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Effect of geometric work-hardening and matrix work-hardening on workability and densification of aluminium–3.5% alumina composite during cold upsetting
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Materials & Design
Materials and Design 29 (2008) 1582–1599 www.elsevier.com/locate/matdes
Effect of geometric work-hardening and matrix work-hardening on workability and densification of aluminium–3.5% alumina composite during cold upsetting R. Narayanasamy a,*, V. Anandakrishnan b, K.S. Pandey c a
Department of Production Engineering, National Institute of Technology, Tiruchirappalli 620 015, Tamilnadu, India b School of Mechanical Engineering, SASTRA University, Thanjavur 613 402, Tamilnadu, India c Department of Metallurgical Engineering, National Institute of Technology, Tiruchirappalli 620 015, Tamilnadu, India Received 31 July 2006; accepted 12 November 2007 Available online 8 December 2007
Abstract The densification is a measure of deformation in upset forming of Powder Metallurgy (P/M) processes. A complete experimental investigation on the deformation behaviour of aluminium–3.5% alumina powder composite has been discussed for the case of triaxial stress state condition. Cold upsetting of aluminium–3.5% alumina composite with and with no annealing having different initial preform relative density and with different aspect ratio was carried out and the densification behaviour of the preform under triaxial stress state condition was determined. A new true strain considering the effect of bulging was taken into account for the case of determination of the hoop strain. Plots made for different preforms were analyzed for the densification behaviour of preforms considering the effect of both geometrical work-hardening (GWH) and the matrix work-hardening (MWH). Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Densification; Geometric work-hardening; Matrix work-hardening; Workability; Formability stress index; Cold upsetting
1. Introduction Powder metallurgy (P/M) products are widely used from the automotive industry through to aerospace, ordnance, power tools, electronics, business machines, household appliances, garden equipment and much more. Moreover powder metallurgy has become widely recognized as a superior way of producing high-quality parts for a variety of important applications. This success is due to the advantages the process offers over other metal forming technologies such as forging and metal casting, advantages in material utilization, shape complexity, near-net shape dimensional control, among others. These, in turn, yield benefits in lower costs and greater versatility. The process *
Corresponding author. Tel.: +91 431 250 1801; fax: +91 431 250 0133. E-mail addresses: [email protected] (R. Narayanasamy), vanandakrishnan@rediffmail.com (V. Anandakrishnan), [email protected] (K.S. Pandey). 0261-3069/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2007.11.006
of producing powder metallurgy (P/M) components is basically a combination of two conventional processes, namely powder metallurgy and forging. The vast application of powder materials in various branches of industries stipulates the increase of interest to analysis of powder performs behaviour under the forming process. The P/M preform has a subsequent lateral flow of material during any upsetting operation as a result of the induced height strain. However, a spherical pore would undergo flattening and simultaneous elongation in the direction of the lateral flow, i.e. normal to the direction of loading. This leads to the situation where a relative motion between the opposite sides of the collapsed pore, due to the presence of shear stress, becomes feasible and the mechanical rupturing of the oxide film takes place. Virgin metal is now exposed for bond formation across the collapsing pore surfaces. It has been established [1–3] that the flow of material during upsetting and densification produces fibering of inclusions in the lateral direction.
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The strain-hardening exponent is the parameter which explains about the behaviour of strain-hardening of materials, which is the phenomenon that takes place at room temperature because of the plastic deformation of metals. The characteristics of stress strain curves such as workhardening exponent and stress ratio are expected to play a vital role in carrying out the plastic deformation of porous material. Narayanasamy and Pandey [4] investigated the work-hardening characteristics of sintered aluminium–iron composite preforms and have established that the initial geometry of the P/M preforms has played a predominant role in influencing both the strain-hardening exponent (n) and the strength coefficient (K). Inigoraj et al. [5] studied the strain-hardening behaviour of sintered aluminium–3.5% alumina composite preforms with and with no annealing have established an empirical relationship among the material parameters, namely, the strainhardening exponent (n), the strength coefficient (K) and the ratio of the initial preform densities to the theoretical density. Selvakumar et al. [6] investigated the strain-hardening behaviour of sintered aluminium preforms under uniaxial upsetting and determined the instantaneous strain-hardening exponent (ni) and the instantaneous strength coefficient (Ki) and concluded that the value of the parameters during the deformation of preforms with lower value of the preform density found to be the maximum. Narayanasamy et al. [7] has done an experimental study on the strain-hardening behaviour of aluminium–3.5% alumina composite material under three-dimensional stress conditions, namely, uniaxial, plane and triaxial. The instantaneous strain-hardening exponent (ni) and the instantaneous strength coefficient (Ki) were investigated for three different stress state conditions on the preforms with three different aspect ratios and two different initial preform densities. Porous P/M material during cold working not only experiences the usual strain hardening but also experiences ‘geometrical work-hardening’, due to a continued increase in density which leads to the enhancement in area of the cross-section. Thus, the total work-hardening in P/M preforms is due to densification as well as cold working of the base material surrounding the pores [8,9,6,10,11]. During the elastic deformation of fully dense material, Poisson’s ratio remains constant and is a property of the material, during the plastic deformation of conventional materials [12], this ratio being 0.5 for all materials that conform to volume constancy. However, in the plastic deformation of sintered P/M preforms, density changes occur resulting in Poisson’s ratio remaining less than 0.5 and only approaching to 0.5 in the near vicinity of the theoretical density. It has been well established [13] that increase in the volume of the voids decreases the relative critical pressure and vice-versa. However, the closing and opening of voids occur with tri-axial compression and tension respectively, with the absolute values of the relative critical pressure remaining the-same in each case. Whenever the relative critical pressure [14] is exceeded, a void of given
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geometry begins to open faster than it would close under tri-axial compression of the same magnitude. The factors determining [15] the geometry change of the pore are the pattern and the level of the plastic deformation. Thus, it is obvious that the beginning and the continuation of pore closure can be accomplished at a comparatively lower pressure when the material is subjected to plastic deformation. Workability is a term used to evaluate the capacity of a material to withstand the induced internal stresses of forming prior to the splitting of material occurs. It is a difficult technological concept that depends not only on the material but also the various process parameters such as stress, strain rate, temperature, friction, etc. The uniaxial state of compressive stress using aluminium alloy and the yield surfaces for the various density levels were generated in a three-dimensional principal stress space with the help of computer graphics. Workability criterion of P/M compacts was discussed by Abdel-Rahman et al. [16]. They investigated the effect of relative density on the forming limit of P/M compacts in upsetting. They also proposed a workability factor (b) for describing the effect of the mean stress and the effective stress with the help of two theories and the effect of relative density was discussed. A new yield function for compressible P/M materials was suggested by Doraivelu et al. [17], and it has been verified experimentally. Gouveia et al. [18] conducted an experimental and theoretical research work in which test samples of various geometry are used for the determination of critical damage at fracture under several loading conditions. The reduction in porosity during forging results in the decrease of preform volume. The yielding of porous materials thus does not follow the laws of volume constancy and the material parameters undergo a variation along with a change in porosity. A mathematical theory of plasticity for compressible powder metallurgy materials was developed by Narayansamy and Ponalagusamy [19]. Adoption of latest technology for evaluating the parameters of ductile fracture is a common one. Ko et al. [20] proposed a methodology for the preform design considering workability by Artificial Neural Networks (ANN) and Taguchi method and also using the numerical approximate solution tool called Finite Element Simulation for the prediction of ductile fracture. Wifi et al. [21] developed a Computer Aided workability evaluation system and coupled to an elasto-plastic large strain finite element package to check for ductile fracture in bulk formed work pieces using different workability criteria. Investigation on the appearance of fracture in the cold forming of brass during axisymmetric and non-axi-symmetric conditions was done by Sljapic et al. [22]. The fracture of cylindrical specimen with longitudinal surface notch compression test were experimentally studied by Petruska and Janicek [23] and it evaluates the ability of various ductile fracture criteria of different geometry and end conditions to predict the failure initiation. Bao [24] established a relationship between the stress triaxiality and equivalent strain to crack forma-
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tion. Further it was observed that the equivalent strain and the stress triaxiality were the important parameters which governed the crack formation. The secondary effects are induced primarily by the stress and strain ratio parameters. The present investigation deals with densification behaviour of aluminium–3.5% alumina powder composite with and with no annealing considering the effect of both geometrical work-hardening (GWH) and the matrix workhardening (MWH). The densification behaviour during cold upsetting under triaxial stress state condition has been evaluated and used for the discussion. 2. Experimental details Atomized aluminium powder of 180 lm size was obtained from M/s. The Metal Powder Company, Madurai, India. Upon analysis it was found to contain 0.37% of insoluble impurities. The production of alumina powder is explained elsewhere [25]. The characterization of aluminium and alumina powders was studied by determining the flow rate, the apparent density, tap density and the particle size distribution. Fig. 1 shows the SEM photograph of aluminium powder. This powder was partially oxidized at 300 ± 10 °C to contain nearly 3.5% alumina. The characteristic features of the aluminium and the aluminium– 3.5% alumina powders are given in Table 1. Compacts of 0.35, 0.56 and 0.72 initial aspect ratios as given in Table 2 were prepared from the aforesaid powder on a 0.60 MN hydraulic press, the compacting pressures being 93 ± 5, 135 ± 5 and 210 ± 10 MPa in order to obtain preforms of initial densities of 0.72, 0.80 and 0.90, respectively. Molybdenum disulphide was used as a lubricant during compaction. An indigenously developed ceramic coating [26] was applied on the free surfaces of the compacts and
dried under room-temperature conditions for a period of 9 h. A second coating was applied at 90° to the first coating and allowed to dry for a further period of 9 h under the same conditions as stated above. The ceramic-coated compacts were sintered in an electric muffle furnace in the range of 550 ± 10 °C for a period of 100 min and ultimately cooled to room temperature in the furnace itself. The sintered preforms were cleaned of the ceramic coating residues and measurements such as initial height, diameter and densities were carried out before and after each deformations, the latter being carried out between two flat dies heat treated to Rc 52-55 and tempered to Rc 46-48. The die surfaces were mirror polished. During the cold compression test, molybdenum disulphide lubricant was used at both of the die contact surfaces, which ensured the absolute minimum friction, resulting in quite homogeneous deformation. In general, each preform was subjected to compressive incremental loading in steps of 0.02 MN until fine cracks appeared on its free surface. After each interval of loading dimensional changes in the specimen such as height after deformation (hf), top contact diameter (DTC), bottom contact diameter (DBC), bulged diameter (DB) and density of the preform (qf) were measured following the procedures, described elsewhere [27]. The density of forged preforms was determined using the Archimedes principle. The above measurements were made before and after deformation as illustrated in Fig. 2. Experimental measurements were also used to calculate the true axial stresses and strains which provided the basis for the determination of instantaneous strain-hardening exponent or index value. In order to study some extent the effect of only the geometric work hardening and for the known amount of matrix work-hardening, one set of P/M preforms (after sintering) were annealed at 200 °C for a period of 30 minutes after each deformation during incremental loading. After annealing at each stage, the matrix work-hardening becomes almost negligible value compared to matrix work-hardening. 3. Theoretical analysis The present investigation is based on the analytical determination of the deformed density, strains, stresses, a new version of Poisson’s ratio, formability stress index and strain-hardening index. In the present investigation, the material under consideration is porous. As explained elsewhere [3], the expression for the bulging of a cylindrical preform of fully dense material can be written as follows, provided that the bulging contour assumes the form of a circular arc: p 2 p D h0 ¼ ð2D2b þ D2c Þhf ð1Þ 4 0 12
Fig. 1. The SEM photograph of aluminium powder.
This equation is based on the volume-constancy principle. However, the same is not the case for porous preform upsetting, instead the mass-constancy principle is employed. Applying the mass-constancy principle before and after deformation, Eq. (1) converts to
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Table 1 Characteristics of aluminium and alumina powder Characteristics
Aluminium
Alumina
Apparent density (g/cc) Theoretical density (qth) (g/cc)
1.0284 2.73
1.0028
Sieve analysis Sieve size (lm) Percent distribution (weight)
+150 3.60
+125 3.60
+106 2.50
+ 90 0.71
+ 75 8.30
+ 63 9.20
+ 53 16.70
+ 45 15.80
+ 38 3.62
38 35.71
Table 2 Characteristics of aluminium–3.5% alumina powder compacts Aspect ratio
Height (H0) mm
Diameter (D0) mm
Initial preform density %
Compressive load (MN)
0.35
9.50 9.75 9.80 16.20 15.40 15.00 20.00 20.30 19.65
27.80 27.75 27.75 27.80 27.75 27.75 27.80 27.75 27.75
90 80 72 90 80 72 90 80 72
0.135 0.080 0.055 0.135 0.080 0.055 0.135 0.080 0.055
0.56
H0
0.72
D0
qf ln qth
Fig. 2. Upset forging test preforms before and after deformation.
p 2 q p qf D0 h0 0 ¼ ð2D2b þ D2c Þhf 4 12 qth qth
ð2Þ
where D0 is the initial diameter; h0 is the initial height of the cylindrical preform; q0 is the initial preform density of the cylinder; Db is the bulged diameter; Dc is the contact diameter; hf is the height of the barrelled cylinder after deformation; qf is the density of the preform after deformation; and qth is the theoretical density of the fully dense material rearranged as below qf q h0 3D20 ¼ 0 qth qth hf 2D2b þ D2c
ð3Þ
Taking natural logarithms of both sides of Eq. (3), the following expression is obtained:
q0 ¼ ln qth
2 h0 2Db þ D2c þ ln ln hf 3D20
ð4Þ
However, Eq. (4) can be further simplifies to qf q0 ðez eh Þ ¼ ð5Þ e qth qth 2 2 2Db þDc where ez ¼ ln hh0f and eh ¼ ln 3D . 2 0 q0 Since the ratio qth is taken as constant, Eq. (5) shows an exponential relationship between the relative density of and the difference fractional theoretical density qqthf between the two true strains ez and eh. Now defining a new Poisson’s ratio (c1) based on the contact and bulge diameters as given below eh c1 ¼ ð6Þ 2ez enables Eq. (6) to be written us 2 2 2Db þDc ln 3D 2 1 0 c ¼ 2 lnðh0 =h2f Þ
ð7Þ
According to Narayanasamy et al. [28] the state of stress in a triaxial stress condition is given as follows: ð2 þ R2 Þrh R2 ðrz þ 2rh Þ a¼ ð8Þ ð2 þ R2 Þrz R2 ðrz þ 2rh Þ From the above Eq. (8), for the known values of Poisson’s ratio (a), relative density (R) and axial stress (rz) the Hoop stress component (rh) can be determined as given below
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rh ¼
2a þ R2 rz 2 R2 þ 2R2 a
ð9Þ
or
In the above Eq. (9), the relative density (R) plays a major role in finding the hoop stress component (rh). (For axisymmetric). The hydrostatic stress is given by rm ¼
ðr1 þ r2 þ r3 Þ ðrz þ rr þ rh Þ ¼ 3 3
ð10aÞ
The Eq. (10a) can be written as follows for axisymmetric condition: rm ¼
ðrz þ 2rh Þ 3
ðr21 þ r22 þ r23 R2 ðr1 r2 þ r2 r3 þ r3 r1 ÞÞ ¼ ð2R2 1Þr2eff
ð10bÞ
(Here, for axisymmetric case rr = rh.) The effective stress can be determined from the following expression as explained elsewhere [17]:
r2eff ¼
r2z þ r2h þ r2r R2 ðrz rh þ rh rr þ rr rz Þ ð2R2 1Þ
ð11Þ
Since rh = rr (for axisymmetric case) r2eff ¼
r2z þ 2r2h R2 ðrz rh þ r2h þ rz rh Þ ð2R2 1Þ
ð12Þ
The Eq. (12) can be rearranged as below for the determination of the effective stress, reff reff ¼
2 0:5 rz þ 2r2h R2 ðr2h þ 2rz rh Þ ð2R2 1Þ
Fig. 3a. Flow chart for the determination of workability parameter (b) under triaxial stress state condition.
ð13Þ
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As an evidence of experimental investigation implying the importance of the spherical component of the stress state on fracture, Vujovic and Shabaik [29] proposed a parameter called a formability stress index ‘b’ is given by 3rm b¼ ð14Þ reff This index determines the fracture limit as explained in the reference [30]. Fig. 3a shows the flow chart for computing the stress formability index (b), under triaxial stress state condition, expressed in the Eq. (14). The axial true strain (ez) is expressed as given below ez ¼ lnðh0 =hf Þ
ð15Þ
The hoop strain (eh) can be determined by the following expression as described elsewhere [25]:
eh ¼ ln
2 2Db þ D2c 3D20
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ð16Þ
Since the radial strain (er) = eh in case of triaxial stress state condition, the effective strain is given by 2 eeff ¼ ½ðez eh Þ2 þ ðeh ez Þ2 3ð2 þ RÞ ! )0:5 2 ðez þ 2eh Þ 2 þ ½1 R ð17Þ 3 The strain-hardening exponent value (n) was determined employing the conventional Ludwik equation [17] r ¼ Ken
ð18Þ
where r is the true effective stress, K is the strength coefficient, e is the true effective strain.
Fig. 3b. Flow chart for the determination of instantaneous strain-hardening exponent (ni) under triaxial stress state condition.
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It is assumed that the consecutive compressive loads were specified as 1, 2, 3, . . . , (m 1), m. Now Eq. (18) can be rewritten as: rðm1Þ ¼ K½eðm1Þ n rm ¼ K½em
n
and different aspect ratios namely 0.35, 0.56 and 0.72. These plots shows the variation of instantaneous strainhardening index value (ni) with respect to relative density (R) for both cases namely with and with no annealing. When the preforms are annealed after each loading or deformation the composite material softens and strain hardening due to matrix work-hardening is almost negligible or takes a constant value depending upon the level of porosity. Therefore, the behaviour of geometrical workhardening can be studied arbitrarily. Because both geometrical work-hardening and matrix work-hardening takes place during deformation. As shown in Fig. 4a (for initial preform relative density 0.9 and aspect ratio of 0.35) the instantaneous strain hardening index value (ni) decreases rapidly from higher value for both cases of with and with no annealing, due to the reason that load is taken for closing pores, So the strain softening takes place till the relative density (R) value becomes 0.95. For the case of annealed one the strain softening continues till the end of deformation due to the reason that the load is taken for closing more and more pores. Whereas in the case of with no annealing the matrix work-hardening increases with increase in deformation and therefore instantaneous strain-hardening index (ni) value keeps increases from the relative density (R) of 0.95. The instantaneous strain hardening index (ni) value reaches the peak value and then decreases due to the reason of flow softening towards the end of deformation. The reaching of peak value is due to the effect of both matrix and geometrical work-hardening. The variation of instantaneous strain-hardening index (ni) value with respect to relative density (R) for initial preform relative density of 0.9 and aspect ratio 0.56 is described in Fig. 4b. This figure clearly shows that the instantaneous strain-hardening index (ni) value decreases
ð19Þ ð20Þ
Now by dividing Eq. (20) by Eq. (19) the following expression can be obtained: n rm em ¼ ð21Þ rðm1Þ eðm1Þ Taking natural logarithms on both sides of Eq. (21) rm em ln ¼ n ln ð22Þ rðm1Þ eðm1Þ and the instantaneous strain-hardening exponent(ni) can be obtained as ni ¼
lnðrm =rðm1Þ Þ lnðem =eðm1Þ Þ
ð23Þ
Using Eq. (23), the instantaneous strain-hardening exponent (ni) can be determined. Further, it makes it feasible to evaluate the strain-hardening parameters for triaxial stress state conditions. Fig. 3b shows the flowchart for the determination of instantaneous strain-hardening index value. 4. Results and discussion 4.1. Work-hardening Figs. 4a–i have been plotted between the instantaneous strain hardening index (ni) value and relative density (R) for the initial preform relative density of 0.9, 0.8 and 0.72
1.0
Instantaneous Strain Hardening Index (ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.9 ASPECT RATIO 0.35
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.90
0.95
1.00
Relative Density (R)
Fig. 4a. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.9 for aspect ratio 0.35.
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1.0
Instantaneous Strain Hardening Index (ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.9 ASPECT RATIO 0.56
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.90
0.95
1.00
Relative Density (R)
Fig. 4b. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.9 for aspect ratio 0.56.
Instantaneous Strain Hardening Index (ni)
1.0
Without Annealing
INITIAL PREFORM RELATIVE DENSITY - 0.9 ASPECT RATIO 0.72
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.90
0.95
1.00
Relative Density (R)
Fig. 4c. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.9 for aspect ratio 0.72.
from higher value up to the relative density (R) of 0.915, then maintains almost a constant value and finally decreases for the case of with no annealing. This behaviour is slightly different form the previous one (Fig. 4a) particularly at the end of deformation. Because of more porous bed height, no matrix work-hardening takes place beyond the relative density (R) value of 0.95 and instead flow softening takes place for the case of with no annealing. Whereas for the case of annealing the instantaneous strain-hardening index (ni) value reaches a peak value when the relative density (R) value is 0.93 and this value keep
decreasing with increasing deformation due to more flow softening. This implies that load is considered for closing more and more pores and the geometrical work-hardening increases. Fig. 4c shows the variation of instantaneous strainhardening index (ni) with respect to relative density (R) for initial preform relative density 0.9 and aspect ratio 0.72 for both cases namely with and with no annealing. In the case of with no annealing the instantaneous strain-hardening index (ni) value keeps decreasing due to flow softening (more geometrical work-hardening),
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Instantaneous Strain Hardening Index (ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.8 ASPECT RATIO 0.35
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4d. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.8 for aspect ratio 0.35.
1.0
Instantaneous Strain Hardening Index (ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.8 ASPECT RATIO 0.56
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4e. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.8 for aspect ratio 0.56.
reaches the lowest value, then increases, reaches the peak value and then finally decreases towards the end of deformation. When the instantaneous strain-hardening index (ni) value reaches the peak value, more matrix work-hardening takes place together with less geometrical workhardening. Towards the end of deformation flow softening takes place and due to this reason the instantaneous strain-hardening index (ni) value decreases rapidly. For the case of annealed one the instantaneous strain-hardening index (ni) value slowly decreases, reaches the lowest value due to more flow softening or more geometrical
work-hardening, then keeps increasing, reaches peak values and then decreases rapidly towards the end of deformation. When the instantaneous strain-hardening index (ni) value reaches the peak value there is a small percentage increase in matrix work-hardening. Towards the end of deformation more flow softening takes place and due to this reason the instantaneous strain-hardening index (ni) value drops rapidly. Fig. 4d have been plotted between the instantaneous strain-hardening index (ni) value and the relative density (R) for initial preform relative density of 0.8 and aspect
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1.0
Instantaneous Strain Hardening Index (ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.8 ASPECT RATIO 0.72
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4f. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.8 for aspect ratio 0.72.
1.0
Instantaneous Strain Hardening Index(ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.72 ASPECT RATIO 0.35
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.75
0.80
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4g. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.72 for aspect ratio 0.35.
ratio 0.35 for both cases namely with and with no annealing. As shown in this figure the instantaneous strain-hardening index (ni) value decreases rapidly from higher value due to flow softening or more geometrical work-hardening, reaches the lowest value, then the value increases, reaching the peak value when the relative density (R) is in the order of 0.935 and then rapidly decreases due to flow softening at the end of deformation. Whereas for the case of annealed one the work-hardening index value is more or less a constant value and this implies that the load is considered only for closing pores.
The variation of instantaneous strain-hardening index (ni) value with respect to relative density (R) for initial preform relative density of 0.8 and aspect ratio 0.56 for both cases namely with and with no annealing is shown in Fig. 4e. For the case of annealed one the instantaneous strain-hardening index (ni) value is almost constant through out deformation and then finally increases at the end of deformation, due to the reason that there is a small percentage increase in the matrix work-hardening. Whereas in the case of with no annealing the instantaneous strainhardening index (ni) value maintains almost a constant
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Without Annealing
Instantaneous Strain Hardening Index (ni)
INITIAL PREFORM RELATIVE DENSITY - 0.72 ASPECT RATIO 0.56
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.75
0.80
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4h. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.72 for aspect ratio 0.56.
Instantaneous Strain Hardening Index (ni)
1.0
Without Annealing
INITIAL PREFORM RELATIVE DENSITY - 0.72 ASPECT RATIO 0.72
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.75
0.80
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4i. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.72 for aspect ratio 0.72.
value in the beginning, then increases reaching the peak value when the relative density (R) equals 0.925, and then decreases slowly due to flow softening towards the end of deformation. When instantaneous strain-hardening index (ni) value reaches the peak value the matrix work-hardening takes place to some extend though geometrical workhardening also takes place and is very much dominant. Fig. 4f shows the variation of instantaneous strain-hardening index (ni) value with respect to relative density (R) for initial preform relative density of 0.8 and aspect ratio 0.72 for both cases namely with and with no annealing. For the case of with no annealing the instantaneous strain-hardening index (ni) value maintains almost constant value and
decreases towards the end of deformation due to more flow softening. Whereas for the case of annealed one the instantaneous strain-hardening index (ni) value keeps decreasing, reaches the lowest value, then keep increasing reaching the peak value and then decreases due to the flow softening towards the end of deformation. When the instantaneous strain-hardening index (ni) value decreases, touching the lowest value implying that load is taken for only closing the pores and the geometrical work-hardening is more. As shown in Figs. 4g–i the instantaneous strain-hardening index (ni) value keeps increasing with increasing relative density (R) due to the reason that more amount of geometrical work-hardening takes place and very less amount of
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and hoop for various initial preform relative density with different aspect ratios for both cases namely with and with no annealing. The parameter eðez eh Þ can be determined as per the Eq. (5) provided in the theoretical discussion. Fig. 5a shows the variation of relative density (R) with respect to eðez eh Þ for the initial preform relative density of 0.9 with annealing. This clearly indicates that the rate of densification is very much faster for the case of lower aspect ratio (0.35) and the densification is slower for the case of higher aspect ratio (0.72). The curve fitted result shows that the slope value obtained between relative density (R) and the eðez eh Þ is
matrix work-hardening may be the reason. The nature of variation of the instantaneous strain hardening index (ni) value with respect to the relative density (R) is more or less same for both cases of with and with no annealing, due to the reason that only geometrical work-hardening takes place and matrix work-hardening is almost negligible. 4.2. Fractional theoretical density ratio Figs. 5a–f have been plotted between the fractional theoretical density ratio or relative density (R) with respect to the exponential of the difference in strains between axial 1.00
ρf / ρth
INITIAL PREFORM RELATIVE DENSITY - 0.9 WITH ANNEALING
0.95
SLOPE
0.90 1.00
1.05
Aspect Ratio 0.35
-
2.50
Aspect Ratio 0.56
-
2.23
Aspect Ratio 0.72
-
1.54
1.10
1.15
(ε z - ε θ)
e
Fig. 5a. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.9 with annealing.
1.00
ρf / ρth
INITIAL PREFORM RELATIVE DENSITY - 0.9 WITHOUT ANNEALING
0.95
SLOPE
0.90 1.00
1.05
Aspect Ratio 0.35
-
1.22
Aspect Ratio 0.56
-
1.00
Aspect Ratio 0.72
-
1.03
1.10
1.15
(εz - εθ)
e
Fig. 5b. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.9 without annealing.
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R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599 1.00
INITIAL PREFORM RELATIVE DENSITY - 0.8 WITH ANNEALING
ρf / ρth
0.95
0.90
SLOPE 0.85
0.80 1.00
1.05
1.10
Aspect Ratio 0.35
-
1.07
Aspect Ratio 0.56
-
0.83
Aspect Ratio 0.72
-
0.84
1.15
1.20
(εz - ε θ)
e
Fig. 5c. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.8 with annealing.
1.00
INITIAL PREFORM RELATIVE DENSITY - 0.8 WITHOUT ANNEALING
ρf / ρth
0.95
0.90
SLOPE
0.85
0.80 1.00
1.05
1.10
Aspect Ratio 0.35
-
0.84
Aspect Ratio 0.56
-
0.84
Aspect Ratio 0.72
-
0.84
1.15
1.20
(εz - ε θ)
e
Fig. 5d. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.8 without annealing.
greater for lower aspect ratio (0.35) and lower for the higher aspect ratio (0.72). This indicated that both matrix workhardening and geometrical work-hardening is very much dominant for lower aspect ratio (0.35). When the aspect ratio increased to 0.72 the geometrical work-hardening is very much dominant and matrix work-hardening is almost nil due to the presence of more pores bed height. Due to this reason the above slope value decreases with increasing aspect ratios, implying that the geometrical work-hardening is more and more dominant in the case of higher aspect ratios.
Fig. 5b have been plotted between the relative density (R) and eðez eh Þ for initial preform relative density 0.9 with no annealing. As observed in the previous case (with annealing) the rate of densification is faster for the lower aspect ratio (0.35) and the rate of densification is slower for the higher aspect ratio (0.72). The slope value for plots between the relative density (R) and eðez eh Þ decreases with increasing aspect ratios. The difference in the above slope value between lower aspect ratio and higher aspect ratio is not significant in the case of with no annealing when comparing with the case of annealing. Further it is noticed that the above slope value
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1.00
INITIAL PREFORM RELATIVE DENSITY - 0.72 WITH ANNEALING 0.96
0.92
ρf / ρth
0.88
0.84
0.80 SLOPE 0.76
0.72 1.00
1.05
Aspect Ratio 0.35
-
1.08
Aspect Ratio 0.56
-
1.09
Aspect Ratio 0.72
-
0.96
1.10
1.15
(εz - ε θ)
e
Fig. 5e. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.72 with annealing. 1.00
INITIAL PREFORM RELATIVE DENSITY - 0.72 WITHOUT ANNEALING SLOPE
0.96
Aspect Ratio 0.35
-
0.98
Aspect Ratio 0.56
-
0.80
Aspect Ratio 0.72
-
0.72
0.92
ρf / ρth
0.88
0.84
0.80
0.76
0.72 1.00
1.05
1.10
1.15
(ε z - εθ)
e
Fig. 5f. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.72 without annealing.
is in the order of 1.00 to 1.22 for the case of with no annealing and 1.54 to 2.50 for the case of with annealing. As described earlier the geometrical work-hardening is more dominant when the aspect ratio increases to higher value. When comparison is made between with and with no annealing, the densification rate is faster in the case of with annealing because the above slope value is greater for with annealing, which implies that more pores will be close when intermediate annealing is carried out during deformation. This means that more amount of geometrical work-hardening takes place in the case of with annealing when compared with the case of with no annealing.
Figs. 5c and d show the variation of the relative density (R) with respect to eðez eh Þ for initial preform relative density 0.8 for both cases namely with and with no annealing. As observed in the previous case the rate of densification is higher for lower aspect ratio when compared with higher aspect ratio. However, the rate of densification has no significant difference for lower and higher aspect ratios due to the reason that the above slope values for both cases namely with and with no annealing are almost same. The reason is due to the fact that the geometrical work-hardening is very much dominant and the matrix work-hardening is almost nil when the initial preform relative density
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As described in the figures the above slope value obtained for both cases namely with and with no annealing is almost same due to the reason that only geometrical work-hardening is dominant and no matrix work-hardening takes place.
decreases to lower value (0.8) for both cases namely with and with no annealing. There is no significant difference between the above slope values for both cases namely with and with no annealing. This may be attributed due to the reason that there is no matrix work-hardening and only geometrical work-hardening is very much dominant and prevails. Figs. 5e and f have been plotted between the relative density (R) and eðez eh Þ for initial preform relative density 0.72 for both cases namely with and with no annealing.
4.3. Forming limit diagram Figs. 6a–f have been plotted between the formability stress index (b) value determined under triaxial stress state condition and the relative density (R) for both cases namely
0.6
Aspect Ratio 0.35 INITIAL PREFORM RELATIVE DENSITY - 0.9
Aspect Ratio 0.56
WITH ANNEALING
0.5
Aspect Ratio 0.72
SLOPE = 1.74 0.4 SLOPE = 5.30
0.2
STAGE - 2
0.011
0.061
STAGE - 3
0.3
STAGE - 1
Formability Stress Index (β)
SLOPE = 4.33
0.1
0.0 0.88
0.90
0.92
0.010
0.94
0.96
0.98
1.00
Relative Density (R)
Fig. 6a. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.9 with annealing.
0.6
INITIAL PREFORM RELATIVE DENSITY - 0.9 WITHOUT ANNEALING
Aspect Ratio 0.35 Aspect Ratio 0.56
Formability Stress Index (β)
0.5
SLOPE = 4.28
Aspect Ratio 0.72 SLOPE = 1.95
0.4 SLOPE = 5.65 0.3
0.2
STAGE - 1
STAGE - 2
STAGE - 3
0.1 0.056
0.015 0.0 0.88
0.90
0.92
0.94
0.012
0.96
0.98
1.00
Relative Density (R)
Fig. 6b. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.9 without annealing.
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0.6
Aspect Ratio 0.35
INITIAL PREFORM RELATIVE DENSITY - 0.8
Aspect Ratio 0.72 Slope = 3.52
0.4
Slope = 1.70 0.3
STAGE - 3
Slope = 6.38 0.2
0.1
STAGE - 1
Formability Stress Index (β)
Aspect Ratio 0.56
WITH ANNEALING
0.5
STAGE - 2
0.115
0.013 0.0 0.80
0.011
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 6c. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.8 with annealing.
0.6
Aspect Ratio 0.35 0.5
INITIAL PREFORM RELATIVE DENSITY - 0.8 WITHOUT ANNEALING
Aspect Ratio 0.56 Aspect Ratio 0.72
0.4 SLOPE = 1.77
STAGE - 3
0.3
SLOPE = 3.71 0.2
0.1
STAGE - 2
STAGE - 1
Formability Stress Index (β)
SLOPE = 3.22
0.016 0.0 0.80
0.103
0.85
0.017
0.90
0.95
1.00
Relative Density (R)
Fig. 6d. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.8 without annealing.
with and with no annealing for preforms having different initial relative densities. As observed in Figs. 6a and b, the densification takes place in three different stages namely stage-1, stage-2 and stage-3. During stage-1 the formability stress index (b) value rapidly increases, maintains almost constant slope value during stage-2 and then increases towards the end of deformation during stage-3. The slope value are different for three different stages and stage-2 representing constant rate of pore closer, because the formability stress index (b) value steadily increases with increase in deforma-
tion. The slope depends on the nature of work-hardening (namely geometric and matrix work-hardening) taking place during deformation. The slope value observed for the stage-2 are different for with and with no annealing. The slope value is less for the case of with annealing, implying that more pores are being closed during deformation. The range of plasticity is wider for the case of annealed comparing with the case of with no annealing. For the case of with no annealing, the matrix work-hardening is dominant to some extend comparing with the geometric work-hardening.
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0.5
Aspect Ratio 0.35
INITIAL PREFORM RELATIVE DENSITY - 0.72 WITH ANNEALING
Aspect Ratio 0.56
0.4
Slope = 1.51
0.3
0.2 Slope = 2.78
0.1 0.020
NO STAGE - 3
STAGE - 2
STAGE - 1
Formability Stress Index (β)
Aspect Ratio 0.72
0.111
0.0 0.75
0.80
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 6e. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.72 with annealing.
0.6
INITIAL PREFORM RELATIVE DENSITY - 0.72 WITHOUT ANNEALING
Aspect Ratio 0.35 Aspect Ratio 0.56
0.5
0.4
0.3 Slope = 1.19 0.2
0.1
0.0 0.75
Slope = 2.76 STAGE - 1
Formability Stress Index (β)
Aspect Ratio 0.72
STAGE - 2
0.022
0.067
0.80
NO STAGE - 3
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 6f. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.72 without annealing.
Figs. 6c and d show the variation of formability stress index (b) value determined under triaxial stress state condition with respect to the relative density (R) for both cases with and with no annealing when for initial preform relative density is 0.8. As observed in the previous case the formability stress index (b) value shows three different densification regions namely stage-1, stage-2 and stage-3. The stage-2 represents constant slope between the formability stress index (b) value and the relative density (R) representing the plastic deformation behaviour of porous metals. The plasticity range for stage-2 increases to higher value
when the initial preform density decreases to lower value (0.8). Further it is observed that the slope values obtained for the above plots decrease, due to the fact that the geometrical work-hardening is very much dominant when the initial preform relative density decreases to lower value (0.8). When comparison is made between with and with no annealing, the plasticity range for state-2 increases with annealing case. The above slope value decreases for the case of with annealing compared with no annealing, implying that more and more geometrical work-hardening takes place and the matrix work-hardening is nil.
R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599
As shown in Figs. 6e and f the stage-3 disappears when the initial preform relative density decreases to lower value (0.72) for both cases namely with and with no annealing. It is further observed that the slope value of above plots decreases to lower value (in the order of 1.50), implying that only geometrical work-hardening takes place during deformation. When the initial preform relative density is decreasing to lower value (0.72) the formability stress index (b) value also decreases. The plasticity range for the stage-2 obtained is greater for with annealing compared with no annealing. 5. Conclusion The following conclusions can be drawn from the above results and discussion: The matrix work-hardening may be dominant when initial preform relative density is 0.9 and aspect ratio is lower (0.35). As the initial preform relative density decreases to lower value, the geometrical work-hardening is dominant and the matrix work-hardening is almost nil. The rate of densification depends on the amount of matrix work-hardening (if the preforms are annealed or not) particularly when initial preform relative density is 0.9. The rate of densification is lower when initial preform relative density is in the order of 0.72 and 0.8, due to the reason that only geometrical work-hardening is dominant. The plasticity range for stage-2 depends on the initial preform relative density and the nature of deformation (annealed or not) which depends on the nature of matrix and geometric work-hardening values.
References [1] Bockstiegel G. A study of the work of compaction in powder pressing. Powder Met Int 1971;3(1):108–12. [2] Narayanasamy R, Pandey KS. Some studies on barrelling of powder preforms during upsetting. Int J Powder Met Sci Technol 1990;1(2):10–23. [3] Narayanasamy R, Pandey KS. Some aspects of barrelling of porous solid cylinders under axial compression. Powder Met Sci Technol 1990;1(2):24–34. [4] Narayanasamy R, Pandey KS. Some aspects of work hardening in sintered aluminium–iron composite preforms during cold axial forming. J Mater Process Technol 1998;84(1–3):136–42. [5] Inigoraj AJR, Narayanasamy R, Pandey KS. Strain-hardening behaviour in sintered aluminium–3.5% alumina composite preforms during axial compression with and without annealing. J Mater Process Technol 1998;84(1–3):143–8. [6] Selvakumar N, Narayanasamy R. Phenomenon of strain hardening behaviour of sintered aluminium preforms during cold axial forming. J Mater Process Technol 2003;142(2):347–54.
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[7] Narayanasamy R, Ramesh T, Pandey KS. An experimental investigation on strain hardening behaviour of aluminium–3.5% alumina powder metallurgy composite preform under various stress states during cold upset forming. Mater Des 2007;28(4):1211–23. [8] Kuhn HA, Lawley A. Powder metallurgy processing. New York: Academic; 1978. p. 99–112. [9] Fischmeister HF, Aren B, Easterling KE. Deformation and densification of porous preform in hot forging. Powder Met 1971;14(27):144–63. [10] Hewitt RL, Wallace W, De Malherbe MC. Strain-hardening hollow sphere compaction model applied to aluminum. Int J Powder Metall Powder Technol 1974;10(2):131–4. [11] Kuhn KA, Downey CL. Material behaviour in powder preform forging. Trans ASME Eng Mater Technol 1973;95:41–6. [12] Samuel B. Source book on powder metallurgy. Hoeganeaes Corporation; 1979. p. 151–8. [13] Kahlow KJ. Void behaviour as influenced by pressure and plastic deformation, Institute for Metal Forming Report, Lehigh University, 1971: p. 10–6. [14] Gaiger HL, Lawley A. Changes during the densification of P/M preforms. Int J Powder Met Powder Technol 1974;10(1):21–31. [15] Ante HW, Deformation of Porous Materials, A Report, Hoeganaes Corporation, Riverton, NJ, p. 1–20. [16] Abdel-Rahman M, El-Sheikh MN. Workability in forging of powder metallurgy compacts. J Mater Process Technol 1995;54(1-4):97–102. [17] Doraivelu SM, Gegel HL, Gunasekera JS, Malas JC, Morgan JT, Thomas Jr JF. New yield function for compressible P/M materials. Int J Mech Sci 1984;26(9-10):527–35. [18] Gouveia BPPA, Rodrigues JMC, Martins PAF. Ductile fracture in metalworking: experimental and theoretical research. J Mater Process Technol 2000;101(1):52–63. [19] Narayanasamy R, Ponalagusamy R, Subramanian KR. Generalized yield criteria of porous sintered powder metallurgy metals. J Mater Process Technol 2001;110(2):182–5. [20] Dae-Cheol Ko, Byung-Min Kim, Jae-Chan Choi. Prediction of surface-fracture initiation in the axisymmetric extrusion and simple upsetting of an aluminum alloy. J Mater Process Technol 1996; 62(1–3):166–74. [21] Wifi AS, Abdel-Hamid A, El-Abbasi N. Computer-aided evaluation of workability in bulk forming processes. J Mater Process Technol 1998;300(3-4):285–93. [22] Sljapic V, Hartley P, Pillinger I. Observations on fracture in axisymmetric and three-dimensional cold upsetting of brass. J Mater Process Technol 2002;125–126:267–74. [23] Petruska J, Janicek L. Computationally-experimental workability determination of compressed cylindrical specimen with surface defect. J Mater Process Technol 1998;80–81:572–8. [24] Bao Yingbin. Dependence of ductile cracks formation in tensile tests on stress triaxiality stress and strain ratios. Eng Fract Mech 2005;72:505–22. [25] Narayansamy R, Pandey KS. A study on the barrelling of sintered iron preforms during hot upset forging. J Mater Process Technol 2000;100(1–3):87–94. [26] Pandey KS. Indigenous ceramic coating, Regional Engineering College, Tiruchirappalli-620 015, India, 1983. [27] Moyer KH. Measuring density of P/M materials with improved precision. Int J Powder Met Powder Technol 1979;15:33–42. [28] Narayanasamy R, Ramesh T, Pandey KS. Workability studies on cold upsetting of Al–Al2O3 composite material. Mater Des 2006;27(7):566–75. [29] Vujovic V, Shabaik AH. New workability criterion for ductile metals. J Eng Mater Technol Trans ASME 1986;108(3):245–9. [30] Narayanasamy R, Ramesh T, Pandey KS. Some aspects on workability of aluminium–iron powder metallurgy composite during cold upsetting. Mater Sci Eng A 2005;391(1–2):418–26.
Materials & Design
Materials and Design 29 (2008) 1582–1599 www.elsevier.com/locate/matdes
Effect of geometric work-hardening and matrix work-hardening on workability and densification of aluminium–3.5% alumina composite during cold upsetting R. Narayanasamy a,*, V. Anandakrishnan b, K.S. Pandey c a
Department of Production Engineering, National Institute of Technology, Tiruchirappalli 620 015, Tamilnadu, India b School of Mechanical Engineering, SASTRA University, Thanjavur 613 402, Tamilnadu, India c Department of Metallurgical Engineering, National Institute of Technology, Tiruchirappalli 620 015, Tamilnadu, India Received 31 July 2006; accepted 12 November 2007 Available online 8 December 2007
Abstract The densification is a measure of deformation in upset forming of Powder Metallurgy (P/M) processes. A complete experimental investigation on the deformation behaviour of aluminium–3.5% alumina powder composite has been discussed for the case of triaxial stress state condition. Cold upsetting of aluminium–3.5% alumina composite with and with no annealing having different initial preform relative density and with different aspect ratio was carried out and the densification behaviour of the preform under triaxial stress state condition was determined. A new true strain considering the effect of bulging was taken into account for the case of determination of the hoop strain. Plots made for different preforms were analyzed for the densification behaviour of preforms considering the effect of both geometrical work-hardening (GWH) and the matrix work-hardening (MWH). Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Densification; Geometric work-hardening; Matrix work-hardening; Workability; Formability stress index; Cold upsetting
1. Introduction Powder metallurgy (P/M) products are widely used from the automotive industry through to aerospace, ordnance, power tools, electronics, business machines, household appliances, garden equipment and much more. Moreover powder metallurgy has become widely recognized as a superior way of producing high-quality parts for a variety of important applications. This success is due to the advantages the process offers over other metal forming technologies such as forging and metal casting, advantages in material utilization, shape complexity, near-net shape dimensional control, among others. These, in turn, yield benefits in lower costs and greater versatility. The process *
Corresponding author. Tel.: +91 431 250 1801; fax: +91 431 250 0133. E-mail addresses: [email protected] (R. Narayanasamy), vanandakrishnan@rediffmail.com (V. Anandakrishnan), [email protected] (K.S. Pandey). 0261-3069/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2007.11.006
of producing powder metallurgy (P/M) components is basically a combination of two conventional processes, namely powder metallurgy and forging. The vast application of powder materials in various branches of industries stipulates the increase of interest to analysis of powder performs behaviour under the forming process. The P/M preform has a subsequent lateral flow of material during any upsetting operation as a result of the induced height strain. However, a spherical pore would undergo flattening and simultaneous elongation in the direction of the lateral flow, i.e. normal to the direction of loading. This leads to the situation where a relative motion between the opposite sides of the collapsed pore, due to the presence of shear stress, becomes feasible and the mechanical rupturing of the oxide film takes place. Virgin metal is now exposed for bond formation across the collapsing pore surfaces. It has been established [1–3] that the flow of material during upsetting and densification produces fibering of inclusions in the lateral direction.
R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599
The strain-hardening exponent is the parameter which explains about the behaviour of strain-hardening of materials, which is the phenomenon that takes place at room temperature because of the plastic deformation of metals. The characteristics of stress strain curves such as workhardening exponent and stress ratio are expected to play a vital role in carrying out the plastic deformation of porous material. Narayanasamy and Pandey [4] investigated the work-hardening characteristics of sintered aluminium–iron composite preforms and have established that the initial geometry of the P/M preforms has played a predominant role in influencing both the strain-hardening exponent (n) and the strength coefficient (K). Inigoraj et al. [5] studied the strain-hardening behaviour of sintered aluminium–3.5% alumina composite preforms with and with no annealing have established an empirical relationship among the material parameters, namely, the strainhardening exponent (n), the strength coefficient (K) and the ratio of the initial preform densities to the theoretical density. Selvakumar et al. [6] investigated the strain-hardening behaviour of sintered aluminium preforms under uniaxial upsetting and determined the instantaneous strain-hardening exponent (ni) and the instantaneous strength coefficient (Ki) and concluded that the value of the parameters during the deformation of preforms with lower value of the preform density found to be the maximum. Narayanasamy et al. [7] has done an experimental study on the strain-hardening behaviour of aluminium–3.5% alumina composite material under three-dimensional stress conditions, namely, uniaxial, plane and triaxial. The instantaneous strain-hardening exponent (ni) and the instantaneous strength coefficient (Ki) were investigated for three different stress state conditions on the preforms with three different aspect ratios and two different initial preform densities. Porous P/M material during cold working not only experiences the usual strain hardening but also experiences ‘geometrical work-hardening’, due to a continued increase in density which leads to the enhancement in area of the cross-section. Thus, the total work-hardening in P/M preforms is due to densification as well as cold working of the base material surrounding the pores [8,9,6,10,11]. During the elastic deformation of fully dense material, Poisson’s ratio remains constant and is a property of the material, during the plastic deformation of conventional materials [12], this ratio being 0.5 for all materials that conform to volume constancy. However, in the plastic deformation of sintered P/M preforms, density changes occur resulting in Poisson’s ratio remaining less than 0.5 and only approaching to 0.5 in the near vicinity of the theoretical density. It has been well established [13] that increase in the volume of the voids decreases the relative critical pressure and vice-versa. However, the closing and opening of voids occur with tri-axial compression and tension respectively, with the absolute values of the relative critical pressure remaining the-same in each case. Whenever the relative critical pressure [14] is exceeded, a void of given
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geometry begins to open faster than it would close under tri-axial compression of the same magnitude. The factors determining [15] the geometry change of the pore are the pattern and the level of the plastic deformation. Thus, it is obvious that the beginning and the continuation of pore closure can be accomplished at a comparatively lower pressure when the material is subjected to plastic deformation. Workability is a term used to evaluate the capacity of a material to withstand the induced internal stresses of forming prior to the splitting of material occurs. It is a difficult technological concept that depends not only on the material but also the various process parameters such as stress, strain rate, temperature, friction, etc. The uniaxial state of compressive stress using aluminium alloy and the yield surfaces for the various density levels were generated in a three-dimensional principal stress space with the help of computer graphics. Workability criterion of P/M compacts was discussed by Abdel-Rahman et al. [16]. They investigated the effect of relative density on the forming limit of P/M compacts in upsetting. They also proposed a workability factor (b) for describing the effect of the mean stress and the effective stress with the help of two theories and the effect of relative density was discussed. A new yield function for compressible P/M materials was suggested by Doraivelu et al. [17], and it has been verified experimentally. Gouveia et al. [18] conducted an experimental and theoretical research work in which test samples of various geometry are used for the determination of critical damage at fracture under several loading conditions. The reduction in porosity during forging results in the decrease of preform volume. The yielding of porous materials thus does not follow the laws of volume constancy and the material parameters undergo a variation along with a change in porosity. A mathematical theory of plasticity for compressible powder metallurgy materials was developed by Narayansamy and Ponalagusamy [19]. Adoption of latest technology for evaluating the parameters of ductile fracture is a common one. Ko et al. [20] proposed a methodology for the preform design considering workability by Artificial Neural Networks (ANN) and Taguchi method and also using the numerical approximate solution tool called Finite Element Simulation for the prediction of ductile fracture. Wifi et al. [21] developed a Computer Aided workability evaluation system and coupled to an elasto-plastic large strain finite element package to check for ductile fracture in bulk formed work pieces using different workability criteria. Investigation on the appearance of fracture in the cold forming of brass during axisymmetric and non-axi-symmetric conditions was done by Sljapic et al. [22]. The fracture of cylindrical specimen with longitudinal surface notch compression test were experimentally studied by Petruska and Janicek [23] and it evaluates the ability of various ductile fracture criteria of different geometry and end conditions to predict the failure initiation. Bao [24] established a relationship between the stress triaxiality and equivalent strain to crack forma-
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R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599
tion. Further it was observed that the equivalent strain and the stress triaxiality were the important parameters which governed the crack formation. The secondary effects are induced primarily by the stress and strain ratio parameters. The present investigation deals with densification behaviour of aluminium–3.5% alumina powder composite with and with no annealing considering the effect of both geometrical work-hardening (GWH) and the matrix workhardening (MWH). The densification behaviour during cold upsetting under triaxial stress state condition has been evaluated and used for the discussion. 2. Experimental details Atomized aluminium powder of 180 lm size was obtained from M/s. The Metal Powder Company, Madurai, India. Upon analysis it was found to contain 0.37% of insoluble impurities. The production of alumina powder is explained elsewhere [25]. The characterization of aluminium and alumina powders was studied by determining the flow rate, the apparent density, tap density and the particle size distribution. Fig. 1 shows the SEM photograph of aluminium powder. This powder was partially oxidized at 300 ± 10 °C to contain nearly 3.5% alumina. The characteristic features of the aluminium and the aluminium– 3.5% alumina powders are given in Table 1. Compacts of 0.35, 0.56 and 0.72 initial aspect ratios as given in Table 2 were prepared from the aforesaid powder on a 0.60 MN hydraulic press, the compacting pressures being 93 ± 5, 135 ± 5 and 210 ± 10 MPa in order to obtain preforms of initial densities of 0.72, 0.80 and 0.90, respectively. Molybdenum disulphide was used as a lubricant during compaction. An indigenously developed ceramic coating [26] was applied on the free surfaces of the compacts and
dried under room-temperature conditions for a period of 9 h. A second coating was applied at 90° to the first coating and allowed to dry for a further period of 9 h under the same conditions as stated above. The ceramic-coated compacts were sintered in an electric muffle furnace in the range of 550 ± 10 °C for a period of 100 min and ultimately cooled to room temperature in the furnace itself. The sintered preforms were cleaned of the ceramic coating residues and measurements such as initial height, diameter and densities were carried out before and after each deformations, the latter being carried out between two flat dies heat treated to Rc 52-55 and tempered to Rc 46-48. The die surfaces were mirror polished. During the cold compression test, molybdenum disulphide lubricant was used at both of the die contact surfaces, which ensured the absolute minimum friction, resulting in quite homogeneous deformation. In general, each preform was subjected to compressive incremental loading in steps of 0.02 MN until fine cracks appeared on its free surface. After each interval of loading dimensional changes in the specimen such as height after deformation (hf), top contact diameter (DTC), bottom contact diameter (DBC), bulged diameter (DB) and density of the preform (qf) were measured following the procedures, described elsewhere [27]. The density of forged preforms was determined using the Archimedes principle. The above measurements were made before and after deformation as illustrated in Fig. 2. Experimental measurements were also used to calculate the true axial stresses and strains which provided the basis for the determination of instantaneous strain-hardening exponent or index value. In order to study some extent the effect of only the geometric work hardening and for the known amount of matrix work-hardening, one set of P/M preforms (after sintering) were annealed at 200 °C for a period of 30 minutes after each deformation during incremental loading. After annealing at each stage, the matrix work-hardening becomes almost negligible value compared to matrix work-hardening. 3. Theoretical analysis The present investigation is based on the analytical determination of the deformed density, strains, stresses, a new version of Poisson’s ratio, formability stress index and strain-hardening index. In the present investigation, the material under consideration is porous. As explained elsewhere [3], the expression for the bulging of a cylindrical preform of fully dense material can be written as follows, provided that the bulging contour assumes the form of a circular arc: p 2 p D h0 ¼ ð2D2b þ D2c Þhf ð1Þ 4 0 12
Fig. 1. The SEM photograph of aluminium powder.
This equation is based on the volume-constancy principle. However, the same is not the case for porous preform upsetting, instead the mass-constancy principle is employed. Applying the mass-constancy principle before and after deformation, Eq. (1) converts to
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Table 1 Characteristics of aluminium and alumina powder Characteristics
Aluminium
Alumina
Apparent density (g/cc) Theoretical density (qth) (g/cc)
1.0284 2.73
1.0028
Sieve analysis Sieve size (lm) Percent distribution (weight)
+150 3.60
+125 3.60
+106 2.50
+ 90 0.71
+ 75 8.30
+ 63 9.20
+ 53 16.70
+ 45 15.80
+ 38 3.62
38 35.71
Table 2 Characteristics of aluminium–3.5% alumina powder compacts Aspect ratio
Height (H0) mm
Diameter (D0) mm
Initial preform density %
Compressive load (MN)
0.35
9.50 9.75 9.80 16.20 15.40 15.00 20.00 20.30 19.65
27.80 27.75 27.75 27.80 27.75 27.75 27.80 27.75 27.75
90 80 72 90 80 72 90 80 72
0.135 0.080 0.055 0.135 0.080 0.055 0.135 0.080 0.055
0.56
H0
0.72
D0
qf ln qth
Fig. 2. Upset forging test preforms before and after deformation.
p 2 q p qf D0 h0 0 ¼ ð2D2b þ D2c Þhf 4 12 qth qth
ð2Þ
where D0 is the initial diameter; h0 is the initial height of the cylindrical preform; q0 is the initial preform density of the cylinder; Db is the bulged diameter; Dc is the contact diameter; hf is the height of the barrelled cylinder after deformation; qf is the density of the preform after deformation; and qth is the theoretical density of the fully dense material rearranged as below qf q h0 3D20 ¼ 0 qth qth hf 2D2b þ D2c
ð3Þ
Taking natural logarithms of both sides of Eq. (3), the following expression is obtained:
q0 ¼ ln qth
2 h0 2Db þ D2c þ ln ln hf 3D20
ð4Þ
However, Eq. (4) can be further simplifies to qf q0 ðez eh Þ ¼ ð5Þ e qth qth 2 2 2Db þDc where ez ¼ ln hh0f and eh ¼ ln 3D . 2 0 q0 Since the ratio qth is taken as constant, Eq. (5) shows an exponential relationship between the relative density of and the difference fractional theoretical density qqthf between the two true strains ez and eh. Now defining a new Poisson’s ratio (c1) based on the contact and bulge diameters as given below eh c1 ¼ ð6Þ 2ez enables Eq. (6) to be written us 2 2 2Db þDc ln 3D 2 1 0 c ¼ 2 lnðh0 =h2f Þ
ð7Þ
According to Narayanasamy et al. [28] the state of stress in a triaxial stress condition is given as follows: ð2 þ R2 Þrh R2 ðrz þ 2rh Þ a¼ ð8Þ ð2 þ R2 Þrz R2 ðrz þ 2rh Þ From the above Eq. (8), for the known values of Poisson’s ratio (a), relative density (R) and axial stress (rz) the Hoop stress component (rh) can be determined as given below
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rh ¼
2a þ R2 rz 2 R2 þ 2R2 a
ð9Þ
or
In the above Eq. (9), the relative density (R) plays a major role in finding the hoop stress component (rh). (For axisymmetric). The hydrostatic stress is given by rm ¼
ðr1 þ r2 þ r3 Þ ðrz þ rr þ rh Þ ¼ 3 3
ð10aÞ
The Eq. (10a) can be written as follows for axisymmetric condition: rm ¼
ðrz þ 2rh Þ 3
ðr21 þ r22 þ r23 R2 ðr1 r2 þ r2 r3 þ r3 r1 ÞÞ ¼ ð2R2 1Þr2eff
ð10bÞ
(Here, for axisymmetric case rr = rh.) The effective stress can be determined from the following expression as explained elsewhere [17]:
r2eff ¼
r2z þ r2h þ r2r R2 ðrz rh þ rh rr þ rr rz Þ ð2R2 1Þ
ð11Þ
Since rh = rr (for axisymmetric case) r2eff ¼
r2z þ 2r2h R2 ðrz rh þ r2h þ rz rh Þ ð2R2 1Þ
ð12Þ
The Eq. (12) can be rearranged as below for the determination of the effective stress, reff reff ¼
2 0:5 rz þ 2r2h R2 ðr2h þ 2rz rh Þ ð2R2 1Þ
Fig. 3a. Flow chart for the determination of workability parameter (b) under triaxial stress state condition.
ð13Þ
R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599
As an evidence of experimental investigation implying the importance of the spherical component of the stress state on fracture, Vujovic and Shabaik [29] proposed a parameter called a formability stress index ‘b’ is given by 3rm b¼ ð14Þ reff This index determines the fracture limit as explained in the reference [30]. Fig. 3a shows the flow chart for computing the stress formability index (b), under triaxial stress state condition, expressed in the Eq. (14). The axial true strain (ez) is expressed as given below ez ¼ lnðh0 =hf Þ
ð15Þ
The hoop strain (eh) can be determined by the following expression as described elsewhere [25]:
eh ¼ ln
2 2Db þ D2c 3D20
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ð16Þ
Since the radial strain (er) = eh in case of triaxial stress state condition, the effective strain is given by 2 eeff ¼ ½ðez eh Þ2 þ ðeh ez Þ2 3ð2 þ RÞ ! )0:5 2 ðez þ 2eh Þ 2 þ ½1 R ð17Þ 3 The strain-hardening exponent value (n) was determined employing the conventional Ludwik equation [17] r ¼ Ken
ð18Þ
where r is the true effective stress, K is the strength coefficient, e is the true effective strain.
Fig. 3b. Flow chart for the determination of instantaneous strain-hardening exponent (ni) under triaxial stress state condition.
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It is assumed that the consecutive compressive loads were specified as 1, 2, 3, . . . , (m 1), m. Now Eq. (18) can be rewritten as: rðm1Þ ¼ K½eðm1Þ n rm ¼ K½em
n
and different aspect ratios namely 0.35, 0.56 and 0.72. These plots shows the variation of instantaneous strainhardening index value (ni) with respect to relative density (R) for both cases namely with and with no annealing. When the preforms are annealed after each loading or deformation the composite material softens and strain hardening due to matrix work-hardening is almost negligible or takes a constant value depending upon the level of porosity. Therefore, the behaviour of geometrical workhardening can be studied arbitrarily. Because both geometrical work-hardening and matrix work-hardening takes place during deformation. As shown in Fig. 4a (for initial preform relative density 0.9 and aspect ratio of 0.35) the instantaneous strain hardening index value (ni) decreases rapidly from higher value for both cases of with and with no annealing, due to the reason that load is taken for closing pores, So the strain softening takes place till the relative density (R) value becomes 0.95. For the case of annealed one the strain softening continues till the end of deformation due to the reason that the load is taken for closing more and more pores. Whereas in the case of with no annealing the matrix work-hardening increases with increase in deformation and therefore instantaneous strain-hardening index (ni) value keeps increases from the relative density (R) of 0.95. The instantaneous strain hardening index (ni) value reaches the peak value and then decreases due to the reason of flow softening towards the end of deformation. The reaching of peak value is due to the effect of both matrix and geometrical work-hardening. The variation of instantaneous strain-hardening index (ni) value with respect to relative density (R) for initial preform relative density of 0.9 and aspect ratio 0.56 is described in Fig. 4b. This figure clearly shows that the instantaneous strain-hardening index (ni) value decreases
ð19Þ ð20Þ
Now by dividing Eq. (20) by Eq. (19) the following expression can be obtained: n rm em ¼ ð21Þ rðm1Þ eðm1Þ Taking natural logarithms on both sides of Eq. (21) rm em ln ¼ n ln ð22Þ rðm1Þ eðm1Þ and the instantaneous strain-hardening exponent(ni) can be obtained as ni ¼
lnðrm =rðm1Þ Þ lnðem =eðm1Þ Þ
ð23Þ
Using Eq. (23), the instantaneous strain-hardening exponent (ni) can be determined. Further, it makes it feasible to evaluate the strain-hardening parameters for triaxial stress state conditions. Fig. 3b shows the flowchart for the determination of instantaneous strain-hardening index value. 4. Results and discussion 4.1. Work-hardening Figs. 4a–i have been plotted between the instantaneous strain hardening index (ni) value and relative density (R) for the initial preform relative density of 0.9, 0.8 and 0.72
1.0
Instantaneous Strain Hardening Index (ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.9 ASPECT RATIO 0.35
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.90
0.95
1.00
Relative Density (R)
Fig. 4a. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.9 for aspect ratio 0.35.
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1.0
Instantaneous Strain Hardening Index (ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.9 ASPECT RATIO 0.56
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.90
0.95
1.00
Relative Density (R)
Fig. 4b. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.9 for aspect ratio 0.56.
Instantaneous Strain Hardening Index (ni)
1.0
Without Annealing
INITIAL PREFORM RELATIVE DENSITY - 0.9 ASPECT RATIO 0.72
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.90
0.95
1.00
Relative Density (R)
Fig. 4c. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.9 for aspect ratio 0.72.
from higher value up to the relative density (R) of 0.915, then maintains almost a constant value and finally decreases for the case of with no annealing. This behaviour is slightly different form the previous one (Fig. 4a) particularly at the end of deformation. Because of more porous bed height, no matrix work-hardening takes place beyond the relative density (R) value of 0.95 and instead flow softening takes place for the case of with no annealing. Whereas for the case of annealing the instantaneous strain-hardening index (ni) value reaches a peak value when the relative density (R) value is 0.93 and this value keep
decreasing with increasing deformation due to more flow softening. This implies that load is considered for closing more and more pores and the geometrical work-hardening increases. Fig. 4c shows the variation of instantaneous strainhardening index (ni) with respect to relative density (R) for initial preform relative density 0.9 and aspect ratio 0.72 for both cases namely with and with no annealing. In the case of with no annealing the instantaneous strain-hardening index (ni) value keeps decreasing due to flow softening (more geometrical work-hardening),
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Instantaneous Strain Hardening Index (ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.8 ASPECT RATIO 0.35
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4d. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.8 for aspect ratio 0.35.
1.0
Instantaneous Strain Hardening Index (ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.8 ASPECT RATIO 0.56
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4e. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.8 for aspect ratio 0.56.
reaches the lowest value, then increases, reaches the peak value and then finally decreases towards the end of deformation. When the instantaneous strain-hardening index (ni) value reaches the peak value, more matrix work-hardening takes place together with less geometrical workhardening. Towards the end of deformation flow softening takes place and due to this reason the instantaneous strain-hardening index (ni) value decreases rapidly. For the case of annealed one the instantaneous strain-hardening index (ni) value slowly decreases, reaches the lowest value due to more flow softening or more geometrical
work-hardening, then keeps increasing, reaches peak values and then decreases rapidly towards the end of deformation. When the instantaneous strain-hardening index (ni) value reaches the peak value there is a small percentage increase in matrix work-hardening. Towards the end of deformation more flow softening takes place and due to this reason the instantaneous strain-hardening index (ni) value drops rapidly. Fig. 4d have been plotted between the instantaneous strain-hardening index (ni) value and the relative density (R) for initial preform relative density of 0.8 and aspect
R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599
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1.0
Instantaneous Strain Hardening Index (ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.8 ASPECT RATIO 0.72
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4f. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.8 for aspect ratio 0.72.
1.0
Instantaneous Strain Hardening Index(ni)
Without Annealing INITIAL PREFORM RELATIVE DENSITY - 0.72 ASPECT RATIO 0.35
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.75
0.80
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4g. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.72 for aspect ratio 0.35.
ratio 0.35 for both cases namely with and with no annealing. As shown in this figure the instantaneous strain-hardening index (ni) value decreases rapidly from higher value due to flow softening or more geometrical work-hardening, reaches the lowest value, then the value increases, reaching the peak value when the relative density (R) is in the order of 0.935 and then rapidly decreases due to flow softening at the end of deformation. Whereas for the case of annealed one the work-hardening index value is more or less a constant value and this implies that the load is considered only for closing pores.
The variation of instantaneous strain-hardening index (ni) value with respect to relative density (R) for initial preform relative density of 0.8 and aspect ratio 0.56 for both cases namely with and with no annealing is shown in Fig. 4e. For the case of annealed one the instantaneous strain-hardening index (ni) value is almost constant through out deformation and then finally increases at the end of deformation, due to the reason that there is a small percentage increase in the matrix work-hardening. Whereas in the case of with no annealing the instantaneous strainhardening index (ni) value maintains almost a constant
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Without Annealing
Instantaneous Strain Hardening Index (ni)
INITIAL PREFORM RELATIVE DENSITY - 0.72 ASPECT RATIO 0.56
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.75
0.80
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4h. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.72 for aspect ratio 0.56.
Instantaneous Strain Hardening Index (ni)
1.0
Without Annealing
INITIAL PREFORM RELATIVE DENSITY - 0.72 ASPECT RATIO 0.72
With Annealing
0.8
0.6
0.4
0.2
0.0
-0.2 0.75
0.80
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 4i. The variation of the instantaneous strain-hardening index (ni) with respect to relative density (R) for initial preform relative density 0.72 for aspect ratio 0.72.
value in the beginning, then increases reaching the peak value when the relative density (R) equals 0.925, and then decreases slowly due to flow softening towards the end of deformation. When instantaneous strain-hardening index (ni) value reaches the peak value the matrix work-hardening takes place to some extend though geometrical workhardening also takes place and is very much dominant. Fig. 4f shows the variation of instantaneous strain-hardening index (ni) value with respect to relative density (R) for initial preform relative density of 0.8 and aspect ratio 0.72 for both cases namely with and with no annealing. For the case of with no annealing the instantaneous strain-hardening index (ni) value maintains almost constant value and
decreases towards the end of deformation due to more flow softening. Whereas for the case of annealed one the instantaneous strain-hardening index (ni) value keeps decreasing, reaches the lowest value, then keep increasing reaching the peak value and then decreases due to the flow softening towards the end of deformation. When the instantaneous strain-hardening index (ni) value decreases, touching the lowest value implying that load is taken for only closing the pores and the geometrical work-hardening is more. As shown in Figs. 4g–i the instantaneous strain-hardening index (ni) value keeps increasing with increasing relative density (R) due to the reason that more amount of geometrical work-hardening takes place and very less amount of
R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599
1593
and hoop for various initial preform relative density with different aspect ratios for both cases namely with and with no annealing. The parameter eðez eh Þ can be determined as per the Eq. (5) provided in the theoretical discussion. Fig. 5a shows the variation of relative density (R) with respect to eðez eh Þ for the initial preform relative density of 0.9 with annealing. This clearly indicates that the rate of densification is very much faster for the case of lower aspect ratio (0.35) and the densification is slower for the case of higher aspect ratio (0.72). The curve fitted result shows that the slope value obtained between relative density (R) and the eðez eh Þ is
matrix work-hardening may be the reason. The nature of variation of the instantaneous strain hardening index (ni) value with respect to the relative density (R) is more or less same for both cases of with and with no annealing, due to the reason that only geometrical work-hardening takes place and matrix work-hardening is almost negligible. 4.2. Fractional theoretical density ratio Figs. 5a–f have been plotted between the fractional theoretical density ratio or relative density (R) with respect to the exponential of the difference in strains between axial 1.00
ρf / ρth
INITIAL PREFORM RELATIVE DENSITY - 0.9 WITH ANNEALING
0.95
SLOPE
0.90 1.00
1.05
Aspect Ratio 0.35
-
2.50
Aspect Ratio 0.56
-
2.23
Aspect Ratio 0.72
-
1.54
1.10
1.15
(ε z - ε θ)
e
Fig. 5a. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.9 with annealing.
1.00
ρf / ρth
INITIAL PREFORM RELATIVE DENSITY - 0.9 WITHOUT ANNEALING
0.95
SLOPE
0.90 1.00
1.05
Aspect Ratio 0.35
-
1.22
Aspect Ratio 0.56
-
1.00
Aspect Ratio 0.72
-
1.03
1.10
1.15
(εz - εθ)
e
Fig. 5b. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.9 without annealing.
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R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599 1.00
INITIAL PREFORM RELATIVE DENSITY - 0.8 WITH ANNEALING
ρf / ρth
0.95
0.90
SLOPE 0.85
0.80 1.00
1.05
1.10
Aspect Ratio 0.35
-
1.07
Aspect Ratio 0.56
-
0.83
Aspect Ratio 0.72
-
0.84
1.15
1.20
(εz - ε θ)
e
Fig. 5c. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.8 with annealing.
1.00
INITIAL PREFORM RELATIVE DENSITY - 0.8 WITHOUT ANNEALING
ρf / ρth
0.95
0.90
SLOPE
0.85
0.80 1.00
1.05
1.10
Aspect Ratio 0.35
-
0.84
Aspect Ratio 0.56
-
0.84
Aspect Ratio 0.72
-
0.84
1.15
1.20
(εz - ε θ)
e
Fig. 5d. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.8 without annealing.
greater for lower aspect ratio (0.35) and lower for the higher aspect ratio (0.72). This indicated that both matrix workhardening and geometrical work-hardening is very much dominant for lower aspect ratio (0.35). When the aspect ratio increased to 0.72 the geometrical work-hardening is very much dominant and matrix work-hardening is almost nil due to the presence of more pores bed height. Due to this reason the above slope value decreases with increasing aspect ratios, implying that the geometrical work-hardening is more and more dominant in the case of higher aspect ratios.
Fig. 5b have been plotted between the relative density (R) and eðez eh Þ for initial preform relative density 0.9 with no annealing. As observed in the previous case (with annealing) the rate of densification is faster for the lower aspect ratio (0.35) and the rate of densification is slower for the higher aspect ratio (0.72). The slope value for plots between the relative density (R) and eðez eh Þ decreases with increasing aspect ratios. The difference in the above slope value between lower aspect ratio and higher aspect ratio is not significant in the case of with no annealing when comparing with the case of annealing. Further it is noticed that the above slope value
R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599
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1.00
INITIAL PREFORM RELATIVE DENSITY - 0.72 WITH ANNEALING 0.96
0.92
ρf / ρth
0.88
0.84
0.80 SLOPE 0.76
0.72 1.00
1.05
Aspect Ratio 0.35
-
1.08
Aspect Ratio 0.56
-
1.09
Aspect Ratio 0.72
-
0.96
1.10
1.15
(εz - ε θ)
e
Fig. 5e. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.72 with annealing. 1.00
INITIAL PREFORM RELATIVE DENSITY - 0.72 WITHOUT ANNEALING SLOPE
0.96
Aspect Ratio 0.35
-
0.98
Aspect Ratio 0.56
-
0.80
Aspect Ratio 0.72
-
0.72
0.92
ρf / ρth
0.88
0.84
0.80
0.76
0.72 1.00
1.05
1.10
1.15
(ε z - εθ)
e
Fig. 5f. The variation of the fractional theoretical density ratio (qf/qth) with respect to eðez eh Þ for initial preform relative density 0.72 without annealing.
is in the order of 1.00 to 1.22 for the case of with no annealing and 1.54 to 2.50 for the case of with annealing. As described earlier the geometrical work-hardening is more dominant when the aspect ratio increases to higher value. When comparison is made between with and with no annealing, the densification rate is faster in the case of with annealing because the above slope value is greater for with annealing, which implies that more pores will be close when intermediate annealing is carried out during deformation. This means that more amount of geometrical work-hardening takes place in the case of with annealing when compared with the case of with no annealing.
Figs. 5c and d show the variation of the relative density (R) with respect to eðez eh Þ for initial preform relative density 0.8 for both cases namely with and with no annealing. As observed in the previous case the rate of densification is higher for lower aspect ratio when compared with higher aspect ratio. However, the rate of densification has no significant difference for lower and higher aspect ratios due to the reason that the above slope values for both cases namely with and with no annealing are almost same. The reason is due to the fact that the geometrical work-hardening is very much dominant and the matrix work-hardening is almost nil when the initial preform relative density
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As described in the figures the above slope value obtained for both cases namely with and with no annealing is almost same due to the reason that only geometrical work-hardening is dominant and no matrix work-hardening takes place.
decreases to lower value (0.8) for both cases namely with and with no annealing. There is no significant difference between the above slope values for both cases namely with and with no annealing. This may be attributed due to the reason that there is no matrix work-hardening and only geometrical work-hardening is very much dominant and prevails. Figs. 5e and f have been plotted between the relative density (R) and eðez eh Þ for initial preform relative density 0.72 for both cases namely with and with no annealing.
4.3. Forming limit diagram Figs. 6a–f have been plotted between the formability stress index (b) value determined under triaxial stress state condition and the relative density (R) for both cases namely
0.6
Aspect Ratio 0.35 INITIAL PREFORM RELATIVE DENSITY - 0.9
Aspect Ratio 0.56
WITH ANNEALING
0.5
Aspect Ratio 0.72
SLOPE = 1.74 0.4 SLOPE = 5.30
0.2
STAGE - 2
0.011
0.061
STAGE - 3
0.3
STAGE - 1
Formability Stress Index (β)
SLOPE = 4.33
0.1
0.0 0.88
0.90
0.92
0.010
0.94
0.96
0.98
1.00
Relative Density (R)
Fig. 6a. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.9 with annealing.
0.6
INITIAL PREFORM RELATIVE DENSITY - 0.9 WITHOUT ANNEALING
Aspect Ratio 0.35 Aspect Ratio 0.56
Formability Stress Index (β)
0.5
SLOPE = 4.28
Aspect Ratio 0.72 SLOPE = 1.95
0.4 SLOPE = 5.65 0.3
0.2
STAGE - 1
STAGE - 2
STAGE - 3
0.1 0.056
0.015 0.0 0.88
0.90
0.92
0.94
0.012
0.96
0.98
1.00
Relative Density (R)
Fig. 6b. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.9 without annealing.
R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599
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0.6
Aspect Ratio 0.35
INITIAL PREFORM RELATIVE DENSITY - 0.8
Aspect Ratio 0.72 Slope = 3.52
0.4
Slope = 1.70 0.3
STAGE - 3
Slope = 6.38 0.2
0.1
STAGE - 1
Formability Stress Index (β)
Aspect Ratio 0.56
WITH ANNEALING
0.5
STAGE - 2
0.115
0.013 0.0 0.80
0.011
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 6c. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.8 with annealing.
0.6
Aspect Ratio 0.35 0.5
INITIAL PREFORM RELATIVE DENSITY - 0.8 WITHOUT ANNEALING
Aspect Ratio 0.56 Aspect Ratio 0.72
0.4 SLOPE = 1.77
STAGE - 3
0.3
SLOPE = 3.71 0.2
0.1
STAGE - 2
STAGE - 1
Formability Stress Index (β)
SLOPE = 3.22
0.016 0.0 0.80
0.103
0.85
0.017
0.90
0.95
1.00
Relative Density (R)
Fig. 6d. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.8 without annealing.
with and with no annealing for preforms having different initial relative densities. As observed in Figs. 6a and b, the densification takes place in three different stages namely stage-1, stage-2 and stage-3. During stage-1 the formability stress index (b) value rapidly increases, maintains almost constant slope value during stage-2 and then increases towards the end of deformation during stage-3. The slope value are different for three different stages and stage-2 representing constant rate of pore closer, because the formability stress index (b) value steadily increases with increase in deforma-
tion. The slope depends on the nature of work-hardening (namely geometric and matrix work-hardening) taking place during deformation. The slope value observed for the stage-2 are different for with and with no annealing. The slope value is less for the case of with annealing, implying that more pores are being closed during deformation. The range of plasticity is wider for the case of annealed comparing with the case of with no annealing. For the case of with no annealing, the matrix work-hardening is dominant to some extend comparing with the geometric work-hardening.
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R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599 0.6
0.5
Aspect Ratio 0.35
INITIAL PREFORM RELATIVE DENSITY - 0.72 WITH ANNEALING
Aspect Ratio 0.56
0.4
Slope = 1.51
0.3
0.2 Slope = 2.78
0.1 0.020
NO STAGE - 3
STAGE - 2
STAGE - 1
Formability Stress Index (β)
Aspect Ratio 0.72
0.111
0.0 0.75
0.80
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 6e. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.72 with annealing.
0.6
INITIAL PREFORM RELATIVE DENSITY - 0.72 WITHOUT ANNEALING
Aspect Ratio 0.35 Aspect Ratio 0.56
0.5
0.4
0.3 Slope = 1.19 0.2
0.1
0.0 0.75
Slope = 2.76 STAGE - 1
Formability Stress Index (β)
Aspect Ratio 0.72
STAGE - 2
0.022
0.067
0.80
NO STAGE - 3
0.85
0.90
0.95
1.00
Relative Density (R)
Fig. 6f. The variation of the formability stress index (b) under triaxial stress state condition with respect to relative density (R) for initial preform relative density 0.72 without annealing.
Figs. 6c and d show the variation of formability stress index (b) value determined under triaxial stress state condition with respect to the relative density (R) for both cases with and with no annealing when for initial preform relative density is 0.8. As observed in the previous case the formability stress index (b) value shows three different densification regions namely stage-1, stage-2 and stage-3. The stage-2 represents constant slope between the formability stress index (b) value and the relative density (R) representing the plastic deformation behaviour of porous metals. The plasticity range for stage-2 increases to higher value
when the initial preform density decreases to lower value (0.8). Further it is observed that the slope values obtained for the above plots decrease, due to the fact that the geometrical work-hardening is very much dominant when the initial preform relative density decreases to lower value (0.8). When comparison is made between with and with no annealing, the plasticity range for state-2 increases with annealing case. The above slope value decreases for the case of with annealing compared with no annealing, implying that more and more geometrical work-hardening takes place and the matrix work-hardening is nil.
R. Narayanasamy et al. / Materials and Design 29 (2008) 1582–1599
As shown in Figs. 6e and f the stage-3 disappears when the initial preform relative density decreases to lower value (0.72) for both cases namely with and with no annealing. It is further observed that the slope value of above plots decreases to lower value (in the order of 1.50), implying that only geometrical work-hardening takes place during deformation. When the initial preform relative density is decreasing to lower value (0.72) the formability stress index (b) value also decreases. The plasticity range for the stage-2 obtained is greater for with annealing compared with no annealing. 5. Conclusion The following conclusions can be drawn from the above results and discussion: The matrix work-hardening may be dominant when initial preform relative density is 0.9 and aspect ratio is lower (0.35). As the initial preform relative density decreases to lower value, the geometrical work-hardening is dominant and the matrix work-hardening is almost nil. The rate of densification depends on the amount of matrix work-hardening (if the preforms are annealed or not) particularly when initial preform relative density is 0.9. The rate of densification is lower when initial preform relative density is in the order of 0.72 and 0.8, due to the reason that only geometrical work-hardening is dominant. The plasticity range for stage-2 depends on the initial preform relative density and the nature of deformation (annealed or not) which depends on the nature of matrix and geometric work-hardening values.
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