Upper bound analysis and experimental investigations of dynamic effects during sinter-forging of irregular polygonal preforms

Journal of Materials Processing Technology 194 (2007) 134–144

Upper bound analysis and experimental investigations of dynamic effects during sinter-forging of irregular polygonal preforms S. Singh a,∗ , A.K. Jha b , S. Kumar a a

b

Department of Production Engineering, Birla Institute of Technology, Mesra, Ranchi 835215, Jharkhand, India Department of Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi 221005, UP, India Received 15 September 2006; received in revised form 19 December 2006; accepted 11 April 2007

Abstract The present paper investigates the effect of die velocity, i.e. dynamic effects on various deformation characteristics like flow of material, densification, strain rates, energy dissipations and die load during sinter-forging of irregular polygonal preforms. The expressions for exponential velocity fields, strain rates, energy dissipations and average die load have been established based on upper bound approach. The experiments have been conducted using compacted aluminium metal powder preforms to investigate their flow behavior. Based on the preform shape geometry and observed flow characteristics, it have been found that there was no velocity discontinuities or jumps across the common boundaries of small zones assumed over the surface of irregular polygonal preforms. To investigate the dynamic effects, two important parameters; energy and load factors have been introduced in the present study. The design of experiment (DOE) and response surface methodology (RSM) techniques have been used to demonstrate the interaction of various processing parameters on energy and load factors. The preform height reduction, relative density, axial strain rate and inertia energy dissipation were found to increase, where as die load was found to decrease with die velocity. The dynamic effects were found to be more pronounced during sinter-forging process at higher die velocities. It is expected that the present research work will be highly useful for understanding the dynamic effects during sinter-forging of complicate shape preforms, e.g. irregular polygonal shape. © 2007 Published by Elsevier B.V. Keywords: Dynamic effect; Sinter-forging; Irregular polygonal preform; Inertia factor; Load factor

1. Introduction Sinter-forging is a rapidly developing net-shape mass production technology, in which compacted metal powder preforms are forged within the closed dies to produce precise, high performance and economically competitive engineering components. The final density of such sinter-forged products compares favorably with that of wrought products [1,2]. The technology has extensive applications in the field of automobiles, aerospace, defense and other household products, e.g. connecting rods, gears, crankshafts, engine valves, etc. [3–5]. The sinter-forging technology is entirely different from conventional wrought metal forging, as characteristics of porous materials during compression have to be taken into consideration. The densification (compaction or closing of pores) and compression (change in

∗

Corresponding author. Tel.: +91 651 2275044; fax: +91 651 2275401. E-mail addresses: [email protected], [email protected], [email protected] (S. Singh). 0924-0136/$ – see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.jmatprotec.2007.04.108

shape) of sintered preform takes place simultaneously during sinter-forging process and hence volumetric constancy principle is not valid, as density of preform changes due to closing of inter-particle pores. Thus, the sensitivity of yielding on hydrostatic stress component necessitates the use of suitable yield criterion, which is dependent on relative density of preform [6,7]. The high interfacial pressure coupled with severe deformation modes breaks the die-workpiece interfacial lubricant film and creates conditions essential for adhesion friction in addition to sliding friction. Such interfacial frictional conditions are modeled as composite friction including both sliding and sticking frictions [8,9]. Many solutions to problems of forging of conventional and sintered materials have been reported from various aspects using simple generic shapes, but not much systematic attempt has been made so far to study the dynamic effects during deformation of complex shaped non-axisymmetric components [10–25]. The current paper aims at studying and analyzing one such component, i.e. four sided irregular polygonal preform fabricated from sintered aluminium metal powder, which is subjected to

S. Singh et al. / Journal of Materials Processing Technology 194 (2007) 134–144

Nomenclature a, b, c, d preform dimensions aij associated acceleration field Cq shape-complexity factor H0 preform height J total external energy L length of rectangular zone n constant quantity (1) P die pressure Pav average die pressure sticking zone radius rm R0 width of rectangular zone R1 , R2 radius of imaginary circles S preform surface area U die velocity Uij associated velocity field U˙ die acceleration U die-workpiece relative velocity V preform volume Wa inertia energy dissipation frictional energy dissipation Wf Wi internal energy dissipation xm sticking zone axial distance Greek letters β barreling parameter ε˙ ij associated strain rate field ζ load factor η constant function of ‘ρ0 ’ θ 1 , θ 2 , θ 3 , θ 4 internal angles of zones μ coefficient of friction ξ energy factor ρi initial density of perform ρ0 relative density of perform σm hydrostatic stress σ0 flow stress of sintered material τ interfacial shear stress φ0 specific cohesion ψ, χ constants and function of ‘η’ Subscripts x axial y longitudinal z vertical r radial θ circumferential

open-die forging between two rigid, parallel and flat die platens at cold conditions. The present work investigates the effect of die velocity, i.e. dynamic effects on various deformation characteristics like flow of material, densification, strain rates, energy dissipations and die load during sinter-forging of irregular polygonal preforms. The surface of irregular polygonal preforms was divided into

135

number of small zones based on shape geometry and flow characteristics, such that velocity discontinuities or jumps across their common boundaries are zero. Experiments were conducted on aluminium powder sintered polygonal preforms and flow characteristics were analyzed to confirm the validity of this division. Also, various sinter-forging characteristics, e.g. flow of material, densification under lubricated friction conditions, barreling and strain at free surface were studied and measured experimentally. Subsequently, the expressions for exponential velocity fields, strain rates and various energy dissipations were formulated for these zones. Finally, these expressions were clubbed together to evaluate the estimate of average forging load. The interrelationships between important process parameters and their influence on energy and load factors were done using DOE and RSM techniques and it is expected that the present work will be highly useful for assessment of dynamic effects during processing of sintered materials. 2. Experimental work In order to investigate the effect of die-speed, i.e. dynamic effects and to establish the related plasticity characteristics during sinter-forging process, experiments have been conducted using sintered aluminium metal powder irregular polygonal preforms under cold conditions. It includes fabrication of powder compaction die, fabrication of powder preforms and investigation and measurement of various deformation characteristics, e.g. flow behavior, densification, height reduction, strain rates, forging load with die-speed during sinter-forging of irregular polygonal preforms.

2.1. Fabrication of dies and powder preforms The closed square-cavity powder compaction die of alloy steel having good surface finish with cavity side length equal to 35 mm have been fabricated for the production of irregular polygonal preforms. The die set consists of three separate parts, namely flat upper punch, lower counter punch with obtrude recess and a central container to house punch and counter-punch. The loose aluminium powder (refer Table 1 for physical and chemical properties) has been compacted at compacting pressure of about 20 t. The green square compacts obtained were sintered at about 400 ◦ C for 4 h in an endothermic sand atmosphere and subsequently, machined to the required dimensions depending upon the shape-complexity factors (refer Table 2). The preform densities of have been obtained indirectly by measuring their dimensions and weight. Subsequently, relative densities have been obtained as a ratio of density of sintered preform and actual density of corresponding solid metal. A 150-t computer controlled mechatronic press has been used to investigate experimentally the dynamic effects during sinter-forging of irregular polygonal preform. The computer control of press enabled the precise control and accurate record of various deformation characteristics during sinter-forging of irregular polygonal preforms.

Table 1 Physical and chemical characteristics of aluminium metal powder Particle size (␮m)

wt.%

Chemical analysis

wt.%

118.0–88.0 88.0–65.5 65.5–49.0 49.0–36.0 36.0–27.0 27.0–17.5 17.5–13.0 13.0 and less

1.1 3.4 6.7 9.8 13.2 18.6 21.7 25.5

Aluminium Iron Silicon Zinc Manganese Magnesium Apparent density Tap density

99.500 <0.1700 <0.1313 <0.0053 <0.0023 <0.0016 1.25 g/cm3 1.50 g/cm3

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Table 2 Dimensions of aluminium metal powder irregular polygonal preforms Dimensions of preform (mm)

Specimen 1 Specimen 2 Specimen 3

Cq

A

b

c

d

L

R1

R2

0.85 0.60 0.30

6.5 4.6 2.6

5.5 4.4 2.4

16 15.6 15.4

14 14.4 14.6

0.8 3.6 6.0

9.2 7.5 4.5

10.8 12.5 15.5

no shear flow or velocity jumps or discontinuity across the boundaries of zones considered. Fig. 2(a) and (b) shows the sintered aluminium metal irregular polygonal preform before and after sinter-forging process. The similarity in zones also reduced the mathematical formulations considerably, as energy dissipations will be equal for similar zones. Eqs. (1) and (2) give the radii of imaginary circles with centers ‘O1 ’ and ‘O2 ’, respectively (refer Fig. 1), which can be drawn inscribing the sides of irregular polygonal preform. The deformation behavior of zones I and III may be suitably analyzed under axi-symmetry conditions, whereas zone II may be analyzed under plane strain conditions for all practical conditions R1 = R2 =

 a  tan θ1

 c  tan θ3

= =

 b 

(1)

tan θ2

 d 

(2)

tan θ4

2.3. Interfacial friction, densification and compatability

Fig. 1. Division of irregular polygonal preform surface into different zones.

The flow pattern during sinter-forging suggests that die-workpiece interfacial lubrication film breaks at the points of severe pressure and conditions essential for adhesion friction is created. Thus, interfacial friction condition during sinter-forging is essentially composite in nature and includes both sliding and sticking frictions. Such composite interfacial friction conditions may be modeled using two interfacial friction zones, i.e. an inner sticking interfacial friction zone and an outer sliding interfacial friction zone. The shear stress equation for axi-symmetric and plane strain deformation condition respectively may be given as [26,27]

2.2. Flow behaviour

τi = μ

The irregular shape of polygonal preforms rules out the formulation of single kinematically admissible velocity field for entire deforming preform, which precludes the application of upper bound theorem. The familiar way of dealing such problems is division of preform into smaller zones in such a manner that the rules of upper bound theorem may be applied successfully. The four-sided irregular polygonal preform ‘ABCD’ has been divided into three zones, e.g. zone I (ABGO1 E), zone II (EO1 GHO2 J) and zone III (JO2 HCD) as shown in Fig. 1, which have been obtained by drawing angle bisectors from various internal corner angles and then drawing perpendiculars to various sides of irregular polygonal preform from the intersection points of angle bisectors. Each zone has two pairs of symmetrical regions as: • Zone I: AO1 E (1) ∼ = AO1 F (2) and BO1 F (3) ∼ = BO1 G (4) • Zone II: zone O1 O2 GH (5) ∼ = zone O1 O2 EJ (10) • Zone III: CO2 H (6) ∼ = CO2 I (7) and DO2 I (8) ∼ = DO2 J (9) The sinter-forging of the aluminium powder irregular polygonal preform ‘ABCD’ indicated that there has been symmetry in metal flow about the perpendicular planes passing through boundaries of these zones. The planes remain straight throughout the deformation process, which confirmed that there was





P + ρ0 φ0 1 −





τ = μ P + ρ0 φ0 1 − where H0 ln rm = Ri sec θ − 2μ H0 xm = L − ln 2μ



 r − r  m nRi sec θ

 x − x  m



1 √ μ 3

nL

1 √ μ 3

(3) (4)

(5)

(6)

The sinter-forging of powder preforms revealed that compressive forces gradually close down the interparticle pores leading to decrease in volume and subsequent increase in relative density of preforms. Fig. 3 shows the photomicrographs of the sintered preform surfaces at various height reductions justifying the closing of the interparticle pores and consequently increase in its relative density. The variation of relative density with height reduction under lubricated interfacial friction conditions experimentally during sinter-forging of irregular polygonal preform is shown in Fig. 4. It is evident that preform relative density increases with the percent height reduction and becomes comparable to the density of corresponding wrought materials at the end of operation. Thus, it is

Fig. 2. Aluminium metal powder irregular polygonal preforms before and after sinter-forging under lubricated interfacial friction conditions.

S. Singh et al. / Journal of Materials Processing Technology 194 (2007) 134–144

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Fig. 3. Photomicrographs of sintered preforms at various stages of height reductions. apparent that compression (deformation) of preform takes place along with the compaction (densification) and volume of preform does not remain constant. Hence, yielding of sintered materials is sensitive to the compressive hydrostatic stress component imposed during deformation. Therefore, an appropriate yield criterion for sintered porous materials has been considered during present investigations, which is as [28] √ ρik σ0 = 3J2 − 3ησm (7)

ε˙ rr +

 (1 − 2η)  2(1 + η)

ε˙ xx +

ε˙ zz = 0



(1 + 2η2 ) − 2η

(9) 3(1 − η2 )

(1 − 4η2 )

ε˙ zz = 0

(10)

3. Upper bound analysis

where η = 0.54(1 − ρ0 )1.2

on the yield criterion as

(8)

The compatibility equations [29] incorporating densification of preform has been derived for axi-symmetric and plane strain deformation conditions based

The present upper bound analysis considers following important assumptions: • Interfacial friction during sinter-forging operation is composite in nature and the interfacial sticking friction is a function of relative density, which increases asymptotically with real area of contact. • The deformation and compaction, i.e. densification of sintered porous preform takes place simultaneously and thus yielding is sensitive to hydrostatic stress component. • The width of rectangular zones O1 O2 GH and O1 O2 EJ is considered to be an average measure of radii of imaginary circles, keeping the computed energy dissipations reasonably accurate.

Fig. 4. Variation of relative density with height reduction under lubricated interfacial friction conditions during sinter-forging of irregular polygonal preform.

Consider sinter-forging of four-sided irregular polygonal preform between two perfectly flat, parallel and rigid die platens with lower die platen moving upwards with velocity ‘U’ and upper die platen stationary as shown in Fig. 5. As discussed earlier, the preform surface has been divided into three zones, where zones I (ABGO1 E) and III (JO2 HCD) can be analyzed under axi-symmetric deformation condition, where as zone II

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3.2. Zone II The internal, frictional shear and inertia energy dissipation for zone O1 O2 HG (5) during sinter-forging of irregular polygonal preform are formulated by substituting respective velocity field and strain rates into Eq. (13) (refer Appendix A) and are given as   2σ0 U(1 + χ2 ) √ Wi5 = 3χβ ⎧     2  ⎪ ⎨  χβR0 χβR0   × + 1+ ⎪ 2H0 1 + χ2 ⎩ 2H0 1 + χ2 ⎡ + ln ⎣

Fig. 5. Schematic diagram for sinter-forging of irregular polygonal preform between two flat die platens.

χβR0  2H0 1 + χ2



    + 1+

⎤⎫ ⎬ χβR0 ⎦  ⎭ 2H0 1 + χ2 (17)

(EO1 GHO2 J) can be analyzed under plane strain deformation condition. The boundary conditions during sinter-forging of irregular polygonal preform are given as Uz = U at z = 0

(11)

Uz = 0 at z = H0

(12)

The external energy ‘J’ supplied by the die platens based on upper bound approach is given as [30]

 1 2σ0 ε˙ ij ε˙ ij dV + τ|U|dS + ρi (ai Ui ) dV J= √ 3 V 2 S V (13) The first term denotes rate of internal energy dissipation ‘Wi ’, second term denotes frictional shear energy dissipation ‘Wf ’, and last term denotes energy dissipation due to inertia forces ‘Wa ’. 3.1. Zones I and III



μχβ e−β UR20 Wf5 = H0 (1 − e−β )



   3xm − 2R0 Pav + ρ0 φ0 1 − 3nL (18)



W a5

3  2 2 2 −β 2χ β R0 e (1 − e−β ) U = 2ρi LR0 1 − e−β 9H02 (5 + 2χ)    1 + e−β 1+χ + × 5 + 2χ 2    2 1 − e−β + + U U˙ (19) 2

The energy dissipations for zones AO1 F (2), BO1 G (4), CO2 I (7), DO2 J (9) and zone O1 O2 JE (10) are estimated by equating to those of zones AO1 E (1), BO1 F (3), CO2 H (6), DO2 I (8) and zone O1 O2 HG (5) respectively, as they are symmetrical to each other. Thus, total energy dissipation during sinter-forging of irregular polygonal preform is estimated by clubbing all the expressions as % J =2 Wiq + Wfq + Waq (20)

The internal, frictional shear and inertia energy dissipations for zones AO1 E (1), BO1 F (3), CO2 H (6) and DO2 I (8) during sinter-forging of irregular polygonal preform are formulated by substituting respective velocity field and strain rates into Eq. (13) (refer Appendix A) and are obtained by substituting k (zone number) = 1, 2, 3 and 4 into Eqs. (14)–(16) respectively ⎫ ⎧

  3/2

⎬ ⎨ β2 ak2 sec2 θk 8σ0 UH02 ψ3/2 3 1 + Wik = √ − tan θ k ⎭ 4ψH02 tan 2 θk 3 6β2 (ψ − 2)1/2 ⎩

q=1,3,6,8,5

     μ(1 − 2η)β e−β ak3 U 3 2ρ0 φ0 rm tan2 θk Wfk = P 1+ (sec θ +ρ φ tan θ +ln|sec θ + tan θ |) − av 0 0 k k k k 6(1+η)(1−e−β ) tan3 θk H0 4n nak 



  (1 − 2η)2 β2 ak2 U 2 (2 + sec2 θk ) ρi H0 ak2 U (e−β + e−2β )(2 − η) U2β ˙ Wak = −1 − + UU 2 2tan θk (1 + 4η) H0 4(1 + η)3 (1 + 4η)(1 − e−β ) tan2 θk H 3

(14)



0

where a2 = b; a3 = c; a4 = d and ψ = {2 + [2(1 + η)2 /1 − 2η]}.

(15)

(16)

S. Singh et al. / Journal of Materials Processing Technology 194 (2007) 134–144

The average forging load during sinter-forging of irregular polygonal preform may be expressed as

R1 R2

(22)

To illustrate the dynamic effects, i.e. effect of die velocity on relative magnitudes of various energy dissipations and average die load involved during sinter-forging of irregular polygonal preform, concepts of inertia factor ‘ξ’ and load factor ‘ζ’ (refer Eqs. (23) and (24)) respectively has been introduced. The inertia factor is defined as the ratio of inertia energy dissipation to total energy dissipated (in percent), whereas the load factor is defined as the ratio of difference in die load with and without dynamic effects to die load with dynamic effects (in percent) for maximum allowable reduction of preform height ξ (%) =

Factor description

Levels

(21)

The effect of preform shape variations on dynamic effects during sinter-forging of irregular polygonal preform is investigated using shape-complexity factor ‘’Cq ’, which is defined as ratio of radii of imaginary circles considered during division of preform surface into various zones. The factor tends to unity, when R1 ∼ = R2 , i.e. shape of irregular polygonal preform approaches to that of square and tends to reduce to zero, when R1 ∼ = 0, i.e. shape of irregular polygonal preform approaches to that of a triangle and expressed as Cq =

Table 3 Factor levels during DOE analysis Factors

Fav = J(U)−1 Aav

139

Wa × 100 J

(23)

& & & |Fav |with dynamic effects −|Fav |without dynamic effects & & × 100 & ζ (%)= & & |F | av with dynamic effects

A B C D

Shape-complexity factor (Cq ) Barrelling parameter (β) Initial relative density (ρ0 ) Die velocity (U)

Low

High

0.30 0.30 0.70 0.01 m/s

0.85 0.40 0.90 10 m/s

4. Design of experiment (DOE) analysis [31–35] The two level (2n ) full factorial randomized DOE and RSM techniques has been used to study the interrelationship between important deformation characteristics and their influence on dynamic effects during sinter-forging of irregular polygonal preform. The analysis considers four ‘’Factors’, e.g. shapecomplexity factor (A), barrelling parameter (B), initial relative density (C) and die velocity (D) and two ‘Response Variable’, i.e. energy and load factors. The ‘Factor’ levels are shown in Table 3. Table 4 shows the ‘Coefficient of Factor Effect Estimate’, ‘Sum of Squares’ and ‘Percent Contribution’ for each experiment run for energy and load factors. It is evident from the table that ‘Interaction Effects’ between shape-complexity factor and die velocity (AD) and between initial preform relative density and die velocity (CD) are prominent during sinter-forging of irregular polygonal preform. Further, these interactions are investigated using RSM technique and multiple linear regression equations of second order have been formulated mathematically (refer Eqs. (25)–(28)) and three-dimensional graphs have been generated using MATLAB software |YAD |inertia factor = 4.7713 + 24.3425A + 1.77D − 28.767A2

(24)

+ 0.0521D2 − 1.67AD + 0.303A2 D2 (25)

Table 4 Factor effect estimate and percent contribution for energy and load factors Factor effect

A B AB C AC BC ABC D AD BD ABD CD ACD BCD ABCD

Factor effect estimate

Sum of squares

Percent contribution

Energy factor

Load factor

Energy factor

Load factor

Energy factor

Load factor

7.08E+00 3.12E−01 2.68E−01 1.97E+00 1.37E+00 1.41E−01 3.43E−02 1.02E+01 6.88E+00 3.71E−01 3.43E−02 1.95E+00 1.24E+00 6.40E−02 1.04E−01

7.18E+00 2.49E+00 −9.36E−02 4.83E+00 5.75E−01 −5.47E−01 −6.04E−01 −2.38E+01 −4.84E+00 −1.34E+00 −6.04E−01 −3.57E+00 −3.35E−01 3.69E−02 4.56E−01

2.01E+02 3.90E−01 2.87E−01 1.55E+01 7.50E+00 7.90E−02 4.69E−03 4.20E+02 1.89E+02 5.49E−01 4.69E−03 1.53E+01 6.19E+00 1.64E−02 4.31E−02

2.06E+02 2.47E+01 3.51E−02 9.33E+01 1.32E+00 1.20E+00 1.46E+00 2.26E+03 9.37E+01 7.14E+00 1.46E+00 5.10E+01 4.50E−01 5.44E−03 8.33E−01

2.35E+01 4.56E−02 3.36E−02 1.81E+00 8.76E−01 9.23E−03 5.49E−04 4.91E+01 2.21E+01 6.42E−02 5.49E−04 1.79E+00 7.24E−01 1.92E−03 5.03E−03

7.52E+00 9.02E−01 1.28E−03 3.40E+00 4.83E−02 4.36E−02 5.32E−02 8.24E+01 3.42E+00 2.61E−01 5.32E−02 1.86E+00 1.64E−02 1.98E−04 3.04E−02

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|YCD |inertia factor = 46.09 − 75.328C + 1.482D + 10.392C2 − 0.196D2 − 6.778CD + 1.068C2 D2 (26) |YAD |load factor = 18.144 − 12.719A − 0.159D + 27.662A2 − 0.3144D2 + 1.4113AD − 0.196A2 D2 (27) |YCD |load factor = 11.483 − 28.325C − 3.169D + 30.937C2 − 0.416D2 − 3.54CD + 0.208C2 D2

(28)

5. Results and parametric discussion A typical data of deformation characteristics compatible with corresponding experimental work to illustrate the dynamic effects during sinter-forging of irregular polygonal preform has been considered as: H0 = 10 mm; β = 0.30; ρ0 φ0 = 0.30 Pav ; n = 3; ρ0 = 0.80; σ 0 = 6.25 kg/mm2 ; μ = 0.30 (high interfacial friction) and 0.01 (low interfacial friction); U = 0.001–10 m/s. Fig. 6 shows the variation of inertia energy dissipation with die velocity for different shape-complexity factor during sinterforging of irregular polygonal preform. It is apparent from figure that inertia energy dissipation increases exponentially with increase in the die velocity. The curves for high shapecomplexity factors are higher, which indicates that inertia energy dissipation increases, as the shape of irregular polygonal preform approaches to that of an enclosing square. The investigations into dynamic effects on inertia energy dissipation, as a fraction of total external energy supplied is shown in the form of variation of energy factor ‘ξ’ in Fig. 7. It is clearly evident that energy factor increases exponentially with die velocity and hence magnitude of inertia energy dissipation is substantially high at higher die velocity, e.g. 10–20% of total energy dissipation at U = 10 m/s. This indicates that inertia energy dissipation must be considered during analysis for accurate measure of total press power requirement.

The variation of average forging load with die velocity and shape-complexity factor under high interfacial friction conditions for maximum height reductions both theoretically and experimentally during sinter-forging of irregular polygonal preform is shown in Fig. 8. It is clearly observable that forging load decreases rapidly with increase in die velocity and becomes asymptote with x-axis at higher die speeds. This is attributed due to the small contact time between die and preform, which creates conditions essential for load reduction. The small deformation time due to high die velocities does not allow the heat generated during plastic working to dissipate quickly from preform-die interface, leading to decrease in the resistance of sintered preforms against deformation and die-preform interfacial friction constraint. Fig. 9 demonstrate the variation of load factor ‘ζ’ for maximum allowable height reduction with die velocity during sinter-forging of irregular polygonal preform. It is revealed from the figure that load factor decreases with increase in die velocity and becomes asymptote with x-axis at higher die velocities.

Fig. 6. Variation of inertia energy dissipation with die velocity during sinterforging of irregular polygonal preform.

Fig. 8. Variation of forging load with die velocity during sinter-forging of irregular polygonal preform.

Fig. 7. Variation of energy factor with die velocity during sinter-forging of irregular polygonal preform.

S. Singh et al. / Journal of Materials Processing Technology 194 (2007) 134–144

141

Fig. 9. Variation of load factor with die velocity during sinter-forging of irregular polygonal preform.

Thus, higher dynamic effects leads to lower load requirements during sinter-forging operation, especially at higher die speeds, as compared to slow-speed deformation. Fig. 10 show the variation of radial and axial strain rates with percent height reduction and die velocity during sinter-forging of irregular polygonal preforms. It can be seen that both radial and axial strain rates increases exponentially with preform height reduction. Also, increase in the die velocity increases the axial strain rates but correspondingly decreases the radial strain rates. This may be explained from the compatibility condition, i.e. volume of preform changes due to densification. Thus, decrease in lateral strain rate is due to simultaneous increase in relative density and axial strain rate of preforms. Fig. 11(a) and (b) shows the interaction effects ‘A–D’ and ‘C–D’, respectively for inertia factor during sinter-forging of irregular polygonal preform in form of 3D response surfaces. It is evident that inertia factor increases linearly with the shape-complexity factor and die

Fig. 10. Variation of radial and axial strain rates with die velocity during sinterforging of irregular polygonal preform.

Fig. 11. (a) Interaction effect ‘A–D’ with energy factor. (b) Interaction effect ‘C–D’ with energy factor.

velocity. It is also found to increase with preform relative density and die velocity, which shows that magnitude of inertia energy dissipation becomes comparable with those of internal and frictional shear energy dissipations, especially at higher die velocities. Hence, it must be considered during the analysis for accurate prediction of die loads. It is also evident that interaction effect ‘C–D’ is non-linear. The interaction effects ‘A–D’ and ‘C–D’ with load factor are shown in Fig. 12(a) and (b), respectively. It has been apparent that load factor decreases exponentially with increase in die velocity and increases slightly with increase in shapecomplexity factor and preform relative density. The interaction effect ‘A–D’ with load factor is nonlinear, whereas interaction effect ‘C–D’ is linear. This suggests that dynamic effects, i.e. effect of die velocity in combination with the preform relative density and shape-complexity factor produces nonlinear effects in case of inertia and load factors respectively. Hence, it may be concluded that die velocity is the most critical deformation characteristics during mechanical processing of sintered materials, whose effects on the inertia energy dissipation and average die load becomes pronounced at higher deformation speeds.

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materials against deformation. Also, the energy dissipation and die load requirement to deform four-sided irregular polygonal preform are lower as compared to the enclosing shapes. The DOE and RSM techniques were employed successfully for the analysis of dynamic effects during sinter-forging of irregular polygonal preform. It was found that both the interaction effects between die velocity and shape-complexity factor, as well as between die velocity and preform relative density are of significant importance. The multiple linear regression equations of second order and the corresponding three-dimensional graphs were used to depict these interaction effects. The paper clearly established that dynamic effects, i.e. effect of die velocity on various deformation characteristic are of paramount importance during sinter-forging of metal powder preforms, especially at higher die speed and therefore, must be considered during such investigations. Appendix A The exponential velocity field and corresponding strain rates satisfying compatibility Eq. (9) and boundary conditions (11) and (12) for zones I and III are given as 

(1 − 2η)β e−βz/H0 Ur Ur = 2(1 + η)(1 − e−β )H0  Uz = −

Fig. 12. (a) Interaction effect ‘A–D’ with load factor. (b) Interaction effect ‘A–D’ with load factor.

6. Conclusions The present work confirmed that any complex shaped nonaxisymmetric preforms can be analyzed using upper bound approach by dividing the preform surface into various symmetric zones provided velocity discontinuities or jumps are zero at its boundaries. The relative density of irregular polygonal preform increases with increase in the magnitude of die velocity, die load and height reduction. The high-speed sinter-forging process is characterized by very high magnitude of strain rates up to the order of 2000 s−1 . The inertia energy dissipations have been found to increase with the die velocity and become comparable with other energy dissipations at higher die speeds. This effect of dynamic effects on the relative magnitudes of various energy dissipations has been illustrated using energy factor ‘ξ’, which has been found to increases exponentially with die velocity and becomes asymptote to y-axis and must be considered for accurate and reasonable measure of die load. The magnitudes of die load and load factor has been found to decrease with die velocity and is appreciably low at higher die speeds. This is due to very small contact time under load at higher die speeds. This short duration of deformation time restricts the internal heat generated during plastic working to dissipate quickly and hence, reduce the resistance of sintered

(e−β − e−βz/H0 )U (1 − e−β )

 (A.1)

 (A.2)

Uθ = 0

(A.3)

  ∂Ur (1 − 2η)β e−βz/H0 U = ε˙ rr = ∂r 2(1 + η)(1 − e−β )H0

(A.4)

ε˙ zz

  ∂Uz β e−βz/H0 U = =− ∂z (1 − e−β )H0

ε˙ θθ =

(A.5)

  (1 − 2η)β e−βz/H0 U Ur = r 2(1 + η)(1 − e−β )H0

1 ε˙ rz = 2



∂Ur ∂Uz + ∂z ∂r





(1 − 2η)β2 e−βz/H0 Ur =− 4(1 + η)(1 − e−β )H02

ε˙ rθ = ε˙ θz = 0

(A.6) (A.7) (A.8)

The internal energy dissipation for zone AO1 E (1) is given as  Wi =  = ×

2σ0 √ 3

'

1 ε˙ ij ε˙ ij 2 V z=H0 θ=θ1

2σ0 √ 3 

z=0

θ=0

 dV r=(R1 sec θ)

r=0

 1 ε˙ 2rr + (˙ε2zz + ε˙ 2rz ) (r dr dθ dz) 2

(A.9)

S. Singh et al. / Journal of Materials Processing Technology 194 (2007) 134–144

Substituting Eqs. (A.4)–(A.8) above and solving, internal energy dissipation is given as

8σ0 UH02 ψ3/2 √ Wi1 = 3 6β2 (ψ − 2)1/2 ⎫ ⎧

 3/2  ⎬ ⎨ β2 a2 sec2 θ1 3 (A.10) − tan θ × 1+ 1 ⎭ ⎩ 4ψH02 tan2 θ1 The frictional shear energy dissipation for zone AO1 E (1) is given as

θ=θ1 r=R1 sec θ Wf = τ|U|dS = τ1 |Ur |z=H0 (r dr dθ) S

where

θ=0



|Ur |z=H0 =

r=0

(A.11)

r −β 2(1 + η)(1 − e )H

and (A.12)

Substituting Eq. (A.12) above and solving, frictional energy dissipation is given as   μ(1 − 2η)β e−β a3 U Wf1 = 6(1 + η)(1 − e−β ) tan3 θ1 H0    3 × Pav + ρ0 φ0 1 + 4n × (sec θ1 tan θ1 + ln|sec θ1 + tan θ1 |)   2ρ0 φ0 rm tan2 θ1 − na

(A.13)

The inertia energy dissipation for zone AO1 E (1) is given

Wa = =

ρi (ai Ui ) dV V

z=H0 θ=θ1 r=R1 sec θ z=0

θ=0

r=0

ρi (ar Ur + az Uz )(r dr dz dθ)

where   ∂Ur ∂Ur ∂Ur ar = Ur + Uz + and ∂r ∂z ∂t   ∂Uz ∂Uz az = Uz + ∂z ∂t

Uy = 0

(A.19)





∂Ux χβ e−βz/H0 Ux = ∂x (1 − e−β )H0   ∂Uz β e−βz/H0 U = =− ∂z (1 − e−β )H0

ε˙ xx =

(A.20)

ε˙ zz

(A.21)

ε˙ xz =

0

    rm − r τ1 = μ P + ρ0 φ0 1 − nR1 sec θ

as

The exponential velocity field and corresponding strain rates satisfying compatibility Eq. (10) and boundary conditions (11) and (12) for zone II are given as   χβ e−βz/H0 Ux (A.17) Ux = (1 − e−β )H0  −β  (e − e−βz/H0 )U Uz = − (A.18) (1 − e−β )

ε˙ yy = 0



(1 − 2η)β e−β U

143

1 2



∂Ux ∂Uz + ∂z ∂x



=−

χβ2 e−βz/H0 Ux ) ( 2 1 − e−β H02

1/2

× [˙ε2xx + ε˙ 2yy + ε˙ 2zz ]

(dx dy dz)

(A.23)

(A.24)

Substituting Eqs. (A.20)–(A.23) above and solving internal energy dissipation is given as   2σ0 U(1 + χ2 ) √ Wi5 = 3χβ ⎧     2  ⎪ ⎨  χβR0 χβR0   × + 1+ ⎪ 2H0 1 + χ2 ⎩ 2H0 1 + χ2 ⎡ + ln ⎣

χβR0  2H0 1 + χ2



    + 1+

⎤⎫ ⎬ χβR0 ⎦  ⎭ 2H0 1 + χ2 (A.25)

The frictional shear energy dissipation for zone O1 O2 HG (5) is given as

x=R0 y=L τ2 |U|(x dx dy) (A.26) Wf = x=0

y=0

Substituting from Eqs. (A.1)–(A.3) above, simplifying and substituting into Eq. (A.14) inertia energy dissipation is given as    

  (1 − 2η)2 β2 a2 U 2 (2 + sec2 θ1 ) (e−β + e−2β )(2 − η) ρi H0 a2 U U2β ˙ Wa1 = + UU −1 − 2 2 tan θ1 (1 + 4η) H0 4(1 + η)3 (1 + 4η)(1 − e−β ) tan2 θ1 H03 where a2 = b; a3 = c; a4 = d and ψ = {2 + [2(1 + η)/(1 − 2η)]2 }.

(A.22)

 where χ = [1 + 2η2 − 2η 3(1 − η2 )/((1 − 4η2 ))]. The internal energy dissipation for zone O1 O2 HG (5) is given as  x=R0 y=L z=H0 2σ0 Wi5 = √ 3 y=0 x=0 z=0

(A.14)

(A.15)



(16a)

144

S. Singh et al. / Journal of Materials Processing Technology 194 (2007) 134–144

where



|U| = |Ux |z=H0 =

χβ e−β U

x



and (1 − e−β )H0     xm − x τ2 = μ P + ρ0 φ0 1 − nL

(A.27)

Substituting Eq. (A.27) above and solving frictional energy dissipation is given as

    μχβ e−β UR20 3xm − 2R0 Wf5 = 1 − P + ρ φ av 0 0 H0 (1 − e−β ) 3nL (A.28) The inertia energy dissipation for zone O1 O2 HG (5) is given as

Wa1 = ρi

x=R0

x=0

y=L z=H0

y=0

z=0

(ax Ux + az Uz )(dx dy dz) (A.29)

where     ∂Ux ∂Ux ∂Uz ∂Ux ∂Uz + Uz + and az = Uz + ax = Ux ∂x ∂z ∂t ∂z ∂t (A.30) Substituting from Eqs. (A.17)–(A.19) above, simplifying and substituting into Eq. (A.29) inertia energy dissipation is given as  3  2 2 2 −β 2χ β R0 e (1 − e−β ) U Wa5 = 2ρi LR0 1 − e−β 9H02 (5 + 2χ)    1 + e−β 1+χ + × 5 + 2χ 2    2 1 − e−β + U U˙ (A.31) + 2

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