will result in the differences of process conditions in each stroke. When the relative radial reduction increases, the plastic deformation penetrates gradually to the centerline of workpiece and simultaneously the contact surface between the workpiece and hammer becomes increased. The former leads to the increase of axial velocity at the centerline of workpiece, the later results in the decrease of axial velocity on the outside surface of workpiece due to frictional effects. Therefore, the net increment of concave–convex depth at the steady state decreases gradually to zero and finally increases in opposite direction, namely, the end cross section tends to change from the concave surface to a convex surface. In summary, when the radial reduction increases, the axial flow inhomogeneity at the preforging state increases, at the steady state the axial flow inhomogeneity with concave mode declines slightly firstly then increases slightly with convex mode. Besides, it is observed from this figure that the net increment of concave–convex depth at the end of pre-forging state is significantly greater than that at the steady state. For example, when the relative radial reduction is 18.6%, the forging lengths during these two states are 19.75 mm and 18 mm respectively, but the depth increments are 0.218 mm and 0.0025 mm respectively. Although the forging lengths are in the same order of magnitude, the differential of depth increments is about two orders of magnitude. The reason is that the neutral plane tends to change from curved surface to a planar surface when radial reduction increases, as shown in Fig. 4, consequently the inhomogeneity in deformation zone decreases. The final concave– convex depth of end surface is the sum of depth increments at both the pre-forging and the steady states, as seen in the curve marked with open square. Comparing with the curve at the pre-forging state, the final depth is reduced by the deformation at the steady state slightly. Although higher value of radial reduction can reduce the axial flow inhomogeneity at the steady state, the positive effect is insufficient to compensate the increase of inhomogeneity at the pre-forging state. According to the above analysis, conclusion can be drawn that the concave profile of end surface is mainly formed at the pre-forging state and it is suggested that a relatively low value of radial reduction is conducive to the axial flow homogeneity.
109
3.3.3. Effects of the axial feed Fig. 7 shows the effects of relative axial feed on the concave– convex depths when the workpiece is forged from an initial radius R0 ¼7.985 mm to the final radius R1 ¼6.59 mm with an axial feed range of Lf ¼0.1–1.1 mm. Similarly, the relative axial feed is defined as a dimensionless ratio of the axial feed Lf to the initial radius R0. The curve marked with open circle and that marked with open triangle illustrate the net increments of concave–convex depths at the pre-forging state and that at the steady state, respectively. The increase of relative axial feed contributes to the penetration of plastic deformation at both the pre-forging and the steady states. Accordingly, the axial flow inhomogeneity between the material on the outside surface and that at the centerline is reduced, so that the axial velocity at the centerline increases gradually and the velocity differential becomes decreased. Therefore, the concave– convex depth at the pre-forging state is decreased. At the steady state, the forging part of hammer contacts fully with the workpiece and the contact surface increases with the increase of axial feed. The axial flow of material on the outside surface is severely restricted due to the frictional effects, so that the end surface prefers to be a convex mode and the axial flow inhomogeneity increase gradually, as shown in the curve marked with open triangle. Compared to Fig. 6, it is also found that the radial reduction has more significant effects than the axial feed on the concave–convex depth of end surface at the pre-forging state, while the influences of them are at the same level at the steady state. For example, the depth increasing rate versus relative radial reduction at the pre-forging state is nearly 10 times of the rate versus relative axial feed. Accordingly, it is concluded that the axial flow inhomogeneity is more sensitive to the radial reduction than the axial feed.
4. Conclusion and future work In the radial forging process, the frictional effects between the hammers and the workpiece lead to redundant deformation characterized by an inhomogeneous distribution of axial metal flow along the radial direction. The surface layers are sheared relative to the center so that the planar sections will no longer remain plane. Therefore, the axial metal flow inhomogeneity can be analyzed from the viewpoint of the end surface concave–convex depth of the workpiece. In order to solve this problem, an upper bound model considering the redundant deformation mode is proposed and
Fig. 7. The effects of relative axial feed Lf /R0 on the concave–convex depths d at different states (R0 ¼ 7.985 mm, R1 ¼6.59 mm, and Lf ¼0.1–1.1 mm).
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validated by comparing the predicted forging loads with published experimental results. Finally, the following conclusions are drawn: (1) Reasonable predictions of radial load can be achieved even when the radial reduction is small since the redundant deformation mode is considered. Besides, it is found that the neutral plane predicted in present study is no longer a planar cross section. (2) During a forging process at the pre-forging state, the concave– convex depth increases gradually because of the accumulation of deformation inhomogeneity and the increasing rate will decline due to the penetration of plastic deformation. (3) The concave or convex profile is mainly formed at the preforging state and finally determined by the sum of depth increments at both the pre-forging and the steady states. At the pre-forging state, the concave–convex depth increases with the increase of radial reduction or the decrease of axial feed. At the steady state, the concave depth decreases to zero and then increases with convex mode with the increases of both the radial reduction and the axial feed. Therefore, in order to achieve a homogeneous distribution of axial metal flow, a relatively low value of radial reduction or a relatively high value of axial feed is required. (4) The axial flow inhomogeneity is more sensitive to the radial reduction than the axial feed. There is also further work to improve the current analysis. As discussed in Section 2.3, numerical methods on the integral of plastic powers should be improved to balance the efficiency and the accuracy of this analysis. Besides, since the front-pull and back-push forces are neglected in the present model, the absolute position of the end surface is accordingly changed. However, the concave–convex depth expressed in Eq. (23), which is the relative difference of the surface contour, is not significantly affected unless the contour of the neutral plane is also changed by these forces. Therefore, the effects of the front-pull and back-push forces on the contour of the neutral plane need to be studied in the future work.
Acknowledgments The authors would like to gratefully acknowledge the support of National Natural Science Foundation of China by Grant no. 51275297. References [1] Hosford WF, Caddell RM. Work balance. Metal forming: mechanics and metallurgy. Cambridge University Press; Cambridge, UK; 2007. p. 76–84. [2] Hou J, Ståhlberg U, Keife H. Bulging and folding-over in plane-strain upset forging. J Mater Process Technol 1992;35:199–212. [3] Yeh W-C, Wu M-C, Hong. J-J. A general upper-bound formulation of stream function of upset forging of ring using a variational approach. Mater Sci Eng: B 2010;172:55–67.
[4] Chin RK, Steif. PS. A computational study of strain inhomogeneity in wire drawing. Int J Mach Tools Manuf 1995;35:1087–98. [5] Majzoobi GH, Saniee FF, Aghili. A. An investigation into the effect of redundant shear deformation in bar drawing. J Mater Process Technol 2008;201:133–7. [6] Mohebbi MS, Akbarzadeh A. Experimental study and FEM analysis of redundant strains in flow forming of tubes. J Mater Process Technol 2010;210:389–95. [7] Lindh E, Hutchinson B, Ueyama S. Effect of redundant deformation on recrystallisation behaviour of copper. Scr Metall Mater 1993;29:347–52. [8] Kamikawa N, Sakai T, Tsuji N. Effect of redundant shear strain on microstructure and texture evolution during accumulative roll-bonding in ultralow carbon IF steel. Acta Mater 2007;55:5873–88. [9] Guo L, Fujita F. Effect of equivalent strain and redundant shear strain on microstructure and texture evolution during hot rolling in Mg–3Al–1Zn alloys. J Mater Sci 2012;47:6213–9. [10] Lahoti GD, Altan T. Analysis of the radial forging process for manufacturing rods and tubes. J Eng Ind 1976;98:265–71. [11] Lahoti GD, Liuzzi L, Altan T. Design of dies for radial forging of rods and tubes. J Mech Work Technol 1977;1:99–109. [12] Ghaei A, Movahhedy MR, Taheri AK. Study of the effects of die geometry on deformation in the radial forging process. J Mater Process Technol 2005;170:156–63. [13] Afrasiab H, Movahhedy MR. Generalized slab method analysis of radial forging with a general curved shape die. In: Proceedings of the ASME 8th Biennial Conference on Engineering Systems Design and Analysis, vol. 4; 2006. p. 883–89. [14] Ghaei A, Taheri AK, Movahhedya. MR. A new upper bound solution for analysis of the radial forging process. Int J Mech Sci 2006:1264–72. [15] Sanjari M, Karimi Taheri A, Ghaei A. Prediction of neutral plane and effects of the process parameters in radial forging using an upper bound solution. J Mater Process Technol 2007;186:147–53. [16] Wu Y, Dong X, Yu. Q. An upper bound solution of axial metal flow in cold radial forging process of rods. Int J Mech Sci 2014;85:120–9. [17] Domblesky JP, Shivpuri R. Development and validation of a finite-element model for multiple-pass radial forging. J. Mater. Process. Technol. 1995;55:432–41. [18] Chen J, Chandrashekhara K, Richards VL, Lekakh. SN. Three-dimensional nonlinear finite element analysis of hot radial forging process for large diameter tubes. Mater Manuf Process 2010;25:669–78. [19] Sanjari M, Taheri AK, Movahedi. MR. An optimization method for radial forging process using ANN and Taguchi method. Int J Adv Manuf Technol 2009;40:776–84. [20] Sanjari M, Saidi P, Karimi Taheri A, Hossein-Zadeh M. Determination of strain field and heterogeneity in radial forging of tube using finite element method and microhardness test. Mater Des 2012;38:147–53. [21] Tseng AA, Tong SX, Chen TC, Hashemi J. Thermomechanical simulation of a radial forging process. Mater Des 1994;15:87–98. [22] Hwang Y-M, Hsu H-H. Analysis of plate rolling using the dual-stream function method and cylindrical coordinates. Int J Mech Sci 1998;40:371–85. [23] Hwang Y-M, Hwang. T-F. An investigation into the plastic deformation behavior within a conical die during composite rod extrusion. J Mater Process Technol 2002;121:226–33. [24] Rejaeian M, Aghaie-Khafri M. Study of ECAP based on stream function. Mech Mater 2014;76:27–34. [25] Avitzur B, Sauerwine. F. Limit analysis of hollow disk forging—Part 1: upper bound. J Manuf Sci Eng 1978;100:340–4. [26] Aksakal B, Sezek S, Can Y. Forging of polygonal discs using the dual stream functions. Mater Des 2005;26:643–54. [27] Singh S, Jha AK, Kumar S. Upper bound analysis and experimental investigations of dynamic effects during sinter-forging of irregular polygonal preforms. J Mater Process Technol 2007;194:134–44. [28] Uhlig A. Investigation of the motions and the forces in radial swaging. Hannover: Technical University; 1964. [29] Ameli A, Movahhedy. MR. A parametric study on residual stresses and forging load in cold radial forging process. Int J Adv Manuf Technol 2007;33:7–17. [30] Ghaei A, Movahhedy MR, Taheri AK. Finite element modelling simulation of radial forging of tubes without mandrel. Mater Des 2008:867–72. [31] Vanyo JP. Chapter 3 – Potential (inviscid) flow. In: Vanyo JP, editor. Rotating fluids in engineering and science. Boston: Butterworth-Heinemann; 1993. p. 43–55.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Upper bound analysis of axial metal flow inhomogeneity in radial forging process Yunjian Wu, Xianghuai Dong n, Qiong Yu National Die & Mold CAD Engineering Research Center, Shanghai Jiao Tong University, 1954 Hua Shan Road, Shanghai 200030, China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 October 2014 Received in revised form 24 December 2014 Accepted 19 January 2015 Available online 28 January 2015
An axisymmetric upper bound method model is proposed to analyze the inhomogeneity of axial metal flow in the radial forging process from the viewpoint of end profile in the meridian plane of the workpiece. The velocity fields are newly derived using stream function approach so that they can automatically satisfy the volume constancy as well as the velocity boundary conditions on the contact surface between the hammers and the workpiece. The assumed stream function has considered the inhomogeneous deformation mode. As a result, compared to the parallel velocity fields proposed in previous studies, reasonable predictions of radial load can be achieved even though the value of radial reduction is relatively low. Besides, the axial flow inhomogeneity along the radial direction can be described by this model because the axial velocity is no longer independent of the radial position. This model is verified by comparing the predicted forging load with published experimental data. Finally, the influences of axial feed and radial reduction on the end profile in the meridian plane of the workpiece are investigated. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Radial forging Inhomogeneous deformation Stream function Concave–convex depth
1. Introduction In industrial metal forming processes, redundant or unwanted deformation such as bulged profiles of upset forging occurs commonly. The material undergoes more strain than required for the area reduction and consequently strain hardens more and is less ductile [1]. To obtain the optimum conditions for the minimization of increased forming energy or the improvement of inhomogeneous strain distribution due to the redundant work, investigations on various processes such as upset forging [2,3], bar or wire drawing [4,5], flow forming [6], are conducted. On the other hand, by affecting the kinetics of recrystallization and the microstructure during subsequent annealing, such redundant deformation can also influence the mechanical properties of materials [7–9]. Deformation in radial forging results from a large number of shortstroke and high speed pressing operations by usually four hammer dies, arranged radially around the workpiece, as illustrated in Fig. 1. The workpiece is enclosed by forging dies and is therefore only able to flow axially, thereby avoiding or reducing greatly the redundant work spending on lateral spread. However, the frictional effects between the hammers and the workpiece still lead to redundant deformation characterized by an inhomogeneous distribution of axial metal flow in the radial direction. The surface layers are sheared relative to the
n
Corresponding author. Tel./fax: þ 86 21 62813435. E-mail address: [email protected] (X. Dong).
http://dx.doi.org/10.1016/j.ijmecsci.2015.01.012 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
center so that the plane sections will no longer remain plane [1]. As a result, concave or sometimes convex profiles occur in the meridian plane of the workpiece, which usually have to be cut off in industrial production. Therefore, it is necessary to study the inhomogeneity of axial metal flow to avoid excessive concave or convex profiles in the meridian plane. Radial forging process is generally studied by two approaches: theoretical analyses such as the slab method and upper bound method, and numerical analyses represented by the finite element method. It is known that slab analysis is based on making a force balance on a differentially thick slab of material, thus it is employed to predict the forging load or pressure distribution. For example, Lahoti et al. [10,11] developed an analytical model which takes into account the strain, strain rate and temperature effects on the material for radial forging of rods and tubes. Ghaei et al. [12,13] studied the effects of hammer geometry on the radial pressure and consequently drew conclusions for forging dies design. In essence, slab analysis cannot solve displacement field, so it is incapable of solving inhomogeneous metal flow problems. Upper bound analyses for the radial forging process are usually conducted by assuming a kinematically admissible velocity field and then minimizing the deformation power with regards to the location of the neutral plane. In this way, Ghaei et al. [14] and Sanjari et al. [15] investigated the influence of process parameters on forging load in the case of radial forging of rods and tubes. Wu et al. [16] studied the axial metal flow, especially the forward flow in the cold radial forging process of rods. However, the aforementioned velocity fields
Y. Wu et al. / International Journal of Mechanical Sciences 93 (2015) 102–110
103
Fig. 1. Schematic representation of the radial forging process.
did not consider the redundant deformation mode. They are called the parallel velocity fields since the axial velocity was assumed to be completely independent of the radial position. Above all, an available theoretical model for the study of axial metal flow inhomogeneity in radial forging process has not been proposed. Comparing with the solutions of theoretical analyses, the finite element method can be more practical and more accurate, especially for the prediction of strain fields. Domblesky and Shivpuri [17] developed a finite element model to study the plastic strain distribution along the radial direction for multiple-pass radial forging. For the prediction of plastic strain and temperature along the axial and hoop directions in the deformed workpiece, Chen et al. [18] presented a 3D nonlinear thermo-mechanical coupled model. Furthermore, utilizing the finite element method verified by the microhardness test, Sanjari et al. [19,20] studied the effects of process conditions on heterogeneity of the strain field, which was defined as an inhomogeneity factor. In summary, numerical method is available for the study of axial metal flow inhomogeneity, but it has not yet been discussed from the viewpoint of end profile in the meridian plane in the literature. In the present work, an axisymmetric upper bound method model is proposed to simulate the axial metal flow from the viewpoint of end profile in the meridian plane in the radial forging process. The velocity fields are derived from a specific stream function considering inhomogeneous deformation mode. Consequently, it can automatically satisfy the volume constancy as well as the velocity boundary conditions on the contact surface between the hammers and the workpiece. This model is verified by comparing the predicted forging load with published experimental data. Then the influences of axial feed and radial reduction on the axial metal flow inhomogeneity are investigated.
2. Mathematic modeling In the following analysis, the upper bound method model is based on the assumptions that (a) the deformation mode is simplified to be axisymmetric [10,14,15]; (b) the material is rigid-plastic and strainhardened; (c) shear frictional law is applied on the contact surface; and (d) the front-pull and back-push forces are neglected. Compared to the previous studies [10,21], it is worth noting that the end surface of workpiece no longer remains planar because the distribution of axial metal flow along the radial direction is actually inhomogeneous. The concave or convex depth of end surface is defined as the difference between the axial distance of particles on the outside surface and that at the centerline of the workpiece, as marked with d in Fig. 2(a). A positive value of d means that the end profile of the workpiece is concave. A negative value indicates a convex surface. The concave or convex depth reflects the degree of
Fig. 2. Schematic diagram of the upper bound method model: (a) the pre-forging state and (b) the steady state of the radial forging process.
deformation inhomogeneity: the larger the absolute depth, the more inhomogeneous the deformation. In the present model, a redundant deformation mode has been considered, so the concave or convex depth can be solved and it is adopted to analyze the axial metal flow inhomogeneity in the radial forging process. In general, the analytical models are built at the steady state of the radial forging process. However, in order to solve the end profile in the meridian plane, it needs to be modeled since the first stroke of the hammers. Fig. 2 shows the schematic diagram of this model in meridian plane. At the beginning of each pass, the workpiece is fed axially; therefore the contact surface between the workpiece and the pre-forging part of hammers increases gradually until the workpiece is fed to touch the sizing part of hammers. This process usually contains first several or dozens of strokes and it is called the pre-forging state of radial forging in this paper, as seen in Fig. 2(a). Then, the hammers are in full contact with the workpiece at the end of each subsequent
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stroke. Accordingly it is called the steady state of radial forging, as seen in Fig. 2(b). At the steady state, the plastic zone of workpiece contains two distinct regions: the sinking zone II and the forging zone III. Zone I and zone IV are generally considered as rigid zones since they are assumed to be long enough. In this model, zone I is formed at the beginning of the steady state and expands gradually due to the axial feed. Therefore, it is taken as a plastic zone within a certain range of forging length. At the pre-forging state, in contrast, there is only the forging zone III and a rigid zone IV. 2.1. Velocity fields In Fig. 2, extending the hammer contour to intersect with the centerline of the workpiece at point O, a cylindrical coordinate system is defined with point O as origin and with radial, circumferential, axial directions as r, θ, and z axis, respectively. The contour plotted in dashed line illustrates the deformation state at the beginning of present stroke, namely the end of previous stroke. The solid line illustrates the current position of the hammer and workpiece. 2.1.1. The forging zone A kinematically admissible velocity field must satisfy the volume constancy and the velocity boundary condition. Besides, it is generally considered that there is a neutral plane located in the forging zone and deforming metal flows axially away from it to both sides [10]. In the following analysis, not only the preceding conditions are satisfied, but also a redundant deformation mode which has never been considered is presented in present velocity field. For incompressible materials, the velocity fields derived from a stream function which is a powerful method in fluid mechanic can automatically satisfy the volume constancy [22–24]. The axial velocity u_ z .and radial velocity u_ r in cylindrical coordinates can be expressed by a stream function ψ as follows: 1 ∂ψ ðz; r Þ r ∂r 1 ∂ψ ðz; r Þ u_ r ðz; r Þ ¼ r ∂z u_ z ðz; r Þ ¼
ð1Þ
Obviously, they satisfy the continuity condition in cylindrical coordinates expressed as ∂ðr u_ z ðz; r ÞÞ ∂ðr u_ r ðz; r ÞÞ þ ¼0 ∂z ∂r
where φz ðz; r Þ and φr ðz; r Þ are the first derivative of φðz; r Þ with respect to the coordinates z and r, respectively. The position of the neutral plane is defined by u_ z ¼ 0, so it is found that the neutral plane is affected by both parameters x1, x2 and expression φðz; r Þ, and parameter x1 plays a leading role. Namely, parameter x1 mainly determines the approximate location of the neutral plane. On the other hand, due to the introduction of expression φðz; r Þ, the axial velocity is a function of both the axial and the radial coordinates while axial velocities proposed in other studies are generally with respect to only the axial coordinate. Accordingly, redundant deformation characterized by a concave or convex contour occurred on the end cross sections of the workpiece can be analyzed. Furthermore, φðz; r Þ is associated with the end profile in the meridian plane and the coefficient x2 reflects the extent of concave or convex profile. When x2 is equal to zero, this velocity field correspondingly degenerates to a parallel velocity field proposed in [16]. Substituting Eq. (5) into Eq. (3), the expressions of φðz; r Þ can be obtained as zφz ðz; z tan αÞ þ r φr ðz; z tan αÞ ¼ 0
φðz; z tan αÞ ¼ 0
ð6Þ
The general solution of the first-order partial differential equation (Eq. (6)) is φðz; rÞ ¼ η zr
ηð tan αÞ ¼ 0
ð7Þ
where ηðxÞ is an arbitrary function satisfying the condition ηð tan αÞ ¼ 0. Considering the general use of exponential function in the simulation of bulged profile [25–27], φðz; r Þ is defined as ! r cot α 2 φðz; r Þ ¼ exp 1 1 ð8Þ z
ð2Þ
According to the boundary conditions on the contact surface and that at the centerline, as illustrated in Fig. 2(b), the following equations are obtained. u_ z ðz; z tan αÞ sin α u_ r ðz; z tan αÞ cos α ¼ u_ cos α u_ r ðz; 0Þ ¼ 0
Referring to Eqs. (1) and (4), the forms of the axial velocity and the radial velocity can be obtained. 1 x2 x2 u_ z ðz; r Þ ¼ u_ cot α 1 21 1 þ x2 φðz; r Þ þ x2 u_ cot α 1 21 r φr ðz; r Þ 2 z z 1 x2 x2 u_ r ðz; r Þ ¼ u_ cot α 31 r 1 þ x2 φðz; r Þ x2 u_ cot α 1 21 r φz ðz; r Þ 2 z z ð5Þ
ð3Þ
where u_ z ðz; z tan αÞ and u_ r ðz; z tan αÞ are the axial and radial velocities on the contact surface between the hammer and the workpiece respectively; u_ r ðz; 0Þ is the radial velocity at the centerline of the workpiece; u_ is the instantaneous radial velocity and α is the inlet angle of hammer, respectively. In order to model the metal flow accurately, deforming characteristics of the metal flow should be considered as many as possible. For example, the existence of the neutral plane discussed by many studies and the inhomogeneity of axial metal flow proposed in this analysis are both included in the stream function given as x2 1 ψ ðz; rÞ ¼ u_ cot α 1 21 r2 1 þx2 φðz; rÞ ð4Þ 2 z where x1 and x2 are undetermined parameters and they will be finally determined by the minimization of dissipated power in each solving procedure. φðz; r Þ is an expression to be determined later.
2.1.2. The sizing zone Compared with the pre-forging state, analysis of velocity field in the sizing zone also needs to be performed at the steady state of the radial forging process. In Fig. 2(b), the radial velocity in sizing zone II is assumed to change linearly from u_ on the outside surface to zero at the centerline of the workpiece [16]. It is expressed as r u_ IIr ðz; r Þ ¼ u_ R
ð9Þ
where R is the current radius of sizing zone. According to the geometrical relations in Fig. 2, it can be derived as R ¼ R1 þ Lf tan α h
ð10Þ
where Lf is the axial feed and h is the current reduction in the radial direction. The velocity boundary condition on the boundary between zone II and zone III (hereinafter referred to as boundary II–III) is u_ IIz ðR cot α; r Þ ¼ u_ z ðR cot α; r Þ
ð11Þ
and u_ z are the axial velocities in the sizing zone II and where that in the forging zone III, respectively. Then the velocity field in the sizing zone can be obtained by solving the simultaneous Eqs. (2), (9) and (11). u_ IIz
Y. Wu et al. / International Journal of Mechanical Sciences 93 (2015) 102–110
2.1.3. Zone I Because the material in zone I has been forged to the required dimensions in current pass, there is no deformation along the radial direction. Besides, according to the velocity boundary condition on boundary I–II, the velocity field in this zone can be written as u_ Ir ¼ 0 u_ Iz
¼ u_ IIz R u_ Ir
cot α Lf ; r
ð12Þ
u_ Iz
and are the radial velocity and axial velocity in zone I, where respectively. Eq. (12) indicates that the metal in this zone shears only along the axial direction and it degenerates to a rigid zone if the axial velocity on boundary I–II is equal to a constant. 2.2. Dissipated powers At the pre-forging and the steady states, the required total power is divided into three components: the plastic power, the shear power and the frictional power. Because the compositions of the deformation zones at these two states are different, slight differences between the two states for the calculation of each kind of dissipative power exist as well. Although the following expressions are derived for the model at the steady state, as seen in Fig. 2(b), they can also be applied to the pre-forging state when some corrections are made. Referring to the strain–displacement relations, effective plastic strain rates can be derived from the velocity fields, and finally plastic power can be computed as Z Z Z _ p¼ W σ I ε_ I dV þ σ II ε_ II dV þ σ III ε_ III dV ð13Þ VI
V II
V III
where σ and ε_ are the average effective stress and the effective plastic strain rate respectively. The subscripts “I, II, III” denote zones I, II and III, respectively. The total shear power results from the velocity discontinuities existed on the boundaries between every two connected zones. It can be derived as follows: Z Z Z _t¼ W K I;II Δu_ I II dΓ þ K II;III Δu_ II III dΓ þ K III;IV Δu_ III IV dΓ Γ I II
Γ II III
Γ I IV
ð14Þ where K is the average yield shear stress and Δu_ is the velocity discontinuity. They are written as pffiffiffi K ¼ σ= 3 Δu_ I II ¼ u_ IIr R cot α Lf ; r Δu_ II III ¼ u_ r ðR cot α; rÞ u_ IIr ðR cot α; r Þ Δu_ III IV ¼ u_ r ðR0 cot α; r Þ ð15Þ where the subscripts “I–II” denote boundaries I–II, etc. The frictional power dissipated on the contact surface between the hammer and the workpiece is computed by Z Z _f¼ τII Δu_ II dΓ þ τIII Δu_ III dΓ ð16Þ W Γ f II
Γ f III
where τ is the shear stress on the frictional surface. According to the assumption of shear frictional law, τ is expressed as Eq. (17). Δu_ is the velocity discontinuity written as Eq. (18).
τ ¼ mK
ð17Þ
where m is the friction factor.
Δu_ II ¼ u_ IIz ðz; RÞ
Δu_ III ¼ u_ z ðz; z tan αÞ cos α þ u_ r ðz; z tan αÞ sin α þ u_ sin α
ð18Þ
105
According to the upper bound theorem, an inequality for the radial load is established, as shown in Eq. (19). The total power obtained by adding all the preceding powers is expressed as a function of the undetermined parameters x1 and x2 . _ t þW _ ¼ f ðx1 ; x2 Þ _ p þW F u_ r W f
ð19Þ
For the pre-forging state, the dissipated powers associated with zone I and the sizing zone II should be equal to zero because it does not exist actually. Hence, when the first two terms on the right of Eqs. (13) and (14), the first term on the right of Eq. (16) are corrected to be zero, the equations are available for the pre-forging state as well. 2.3. Solving procedure Based on the upper bound theorem, the radial load can be solved by minimizing the power with respect to x1 and x2 in the current step. Taking these results as the initial condition of next step and assuming that physical quantities such as the velocity and the flow stress are constant, the whole process at both the preforging and the steady states can be solved step by step. The average effective stress in next step σ i þ 1 is updated by ð20Þ σ i þ 1 ¼ σ εai þ 1 ; ε_ ai ; T i where the subscript denotes the step number. σ εai ; ε_ ai ; T i is the mapping relationship between the average flow stress and other parameters. It can be a fitting model such as the Swift equation, the Voce equation, the Johnson–Cook law and their improved models; or it is just the interpolation of the original experimental data. T is the average temperature of the workpiece. εa and ε_ a are the average effective plastic strain and strain rate, respectively. They are defined as R ε_ dV ε_ ai ¼ R i dV εai þ 1 ¼ εai þ ε_ ai Δt ð21Þ where Δt is the time increment between two steps. In this model, the integration domain of the plastic zone is regular, so the integral of the plastic power is implemented directly on the whole domain without any discretization. As seen in Eq. (21), the average effective plastic strain rate is correspondingly calculated on the whole domain and it finally results in an average flow stress. As a result, the computational efficiency is consequently improved, but the flow stress and the yield shear stress are simplified to distribute homogeneously on the whole domain. If both sides of Eq. (19) are divided by σ , it is obvious that on the left side of the equation, the radial load is affected by the flow stress; on the right side of the equation, the three terms are independent of the flow stress. Therefore, the flow stress model affects slightly the velocity field of the workpiece. It is the limitation of both the current model and the traditional upper bound method. To balance the efficiency and the accuracy of the theoretical model, numerical methods on the integral of plastic powers will be improved in future work. Finally, the contour of end profile can be predicted by tracing the position of particles. For a certain particle, the final position in absolute coordinate system is obtained from the vector sum of displacement vectors in all previous strokes. Take a particle on the outside surface of the workpiece for example, when it moves form position P0 at the beginning of present stroke (namely the end of previous stroke) to position P at the end of present stroke (namely the beginning of next stroke), as shown in Fig. 2(a), the displace! ment vector u P 0 P is expressed as ! !0 u P P ¼ u_ P 0 ðzP0 ; r P 0 ÞΔt
ð22Þ
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! where u_ P 0 is the velocity vector of the particle at position P0 at the beginning of present stroke. Δt is the time increment between two strokes. In order to take advantages of the remarkable properties of symbolic mathematics and optimization algorithms, the entire procedure is coded in MATLAB. Similar to the authors' previous study [16], the results are not sensitive to the number of steps. In addition, when the radial reduction of hammer in each step is less than about 3.3% of the whole stroke, the fractional error caused by the number of steps is less than 1%.
3. Results and discussions 3.1. Radial load The predicted radial load was compared with the experimental measurement of cold swaging of AISI 1015 steel rods by Uhlig [28], which was also used by other researchers to verify their predictions [10,14,29,30]. In Uhlig's work, the stress–strain relationship of the material is assumed as σ ¼ σ 0 εn . The basic experiment parameters are listed in Table 1 and the results are shown in Table 2. It is found that the fraction errors between the predicted results and the experimental data are lower than 5%. Hence the availability of the velocity fields is verified. It is worth noting that the prediction in experimental case no. 5 is reasonable, while this prediction is not available in the authors' previous model [16]. For a parallel velocity field, the axial velocity is assumed to be uniform across a section and the radial velocity distributes linearly from the outside surface to the centerline of the workpiece. Accordingly, if the radial reduction is small, plastic deformation concentrates near the outside surface of the workpiece, so that deviations of velocity distributions between the assumed velocity field and the experiment increases. The present velocity field considering the inhomogeneous deformation mode is closer to the reality, so reasonable prediction can be obtained even though the radial reduction is relatively small. In order to confirm this viewpoint, contour plots of the stream function, respectively corresponding to the inhomogeneous velocity field in present paper and the formerly adopted homogeneous one, are compared under forging conditions of experimental case no. 5 in Table 2. According to the stream function expressed as Eq. (4), every position in forging zone corresponds to a specific value of stream function ψ and they are plotted with different colors depending on these values. Besides, as discussed earlier, a homogeneous deformation mode can be obtained by setting the undetermined parameter x2 to be zero. In this way, contour plots of stream functions are drawn in Fig. 3. Lines of ψ ¼ constant are streamlines and show the plastic flow patterns, since the instantaneous velocity vector of a particle is along the tangent direction of the streamline passing through it [31]. At the beginning of the stroke, comparing with Fig. 3(a) and (b), it is obvious that the deformation zone predicted by an inhomogeneous deformation mode is much closer to the surface. In contrast, the deformation predicted by a homogeneous deformation mode is distributed almost along the entire radial direction on the longitudinal section of the workpiece. As a result, the left boundary of the forging zone in Fig. 3(a) presents to be a concave contour while that in (b) remains a vertical line. At the end of the stroke, Fig. 3(c) and (d) shows a similar distribution of deformation. It should be noted that Table 1 Process parameters of Uhlig's experiment. Material
K (MPa)
n
α (deg)
m
Lf (mm)
AISI 1015
618.14
0.1184
4.3
0.15
0.37
Table 2 Comparison of predicted radial loads with the experimental results. No.
1 2 3 4 5
Diameter (mm)
Radial load (kN)
Workpiece
Product
Experiment
Analytical model
15.97 15.97 15.03 13.99 13.17
13.18 13.25 13.11 13.03 13.01
172 167 124 74 54
166.05 163.05 120.43 75.41 52.97
Error (%)
3.46 2.37 2.88 1.91 1.9
the boundaries in this figure represent only the profiles of the forging zone in the current step; the final profiles of the end surface are the accumulation of deformations in all steps. In summary, during the forging process of a stroke, the deformation predicted by present model penetrates gradually to the centerline of the workpiece, while the deformation predicted by the previous parallel velocity field is always distributed along the entire radial direction. This difference finally results in the different predictions of radial load for experimental case no. 5. 3.2. The neutral plane Generally, the velocity fields are solved by minimizing the total power with respect to an undetermined parameter in theoretical analyses of the radial forging process. This parameter physically represents the position of the neutral plane. The material which is exactly on this neutral plane is only deformed radially and does not flow axially. In the present model, the neutral plane can also be obtained according to two parameters defined in the stream function Eq. (4). Expression of axial velocity can be derived from the stream function, and then solutions of the equation u_ z ¼ 0 illustrate exactly the position of the neutral plane. It is marked with dot dash line in Fig. 4. On the other hand, as shown in Fig. 4, contour plots of the stream function are drawn under certain forging conditions. Since the velocity vector of a particle is always along the tangent of the streamline passing through it, and the axial velocities of particles on the neutral plane are equal to zero, so the neutral plane is characterized by vertical streamline, and its position can be roughly estimated. In Fig. 4(a), the profile of “the neutral plane” is actually not a planar cross section of the workpiece. It is ascribed to the fact that the assumed velocity field has considered the inhomogeneous deformation mode. The axial velocity is no longer evenly distributed along the radial direction, especially at the pre-forging state. With the increase of the radial reduction rate from 2.1% in Fig. 4(a) to 4.7% in (b), “the neutral plane” tends to be a plane. Therefore, it is reasonable for Lahoti [10,11] and other researchers [12,14,15] to assume that the position on which the material is only deformed radially and does not flow axially, is exactly a planar section which is called the neutral plane. It is also found in Fig. 4 that (a) the streamlines are distributed on both sides of the neutral plane, and the farther from the neutral plane or the centerline of the workpiece, the higher the density of the streamlines; (b) the streamlines are concentrated near to the outside surface when the current radius is relatively large. The former indicates that the deforming material flows axially away from the neutral plane to both sides simultaneously; the velocity increases with an increasing distance to both the position of the neutral plane and the centerline of the workpiece, because the velocity is inversely proportional to the spacing of the streamlines. The later shows that as the radial reduction increases, the plastic deformation penetrates gradually to the centerline of the workpiece. In summary, the above corollaries are consistent with the conclusions of previous studies [10,16] and empirical knowledge; therefore the present model can be validated qualitatively as well.
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107
Fig. 3. Contour plots of the stream functions which have considered (a) inhomogeneous, (b) homogeneous deformation modes at the beginning of the stroke and (c) inhomogeneous, (d) homogeneous deformation modes at the end of the stroke. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
3.3. Concave–convex depth The end surface of the workpiece keeps planar if the metal flows homogeneously. Actually, the distribution of axial metal flow along the radial direction is inhomogeneous and consequently concave or sometimes convex profiles occur on the end surface of the workpiece, as indicated in Fig. 2. The concave or convex depth of end surface reflects the degree of deformation inhomogeneity: the larger the depth, the more inhomogeneous the deformation. According to the displacements of particles at different radial positions, as expressed by Eq. (22), the concave or convex depth of end surface can be obtained d ¼ z Q zP
ð23Þ
where zQ and zP are the axial coordinates of position at the centerline and that on the outside surface of the workpiece, respectively. In the following analysis, the concave or convex depth is analyzed under various process parameters listed in Table 3. The properties of material, geometry of hammer and friction factor are the same as those adopted in Section 3.1. 3.3.1. History plots of the end surface profile Fig. 5(a) shows the history plots of end surface profiles when the workpiece is forged from an initial radius R0 ¼7.985 mm to the final radius R1 ¼6.5 mm with a constant axial feed Lf ¼0.5 mm in a single pass. Each curve in this figure corresponds to one stroke shown in Fig. 2. It is assumed that deformation in radial forging results from the accumulation of successive short-strokes. The plastic deformation is distributed unevenly along the radial direction because a short stroke tends to make the plastic deformation concentrate mainly near the outside surface rather than penetrate to the centerline of the workpiece. The axial velocity of material is decreased with a decreasing distance to the centerline of workpiece. As a result, the end surface presents to be a concave profile. As the hammer presses radially, the concave depth increases gradually. If the curves are rotated anticlockwise 90º, they coincide with the end surface profile illustrated in the schematic diagram at the top left corner of Fig. 5(a). History plot of the concave–convex depth during this process, namely the depth of each curve in Fig. 5(a), is drawn in (b). In this
figure, the abscissa axis, forging length, means the length of the deformed workpiece. The curve on both sides of the dashed line represents the history plot of concave–convex depth at the preforging state and that at the steady state, respectively. At the preforging state, due to the accumulation of deformation inhomogeneity during the successive strokes, the concave–convex depth of end surface becomes increased. On the other hand, when the radial reduction increases, the plastic deformation zone expands gradually to the centerline of the workpiece and consequently the difference between the velocity on the outside surface and that at the centerline of the workpiece decreases. Finally, the depth increasing rate will decline and it is proved in Fig. 5(b). It is observed in Fig. 5(b) that at the steady state the concave–convex depth tends to be a stable value. To show detailed variation of concave–convex depth at the steady state, the enlarged history plot of this state is redrawn in Fig. 5(c). As mentioned in Section 2, the calculation at the steady state is performed within a certain range of forging length; if the forging length, and then the length of zone I defined in Fig. 2(b), is long enough, the plastic deformation zone could not extend to the end surface. In the following analyses, for the sake of convenience, the maximal forging length at the steady state is assumed to be equal to the sizing part length of the hammer, namely L in Table 3. As a result, the concave– convex depth decreases gradually to a stable value. It indicates that zone I will change from a plastic zone to a rigid zone within a forging length of L. Accordingly, it is reasonable for other researchers to assume that zone I is a rigid zone at the steady state.
3.3.2. Effects of the radial reduction In order to study the effects of radial reduction on the concave– convex depth, calculation instances are performed, in which the workpiece is forged form an initial radius R0 ¼7.985 mm to the final radius range of R1 ¼6.0–7.5 mm with a constant axial feed Lf ¼0.5 mm, as listed in Table 3. In Fig. 6, the relative radial reduction is defined as a dimensionless ratio of the radial reduction to the initial radius R0 of the workpiece. The curve marked with open circle and that marked with open triangle illustrate the net increments of concave–convex
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Fig. 4. Streamlines and the neutral plane of the workpiece forged from an initial radius R0 ¼ 7.985 mm to current radii (a) R¼ 7.816 mm and (b) R ¼7.609 mm.
Table 3 Process parameters in present study. R0 (mm)
R1 (mm)
Lf (mm)
L (mm)a
7.985
6–7.5
0.1–1.1
18
a
L is the length of the sizing part of hammer, as illustrated in Fig. 2.
depths at the pre-forging state and that at the steady state, respectively. At the pre-forging state, the workpiece is fed axially and forged by the forging part of the hammer, as seen in Fig. 2(a). During the
Fig. 5. History plots of (a) the end surface profiles, (b) the concave–convex depth d and (c) the enlarged curve of concave–convex depth d at the steady state when the workpiece is forged from an initial radius R0 ¼ 7.985 mm to the final radius R1 ¼ 6.5 mm with a constant axial feed Lf ¼ 0.5 mm.
descent of the hammer, the real radial reduction of the workpiece is lower than the required radial reduction and actually depends on the axial feed per stroke. Consequently, for the forging processes with different radial reductions, although the final radial reductions are different with each other, the real radial reductions of the workpiece
Y. Wu et al. / International Journal of Mechanical Sciences 93 (2015) 102–110
Fig. 6. The effects of relative radial reduction (R0 R1)/R0 on the concave–convex depths d at different states (R0 ¼7.985 mm, R1 ¼ 6.0–7.5 mm, Lf ¼ 0.5 mm).
are identical since the workpieces are forged with the same axial feed. In other words, the radial reduction affects only the number of total strokes rather than the process condition of each stroke. Therefore, with the increase of relative radial reduction, the concave–convex depth at the pre-forging state becomes increased due to the accumulation of strokes. At the steady state, however, the different radial reductions will result in the differences of process conditions in each stroke. When the relative radial reduction increases, the plastic deformation penetrates gradually to the centerline of workpiece and simultaneously the contact surface between the workpiece and hammer becomes increased. The former leads to the increase of axial velocity at the centerline of workpiece, the later results in the decrease of axial velocity on the outside surface of workpiece due to frictional effects. Therefore, the net increment of concave–convex depth at the steady state decreases gradually to zero and finally increases in opposite direction, namely, the end cross section tends to change from the concave surface to a convex surface. In summary, when the radial reduction increases, the axial flow inhomogeneity at the preforging state increases, at the steady state the axial flow inhomogeneity with concave mode declines slightly firstly then increases slightly with convex mode. Besides, it is observed from this figure that the net increment of concave–convex depth at the end of pre-forging state is significantly greater than that at the steady state. For example, when the relative radial reduction is 18.6%, the forging lengths during these two states are 19.75 mm and 18 mm respectively, but the depth increments are 0.218 mm and 0.0025 mm respectively. Although the forging lengths are in the same order of magnitude, the differential of depth increments is about two orders of magnitude. The reason is that the neutral plane tends to change from curved surface to a planar surface when radial reduction increases, as shown in Fig. 4, consequently the inhomogeneity in deformation zone decreases. The final concave– convex depth of end surface is the sum of depth increments at both the pre-forging and the steady states, as seen in the curve marked with open square. Comparing with the curve at the pre-forging state, the final depth is reduced by the deformation at the steady state slightly. Although higher value of radial reduction can reduce the axial flow inhomogeneity at the steady state, the positive effect is insufficient to compensate the increase of inhomogeneity at the pre-forging state. According to the above analysis, conclusion can be drawn that the concave profile of end surface is mainly formed at the pre-forging state and it is suggested that a relatively low value of radial reduction is conducive to the axial flow homogeneity.
109
3.3.3. Effects of the axial feed Fig. 7 shows the effects of relative axial feed on the concave– convex depths when the workpiece is forged from an initial radius R0 ¼7.985 mm to the final radius R1 ¼6.59 mm with an axial feed range of Lf ¼0.1–1.1 mm. Similarly, the relative axial feed is defined as a dimensionless ratio of the axial feed Lf to the initial radius R0. The curve marked with open circle and that marked with open triangle illustrate the net increments of concave–convex depths at the pre-forging state and that at the steady state, respectively. The increase of relative axial feed contributes to the penetration of plastic deformation at both the pre-forging and the steady states. Accordingly, the axial flow inhomogeneity between the material on the outside surface and that at the centerline is reduced, so that the axial velocity at the centerline increases gradually and the velocity differential becomes decreased. Therefore, the concave– convex depth at the pre-forging state is decreased. At the steady state, the forging part of hammer contacts fully with the workpiece and the contact surface increases with the increase of axial feed. The axial flow of material on the outside surface is severely restricted due to the frictional effects, so that the end surface prefers to be a convex mode and the axial flow inhomogeneity increase gradually, as shown in the curve marked with open triangle. Compared to Fig. 6, it is also found that the radial reduction has more significant effects than the axial feed on the concave–convex depth of end surface at the pre-forging state, while the influences of them are at the same level at the steady state. For example, the depth increasing rate versus relative radial reduction at the pre-forging state is nearly 10 times of the rate versus relative axial feed. Accordingly, it is concluded that the axial flow inhomogeneity is more sensitive to the radial reduction than the axial feed.
4. Conclusion and future work In the radial forging process, the frictional effects between the hammers and the workpiece lead to redundant deformation characterized by an inhomogeneous distribution of axial metal flow along the radial direction. The surface layers are sheared relative to the center so that the planar sections will no longer remain plane. Therefore, the axial metal flow inhomogeneity can be analyzed from the viewpoint of the end surface concave–convex depth of the workpiece. In order to solve this problem, an upper bound model considering the redundant deformation mode is proposed and
Fig. 7. The effects of relative axial feed Lf /R0 on the concave–convex depths d at different states (R0 ¼ 7.985 mm, R1 ¼6.59 mm, and Lf ¼0.1–1.1 mm).
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validated by comparing the predicted forging loads with published experimental results. Finally, the following conclusions are drawn: (1) Reasonable predictions of radial load can be achieved even when the radial reduction is small since the redundant deformation mode is considered. Besides, it is found that the neutral plane predicted in present study is no longer a planar cross section. (2) During a forging process at the pre-forging state, the concave– convex depth increases gradually because of the accumulation of deformation inhomogeneity and the increasing rate will decline due to the penetration of plastic deformation. (3) The concave or convex profile is mainly formed at the preforging state and finally determined by the sum of depth increments at both the pre-forging and the steady states. At the pre-forging state, the concave–convex depth increases with the increase of radial reduction or the decrease of axial feed. At the steady state, the concave depth decreases to zero and then increases with convex mode with the increases of both the radial reduction and the axial feed. Therefore, in order to achieve a homogeneous distribution of axial metal flow, a relatively low value of radial reduction or a relatively high value of axial feed is required. (4) The axial flow inhomogeneity is more sensitive to the radial reduction than the axial feed. There is also further work to improve the current analysis. As discussed in Section 2.3, numerical methods on the integral of plastic powers should be improved to balance the efficiency and the accuracy of this analysis. Besides, since the front-pull and back-push forces are neglected in the present model, the absolute position of the end surface is accordingly changed. However, the concave–convex depth expressed in Eq. (23), which is the relative difference of the surface contour, is not significantly affected unless the contour of the neutral plane is also changed by these forces. Therefore, the effects of the front-pull and back-push forces on the contour of the neutral plane need to be studied in the future work.
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