Citation: Alexandrov, S.; Hwang,

Y.-M.; Tsui, H.S.R. Determining the

Drawing Force in a Wire Drawing

Process Considering an Arbitrary

Hardening Law. Processes 2022, 10,

1336. https://doi.org/10.3390/

pr10071336

Academic Editors: Antonino Recca

and Chin-Hyung Lee

Received: 7 June 2022

Accepted: 5 July 2022

Published: 8 July 2022

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processes

Article

Determining the Drawing Force in a Wire Drawing ProcessConsidering an Arbitrary Hardening LawSergei Alexandrov 1, Yeong-Maw Hwang 2,* and Hiu Shan Rachel Tsui 2

1 Federal State Autonomous Educational Institution of Higher Education, South Ural State University(National Research University), 454080 Chelyabinsk, Russia; sergei_alexandrov@yahoo.com

2 Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-sen University,Lien-Hai Rd., Kaohsiung 804, Taiwan; d103020003@student.nsysu.edu.tw

* Correspondence: ymhwang@mail.nsysu.edu.tw

Abstract: The present paper considers the wire drawing process through a conical die and providesan approximate solution for the drawing force, assuming an arbitrary hardening law. The solution isbased on the upper bound theorem. However, this theorem does not apply to the stationary flow ofstrain-hardening materials. Therefore, an adopted engineering approach is to replace the originalmaterial model with a non-homogeneous, perfectly plastic model. The kinematically admissiblevelocity field is derived from an exact semi-analytical solution for the flow through an infinite channel,which increases the accuracy of the solution. The solution for the homogeneous perfectly plasticmaterial compares with an available solution. The general solution is valid for any die angle. Thenumerical example focuses on the range of angles used in wire drawing. Since the material model ispressure-independent, it is straightforward to adopt the solution for calculating the force in extrusion.

Keywords: wire drawing; strain-hardening; upper bound theorem; engineering approach

1. Introduction

Wire drawing is a commonly used metal forming process characterized by its highdeformation rate and excellent surface quality [1–3]. During the process, metal or alloybillets with high tensile strength, such as copper, aluminum, steel, etc., are reshaped throughthe conical drawing die. The products are widely used in different industry sectors, such asconstruction, electrical, chemical, and automotive [4–7]. Papers [8–11] have pointed outthe importance and influence of some wire drawing process parameters, including thereduction ratio, die angle, drawing speed, and friction.

Several approximate analytical methods are available for calculating wire drawingprocess parameters. One of these methods employs analytical solutions for material flowin infinite converging conical channels. The first solution of this class has been derivedin [12]. This rigid perfectly plastic solution is valid for any isotropic pressure-independentyield criterion. Paper [13] has studied some mathematical features of the solution [12]for the von Mises and Tresca yield criteria. Paper [14] has extended the solution [12] tolinear hardening materials, assuming the von Mises yield criterion. An approach to usingradial flows to analyze drawing processes has been developed in [15]. The correspondingapproximate solutions are valid for any hardening law, but small die angle and friction.

Another widely used method for calculating wire drawing process parameters is basedon the upper bound theorem. The first solution of this class has been presented in [16].Other solutions have been derived in [17–19]. A distinguishing feature of the solution [17]is that the kinematically admissible velocity field is continuous. Paper [18] has focused onthe formation of a rigid region near the friction surface. The singular velocity field in thevicinity of maximum friction surfaces derived in [20] has been employed in [19]. Upperbound solutions are also available for non-conical dies (see, for example, [21–23]).

Processes 2022, 10, 1336. https://doi.org/10.3390/pr10071336 https://www.mdpi.com/journal/processes

Processes 2022, 10, 1336 2 of 14

A disadvantage of the upper bound method is that it does not apply to stationaryprocesses for strain-hardening materials. Several papers have calculated the yield stressat the exit from the die by replacing the equivalent strain in the hardening law with theextrusion ratio [24–26]. An extension of this method allows average yield stress in eachcross-section of the wire to be calculated using geometric parameters [27,28]. This methodallows one to replace a strain-hardening material model with the corresponding modelof non-homogeneous perfectly plastic material. The upper bound theorem is valid forthe latter model. The present paper adopts this approach to develop a new approximatesolution for the drawing force. In addition, the velocity field found in [12] is used as akinematically admissible velocity field in the plastic region. Therefore, the stress equationsare satisfied in this region in the case of homogeneous perfectly plastic material, whichincreases the accuracy of the solution. A comparison with the solution [16] is made.

2. Statement of the Problem

A schematic diagram of the wire drawing process through a conical die is shown inFigure 1. The initial and final radii of the wire are denoted as R1 and R2, respectively. Thedie angle is 2ϕ0, and the drawing speed is V2. It is required to calculate the drawing force F.

Processes 2022, 10, x FOR PEER REVIEW 2 of 14

[19]. Upper bound solutions are also available for non-conical dies (see, for example, [21–23]).

A disadvantage of the upper bound method is that it does not apply to stationary processes for strain-hardening materials. Several papers have calculated the yield stress at the exit from the die by replacing the equivalent strain in the hardening law with the extrusion ratio [24–26]. An extension of this method allows average yield stress in each cross-section of the wire to be calculated using geometric parameters [27,28]. This method allows one to replace a strain-hardening material model with the corresponding model of non-homogeneous perfectly plastic material. The upper bound theorem is valid for the latter model. The present paper adopts this approach to develop a new approximate so-lution for the drawing force. In addition, the velocity field found in [12] is used as a kine-matically admissible velocity field in the plastic region. Therefore, the stress equations are satisfied in this region in the case of homogeneous perfectly plastic material, which in-creases the accuracy of the solution. A comparison with the solution [16] is made.

2. Statement of the Problem A schematic diagram of the wire drawing process through a conical die is shown in

Figure 1. The initial and final radii of the wire are denoted as 푅 and 푅 , respectively. The die angle is 2휑 , and the drawing speed is 푉 . It is required to calculate the drawing force F.

Figure 1. Schematic diagram of the process.

The material is regarded as rigid/plastic, and incompressible. In the case of perfectly plastic material, the von Mises yield criterion reads:

휎 = 휎 . (1)

Here 휎 is the equivalent stress and 휎 is the yield stress in tension. The latter is a constitutive parameter. The friction stress between the conical die and material is taken as:

휏 = 푚√

. (2)

Here 푚 is the constant friction factor. Its value may vary in the range 0 ≤ 푚 ≤ 1. A consequence of the incompressibility equation is:

푉 푅 = 푉 푅 . (3)

Here 푉 is the velocity of the wire before entering the die. The solution is based on the upper bound theorem [29]. Therefore, strictly speaking,

strain-hardening models cannot be considered because the process is stationary. The von Mises yield criterion of strain-hardening materials is represented as:

휎 = 휎 훷 휀 (4)

Figure 1. Schematic diagram of the process.

The material is regarded as rigid/plastic, and incompressible. In the case of perfectlyplastic material, the von Mises yield criterion reads:

σeq = σ0. (1)

Here σeq is the equivalent stress and σ0 is the yield stress in tension. The latter is aconstitutive parameter. The friction stress between the conical die and material is taken as:

τf = mσ0√

3. (2)

Here m is the constant friction factor. Its value may vary in the range 0 ≤ m ≤ 1. Aconsequence of the incompressibility equation is:

V1R21 = V2R2

2. (3)

Here V1 is the velocity of the wire before entering the die.The solution is based on the upper bound theorem [29]. Therefore, strictly speaking,

strain-hardening models cannot be considered because the process is stationary. The vonMises yield criterion of strain-hardening materials is represented as:

σeq = σ0Φ(εeq)

(4)

where εeq is the equivalent strain and Φ(εeq)

is an arbitrary function of its argument

satisfying the conditions Φ(0) = 1 anddΦ(εeq)

dεeq≥ 0 for all εeq. An engineering approach to

account for strain hardening has been proposed in [28] for linear strain-hardening materials.

Processes 2022, 10, 1336 3 of 14

According to this approach, the equivalent strain in the hardening law is replaced with anaverage strain defined as:

εa = ln(

A0

A

)(5)

where A0 is the cross-sectional area of the wire before entering the die and A is the cross-sectional area of the wire at any distance from the entrance. It is evident from (5) that εa = 0at A = A0. Therefore, it follows from (4) that σeq = σ0 at the entrance to the die. Paper [28]assumes that Φ

(εeq)

is a linear function of its argument. However, it is straightforward toextend this approach to any function. Then, Equation (4) becomes:

σeq = σ0Φ(εa). (6)

Since εa is a known function of position, the corresponding model can be regardedas rigid perfectly plastic with a known nonuniform yield stress distribution. The upperbound theorem applies to such a model. Equation (2) becomes:

τf = mΦ(εa)√

3. (7)

3. Kinematically Admissible Velocity Field

It is natural to use a spherical coordinate system (r, ϕ, φ). The solution is independentof φ. The shear stress in the spherical coordinate system is denoted as σrϕ. The equation ofthe die surface is ϕ = ϕ0 (Figure 2). The boundary conditions (2) and (7) become:

σrϕ = mσ0√

3and σrϕ = m

σ0Φ(εa)√3

(8)

for ϕ = ϕ0, respectively. Axial symmetry demands

σrϕ = 0 (9)

for ϕ = 0. The spherical coordinate system and the general structure of the kinematicallyadmissible velocity field are shown in Figure 2. The kinematically admissible velocity fieldconsists of a plastic region and two rigid regions. The rigid/plastic boundaries are velocitydiscontinuity lines in meridian planes.

Processes 2022, 10, x FOR PEER REVIEW 4 of 14

Figure 2. General structure of the kinematically admissible velocity field.

3.1. Kinematically Admissible Velocity Field in the Plastic Region Choosing a kinematically admissible velocity field that satisfies the stress equations

when combined with the associated flow rule is advantageous. Semi-analytical solutions to boundary value problems supply such velocity fields. In particular, the solution pre-sented in [12] can be used in the case under consideration. This solution is valid for any pressure-independent yield criterion. The present paper employs the solution for the von

Mises yield criterion. The dimensionless shear stress 휏 = √ satisfies the following Equation [12]:

+ 휏 푐표푡 휑 + 2√3√1 − 휏 = 푐. (10)

Here 푐 is constant. Its value should be found from the solution. It is convenient to put:

휏 = 푠푖푛 휓 and √1 − 휏 = 푐표푠 휓. (11)

Equations (10) and (11) combine to give:

= √ . (12)

The first equation in (8) and (9) become 휏 = 푚 for 휑 = 휑 and 휏 = 0 for 휑 = 0, re-spectively. Using (11), one can transform these boundary conditions to:

휓 = 0 (13)

for 휑 = 0 and:

휓 = 휓 = 푎푟푐푠푖푛 푚 (14)

for 휑 = 휑 . It is seen from (13) that the term 푠푖푛 휓 푐표푡 휑 involved in (12) reduces to the expression at 휑 = 0. Assume that 휓 is a linear function of 휑 in the vicinity of 휑 = 0.

Figure 2. General structure of the kinematically admissible velocity field.

Processes 2022, 10, 1336 4 of 14

3.1. Kinematically Admissible Velocity Field in the Plastic Region

Choosing a kinematically admissible velocity field that satisfies the stress equationswhen combined with the associated flow rule is advantageous. Semi-analytical solutions toboundary value problems supply such velocity fields. In particular, the solution presentedin [12] can be used in the case under consideration. This solution is valid for any pressure-independent yield criterion. The present paper employs the solution for the von Mises yield

criterion. The dimensionless shear stress τ =√

3σrϕ

σ0satisfies the following Equation [12]:

dτ

dϕ+ τcotϕ + 2

√3√

1− τ2 = c. (10)

Here c is constant. Its value should be found from the solution. It is convenient to put:

τ = sinψ and√

1− τ2 = cosψ. (11)

Equations (10) and (11) combine to give:

dψ

dϕ=

c− sinψcotϕ− 2√

3cosψ

cosψ. (12)

The first equation in (8) and (9) become τ = m for ϕ = ϕ0 and τ = 0 for ϕ = 0,respectively. Using (11), one can transform these boundary conditions to:

ψ = 0 (13)

for ϕ = 0 and:ψ = ψ f = arcsin m (14)

for ϕ = ϕ0. It is seen from (13) that the term sinψcotϕ involved in (12) reduces to theexpression 0

0 at ϕ = 0. Assume that ψ is a linear function of ϕ in the vicinity of ϕ = 0. Thisfunction must satisfy (13). Substituting it into (12) and expanding the right-hand side ofthe resulting equation in a power series, one obtains

ψ =

(c− 2

√3)

2ϕ + o(ϕ) (15)

as ϕ→ 0 . Equation (12) should be solved numerically using (15). The solution depends onc. The boundary condition (14) supplies the equation to determine c. Figure 3 illustratesthe dependence of c on m and ϕ0. In what follows, it is assumed that ψ is a known functionof ϕ.

Processes 2022, 10, x FOR PEER REVIEW 5 of 14

This function must satisfy (13). Substituting it into (12) and expanding the right-hand side of the resulting equation in a power series, one obtains

휓 = √ 휑 + 표(휑) (15)

as 휑 → 0. Equation (12) should be solved numerically using (15). The solution depends on c. The boundary condition (14) supplies the equation to determine c. Figure 3 illustrates the dependence of c on m and 휑 . In what follows, it is assumed that 휓 is a known func-tion of 휑.

Figure 3. Variation of c with the die semi-angle at several friction factors.

The only non-vanishing velocity component is the radial velocity. The latter is given by [12]

푢 = − 푒푥푝 −2√3 ∫ 푑훾 . (16)

Here 훽 > 0 is constant whose value should be found from the solution. In (16) and what follows, the quantity 푉푅 may be represented as 푉 푅 or 푉 푅 due to (3). Using (11), one transforms (16) to

푢 = − 푞(휑) (17)

where

푞(휑) = 푒푥푝 −2√3 ∫ 푡푎푛 휓 푑훾 . (18)

A consequence of the incompressibility equation is:

휋푉푅 = −2휋 ∫ 푢푟 푠푖푛 휑 푑휑. (19)

Eliminating here u using (17), one arrives at the following equation for 훽:

1 = 2훽 ∫ 푞(휑) 푠푖푛 휑 푑휑. (20)

In what follows, it is assumed that 훽 has been calculated. Therefore, the kinemati-cally admissible velocity field in the plastic region has been determined.

3.2. Velocity Discontinuity Lines There are the two velocity discontinuity lines shown in Figure 4 (BB1 and AA1). Two

orthogonal systems of unit vectors associated with line BB1 are depicted in this figure. A similar system of vectors may be associated with line AA1. The derivation below is valid for both lines.

Figure 3. Variation of c with the die semi-angle at several friction factors.

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The only non-vanishing velocity component is the radial velocity. The latter is givenby [12]

u = − βVR2

r2 exp[−2√

3∫ ϕ

ϕ0

τ√1− τ2

dγ

]. (16)

Here β > 0 is constant whose value should be found from the solution. In (16) andwhat follows, the quantity VR2 may be represented as V1R2

1 or V2R22 due to (3). Using (11),

one transforms (16) to

u = − βVR2

r2 q(ϕ) (17)

where

q(ϕ) = exp[−2√

3∫ ϕ

ϕ0

tanψdγ

]. (18)

A consequence of the incompressibility equation is:

πVR2 = −2π∫ ϕ0

0ur2sinϕdϕ. (19)

Eliminating here u using (17), one arrives at the following equation for β:

1 = 2β∫ ϕ0

0q(ϕ)sinϕdϕ. (20)

In what follows, it is assumed that β has been calculated. Therefore, the kinematicallyadmissible velocity field in the plastic region has been determined.

3.2. Velocity Discontinuity Lines

There are the two velocity discontinuity lines shown in Figure 4 (BB1 and AA1). Twoorthogonal systems of unit vectors associated with line BB1 are depicted in this figure. Asimilar system of vectors may be associated with line AA1. The derivation below is validfor both lines.

Processes 2022, 10, x FOR PEER REVIEW 6 of 14

The base vectors of the spherical coordinate system are denoted as 풆풓 and 풆흋. The vectors directed along the normal and tangent to the velocity discontinuity line are de-noted as 풏 and 흉, respectively. Moreover, 풊 is the unit vector directed along the axis of symmetry. It follows from the geometry of Figure 4 that the relationships between the unit vectors are:

풏 = 풆풓 푐표푠 훾 + 풆흋 푠푖푛 훾 , 흉 = −풆풓 푠푖푛 훾 + 풆흋 푐표푠 훾 , 풊 = 풆풓 푐표푠 휑 − 풆흋 푠푖푛 휑. (21)

Here 훾 is the inclination of the vector 풏 to the r-axis, measured clockwise. The ve-locity vectors in the plastic and rigid regions are represented as:

푼풑 = 푢풆풓 and 푼풓 = −푉풊, (22)

respectively. Here, and in what follows, V should be replaced with V1 for line BB1 and with V2 for line AA1. The normal velocity must be continuous across the velocity discontinuity line. Therefore, 푼풑 ⋅ 풏 = 푼풓 ⋅ 풏. Upon substitution from (21) and (22), this equation be-comes:

= − ( ). (23)

The amount of velocity jump across the velocity discontinuity line is [푢] = 푼풓 ⋅ 흉 −푼풑 ⋅ 흉. Upon substitution from (21) and (22), this equation becomes:

[ ] = 푠푖푛 훾 + 푠푖푛(훾 + 휑). (24)

Eliminating here u using (23), one obtains:

[푢]푉

=푠푖푛 휑푐표푠 훾

. (25)

Figure 4. Geometry of velocity discontinuity lines.

An infinitesimal length element of the velocity discontinuity line is:

푑푙 = (푑푟) + (푟푑휑) = 푟 + 1푑휑. (26)

It follows from the geometry of Figure 4 that:

푡푎푛 훾 = − . (27)

Equations (26) and (27) combine to give:

푑푙 = 푑휑. (28)

Using (17), one can transform Equation (23) to:

Figure 4. Geometry of velocity discontinuity lines.

The base vectors of the spherical coordinate system are denoted as er and eϕ. Thevectors directed along the normal and tangent to the velocity discontinuity line are denotedas n and τ, respectively. Moreover, i is the unit vector directed along the axis of symmetry.It follows from the geometry of Figure 4 that the relationships between the unit vectors are:

n = ercosγ + eϕsinγ, τ = −ersinγ + eϕcosγ, i = ercosϕ− eϕsinϕ. (21)

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Here γ is the inclination of the vector n to the r-axis, measured clockwise. The velocityvectors in the plastic and rigid regions are represented as:

Up = uer and Ur = −Vi, (22)

respectively. Here, and in what follows, V should be replaced with V1 for line BB1 and withV2 for line AA1. The normal velocity must be continuous across the velocity discontinuityline. Therefore, Up · n = Ur · n. Upon substitution from (21) and (22), this equation becomes:

uV

= − cos(γ + ϕ)

cosγ. (23)

The amount of velocity jump across the velocity discontinuity line is [u] = Ur · τ −Up · τ. Upon substitution from (21) and (22), this equation becomes:

[u]V

=uV

sinγ + sin(γ + ϕ). (24)

Eliminating here u using (23), one obtains:

[u]V

=sinϕ

cosγ. (25)

An infinitesimal length element of the velocity discontinuity line is:

dl =√(dr)2 + (rdϕ)2 = r

√(dr

rdϕ

)2+ 1dϕ. (26)

It follows from the geometry of Figure 4 that:

tanγ = − drrdϕ

. (27)

Equations (26) and (27) combine to give:

dl =r

cosγdϕ. (28)

Using (17), one can transform Equation (23) to:

βR2

r2 q(ϕ) =cos(γ + ϕ)

cosγ. (29)

Employing trigonometric identities and (27), one rewrites this equation as:

dsdϕ

=2

sinϕ[βq(ϕ)− scosϕ] (30)

where s = r2

R2 . Here R = R1 for line BB1 and R = R2 for line AA1. Equation (30) deter-mines the velocity discontinuity line. This line must pass through point B or A (Figure 4).Therefore, the boundary condition to Equation (30) is:

s =1

sin2 ϕ0(31)

for ϕ = ϕ0. The general solution of Equation (30) is:

s =1

sin2 ϕ

[2β∫ ϕ

0sinχq(χ)dχ + s0

]. (32)

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Here s0 is constant. The value of s must be finite at ϕ = 0. Otherwise, the velocitydiscontinuity line does not reach the symmetry axis. Therefore, it is necessary to put s0 = 0.Then, Equation (32) becomes:

s =2β

sin2 ϕS(ϕ), S(ϕ) =

∫ ϕ

0sinχq(χ)dχ. (33)

This solution reduces to the expression 00 at ϕ = 0. Applying L’Hospital’s rule leads to:

s = βq0 (34)

at ϕ = ϕ0. Here q0 is the value of q at ϕ = 0. It follows from (18) that:

q0 = exp[

2√

3∫ ϕ0

0tanψdγ

]. (35)

It is seen from (20) and (33) that (31) is satisfied.An infinitesimal area element of the velocity discontinuity surface is determined from

(28) as:

dΣ =r2sinϕ

cosγdϕdφ. (36)

Using the definition for s and (33), one transforms this equation to:

dΣ =2βR2S(ϕ)

cosγsinϕdϕdφ. (37)

4. Plastic Work Rate

It is necessary to calculate the plastic work rate (i) in the plastic region, (ii) at twovelocity discontinuity surfaces, and (iii) at the friction surface.

4.1. Plastic Work Rate in the Plastic Region

The infinitesimal volume element is dΩ = r2sinϕdrdϕdφ. Using the definition for s,one transforms this equation to:

dΩ =R3√ssinϕ

2dsdϕdφ. (38)

The plastic work rate is defined as:

WΩ =y

Ω

σeqξeqdΩ. (39)

Here ξeq is the equivalent plastic strain rate. In the case under consideration, it isgiven by:

ξeq =

√23

√ξ2

rr + ξ2ϕϕ + ξ2

φφ + 2ξ2rϕ. (40)

The strain rate components are expressed through the radial velocity as:

ξrr =∂u∂r

, ξϕϕ = ξφφ =ur

, ξrϕ =12r

∂u∂ϕ

. (41)

Equations (17), (18) and (41) combine to give:

ξrr =2βVR2

r3 q(ϕ), ξϕϕ = ξφφ = − βVR2

r3 q(ϕ), ξrϕ =

√3βVR2

r3 q(ϕ)tanψ. (42)

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Substituting (41) into (40) and using the definition for s yields:

ξeq =2βVq(ϕ)

Rs√

scosψ. (43)

Equations (38), (39) and (43) combine to give:

wΩ =WΩ

πVR2σ0= 2β

∫ ϕ0

0

∫ s2(ϕ)

s1(ϕ)

σeq

σ0

q(ϕ)sinϕ

scosψdsdϕ. (44)

This equation involves both velocity discontinuity lines. Therefore, it is necessary toadopt the same definition for s. Assuming that R = R2 and using (33) results in:

s1(ϕ) =2β

sin2 ϕ

∫ ϕ

0sinχq(χ)dχ and s2(ϕ) =

2β

sin2 ϕ

(R1

R2

)2 ∫ ϕ

0sinχq(χ)dχ. (45)

The first equation determines the velocity discontinuity line through A and the sec-ond through B (Figure 4). The equations in (45) allow the integral in (44) to be eval-uated numerically. In the case of homogeneous properties, Equation (1) is valid, andEquations (44) and (45) yield:

wΩ = 4βln(

R1

R2

) ∫ ϕ0

0

q(ϕ)sinϕ

cosψdϕ. (46)

In the case of non-homogeneous properties, Equation (6) is valid, and Equation (44)yields:

wΩ = 2β∫ ϕ0

0

∫ s2(ϕ)

s1(ϕ)

Φ(εa)q(ϕ)sinϕ

scosψdsdϕ. (47)

In the spherical coordinate system, Equation (5) becomes:

εa = 2ln(

R1

rsinϕ0

). (48)

Using the definition for s, one can transform this equation to:

εa = 2ln(

R1

R2sinϕ0√

s

). (49)

Eliminating εa in (47) using (49), one represents the integrand as a function of s and ϕ.

4.2. Plastic Work Rate at the Velocity Discontinuity Surfaces

The plastic work rate at each velocity discontinuity surface is given by:

Wd =1√3

x

Σ

σeq[u]dΣ. (50)

Using (25) and (37), one transforms this equation to:

wd =Wd

πVR2σ0=

4β√3

∫ ϕ0

0

σeq

σ0cos2γS(ϕ)dϕ. (51)

This equation is valid for both velocity discontinuity surfaces. In the case of homoge-neous properties, Equation (51) becomes:

wd =4β√

3

∫ ϕ0

0

S(ϕ)

cos2γdϕ (52)

Processes 2022, 10, 1336 9 of 14

for each velocity discontinuity surface. In the case of non-homogeneous properties,Equation (51) becomes:

wd =4β√

3

∫ ϕ0

0

Φ(εa)

cos2γS(ϕ)dϕ. (53)

Here εa should be regarded as a function of ϕ. One derives this function by replacings in (49) with s1(ϕ) or s2(ϕ). Therefore, the plastic work rate is not the same at the twovelocity discontinuity surfaces.

Equations (52) and (53) involve cos2γ. It follows from (27) and (30) that:

tanγ =scosϕ− βq(ϕ)

ssinϕ. (54)

The right-hand side of this equation is a known function of ϕ due to (33). Therefore,cos2γ is determined as a function of ϕ using (54) and the identity cos2γ =

(1 + tan2γ

)−1.

4.3. Plastic Work Rate at the Friction Surface

The radial velocity at the friction surface is determined from (17) and (18) as:

u f = −βVR2

r2 . (55)

The plastic work rate is:

W f =2πm√

3

∫ R1sinϕ0

R2sinϕ0

σeq

∣∣∣u f

∣∣∣rsinϕ0dr. (56)

Equations (55) and (56) combine to give:

w f =W f

πVR2σ0=

2mβsinϕ0√3

∫ R1sinϕ0

R2sinϕ0

σeq

σ0

drr

. (57)

In the case of homogeneous properties, this equation becomes:

w f =2mβsinϕ0√

3ln(

R1

R2

). (58)

In the case of non-homogeneous properties, Equation (57) becomes:

w f =2mβsinϕ0√

3

∫ R1sinϕ0

R2sinϕ0

Φ(εa)

rdr. (59)

Here one should eliminate εa using (48).

5. Drawing Force

Let F be the drawing force. It follows from the upper bound theorem that:

FV2 ≥WΩ + Wd1 + Wd2 + W f . (60)

Here Wd1 is Wd calculated at the velocity discontinuity line through A, and Wd2 is Wdcalculated at the velocity discontinuity line through B (Figure 4). Using (44), (51) and (57),one transforms Equation (60) to:

fu =Fu

πR22σ0

= wΩ + wd1 + wd2 + w f . (61)

Here Fu is the upper bound on F and fu is its dimensionless representation. Theright-hand side of Equation (61) has been calculated in the previous section for both

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homogeneous and non-homogeneous materials. Quantitative results can be obtained afterprescribing the function Φ

(εeq)

and evaluating the integrals numerically.

6. Numerical Examples

The right-hand side of Equation (61) has been calculated using the correspondingbuild-in commands in Wolfram Mathematica (version 11.3). The present section illustratesthese results and emphasizes the effect of the reduction ratio, the die angle, and the frictionfactor on the drawing force. Even though the general solution is valid for any values ofthese parameters, the quantitative results focus on their typical ranges in the drawing ofthin wires. In the case of strain-hardening material, the stress–strain curve of AISI-316stainless steel is approximated by the equation:

Φ(εeq)=(1 + 84.75εeq

)0.52. (62)

The constitutive parameter σ0 involved in (4) and required to calculate the drawingforce F after determining f is σ0 = 119 MPa.

Figure 5 compares the present solution for rigid perfectly plastic material and the solu-tion provided in [16]. The former is shown by the solid lines, and the latter by the brokenlines. The difference is invisible in the figure. It is worthy of note that the solution [16]has been specifically derived for small die angles. The present solution is more generaland is valid for any die angle. Figure 5 demonstrates that this advantageous feature of thesolution does not affect its accuracy.

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factor on the drawing force. Even though the general solution is valid for any values of these parameters, the quantitative results focus on their typical ranges in the drawing of thin wires. In the case of strain-hardening material, the stress–strain curve of AISI-316 stainless steel is approximated by the equation:

훷 휀 = 1 + 84.75휀 . . (62)

The constitutive parameter 휎 involved in (4) and required to calculate the drawing force F after determining f is 휎 = 119 MPa.

Figure 5 compares the present solution for rigid perfectly plastic material and the solution provided in [16]. The former is shown by the solid lines, and the latter by the broken lines. The difference is invisible in the figure. It is worthy of note that the solution [16] has been specifically derived for small die angles. The present solution is more general and is valid for any die angle. Figure 5 demonstrates that this advantageous feature of the solution does not affect its accuracy.

Figures 6–9 show the effect of process parameters on the dimensionless drawing force. Each figure shows the effect of the die angle for several friction factor values. The effect of the reduction ratio is revealed by comparing the results depicted in all these fig-ures. The effect of strain hardening is revealed by comparing the results depicted in Figures 5 and 6. All results coincide with physical expectations. In particular, the pro-cessing of strain-hardening materials requires a larger force than materials with no hard-ening. In the case under consideration, this difference is considerable. It is because of the hardening law (62). It is seen from each figure that an optimal die angle (i.e., the drawing force attains a minimum at this angle) exists if the other process parameters are kept con-stant. Its value depends on these other process parameters and may be outside the typical ranges considered. The drawing force increases as the friction factor increases.

Figure 5. Variation of the dimensionless drawing force with the die semi-angle at = 1.07 and several friction factors for perfectly plastic material. The solid curves correspond to the present so-lution and the broken curves to the solution [16].

Figure 5. Variation of the dimensionless drawing force with the die semi-angle at R1R2

= 1.07 andseveral friction factors for perfectly plastic material. The solid curves correspond to the presentsolution and the broken curves to the solution [16].

Figures 6–9 show the effect of process parameters on the dimensionless drawing force.Each figure shows the effect of the die angle for several friction factor values. The effect ofthe reduction ratio is revealed by comparing the results depicted in all these figures. Theeffect of strain hardening is revealed by comparing the results depicted in Figures 5 and 6.All results coincide with physical expectations. In particular, the processing of strain-hardening materials requires a larger force than materials with no hardening. In the caseunder consideration, this difference is considerable. It is because of the hardening law(62). It is seen from each figure that an optimal die angle (i.e., the drawing force attainsa minimum at this angle) exists if the other process parameters are kept constant. Itsvalue depends on these other process parameters and may be outside the typical rangesconsidered. The drawing force increases as the friction factor increases.

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Figure 6. Variation of the dimensionless drawing force with the die semi-angle at = 1.07 and several friction factors for strain-hardening material.

Figure 7. Variation of the dimensionless drawing force with the die semi-angle at = 1.1 and sev-eral friction factors for strain-hardening material.

Figure 8. Variation of the dimensionless drawing force with the die semi-angle at = 1.13 and several friction factors for strain-hardening material.

Figure 6. Variation of the dimensionless drawing force with the die semi-angle at R1R2

= 1.07 andseveral friction factors for strain-hardening material.

Processes 2022, 10, x FOR PEER REVIEW 11 of 14

Figure 6. Variation of the dimensionless drawing force with the die semi-angle at = 1.07 and several friction factors for strain-hardening material.

Figure 7. Variation of the dimensionless drawing force with the die semi-angle at = 1.1 and sev-eral friction factors for strain-hardening material.

Figure 8. Variation of the dimensionless drawing force with the die semi-angle at = 1.13 and several friction factors for strain-hardening material.

Figure 7. Variation of the dimensionless drawing force with the die semi-angle at R1R2

= 1.1 andseveral friction factors for strain-hardening material.

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Figure 6. Variation of the dimensionless drawing force with the die semi-angle at = 1.07 and several friction factors for strain-hardening material.

Figure 7. Variation of the dimensionless drawing force with the die semi-angle at = 1.1 and sev-eral friction factors for strain-hardening material.

Figure 8. Variation of the dimensionless drawing force with the die semi-angle at = 1.13 and several friction factors for strain-hardening material.

Figure 8. Variation of the dimensionless drawing force with the die semi-angle at R1R2

= 1.13 andseveral friction factors for strain-hardening material.

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Figure 9. Variation of the dimensionless drawing force with the die semi-angle at = 1.16 and several friction factors for strain-hardening material.

7. Discussion The present paper has developed an engineering approach that allows one to use the

upper bound technique for analyzing the stationary metal forming processes for strain-hardening materials. This development is important because the upper bound solutions are in good agreement with finite element solutions when the upper bound theorem is valid. It has been demonstrated in [30] in the case of wire drawing processes. It is unsur-prising because the upper bound theorem provides an accurate result, even from compar-atively crude kinematically admissible velocity fields [31]. The present solution is based on the kinematically admissible velocity field that, in conjunction with the associated flow rule, satisfies all the fundamental and constitutive equations in the plastic region in the case of rigid perfectly plastic material. This kinematically admissible velocity field feature increases the solution’s accuracy.

The solution found shows the effect of process parameters on the drawing force. It is straightforward to use this solution for other applications. Examples of such applications are the prediction of central bursting defects using the approaches proposed in [32,33], and the prediction of fracture initiation using the extended Bernoulli’s theorem [34] in conjunction with uncoupled ductile fracture criteria. A review of such criteria is available, for example, in [35].

8. Conclusions The process of wire drawing through a conical die has been investigated. A new the-

oretical solution for the drawing force has been derived, assuming an arbitrary strain- hardening law. The solution is based on the upper bound theorem, although it is not an upper bound solution for the standard model of rigid plastic strain-hardening material. The reason for it is that the theorem does not apply to the stationary flow of rigid plastic strain-hardening materials. An engineering approach has been used to overcome this dif-ficulty. In particular, the equivalent strain in the strain-hardening law is replaced with an average strain in each cross-section of the wire. The latter is calculated using geometric parameters of the die. As a result, the original material model is replaced with the rigid perfectly plastic model with a non-homogeneous yield stress distribution. The upper bound theorem applies to such a model.

The solution has been reduced to several ordinary integrals that should be evaluated numerically. Section 6 illustrates the effect of process parameters on the dimensionless drawing force, assuming the hardening law (62). The behavior of the solution agrees with physical expectations. The solution practically coincides with solution [16] (Figure 5). The latter has been developed for small die angles. The new solution is valid for any die angle.

Figure 9. Variation of the dimensionless drawing force with the die semi-angle at R1R2

= 1.16 andseveral friction factors for strain-hardening material.

7. Discussion

The present paper has developed an engineering approach that allows one to use theupper bound technique for analyzing the stationary metal forming processes for strain-hardening materials. This development is important because the upper bound solutions arein good agreement with finite element solutions when the upper bound theorem is valid.It has been demonstrated in [30] in the case of wire drawing processes. It is unsurprisingbecause the upper bound theorem provides an accurate result, even from comparativelycrude kinematically admissible velocity fields [31]. The present solution is based on thekinematically admissible velocity field that, in conjunction with the associated flow rule,satisfies all the fundamental and constitutive equations in the plastic region in the caseof rigid perfectly plastic material. This kinematically admissible velocity field featureincreases the solution’s accuracy.

The solution found shows the effect of process parameters on the drawing force. It isstraightforward to use this solution for other applications. Examples of such applicationsare the prediction of central bursting defects using the approaches proposed in [32,33],and the prediction of fracture initiation using the extended Bernoulli’s theorem [34] inconjunction with uncoupled ductile fracture criteria. A review of such criteria is available,for example, in [35].

8. Conclusions

The process of wire drawing through a conical die has been investigated. A newtheoretical solution for the drawing force has been derived, assuming an arbitrary strain-hardening law. The solution is based on the upper bound theorem, although it is not anupper bound solution for the standard model of rigid plastic strain-hardening material.The reason for it is that the theorem does not apply to the stationary flow of rigid plasticstrain-hardening materials. An engineering approach has been used to overcome thisdifficulty. In particular, the equivalent strain in the strain-hardening law is replaced withan average strain in each cross-section of the wire. The latter is calculated using geometricparameters of the die. As a result, the original material model is replaced with the rigidperfectly plastic model with a non-homogeneous yield stress distribution. The upper boundtheorem applies to such a model.

The solution has been reduced to several ordinary integrals that should be evaluatednumerically. Section 6 illustrates the effect of process parameters on the dimensionlessdrawing force, assuming the hardening law (62). The behavior of the solution agrees withphysical expectations. The solution practically coincides with solution [16] (Figure 5). Thelatter has been developed for small die angles. The new solution is valid for any die angle.

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An advantage of this solution is that the kinematically admissible velocity field is takenfrom the exact solution for material flow through an infinite channel [12].

Since the material is pressure-independent, it is straightforward to adopt the solutionprovided for calculating the extrusion force. To this end, it is necessary to add the samehydrostatic stress to all normal stresses such that the drawing force vanishes. Front andback forces can be included similarly.

Author Contributions: Conceptualization, S.A. and Y.-M.H.; formal analysis, H.S.R.T.; writing—review and editing, S.A. and Y.-M.H. All authors have read and agreed to the published version ofthe manuscript.

Funding: This research was funded by the Ministry of Science and Technology of the Republic ofChina, grant number MOST 108-2923-E-110-002-MY3.

Data Availability Statement: Not applicable.

Conflicts of Interest: The authors declare no conflict of interest.

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