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Some aspects of workability studies on hot forging of sintered high strength 4% titanium carbide composite steel preforms
Materials Science and Engineering A 425 (2006) 121–130
Some aspects of workability studies on hot forging of sintered high strength 4% titanium carbide composite steel preforms R. Narayanasamy a,∗ , V. Senthilkumar b , K.S. Pandey c a
b
Department of Production Engineering, National Institute of Technology, Tiruchirappalli 620015, Tamil Nadu, India Department of Mechanical Engineering, Jayaram College of Engineering & Technology, Tiruchirappalli 621014, Tamil Nadu, India c Department of Metallurgical Engineering, National Institute of Technology, Tiruchirappalli 620015, Tamil Nadu, India Received 11 January 2006; accepted 13 March 2006
Abstract The aim of the paper is the study of workability of Fe–1.0% C–4% TiC steel composite during hot upsetting. Ductile fracture is the most common failure acting in bulk forming process. The formability during hot upsetting depends on the temperature, strain and strain rate. A complete experimental investigation of the workability behaviour of Fe–1.0% C–4% TiC steel composite was performed under the triaxial stress state condition. Hot upsetting of Fe–1.0% C–4% TiC steel composite preforms was carried out at a temperature of 1120 ◦ C and the formability behaviour of the same at triaxial stress state condition was determined. The curves plotted for different preforms were analysed and the relationship between the axial strain and the formability stress index were obtained. A relationship between the relative density and the axial strain was also established. A particular attempt was made to relate the various stress ratio parameters, namely, (σ θ /σ eff ), (σ m /σ eff ) and (σ z /σ eff ) with the relative density. © 2006 Elsevier B.V. All rights reserved. Keywords: Workability; Triaxial stress; Formability stress index; Relative density; Axial strain; Stress ratio
1. Introduction Workability refers to the relative ease with which a material can be shaped through plastic deformation. Several approaches have been proposed to understand the constitutive response of the materials during forming operations [1–3]. However, understanding the influence of the process related parameters such as friction and die geometry on the limits of deformation in metalworking has been a difficult problem all along. Ductile fracture is a major factor influencing the limit to workability in many forming operations. Many criteria have been developed to predict ductile fracture. Wifi et al. [4] have presented some of the criteria commonly used to predict ductile fracture. Kuhn and Downey [5] investigated the deformation characteristics and plasticity theory of sintered powder materials and studied the basic deformation behaviour of sintered iron powder performing a simple homogeneous compression tests and also proposed a plasticity theory relating yield stress and Pois-
∗
Corresponding author. Tel.: +91 4312501801; fax: +91 4312500133. E-mail addresses: [email protected] (R. Narayanasamy), [email protected] (V. Senthilkumar). 0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.03.035
son’s ratio to the density. Doraivelu et al. [6] proposed a new yield function for compressible P/M materials, conducting a test under uniaxial state of compressive stress using the P/M aluminium alloy as a model material. Further, yield surfaces for various density levels had been generated in three dimensional principal stress spaces. A new form of yield criterion for porous metals was proposed by Lee and Kim [7] conducting combined tension and torsion tests of sintered porous metals with high degree of accuracy and suggested zero yield stress only at zero relative density. Shima and Oyane [8] suggested a plasticity theory for porous metals and calculated the stress and strain for pore free copper utilizing the basic equations. The proposed equations were applied to frictionless closed die compression and the stress in the direction of compression was evaluated in relation to the relative density and further compared with the experimental results. Gurson [9] discussed the continuum theory of ductile rupture by void nucleation and growth considering yield criteria and flow rules for porous ductile materials by showing the role of hydrostatic stress in plastic yield and void growth. Various ductile fracture criterions were discussed by Oh [10] and proposed a new ductile fracture criterion considering the theory of fracture toughness under different mean stress by conducting a tensile test. Surface fracture is another mode for
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Nomenclature D0 Db Dc Dc1 Dc2 h0 hf R R1 R22
initial diameter of the preform bulged diameter contact diameter contact diameter of the preform (top surface) contact diameter of the preform (bottom surface) initial perform height deformed height of the preform relative density barrel radius correlation coefficient
Greek symbols εθ true Hoop strain εz true axial strain γ Poisson’s ratio ρ0 initial preform density ρf deformed preform density ρth theoretical preform density σ eff effective stress σm mean or hydrostatic stress σr true radial stress σz true axial stress σθ true Hoop stress
the initiation of the crack in the ductile fracture studies. Ko et al. [11] suggested a scheme to simultaneously accomplish both the prediction of surface fracture initiation and the analysis of deformation in the axisymmetric extrusion and simple upsetting of an aluminium alloy. Atkins [12] pointed out that the criteria for fracture initiation should dependent on hydrostatic stress. This conclusion has been independently arrived at by empirical routes [13–15] by porous plasticity or void growth mechanics modeling [16–19], by continuum damage mechanics [20], and by connecting initiation and propagation toughness mechanics [21–24]. Bao and Wierzbicki [25] have also observed that the mechanism of fracture is different depending on the amount of triaxiality. Abdel-Rahman and El-Sheikh [26] discussed workability criterion of powder metallurgy compacts and they investigated the effect of the relative density on the forming of P/M compacts in upsetting. They proposed a workability factor (β) for describing the effect of the mean stress and the effective stress with the help of the mean stress and the effective stress with the help of two theories and they discussed the effect of relative density. A mathematical equation for the determination of the flow stress in the case of the simple upsetting of P/M sintered performs was proposed by Narayanasamy and Ponalagusamy [27]. Further, they developed a new equation for the determination of hydrostatic stress in the case of simple upsetting of sintered P/M compacts. In addition, a new flow rule with anisotrophic parameter for porous metal was also proposed. Gouveia et al. [28] predicted the initiation of the ductile fracture after conducting experiments on powder compacts with various geometrical shapes such as ring, cylindrical, tapered and flanged under several different loading
conditions. Narayanasamy et al. [29] proposed a generalized yield criterion considering anisotrophic parameters for porous sintered powder metallurgy metals and introduced a new flow rule with anisotrophic parameters for porous metals. Further, they proposed new generalized yield criteria with five parameter constants for porous sintered powder metallurgy metals. A new approach for the next generation of metal matrix composite development has been dealt with as described elsewhere [30]. In this paper, a complete investigation on the workability criteria of Fe–1.0% C–4% TiC powder performs was made. Powder metallurgy preforms with various aspect ratios were discussed for studying the behavior for workability during hot upsetting under various triaxial stress state conditions. 2. Theoretical analysis The mathematical expressions used and proposed for the determination of various upsetting parameters of upsetting for various stress state conditions are discussed below. 2.1. Plane stress conditions According to Abdel-Rahman and El-Sheikh [26], the expression for the axial or height strain (εz ) can be written as follows: hf (1) εz = ln h0 where h0 is the initial height and hf is the deformed height of the preform. According to Narayanasamy and Pandey [31], the expression for the new hoop strain (εθ ) is as follows: 2Db2 + Dc2 (2) εθ = ln 3D02 where Db is the bulge diameter, D0 is the initial diameter and Dc is the average contact diameter of the preform. The average diameter of the preform is expressed as follows: Dc1 + Dc2 2 In the above expression for the average contact diameter of the preform, Dc1 is the contact diameter of the preform at top and Dc2 is the contact diameter of the preform at bottom. Further from the above reference [31], the state of stress in a plane stress condition, associated flow characteristics for porous materials can be expressed as follows: dεθ σθ − γσz (3) = dεz σz − γσθ Dc =
dεz is the plastic strain increment in axial direction, dεθ , the plastic strain increment in the hoop direction, σ z , the true axial stress, σ θ , the true hoop stress and γ is the Poisson’s ratio expressed as below. εθ γ= (4) 2εz
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Eq. (3) can be further simplified as follows: σθ α+γ = σz 1 + αγ
(5)
where α = dεθ /dεz According to Narayansamy and Pandey [32], the expression for the hydrostatic stress (σ m ) can be written as below: σm = 31 (σθ + σz )
(6)
Since σ r is considered to be zero in the case of plane stress condition the Eq. (6) can be written as follows: σm σθ 1 (7) 1+ = σz 3 σz According to Narayanasamy and Pandey [33], the relationship between the effective stress (σ eff ) and the axial stress (σ z ) can be written as follows: 0.5 σeff = (0.5 + α) 3(1 + α + α2 ) σz (8) where σ eff is the effective stress, α is the strain increment ratio (dεθ /dεz ) and σ z is the axial stress.The Eq. (8) can be rearranged as follows: σeff 0.5 = (0.5 + α)[3(1 + α + α2 ] (9) σz from the Eqs. (3) and (7), the expression for the stress ratio (σ eff /σ z ) is as given below. σeff 1 α+γ = (10) σz M 1 + γα where M = (0.5 + α)3(1 + α + α2 )0.5 . As an evidence of experimental investigation implying the importance of the spherical component of the stress state on fracture according to Vujovic and Shabalk [34] proposed a parameter called a formability stress index ‘β’ given by 3σm β= (11) σeff This index determines the fracture limit as explained in [35]. The stress formability index (β) can be expressed from Eqs. (5) and (9) as follows:
[σm /σz ] β=3 (12) 1/[σeff /σz ] 2.1.1. Triaxial stress state condition According to Narayansamy and Pandey [32], for the state of stress under triaxial stress state condition is given as below. α=
A , B
α=
dεθ dεz
(13)
where A = (2 + R2 )σ θ − R2 (σ z + 2σ θ ); 2 2 B = (2 + R )σ z − R (σ z + 2σ θ ), where α is the strain increment ratio and R is the relative density. From the above Eq. (13), for
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the known values of α, R and the axial stress, σ z , the hoop stress component, σ θ , can be determined as given below: 2α + R2 (14) σz σθ = 2 − R2 + 2R2 α Rearranging the above Eq. (14), the expression for the stress ratio (σ θ /σ z ) can be expressed as given below. 2α + R2 σθ = (15) σz 2 − R2 + 2R2 α In the above Eq. (15), relative density (R) plays a major role in finding the hoop stress component (σ θ ). It is assumed that σ r = σ θ for the case of axisymmetric condition. It is known that the expression for the hydrostatic stress (σ m ) can be written as follows: σm = 13 (σ1 + σ2 + σ3 ) or σm = 13 (σz + σr + σθ )
(16)
Since σ r = σ θ , the Eq. (16) can be written as follows: σm = 13 (σz + 2σθ )
(17)
The Eq. (17) can be rearranged as follows for the determination of the Stress ratio (σ m /σ z ) σm 1 2σθ (18) = 1+ σz 3 σz The effective stress can be determined from the following expression as explained elsewhere [33]. 2 σ12 + σ22 + σ32 − R2 (σ1 σ2 + σ2 σ3 + σ3 σ1 ) = (2R2 − 1)σeff
This expression can be written as follows in terms of cylindrical coordinates. 2 σeff =
σz2 + σθ2 + σr2 − R2 (σz σθ + σθ σr + σr σz ) 2R2 − 1
(19)
Since σ r = σ θ for cylindrical axisymmetric upsetting or forging operation, the Eq. (19) is as follows: 2 σeff =
σz2 + 2σθ2 − R2 (σz σθ + σθ2 + σθ σz ) 2R2 − 1
(20)
Eq. (20) can be rearranged as below for the determination of the stress ratio (σ eff /σ z ). 1/2 1 + 2(σθ /σz )2 − R2 (2(σθ /σz ) + (σθ /σz )2 ) σeff = (21) σz 2R2 − 1 The stress formability index (β) provided in the Eq. (11) can be derived for the triaxial stress state condition from the Eqs. (18) and (21) as follows:
[σm /σz ] β=3 (22) [1/[σeff /σz ]]
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Fig. 1. Flow chart for the determination of workability parameter (β) under triaxial stress state condition.
Different stress ratio parameters, namely, (σ θ /σ eff ), (σ m /σ eff ) and (σ z /σ eff ) are expressed as follows:
(σθ /σz ) σθ = (23) σeff (σeff /σz )
(σm /σz ) σm = (24) σeff (σeff /σz ) σz 1 = (25) σeff (σeff /σz )
in the steel. Iron powder and the Fe–1% C–4% TiC powders properties are shown in Table 1. The powders with different aspect ratios were then compacted using a 100 tonnes capacity Universal Testing Machine. The three different aspect ratios are shown in Table 2. The compacting pressure was controlled so as to obtain a density level of 7.612 ± 0.001 g/cc. The compacts were coated
The ratios, namely, (σ θ /σ z ), (σ m /σ z ) and (σ eff /σ z ) can be determined from the Eqs. (15), (18) and (21), respectively. Fig. 1 shows the flow chart for computing the stress formability index (β), expressed in the Eq. (22).
Sieve size (m)
Retained (wt.%)
+125 +106 +90 +63 +53 +37 −37
2.12 26.94 0.00 25.84 21.66 16.77 6.51
3. Experimental details Iron powder (−150 m diameter), extra fine Graphite powder and titanium carbide (−48 m diameter) were mixed to obtain the alloy of Fe–1% C–4% TiC via ball milling. The ball mill was operated for 30 h so as to get a homogenized mixture of alloys
Table 1 Characteristics of iron powder
(a) Flow rate: 24.5 s/50 g; (b) apparent density: 3.26 g/cc; (c) compressibility: 6.20 g/cc at a pressure of 4.1 tonf/cm2 (1 tonf/cm2 = 98.1 MPa). Characteristics of Fe–1.0% C–4 TiC powder: (a) apparent density: 3.438 g/cc; (b) compressibility: 6.346 g/cc at a pressure of 146.67 MPa.
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Table 2 Initial parameters of the Fe–1.0% C–4 TiC sintered P/M preforms Aspect ratio
Initial height (h0 ) (mm)
Initial diameter (D0 ) (mm)
Theoretical perform density (ρth ) (g/cc)
0.45 0.71 1.25
11.69 18.77 32.68
25.94 26.27 26.08
7.612 7.612 7.612
Fig. 3. The upset-forging test preform before and after deformation.
Fig. 2. SEM photograph of iron powder.
on all surfaces with an indigenously developed ceramic coating [36]. This coating was allowed to dry for a period of 6 h at normal atmospheric conditions. Recoating was given to the compacts in the direction 90◦ to that of the earlier coating. The second coating was allowed to dry in the same condition of the first coating for a further period of 12 h. These coatings were necessary to avoid oxidation of compacts during sintering. Ceramic-coated compacts were sintered in an electric muffle furnace at a temperature of 1120 ◦ C for a period of 60 min. After the sintering operation, the preforms were upset-forged at a temperature of 1120 ◦ C to the different levels of height strain using a 100 tonnes capacity Friction Screw Press. The forging operation was carried out with no lubricant because it is very
much easier and the cooling of billets or preforms would not take place. The density of forged preforms was determined using the Archimedes principle. After the above-mentioned forging schedule, the dimensions, namely, the height of forged specimen (hf ), the contact diameter (Dc1 and Dc2 ) of top and bottom surfaces, the bulged diameter (Db ) and the barrel radius (R1 ), were measured. Initial dimensions of the specimen (initial height h0 , initial diameter D0 ) and the initial perform density, ρ0 , were measured for each preform before conducting the experiment. The SEM photograph of the iron powder is shown in Fig. 2. The shape dimensions of initial and deformed preforms measured during the experiment as shown in Fig. 3. 4. Results and discussion Fig. 4 shows the variation of the formability stress index determined under the triaxial stress state with respect to the axial strain. For all the aspect ratio, the axial strain (εz ) increases as increasing value of the stress formability index (β). For the lower aspect ratio, the stress formability index takes a very high value at the fracture strain. For equal height strain level, the stress formability index value (for the smaller aspect ratio) is higher compared to that of larger aspect ratio. The above finding of higher stress formability index value for smaller aspect ratio is due to the closure of very fine pores observed during hot upset-
Fig. 4. The variation of the stress formability index (β) with respect to the axial strain (εz ).
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Table 3 Curve fitting results—axial strain (εz ) vs. formability stress index (β) Formability stress index (β) vs. axial strain (εz )
Table 4 Slope of various densification mechanisms obtained between the relative density (R) and the axial strain (εz )
Name of the curve
Aspect ratio
Fractional perform density
Polynomial equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
β = 4.098ε2z + 2.2199εz + 1.886 β = 2.4043ε2z + 2.3325εz + 1.9675 β = 0.3906ε2z + 2.9057εz + 1.8118
0.9872 0.9981 0.9821
ting. Among the different curve-fitting techniques applied, the polynomial curve of second order was found to be best fitted one for the plot between the axial strain and the formability stress index. As shown in Table 3, the constant for the ε2z term decreases with the increasing aspect ratio, whereas the constant term for the εz term increases with the increasing aspect ratio. Fig. 5(a)–(c) have been developed between the relative density and the axial strain for the hot upsetting of Fe–1.0% C–4% TiC powder preforms of the various aspect ratios under triaxial stress state condition. As the axial strain (εz ) increases, the relative density (R) also increases. For smaller aspect ratio, the relative density, R, is found to be higher for any given axial strain
Relationship
Aspect ratio
Slope
R vs. εz
0.45, 0.71 1.25
0.86, 0.305 and 0.158 0.7, 0.44 and 0.21
compared with the larger aspect ratio. The pore closure mechanism is faster in the case of lower aspect ratio and hence the relative density increases. Whereas, the pore closure mechanism is slow because of higher pores bed height in the larger aspect ratio and therefore densification is slow. It has been observed that three different densification mechanisms are operative during the deformation. During the initial stage of deformation, due to faster rate of pore closure, the rate of densification is faster. The rate of densification is medium for the second stage compared with the first stage. During the final stage of the deformation the rate of densification is further slowed down. However, during the third stage of densification an increase in the axial strain has been observed with no significant improvement in the relative density, as shown in Fig. 5(a). The slope of the different densification mechanisms involved is provided in Table 4. This Table 4 and Fig. 5(a) show that there are three different densification mechanisms are operative during hot upsetting. Two different curve fitting techniques, namely, exponential and polynomial curve fittings, were used, as shown in Fig. 5(b) and (c), and the respective equation determined and the correlation coefficient (R22 ) are shown in Table 5. Polynomial curve fitting with correlation coefficient value, which is very close to 1.0, is found to be fit for relating the above parameters, namely, the axial strain and the relative density. The constant for the εz term is found decreasing with the increase in the aspect ratio value. As shown in Fig. 6, which is plotted between the Poisson’s ratio and the relative density (R), it is observed that the Poisson’s ratio increases with increasing relative density (R). The slope of the straight-line relationship established in the case of each aspect ratio is provided in Table 6.
Table 5 Curve fitting results—relative density (R) vs. axial strain (εz )
Fig. 5. (a) The variation of the relative density (R) with respect to the axial strain (εz ); (b) the variation of the relative density (R) with respect to the axial strain (εz ) (exponential curve fitting); (c) the variation of the relative density (R) with respect to the axial strain (εz ) (polynomial curve fitting).
Relative density (R) vs. axial strain (εz )
Name of the curve
Aspect ratio
Fractional perform density
Exponential equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
R = 0.8501e0.1487 εz R = 0.8498e0.133 εz R = 0.8419e0.1283 εz
0.9241 0.9424 0.9393
Relative density (R) vs. axial strain (εz )
Name of the curve
Aspect ratio
Fractional perform density
Polynomial equation
0.45 0.71 1.25
0.8318 0.8317 0.8318
R = −0.0991ε2z + 0.2521εz + 0.8273 0.9974 R = −0.0855ε2z + 0.2245εz + 0.8315 0.9986 R = −0.069ε2z + 0.1959εz + 0.8266 0.9929
R22
R. Narayanasamy et al. / Materials Science and Engineering A 425 (2006) 121–130
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Fig. 6. The variation of the Poisson’s ratio (γ) with respect to the relative density (R).
Table 6 Curve fitting results—relative density (R) vs. Poisson’s ratio (γ) Relationship
Aspect ratio
Slope
R vs. γ
0.45 0.71 1.25
0.221 0.24 0.176
Table 7 Slope of various densification mechanisms obtained between the stress ratio (σ θ /σ eff ) and the relative density (R) Relationship
Aspect ratio
Slope
(σ θ /σ eff ) vs. R
0.45 0.71 1.25
0.267, 0.554, 1.106 and 2.355 0.267, 0.554 and 1.106 0.267 and 0.554
Table 8 Curve fitting results—stress ratio (σ θ /σ eff ) vs. relative density (R) Stress ratio (σ θ /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Power law equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ θ /σ eff = 2.7798R8.7256 σ θ /σ eff = 2.7223R8.4683 σ θ /σ eff = 2.3082R7.2666
0.9656 0.9735 0.9947
Stress ratio (σ θ /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Exponential equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ θ /σ eff = 0.0002e9.6817R σ θ /σ eff = 0.0002e9.3999R σ θ /σ eff = 0.0007e8.1621R
0.9732 0.9802 0.9962
Stress ratio (σ θ /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Polynomial equation
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ θ /σ eff = 102.93R2 − 173.74R + 73.953 0.9759 σ θ /σ eff = 91.973R2 − 154.39R + 65.432 0.9888 σ θ /σ eff = 43.58R2 − 69.267R + 28.085 0.9945
R22
Fig. 7. (a) The variation of the stress ratio (σ θ /σ eff ) with respect to the relative density (R); (b) the variation of the stress ratio (σ θ /σ eff ) with respect to the relative density (R) (power law curve fitting); (c) the variation of the stress ratio (σ θ /σ eff ) with respect to the relative density (R) (exponential curve fitting); (d) the variation of the stress ratio (σ θ /σ eff ) with respect to the relative density (R) (polynomial curve fitting).
Fig. 7(a)–(d) have been plotted between the various stress ratio parameter (σ θ /σ eff ), and the relative density (R) for the hot upsetting of Fe–1.0% C–4% TiC powder preforms of various aspect ratios under triaxial stress state condition. As the relative density increases, the stress ratio (σ θ /σ eff ) also increases, during the hot upsetting. During the hot upsetting, the hoop stress (σ θ ) keeps increasing because of more and more bulging. Fig. 7(a) shows that three different densification mechanisms are operative during hot upsetting because the slope values are different at different relative density levels. Table 7 shows the slope of the different straight line representing various stages of densification of the preforms during the deformation. Whereas in the case of lower aspect ratio, four different densification mechanisms take place during hot upsetting. Fig. 7(b)–(d) show different curve
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R. Narayanasamy et al. / Materials Science and Engineering A 425 (2006) 121–130 Table 9 Slope of various densification mechanisms obtained between the stress ratio (σ m /σ eff ) and the relative density (R) Relationship
Aspect ratio
Slope
(σ m /σ eff ) vs. R
0.45 0.71 1.25
0.267, 0.577, 1.0355 and 2.475 0.267, 0.577 and 1.0355 0.267 and 0.577
Fig. 8(a)–(d) have been plotted between the various stress ratio parameter (σ m /σ eff ), and the relative density (R) for the hot upsetting of Fe–1.0% C–4% TiC powder preforms for various aspect ratio under triaxial stress state condition. As the relative density increases, the stress ratio (σ m /σ eff ) also increases, during hot upsetting. During upsetting, the hydrostatic stress (σ m ) keeps increasing because of more and more bulging. Fig. 8(a) shows that three different densification mechanisms are operative during hot upsetting because the slope values are different at different relative density level. Whereas in the case of lower aspect ratio, four different densification mechanisms take place during hot upsetting. Table 9 shows the slope values of the different straight-line representing the various stages of densification of the preforms during the deformation. Fig. 8(b)–(d) show different curve fitting techniques employed between the above two parameters, namely, the stress ratio (σ m /σ eff ), and the relative density (R). The respective equation determined for each curve fitting techniques and their corresponding correlation coefficient (R22 ) are shown in Table 10. Polynomial curve fitting with good correlation coefficient value is found to fit for relating the above two parameters. It has been observed that from the equation obtained for the selected polynomial Table 10 Curve fitting results—stress ratio (σ m /σ eff ) vs. relative density (R)
Fig. 8. (a) The variation of the stress ratio (σ m /σ eff ) with respect to the relative density (R); (b) the variation of the stress ratio (σ m /σ eff ) with respect to the relative density (R) (power law curve fitting); (c) the variation of the stress ratio (σ m /σ eff ) with respect to the relative density (R) (exponential curve fitting); (d) the variation of the stress ratio (σ m /σ eff ) with respect to the relative density (R) (polynomial curve fitting).
fitting techniques employed between the above two parameters, namely, the stress ratio (σ θ /σ eff ), and the relative density (R). The respective equation determined for each curve fitting technique for each aspect ratio tested and their corresponding correlation coefficient (R22 ) are shown in Table 8. Polynomial curve fitting with good correlation coefficient value (very close to 1.0) is found to fit for relating the above two parameters. It has been observed that, from the equation obtained for the selected polynomial curve fitting, the constant term of the R2 term and the independent constant are decreasing with the increasing level of aspect ratio.
Stress ratio (σ m /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Power law equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ m /σ eff = 2.7554R8.4942 σ m /σ eff = 2.7095R8.3131 σ m /σ eff = 2.2806R7.0493
0.9612 0.9692 0.9934
Stress ratio (σ m /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Exponential equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ m /σ eff = 0.0002e9.4285R σ m /σ eff = 0.0003e9.230R σ m /σ eff = 0.0009e7.9218R
0.9696 0.9764 0.9959
Stress ratio (σ m /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Polynomial equation
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ m /σ eff = 103.51R2 − 174.97R + 74.589 0.9771 σ m /σ eff = 94.076R2 − 158.27R + 67.223 0.987 σ m /σ eff = 48.422R2 − 78.021R + 32.04 0.9979
R22
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Table 11 Slope of various densification mechanisms obtained between the stress ratio (σ z /σ eff ) and the relative density (R) Relationship
Aspect ratio
Slope
(σ z /σ eff ) vs. R
0.45 0.71 1.25
0.203, 0.6, 0.726 and 1.926 0.203, 0.6 and 0.726 0.203 and 0.6
Table 12 Curve fitting results—stress ratio (σ z /σ eff ) vs. relative density (R)
Fig. 9. (a) The variation of the stress ratio (σ z /σ eff ) with respect to the relative density (R); (b) the variation of the stress ratio (σ z /σ eff ) with respect to the relative density (R) (power law curve fitting); (c) the variation of the stress ratio (σ z /σ eff ) with respect to the relative density (R) (exponential curve fitting); (d) the variation of the stress ratio (σ z /σ eff ) with respect to the relative density (R) (polynomial curve fitting).
curve fitting, the constant term of the R2 term and the independent constant are decreasing with the increasing level of aspect ratio. Fig. 9(a)–(d) have been plotted between the various stress ratio parameter (σ z /σ eff ), and the relative density (R) for the hot upsetting of Fe–1.0% C–4% Tic powder preforms for various aspect ratios under triaxial stress state condition. As the relative density increases, the stress ratio (σ z /σ eff ) also increases. During upsetting, the axial stress (σ z ) keeps increasing because of more load required. Fig. 9(a) shows that three different densification mechanisms are operative during hot upsetting because
Stress ratio (σ z /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Power law equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ z /σ eff = 2.7105R8.059 σ z /σ eff = 2.6851R8.0151 σ z /σ eff = 2.2259R6.6291
0.9498 0.9579 0.9688
Stress ratio (σ z /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Exponential equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ z /σ eff = 0.0004e8.9522R σ z /σ eff = 0.0004e8.9043R σ z /σ eff = 0.0013e7.4568R
0.9595 0.9661 0.9731
Stress ratio (σ z /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Polynomial equation
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ z /σ eff = 104.69R2 − 177.42R + 75.861 0.9782 σ z /σ eff = 98.283R2 − 166.02R + 70.806 0.9814 σ z /σ eff = 58.106R2 − 95.528R + 39.95 0.9722
R22
the slope values are different at different relative density level. Whereas in the case of lower aspect ratio, four different densification mechanisms take place during hot upsetting. Table 11 shows the slope of the different straight-line representing the stages of densification of the preforms during the deformation. Fig. 9(b)–(d) show different curve fittings employed between the above two parameters, namely, the stress ratio (σ z /σ eff ), and the relative density (R), and the respective equation determined for each curve fitting and the corresponding correlation coefficient (R22 ) are shown in Table 12. Polynomial curve fitting with good correlation coefficient value (very close to 1.0) is found to fit for relating the above parameters. It has been observed that from the equation obtained for the selected polynomial curve fitting, the constant term of the R2 term and the independent constant decreases with the increasing level of aspect ratio. 5. Conclusion The following conclusion can be drawn from the present experimental work:
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(1) Four different densification mechanisms during hot upsetting of Fe–1.0% C–4% TiC have been observed in the case of smaller aspect ratio. (2) The first stage of densification mechanism has been observed up to a relative density value of 0.92 where the increase in stress ratio (σ θ /σ eff ) is very less. (3) Second stage of densification has been observed in a wider range of relative density value of 0.92–0.96 in which the stress ratio (σ θ /σ eff ) increases by a small amount. (4) The third stage of densification has been observed in a narrow range of relative density value of 0.96–0.98 with a large of increase in the stress ratio (σ θ /σ eff ) value. (5) The final and fourth stage of densification has been observed in a very narrow range of relative density value of 0.98–0.99 with a very large increase in the stress ratio (σ θ /σ eff ) value. (6) A relationship between the stress ratios, namely, (σ θ /σ eff ), (σ m /σ eff ) and (σ z /σ eff ), under triaxial stress state condition and Relative density (R) was established. (7) The stress formability index (β) was expressed theoretically as a function of axial strain (εz ), and empirical constants for the expressions were found experimentally. References [1] R. Raj, Metall. Trans. A 12 (19811089). [2] Y.V.R.K. Prasad, H.L. Gegel, S.M. Doraivelu, J.C. Malas, J.T. Morgan, K.A. Lark, D.R. Barkar, Metall. Trans. A 15 (1984) 1883. [3] J.S. Gunasekhara, CAD/CAM of Dies, Ellis Horwood, Chicehester, UK, 1989. [4] A.S. Wifi, N. El-Abbasi, A. Abdel Hamid, Materials Processing Defects, Elsevier, Amsterdam, 1995. [5] H.A. Kuhn, C.L. Downey, Int. J. Powder Metall. 7 (1971) 15–25. [6] S.M. Doraivelu, H.L. Gegel, J.S. Gunasekaran, J.C. Malas, J.T. Morugan, Int. J. Mech. Sci. 26 (1984) 527–535. [7] D.S. Lee, H.S. Kim, Int. J. Powder Metall. 35 (1992) 275–279. [8] S. Shima, M. Oyane, Int. J. Mech. Sci. 18 (1976) 285–291. [9] A.L. Gurson, J. Eng. Mater. Technol.: Trans. ASME (1977) 1–9. [10] H.-K Oh, J. Mater. Process Tech. 53 (1995) 582–587. [11] D.-C Ko, B.-M Kim, J.-C Choi, J. Mater. Process Tech. 62 (1996) 166–174.
[12] A.G. Atkins, J. Mater. Process Tech. 56 (1996) 609–618. [13] P. Brozo, B. Deluca, R. Rendina, A new method for the prediction of formability in metal sheet, sheet metal forming and formability, in: Proceedings of the 7th Biennial Conference of the IDDRG, Amsterdam, The Netherlands, 1972. [14] D.M. Norris, J.E. Reaugh, B. Moran, D.F. Quinones, J. Eng. Mater. Technol. 100 (1978) 279–286. [15] M. Oyane, T. Sato, K. Okimoto, S. Shima, J. Mech. Work Technol. 4 (1980) 65–81. [16] F.A. McClintoick, J. Appl. Mech. 35 (1968) 363–371. [17] J.R. Rice, D.M. Tracey, J. Mech. Phys. Solids 17 (1969) 201–217. [18] G. LeRoy, J.D. Embury, G. Edward, M.F. Ashby, Acta Metall. 29 (1981) 1509–1522. [19] J. Sun, Eng. Fracture Mech. 39 (1991) 799–805. [20] G.L. Rosa, G. Mirone, A. Risitano, Eng. Fracture Mech. 68 (2001) 417–434. [21] J. Sun, Z. Deng, M. Tu, Eng. Fracture Mech. 39 (1991) 1051–1060. [22] B.S. Henry, A.R. Luxmoore, Eng. Fracture Mech. 57 (1997) 375– 390. [23] H. Ma, Int. J. Fracture 89 (1998) 143–157. [24] A.G. Atkins, Y.W. Mai, Elastic and Plastic Fracture, Ellis Horwood, Chichester, 1988. [25] Y. Bao, T. Wierzbicki, J. Eng. Mater. Technol. 46 (2004) 81–98. [26] M. Abdel-Rahman, M.N. El-Sheikh, J. Mater. Process. Technol. 54 (1995) 97–102. [27] R. Narayanasamy, R. Ponalagusamy, J. Mater. Process. Technol. 97 (2000) 107. [28] B.P.P.A. Gouveia, J.M.C. Rodrigues, P.A.F. Martins, J. Mater. Process. Technol. 101 (2000) 52–63. [29] R. Narayanasamy, R. Ponalagusamy, K.R. Subramanian, J. Mater. Process. Technol. 110 (2001) 182–185. [30] D.B. Miracle, Compos. Sci. Technol. 65 (15–16) (2005) 21–32. [31] R. Narayanasamy, K.S. Pandey, J. Mater. Process. Technol. 72 (1997) 201–207. [32] R. Narayanasamy, K.S. Pandey, J. Mater. Process. Technol. 70 (1997) 17–20. [33] R. Narayansamy, K.S. Pandey, J. Mater. Process. Technol. 100 (2000) 84–94. [34] V. Vujovic, A.H. Shabalk, J. Eng. Mater. Technol. 108 (1986) 245. [35] R. Narayanasamy, T. Ramesh, K.S. Pandey, Met. Mater. Processes 17 (2005) 1–11. [36] R. Narayan, Ceramic Coatings, IIT, Kanpur, India, private communication, 2001.
Some aspects of workability studies on hot forging of sintered high strength 4% titanium carbide composite steel preforms R. Narayanasamy a,∗ , V. Senthilkumar b , K.S. Pandey c a
b
Department of Production Engineering, National Institute of Technology, Tiruchirappalli 620015, Tamil Nadu, India Department of Mechanical Engineering, Jayaram College of Engineering & Technology, Tiruchirappalli 621014, Tamil Nadu, India c Department of Metallurgical Engineering, National Institute of Technology, Tiruchirappalli 620015, Tamil Nadu, India Received 11 January 2006; accepted 13 March 2006
Abstract The aim of the paper is the study of workability of Fe–1.0% C–4% TiC steel composite during hot upsetting. Ductile fracture is the most common failure acting in bulk forming process. The formability during hot upsetting depends on the temperature, strain and strain rate. A complete experimental investigation of the workability behaviour of Fe–1.0% C–4% TiC steel composite was performed under the triaxial stress state condition. Hot upsetting of Fe–1.0% C–4% TiC steel composite preforms was carried out at a temperature of 1120 ◦ C and the formability behaviour of the same at triaxial stress state condition was determined. The curves plotted for different preforms were analysed and the relationship between the axial strain and the formability stress index were obtained. A relationship between the relative density and the axial strain was also established. A particular attempt was made to relate the various stress ratio parameters, namely, (σ θ /σ eff ), (σ m /σ eff ) and (σ z /σ eff ) with the relative density. © 2006 Elsevier B.V. All rights reserved. Keywords: Workability; Triaxial stress; Formability stress index; Relative density; Axial strain; Stress ratio
1. Introduction Workability refers to the relative ease with which a material can be shaped through plastic deformation. Several approaches have been proposed to understand the constitutive response of the materials during forming operations [1–3]. However, understanding the influence of the process related parameters such as friction and die geometry on the limits of deformation in metalworking has been a difficult problem all along. Ductile fracture is a major factor influencing the limit to workability in many forming operations. Many criteria have been developed to predict ductile fracture. Wifi et al. [4] have presented some of the criteria commonly used to predict ductile fracture. Kuhn and Downey [5] investigated the deformation characteristics and plasticity theory of sintered powder materials and studied the basic deformation behaviour of sintered iron powder performing a simple homogeneous compression tests and also proposed a plasticity theory relating yield stress and Pois-
∗
Corresponding author. Tel.: +91 4312501801; fax: +91 4312500133. E-mail addresses: [email protected] (R. Narayanasamy), [email protected] (V. Senthilkumar). 0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.03.035
son’s ratio to the density. Doraivelu et al. [6] proposed a new yield function for compressible P/M materials, conducting a test under uniaxial state of compressive stress using the P/M aluminium alloy as a model material. Further, yield surfaces for various density levels had been generated in three dimensional principal stress spaces. A new form of yield criterion for porous metals was proposed by Lee and Kim [7] conducting combined tension and torsion tests of sintered porous metals with high degree of accuracy and suggested zero yield stress only at zero relative density. Shima and Oyane [8] suggested a plasticity theory for porous metals and calculated the stress and strain for pore free copper utilizing the basic equations. The proposed equations were applied to frictionless closed die compression and the stress in the direction of compression was evaluated in relation to the relative density and further compared with the experimental results. Gurson [9] discussed the continuum theory of ductile rupture by void nucleation and growth considering yield criteria and flow rules for porous ductile materials by showing the role of hydrostatic stress in plastic yield and void growth. Various ductile fracture criterions were discussed by Oh [10] and proposed a new ductile fracture criterion considering the theory of fracture toughness under different mean stress by conducting a tensile test. Surface fracture is another mode for
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Nomenclature D0 Db Dc Dc1 Dc2 h0 hf R R1 R22
initial diameter of the preform bulged diameter contact diameter contact diameter of the preform (top surface) contact diameter of the preform (bottom surface) initial perform height deformed height of the preform relative density barrel radius correlation coefficient
Greek symbols εθ true Hoop strain εz true axial strain γ Poisson’s ratio ρ0 initial preform density ρf deformed preform density ρth theoretical preform density σ eff effective stress σm mean or hydrostatic stress σr true radial stress σz true axial stress σθ true Hoop stress
the initiation of the crack in the ductile fracture studies. Ko et al. [11] suggested a scheme to simultaneously accomplish both the prediction of surface fracture initiation and the analysis of deformation in the axisymmetric extrusion and simple upsetting of an aluminium alloy. Atkins [12] pointed out that the criteria for fracture initiation should dependent on hydrostatic stress. This conclusion has been independently arrived at by empirical routes [13–15] by porous plasticity or void growth mechanics modeling [16–19], by continuum damage mechanics [20], and by connecting initiation and propagation toughness mechanics [21–24]. Bao and Wierzbicki [25] have also observed that the mechanism of fracture is different depending on the amount of triaxiality. Abdel-Rahman and El-Sheikh [26] discussed workability criterion of powder metallurgy compacts and they investigated the effect of the relative density on the forming of P/M compacts in upsetting. They proposed a workability factor (β) for describing the effect of the mean stress and the effective stress with the help of the mean stress and the effective stress with the help of two theories and they discussed the effect of relative density. A mathematical equation for the determination of the flow stress in the case of the simple upsetting of P/M sintered performs was proposed by Narayanasamy and Ponalagusamy [27]. Further, they developed a new equation for the determination of hydrostatic stress in the case of simple upsetting of sintered P/M compacts. In addition, a new flow rule with anisotrophic parameter for porous metal was also proposed. Gouveia et al. [28] predicted the initiation of the ductile fracture after conducting experiments on powder compacts with various geometrical shapes such as ring, cylindrical, tapered and flanged under several different loading
conditions. Narayanasamy et al. [29] proposed a generalized yield criterion considering anisotrophic parameters for porous sintered powder metallurgy metals and introduced a new flow rule with anisotrophic parameters for porous metals. Further, they proposed new generalized yield criteria with five parameter constants for porous sintered powder metallurgy metals. A new approach for the next generation of metal matrix composite development has been dealt with as described elsewhere [30]. In this paper, a complete investigation on the workability criteria of Fe–1.0% C–4% TiC powder performs was made. Powder metallurgy preforms with various aspect ratios were discussed for studying the behavior for workability during hot upsetting under various triaxial stress state conditions. 2. Theoretical analysis The mathematical expressions used and proposed for the determination of various upsetting parameters of upsetting for various stress state conditions are discussed below. 2.1. Plane stress conditions According to Abdel-Rahman and El-Sheikh [26], the expression for the axial or height strain (εz ) can be written as follows: hf (1) εz = ln h0 where h0 is the initial height and hf is the deformed height of the preform. According to Narayanasamy and Pandey [31], the expression for the new hoop strain (εθ ) is as follows: 2Db2 + Dc2 (2) εθ = ln 3D02 where Db is the bulge diameter, D0 is the initial diameter and Dc is the average contact diameter of the preform. The average diameter of the preform is expressed as follows: Dc1 + Dc2 2 In the above expression for the average contact diameter of the preform, Dc1 is the contact diameter of the preform at top and Dc2 is the contact diameter of the preform at bottom. Further from the above reference [31], the state of stress in a plane stress condition, associated flow characteristics for porous materials can be expressed as follows: dεθ σθ − γσz (3) = dεz σz − γσθ Dc =
dεz is the plastic strain increment in axial direction, dεθ , the plastic strain increment in the hoop direction, σ z , the true axial stress, σ θ , the true hoop stress and γ is the Poisson’s ratio expressed as below. εθ γ= (4) 2εz
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Eq. (3) can be further simplified as follows: σθ α+γ = σz 1 + αγ
(5)
where α = dεθ /dεz According to Narayansamy and Pandey [32], the expression for the hydrostatic stress (σ m ) can be written as below: σm = 31 (σθ + σz )
(6)
Since σ r is considered to be zero in the case of plane stress condition the Eq. (6) can be written as follows: σm σθ 1 (7) 1+ = σz 3 σz According to Narayanasamy and Pandey [33], the relationship between the effective stress (σ eff ) and the axial stress (σ z ) can be written as follows: 0.5 σeff = (0.5 + α) 3(1 + α + α2 ) σz (8) where σ eff is the effective stress, α is the strain increment ratio (dεθ /dεz ) and σ z is the axial stress.The Eq. (8) can be rearranged as follows: σeff 0.5 = (0.5 + α)[3(1 + α + α2 ] (9) σz from the Eqs. (3) and (7), the expression for the stress ratio (σ eff /σ z ) is as given below. σeff 1 α+γ = (10) σz M 1 + γα where M = (0.5 + α)3(1 + α + α2 )0.5 . As an evidence of experimental investigation implying the importance of the spherical component of the stress state on fracture according to Vujovic and Shabalk [34] proposed a parameter called a formability stress index ‘β’ given by 3σm β= (11) σeff This index determines the fracture limit as explained in [35]. The stress formability index (β) can be expressed from Eqs. (5) and (9) as follows:
[σm /σz ] β=3 (12) 1/[σeff /σz ] 2.1.1. Triaxial stress state condition According to Narayansamy and Pandey [32], for the state of stress under triaxial stress state condition is given as below. α=
A , B
α=
dεθ dεz
(13)
where A = (2 + R2 )σ θ − R2 (σ z + 2σ θ ); 2 2 B = (2 + R )σ z − R (σ z + 2σ θ ), where α is the strain increment ratio and R is the relative density. From the above Eq. (13), for
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the known values of α, R and the axial stress, σ z , the hoop stress component, σ θ , can be determined as given below: 2α + R2 (14) σz σθ = 2 − R2 + 2R2 α Rearranging the above Eq. (14), the expression for the stress ratio (σ θ /σ z ) can be expressed as given below. 2α + R2 σθ = (15) σz 2 − R2 + 2R2 α In the above Eq. (15), relative density (R) plays a major role in finding the hoop stress component (σ θ ). It is assumed that σ r = σ θ for the case of axisymmetric condition. It is known that the expression for the hydrostatic stress (σ m ) can be written as follows: σm = 13 (σ1 + σ2 + σ3 ) or σm = 13 (σz + σr + σθ )
(16)
Since σ r = σ θ , the Eq. (16) can be written as follows: σm = 13 (σz + 2σθ )
(17)
The Eq. (17) can be rearranged as follows for the determination of the Stress ratio (σ m /σ z ) σm 1 2σθ (18) = 1+ σz 3 σz The effective stress can be determined from the following expression as explained elsewhere [33]. 2 σ12 + σ22 + σ32 − R2 (σ1 σ2 + σ2 σ3 + σ3 σ1 ) = (2R2 − 1)σeff
This expression can be written as follows in terms of cylindrical coordinates. 2 σeff =
σz2 + σθ2 + σr2 − R2 (σz σθ + σθ σr + σr σz ) 2R2 − 1
(19)
Since σ r = σ θ for cylindrical axisymmetric upsetting or forging operation, the Eq. (19) is as follows: 2 σeff =
σz2 + 2σθ2 − R2 (σz σθ + σθ2 + σθ σz ) 2R2 − 1
(20)
Eq. (20) can be rearranged as below for the determination of the stress ratio (σ eff /σ z ). 1/2 1 + 2(σθ /σz )2 − R2 (2(σθ /σz ) + (σθ /σz )2 ) σeff = (21) σz 2R2 − 1 The stress formability index (β) provided in the Eq. (11) can be derived for the triaxial stress state condition from the Eqs. (18) and (21) as follows:
[σm /σz ] β=3 (22) [1/[σeff /σz ]]
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Fig. 1. Flow chart for the determination of workability parameter (β) under triaxial stress state condition.
Different stress ratio parameters, namely, (σ θ /σ eff ), (σ m /σ eff ) and (σ z /σ eff ) are expressed as follows:
(σθ /σz ) σθ = (23) σeff (σeff /σz )
(σm /σz ) σm = (24) σeff (σeff /σz ) σz 1 = (25) σeff (σeff /σz )
in the steel. Iron powder and the Fe–1% C–4% TiC powders properties are shown in Table 1. The powders with different aspect ratios were then compacted using a 100 tonnes capacity Universal Testing Machine. The three different aspect ratios are shown in Table 2. The compacting pressure was controlled so as to obtain a density level of 7.612 ± 0.001 g/cc. The compacts were coated
The ratios, namely, (σ θ /σ z ), (σ m /σ z ) and (σ eff /σ z ) can be determined from the Eqs. (15), (18) and (21), respectively. Fig. 1 shows the flow chart for computing the stress formability index (β), expressed in the Eq. (22).
Sieve size (m)
Retained (wt.%)
+125 +106 +90 +63 +53 +37 −37
2.12 26.94 0.00 25.84 21.66 16.77 6.51
3. Experimental details Iron powder (−150 m diameter), extra fine Graphite powder and titanium carbide (−48 m diameter) were mixed to obtain the alloy of Fe–1% C–4% TiC via ball milling. The ball mill was operated for 30 h so as to get a homogenized mixture of alloys
Table 1 Characteristics of iron powder
(a) Flow rate: 24.5 s/50 g; (b) apparent density: 3.26 g/cc; (c) compressibility: 6.20 g/cc at a pressure of 4.1 tonf/cm2 (1 tonf/cm2 = 98.1 MPa). Characteristics of Fe–1.0% C–4 TiC powder: (a) apparent density: 3.438 g/cc; (b) compressibility: 6.346 g/cc at a pressure of 146.67 MPa.
R. Narayanasamy et al. / Materials Science and Engineering A 425 (2006) 121–130
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Table 2 Initial parameters of the Fe–1.0% C–4 TiC sintered P/M preforms Aspect ratio
Initial height (h0 ) (mm)
Initial diameter (D0 ) (mm)
Theoretical perform density (ρth ) (g/cc)
0.45 0.71 1.25
11.69 18.77 32.68
25.94 26.27 26.08
7.612 7.612 7.612
Fig. 3. The upset-forging test preform before and after deformation.
Fig. 2. SEM photograph of iron powder.
on all surfaces with an indigenously developed ceramic coating [36]. This coating was allowed to dry for a period of 6 h at normal atmospheric conditions. Recoating was given to the compacts in the direction 90◦ to that of the earlier coating. The second coating was allowed to dry in the same condition of the first coating for a further period of 12 h. These coatings were necessary to avoid oxidation of compacts during sintering. Ceramic-coated compacts were sintered in an electric muffle furnace at a temperature of 1120 ◦ C for a period of 60 min. After the sintering operation, the preforms were upset-forged at a temperature of 1120 ◦ C to the different levels of height strain using a 100 tonnes capacity Friction Screw Press. The forging operation was carried out with no lubricant because it is very
much easier and the cooling of billets or preforms would not take place. The density of forged preforms was determined using the Archimedes principle. After the above-mentioned forging schedule, the dimensions, namely, the height of forged specimen (hf ), the contact diameter (Dc1 and Dc2 ) of top and bottom surfaces, the bulged diameter (Db ) and the barrel radius (R1 ), were measured. Initial dimensions of the specimen (initial height h0 , initial diameter D0 ) and the initial perform density, ρ0 , were measured for each preform before conducting the experiment. The SEM photograph of the iron powder is shown in Fig. 2. The shape dimensions of initial and deformed preforms measured during the experiment as shown in Fig. 3. 4. Results and discussion Fig. 4 shows the variation of the formability stress index determined under the triaxial stress state with respect to the axial strain. For all the aspect ratio, the axial strain (εz ) increases as increasing value of the stress formability index (β). For the lower aspect ratio, the stress formability index takes a very high value at the fracture strain. For equal height strain level, the stress formability index value (for the smaller aspect ratio) is higher compared to that of larger aspect ratio. The above finding of higher stress formability index value for smaller aspect ratio is due to the closure of very fine pores observed during hot upset-
Fig. 4. The variation of the stress formability index (β) with respect to the axial strain (εz ).
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Table 3 Curve fitting results—axial strain (εz ) vs. formability stress index (β) Formability stress index (β) vs. axial strain (εz )
Table 4 Slope of various densification mechanisms obtained between the relative density (R) and the axial strain (εz )
Name of the curve
Aspect ratio
Fractional perform density
Polynomial equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
β = 4.098ε2z + 2.2199εz + 1.886 β = 2.4043ε2z + 2.3325εz + 1.9675 β = 0.3906ε2z + 2.9057εz + 1.8118
0.9872 0.9981 0.9821
ting. Among the different curve-fitting techniques applied, the polynomial curve of second order was found to be best fitted one for the plot between the axial strain and the formability stress index. As shown in Table 3, the constant for the ε2z term decreases with the increasing aspect ratio, whereas the constant term for the εz term increases with the increasing aspect ratio. Fig. 5(a)–(c) have been developed between the relative density and the axial strain for the hot upsetting of Fe–1.0% C–4% TiC powder preforms of the various aspect ratios under triaxial stress state condition. As the axial strain (εz ) increases, the relative density (R) also increases. For smaller aspect ratio, the relative density, R, is found to be higher for any given axial strain
Relationship
Aspect ratio
Slope
R vs. εz
0.45, 0.71 1.25
0.86, 0.305 and 0.158 0.7, 0.44 and 0.21
compared with the larger aspect ratio. The pore closure mechanism is faster in the case of lower aspect ratio and hence the relative density increases. Whereas, the pore closure mechanism is slow because of higher pores bed height in the larger aspect ratio and therefore densification is slow. It has been observed that three different densification mechanisms are operative during the deformation. During the initial stage of deformation, due to faster rate of pore closure, the rate of densification is faster. The rate of densification is medium for the second stage compared with the first stage. During the final stage of the deformation the rate of densification is further slowed down. However, during the third stage of densification an increase in the axial strain has been observed with no significant improvement in the relative density, as shown in Fig. 5(a). The slope of the different densification mechanisms involved is provided in Table 4. This Table 4 and Fig. 5(a) show that there are three different densification mechanisms are operative during hot upsetting. Two different curve fitting techniques, namely, exponential and polynomial curve fittings, were used, as shown in Fig. 5(b) and (c), and the respective equation determined and the correlation coefficient (R22 ) are shown in Table 5. Polynomial curve fitting with correlation coefficient value, which is very close to 1.0, is found to be fit for relating the above parameters, namely, the axial strain and the relative density. The constant for the εz term is found decreasing with the increase in the aspect ratio value. As shown in Fig. 6, which is plotted between the Poisson’s ratio and the relative density (R), it is observed that the Poisson’s ratio increases with increasing relative density (R). The slope of the straight-line relationship established in the case of each aspect ratio is provided in Table 6.
Table 5 Curve fitting results—relative density (R) vs. axial strain (εz )
Fig. 5. (a) The variation of the relative density (R) with respect to the axial strain (εz ); (b) the variation of the relative density (R) with respect to the axial strain (εz ) (exponential curve fitting); (c) the variation of the relative density (R) with respect to the axial strain (εz ) (polynomial curve fitting).
Relative density (R) vs. axial strain (εz )
Name of the curve
Aspect ratio
Fractional perform density
Exponential equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
R = 0.8501e0.1487 εz R = 0.8498e0.133 εz R = 0.8419e0.1283 εz
0.9241 0.9424 0.9393
Relative density (R) vs. axial strain (εz )
Name of the curve
Aspect ratio
Fractional perform density
Polynomial equation
0.45 0.71 1.25
0.8318 0.8317 0.8318
R = −0.0991ε2z + 0.2521εz + 0.8273 0.9974 R = −0.0855ε2z + 0.2245εz + 0.8315 0.9986 R = −0.069ε2z + 0.1959εz + 0.8266 0.9929
R22
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Fig. 6. The variation of the Poisson’s ratio (γ) with respect to the relative density (R).
Table 6 Curve fitting results—relative density (R) vs. Poisson’s ratio (γ) Relationship
Aspect ratio
Slope
R vs. γ
0.45 0.71 1.25
0.221 0.24 0.176
Table 7 Slope of various densification mechanisms obtained between the stress ratio (σ θ /σ eff ) and the relative density (R) Relationship
Aspect ratio
Slope
(σ θ /σ eff ) vs. R
0.45 0.71 1.25
0.267, 0.554, 1.106 and 2.355 0.267, 0.554 and 1.106 0.267 and 0.554
Table 8 Curve fitting results—stress ratio (σ θ /σ eff ) vs. relative density (R) Stress ratio (σ θ /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Power law equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ θ /σ eff = 2.7798R8.7256 σ θ /σ eff = 2.7223R8.4683 σ θ /σ eff = 2.3082R7.2666
0.9656 0.9735 0.9947
Stress ratio (σ θ /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Exponential equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ θ /σ eff = 0.0002e9.6817R σ θ /σ eff = 0.0002e9.3999R σ θ /σ eff = 0.0007e8.1621R
0.9732 0.9802 0.9962
Stress ratio (σ θ /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Polynomial equation
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ θ /σ eff = 102.93R2 − 173.74R + 73.953 0.9759 σ θ /σ eff = 91.973R2 − 154.39R + 65.432 0.9888 σ θ /σ eff = 43.58R2 − 69.267R + 28.085 0.9945
R22
Fig. 7. (a) The variation of the stress ratio (σ θ /σ eff ) with respect to the relative density (R); (b) the variation of the stress ratio (σ θ /σ eff ) with respect to the relative density (R) (power law curve fitting); (c) the variation of the stress ratio (σ θ /σ eff ) with respect to the relative density (R) (exponential curve fitting); (d) the variation of the stress ratio (σ θ /σ eff ) with respect to the relative density (R) (polynomial curve fitting).
Fig. 7(a)–(d) have been plotted between the various stress ratio parameter (σ θ /σ eff ), and the relative density (R) for the hot upsetting of Fe–1.0% C–4% TiC powder preforms of various aspect ratios under triaxial stress state condition. As the relative density increases, the stress ratio (σ θ /σ eff ) also increases, during the hot upsetting. During the hot upsetting, the hoop stress (σ θ ) keeps increasing because of more and more bulging. Fig. 7(a) shows that three different densification mechanisms are operative during hot upsetting because the slope values are different at different relative density levels. Table 7 shows the slope of the different straight line representing various stages of densification of the preforms during the deformation. Whereas in the case of lower aspect ratio, four different densification mechanisms take place during hot upsetting. Fig. 7(b)–(d) show different curve
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R. Narayanasamy et al. / Materials Science and Engineering A 425 (2006) 121–130 Table 9 Slope of various densification mechanisms obtained between the stress ratio (σ m /σ eff ) and the relative density (R) Relationship
Aspect ratio
Slope
(σ m /σ eff ) vs. R
0.45 0.71 1.25
0.267, 0.577, 1.0355 and 2.475 0.267, 0.577 and 1.0355 0.267 and 0.577
Fig. 8(a)–(d) have been plotted between the various stress ratio parameter (σ m /σ eff ), and the relative density (R) for the hot upsetting of Fe–1.0% C–4% TiC powder preforms for various aspect ratio under triaxial stress state condition. As the relative density increases, the stress ratio (σ m /σ eff ) also increases, during hot upsetting. During upsetting, the hydrostatic stress (σ m ) keeps increasing because of more and more bulging. Fig. 8(a) shows that three different densification mechanisms are operative during hot upsetting because the slope values are different at different relative density level. Whereas in the case of lower aspect ratio, four different densification mechanisms take place during hot upsetting. Table 9 shows the slope values of the different straight-line representing the various stages of densification of the preforms during the deformation. Fig. 8(b)–(d) show different curve fitting techniques employed between the above two parameters, namely, the stress ratio (σ m /σ eff ), and the relative density (R). The respective equation determined for each curve fitting techniques and their corresponding correlation coefficient (R22 ) are shown in Table 10. Polynomial curve fitting with good correlation coefficient value is found to fit for relating the above two parameters. It has been observed that from the equation obtained for the selected polynomial Table 10 Curve fitting results—stress ratio (σ m /σ eff ) vs. relative density (R)
Fig. 8. (a) The variation of the stress ratio (σ m /σ eff ) with respect to the relative density (R); (b) the variation of the stress ratio (σ m /σ eff ) with respect to the relative density (R) (power law curve fitting); (c) the variation of the stress ratio (σ m /σ eff ) with respect to the relative density (R) (exponential curve fitting); (d) the variation of the stress ratio (σ m /σ eff ) with respect to the relative density (R) (polynomial curve fitting).
fitting techniques employed between the above two parameters, namely, the stress ratio (σ θ /σ eff ), and the relative density (R). The respective equation determined for each curve fitting technique for each aspect ratio tested and their corresponding correlation coefficient (R22 ) are shown in Table 8. Polynomial curve fitting with good correlation coefficient value (very close to 1.0) is found to fit for relating the above two parameters. It has been observed that, from the equation obtained for the selected polynomial curve fitting, the constant term of the R2 term and the independent constant are decreasing with the increasing level of aspect ratio.
Stress ratio (σ m /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Power law equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ m /σ eff = 2.7554R8.4942 σ m /σ eff = 2.7095R8.3131 σ m /σ eff = 2.2806R7.0493
0.9612 0.9692 0.9934
Stress ratio (σ m /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Exponential equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ m /σ eff = 0.0002e9.4285R σ m /σ eff = 0.0003e9.230R σ m /σ eff = 0.0009e7.9218R
0.9696 0.9764 0.9959
Stress ratio (σ m /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Polynomial equation
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ m /σ eff = 103.51R2 − 174.97R + 74.589 0.9771 σ m /σ eff = 94.076R2 − 158.27R + 67.223 0.987 σ m /σ eff = 48.422R2 − 78.021R + 32.04 0.9979
R22
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Table 11 Slope of various densification mechanisms obtained between the stress ratio (σ z /σ eff ) and the relative density (R) Relationship
Aspect ratio
Slope
(σ z /σ eff ) vs. R
0.45 0.71 1.25
0.203, 0.6, 0.726 and 1.926 0.203, 0.6 and 0.726 0.203 and 0.6
Table 12 Curve fitting results—stress ratio (σ z /σ eff ) vs. relative density (R)
Fig. 9. (a) The variation of the stress ratio (σ z /σ eff ) with respect to the relative density (R); (b) the variation of the stress ratio (σ z /σ eff ) with respect to the relative density (R) (power law curve fitting); (c) the variation of the stress ratio (σ z /σ eff ) with respect to the relative density (R) (exponential curve fitting); (d) the variation of the stress ratio (σ z /σ eff ) with respect to the relative density (R) (polynomial curve fitting).
curve fitting, the constant term of the R2 term and the independent constant are decreasing with the increasing level of aspect ratio. Fig. 9(a)–(d) have been plotted between the various stress ratio parameter (σ z /σ eff ), and the relative density (R) for the hot upsetting of Fe–1.0% C–4% Tic powder preforms for various aspect ratios under triaxial stress state condition. As the relative density increases, the stress ratio (σ z /σ eff ) also increases. During upsetting, the axial stress (σ z ) keeps increasing because of more load required. Fig. 9(a) shows that three different densification mechanisms are operative during hot upsetting because
Stress ratio (σ z /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Power law equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ z /σ eff = 2.7105R8.059 σ z /σ eff = 2.6851R8.0151 σ z /σ eff = 2.2259R6.6291
0.9498 0.9579 0.9688
Stress ratio (σ z /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Exponential equation
R22
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ z /σ eff = 0.0004e8.9522R σ z /σ eff = 0.0004e8.9043R σ z /σ eff = 0.0013e7.4568R
0.9595 0.9661 0.9731
Stress ratio (σ z /σ eff ) vs. relative density (R)
Name of the curve
Aspect ratio
Fractional perform density
Polynomial equation
0.45 0.71 1.25
0.8318 0.8317 0.8318
σ z /σ eff = 104.69R2 − 177.42R + 75.861 0.9782 σ z /σ eff = 98.283R2 − 166.02R + 70.806 0.9814 σ z /σ eff = 58.106R2 − 95.528R + 39.95 0.9722
R22
the slope values are different at different relative density level. Whereas in the case of lower aspect ratio, four different densification mechanisms take place during hot upsetting. Table 11 shows the slope of the different straight-line representing the stages of densification of the preforms during the deformation. Fig. 9(b)–(d) show different curve fittings employed between the above two parameters, namely, the stress ratio (σ z /σ eff ), and the relative density (R), and the respective equation determined for each curve fitting and the corresponding correlation coefficient (R22 ) are shown in Table 12. Polynomial curve fitting with good correlation coefficient value (very close to 1.0) is found to fit for relating the above parameters. It has been observed that from the equation obtained for the selected polynomial curve fitting, the constant term of the R2 term and the independent constant decreases with the increasing level of aspect ratio. 5. Conclusion The following conclusion can be drawn from the present experimental work:
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(1) Four different densification mechanisms during hot upsetting of Fe–1.0% C–4% TiC have been observed in the case of smaller aspect ratio. (2) The first stage of densification mechanism has been observed up to a relative density value of 0.92 where the increase in stress ratio (σ θ /σ eff ) is very less. (3) Second stage of densification has been observed in a wider range of relative density value of 0.92–0.96 in which the stress ratio (σ θ /σ eff ) increases by a small amount. (4) The third stage of densification has been observed in a narrow range of relative density value of 0.96–0.98 with a large of increase in the stress ratio (σ θ /σ eff ) value. (5) The final and fourth stage of densification has been observed in a very narrow range of relative density value of 0.98–0.99 with a very large increase in the stress ratio (σ θ /σ eff ) value. (6) A relationship between the stress ratios, namely, (σ θ /σ eff ), (σ m /σ eff ) and (σ z /σ eff ), under triaxial stress state condition and Relative density (R) was established. (7) The stress formability index (β) was expressed theoretically as a function of axial strain (εz ), and empirical constants for the expressions were found experimentally. References [1] R. Raj, Metall. Trans. A 12 (19811089). [2] Y.V.R.K. Prasad, H.L. Gegel, S.M. Doraivelu, J.C. Malas, J.T. Morgan, K.A. Lark, D.R. Barkar, Metall. Trans. A 15 (1984) 1883. [3] J.S. Gunasekhara, CAD/CAM of Dies, Ellis Horwood, Chicehester, UK, 1989. [4] A.S. Wifi, N. El-Abbasi, A. Abdel Hamid, Materials Processing Defects, Elsevier, Amsterdam, 1995. [5] H.A. Kuhn, C.L. Downey, Int. J. Powder Metall. 7 (1971) 15–25. [6] S.M. Doraivelu, H.L. Gegel, J.S. Gunasekaran, J.C. Malas, J.T. Morugan, Int. J. Mech. Sci. 26 (1984) 527–535. [7] D.S. Lee, H.S. Kim, Int. J. Powder Metall. 35 (1992) 275–279. [8] S. Shima, M. Oyane, Int. J. Mech. Sci. 18 (1976) 285–291. [9] A.L. Gurson, J. Eng. Mater. Technol.: Trans. ASME (1977) 1–9. [10] H.-K Oh, J. Mater. Process Tech. 53 (1995) 582–587. [11] D.-C Ko, B.-M Kim, J.-C Choi, J. Mater. Process Tech. 62 (1996) 166–174.
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