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Steady state creep and material parameters in a rotating disc of Al–SiCP composite
European Journal of Mechanics A/Solids 23 (2004) 335–344
Steady state creep and material parameters in a rotating disc of Al–SiCP composite V.K. Gupta a,∗ , S.B. Singh b , H.N. Chandrawat a , S. Ray c a Department of Mechanical Engineering, T.I.E.T., Patiala-147004, India b School of Basic & Applied Sciences, T.I.E.T., Patiala-147004, India c Department of Metallurgical and Materials Engineering, I.I.T., Roorkee-247667, India
Received 1 April 2003; accepted 26 November 2003
Abstract The steady state creep in a rotating disc made of isotropic aluminium–silicon carbide particulate composite has been investigated in the present study. The creep behaviour of the composite has been described by Sherby’s constitutive model. The creep parameters in the law have been determined using regression equations developed on the basis of available experimental results. The radial and tangential stresses and steady state creep rates in the disc have been calculated and presented for various combinations of material parameters (like particle size and particle content) and temperatures. The study revealed that for given operating conditions, the strain rates in the disc can be controlled by selecting optimum particle content and/or particle size of the reinforcement. 2003 Elsevier SAS. All rights reserved. Keywords: Creep; Particulate composite; Rotating disc
1. Introduction A rotating disc at elevated temperature is a common component in aero-engines, automobiles, turbines, pumps, compressors and in a number of other dynamic applications (Singh and Ray, 2002). A reduced weight of such components, resulting from the use of aluminium/aluminium base alloys, is expected to save power and fuel due to a reduction in the payload. However, the enhanced creep of aluminium and its alloys may be a big hindrance in such applications. Particle- and whisker-reinforced metal matrix composites have shown superior high temperature properties and are finding increasing application in components exposed to higher temperature (Singh and Ray, 2001). Earlier study (Nieh, 1984) reveals that aluminium matrix based composite has better creep resistance compared to the base aluminium alloy. The study pertaining to steady state creep behaviour of Al–SiCP composites under uniaxial condition (Pandey et al., 1992) in the temperature range between 623–723 K for different particle sizes and with varying volume fraction of reinforcement indicated that the composite with finer particle size has better creep resistance than that containing coarser ones. Wahl et al. (1954) theoretically studied steady state creep deformation in a rotating turbine disc made of 12-pct chromium steel using von Mises and Tresca yield criteria and validated his results experimentally. They described creep behaviour by a power function relation of the form ε˙ = kσ n f (t). Arya and Bhatnagar (1979) investigated the effect of orthotropicity in a rotating disc made of wood on stress and strain distributions using Hill yield criterion. Singh et al. (2002) analysed steady * Corresponding author.
E-mail address: [email protected] (V.K. Gupta). 0997-7538/$ – see front matter 2003 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2003.11.005
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Nomenclature a A b |br | DL E f (t) k M n P r t T V Vavg ε˙ ε˙¯ ε˙ r ε˙ θ
inner radius of disc (= 31.75 mm) constant outer radius of disc (= 152.4 mm) magnitude of burgers vector lattice diffusivity Young’s modulus function of time constant creep parameter, (1/E)(ADL λ3 /|br |5 )1/8 constant size of dispersoid, SiCP (µm) radius of disc (a r b) thickness of the disc (= 5 mm) temperature (K) dispersoid (SiCP ) content (vol.%) average particle content in disc (vol.%) strain rate (s−1 ) effective creep rate under multiaxial stress (s−1 ) strain rate (compressive) along radial (r) direction (s−1 ), (¯ε/(2σ¯ ))(2σr − σθ ) strain rate along tangential (θ) direction (s−1 ), (ε˙¯ /(2σ¯ ))(2σθ − σr )
ε˙ z ε˙ rmax ε˙ θmax λ ρ σ σ¯
σr , σθ σθavg σ0 u˙ r u˙ a ω
strain rate along axial (z) direction, (ε˙¯ /(2σ¯ ))(−σr − σθ ) radial strain rate (compressive) at inner radius (s−1 ) tangential strain rate at inner radius (s−1 ) subgrain size density of the composite (kg/m3 ) stress (MPa) effective stress under biaxial state of stress (MPa), = √1 [σθ2 + σr2 + (σr − σθ )2 ]1/2 2 stresses along radial and tangential directions (MPa) average tangential stress over cross section of disc (MPa) threshold stress (MPa) radial strain rate at any radius ‘r’ of disc (s−1 ) rdial strain rate at inner radius ‘a’ of the disc (s−1 ) angular velocity of disc rotating at 15 000 rpm (radian/sec)
state creep using Norton’s creep model in a rotating disc made of aluminium–silicon carbide whiskers using Hill yield criterion. However, it has been observed that application of Norton’s creep model to composite results in very high value of stress exponent as compared to Sherby’s model (Mishra and Pandey, 1990). The survey also revealed that, the effect of size and content of the reinforcement and operating temperature on the steady state creep behaviour of the rotating disc made of composite has not been investigated. In the present study, the analysis of steady state creep in a rotating disc made of isotropic composites containing SiC particles in an aluminium matrix has been carried out. Sherby’s constitutive creep model (Sherby et al., 1977), which is claimed to work better than Norton’s creep model (Mishra and Pandey, 1990) has been used to describe the creep behaviour of the composite. The radial and tangential stress distributions have been determined and the steady state creep rates have been calculated for various combinations of material parameters (like particle size and particle content) and temperatures.
2. Mathematical formulation of creep behaviour A particle reinforced composite disc (Al–SiCP ) of constant thickness with inner and outer radii of a and b respectively, is considered rotating with angular velocity ω (radian/sec). For the purpose of analysis the following assumptions are made: (i) (ii) (iii) (iv)
Steady state condition of stress is assumed. Elastic deformations are small and are neglected as compared to creep deformations. Biaxial state of stress (i.e., stress along axial (z) direction, σz = 0) exist at any point in the disc. The composite shows a steady state creep behaviour, which may be described by following Sherby’s constitutive model of the form (Sherby et al., 1977), ADL λ3 σ¯ − σ0 8 ε˙¯ = E |br |5
or alternatively 8 ε˙¯ = M(σ¯ − σ0 ) .
(1)
(2)
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337
Table 1 Values of material parameters from experimental study Particle size P (µm)
Temperature T (K)
Particle content V (vol.%)
M (s−1/8 /MPa)
σ0 (MPa)
1.7 14.5 45.9
623
10
0.00963 0.01444 0.01897
15.24 11.46 13.65
1.7
623
10 20 30
0.00963 0.00594 0.00518
15.24 24.83 34.32
1.7
623 673 723
20
0.00594 0.00897 0.01295
24.83 24.74 25.72
In a particle-reinforced composite, the values of creep parameters M and σ0 depend on the size and the percentage of dispersed particles (SiCP ) and the temperature of application. For the present work, the values of M and σ0 have been extracted from the creep results experimentally obtained for Al–SiCP composites under uniaxial loading (Pandey et al., 1992) and are shown in Table 1. The following regression equations describe the values of M and σ0 as functions of T , P and V . ln(M) = −35.38 + 0.2077 ln(P ) + 4.98 ln(T ) − 0.622 ln(V ),
(3)
σ0 = −0.03507P + 0.01057T + 1.00536V − 2.11916.
(4)
Taking the reference frame along the axial directions r, θ and z of the disc, the relations for ε˙ r , ε˙ θ and ε˙ z for biaxial state of stress are obtained as (Singh and Ray, 2001), 8 du˙ r (2x(r) − 1) M(σ¯ − σ0 ) , = 2 1/2 dr 2((x(r)) − x(r) + 1) 8 u˙ r (2 − x(r)) M(σ¯ − σ0 ) , = ε˙ θ = r 2((x(r))2 − x(r) + 1)1/2 ε˙ z = −(˙εr + ε˙ θ ),
ε˙ r =
(5) (6) (7)
where x = σr /σθ is the ratio of radial and tangential stresses at any radius r. Eqs. (5) and (6) can be solved to obtain σθ (r) as given below: 1/8
u˙ σθ (r) = a ψ1 (r) + ψ2 (r), M
(8)
where M((b − a)σθavg − ab ψ2 (r) dr) 1/8 , u˙ a = b a ψ1 (r) dr ρω2 (b3 − a 3 ) , σθavg = 3(b − a) ψ(r) ψ1 (r) = , (x(r)2 − x(r) + 1)1/2
(9)
(10)
ψ2 (r) =
σ0 (r) [(x(r))2 − x(r) + 1]1/2
(11)
and
2 {(x(r))2 − x(r) + 1}1/2 ψ(r) = exp r {2 − x(r)}
r a
1 2x(r) − 1 dr . r 2 − x(r)
(12)
In determining the stress distribution in a rotating disc of constant thickness the following equation based on the equilibrium condition of an element of the disc, subjected to radial and tangential stresses, σθ (r) and σr (r), is used (Singh and Ray, 2001), d rσr (r) − σθ + ρω2 r 2 = 0. dr
(13)
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The boundary conditions are σr (a) = σr (b) = 0 at r = a and r = b.
(14)
Using above boundary conditions in equilibrium equation (13), one can obtain the radial stress, σr (r), as given below: σr (r) =
1 r
r σθ (r) dr − a
ρω2 (r 3 − a 3 ) = 0. 3r
(15)
Knowing the radial distribution of σθ (r), values of σr (r) can be obtained from the above equation and thereafter, the strain rates ε˙ r , ε˙ θ and ε˙ z are calculated using Eqs. (5), (6) and (7), respectively.
3. Numerical computations The stress distribution is obtained from the above analysis by an iterative numerical scheme as shown in Fig. 1.To find the first approximation of x(r), [x(r)]1 , to be used in Eq. (12) in the first iteration, it is assumed that σθ (r) = σθavg in Eq. (13)
Fig. 1. Numerical scheme of computation.
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and on integration one gets the first approximation of σr (r), i.e., [σr (r)]1 . The subscript on the first bracket has been used to indicate the iteration number. Then dividing [σr (r)]1 by σθavg one gets [x(r)]1 . Substituting this value of [x(r)]1 for x(r) in Eq. (12), [ψ(r)]1 , to be used in the first iteration, is calculated. Using this [ψ(r)]1 in Eq. (11), [ψ1 (r)]1 and [ψ2 (r)]1 are found which are used in Eq. (9) to find (u˙ a )1 . Using [ψ1 (r)]1 , [ψ2 (r)]1 and (u˙ a )1 thus obtained in Eq. (8), [σθ (r)]1 is obtained. On substituting [σθ (r)]1 for [σθ (r)] in Eq. (15), the second approximation of σr , i.e., [σr (r)]2 is found by numerical integration and from this value, the second approximation of x, i.e., (x)2 is found. The iteration is continued till the process converges yielding the values of stresses at different points of the radius grid. For rapid convergence σθ to be used in the next iteration has been obtained using the following relation; σθNext = 0.25σθPrevious + 0.75σθCurrent . Since the problem has cylindrical symmetry the state of stress is now completely known under condition of creep. The strain rates are calculated now from Eqs. (4), (5) and (6).
4. Results and discussion Numerical calculations based on the analysis presented in this paper have been carried out to obtain the steady-state creep response of the composite disc. The results have been obtained for various combinations of material parameters and temperatures. Before discussing these results, however, it is considered necessary to validate the analysis and the developed computer program. To accomplish this task, the results for tangential and radial creep strains were computed for a rotating steel disc following the same analysis and compared with the available experimental results (Wahl et al., 1954) for the same type of disc and operating conditions as mentioned in Table 2. A good agreement between the theoretical and experimental results is observed, Fig. 2.
Fig. 2. Comparison of theoretical (present study) and experimental strains in a rotating steel disc. Table 2 Values of Wahl’s experimental data Parameters for steel disc: Density of disc material (ρ) = 7823.18 kg/m3 Disc radii: a = 31.75 mm, b = 152.4 mm Creep parameters: M = 4.72 × 10−4 s−1/8 /MPa, σ0 = −54.05 MPa Operating conditions: Disc rpm = 15 000 rpm Operating temperature = 810.78 K Creep duration = 180 hr
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(a)
(b)
(c)
(d)
Fig. 3. Creep response of composite disc for varying particle size of dispersoids (SiCP ). RPM of disc = 15 000, T = 623 K, V = 30 vol%. − − 1.7 µm, · · · 14.5 µm, − · − 45.9 µm.
4.1. Effect of material parameters Fig. 3(a)–(d) shows the creep response of the composite disc for varying particle size of the reinforcement. The variations obtained are similar to that obtained by Wahl and coworkers (Wahl et al., 1954). It is observed that in the region near the inner radius of the disc σθ increases rapidly and reaches a maximum value before decreasing near the outer radius, Fig. 3(a). The effect of particle size on the tangential stress is, however small. The radial stress, Fig. 3(b), also varies in a similar way with the maximum value somewhere near the middle of the disc. The particle size, in this case too, has very little effect. The maximum variation observed in tangential as well as radial stress is about 1%. Fig. 3(c) shows the variation of tangential strain rate with radial distance. The change in particle size of the reinforcement affects the tangential strain rate quite significantly. It is expected that the smaller size particles will be larger in number for the same volume fraction and are able to restrain creep flow more effectively. This is clearly evident from the figure as the creep rate at any radius is reduced by almost three orders of magnitude when the particle size decreases from 45.9 µm to 1.7 µm. Similar effects of particle size have been observed but under uniaxial creep by earlier workers (Pandey et al., 1992). Fig. 3(d) shows the variation of compressive radial strain rate with radial distance. The strain rates decreases with radius up to a certain radius where it becomes minimum. Beyond this value of the radius, the compressive radial strain rate increases slightly. Like
V.K. Gupta et al. / European Journal of Mechanics A/Solids 23 (2004) 335–344
(a)
(b)
(c)
(d)
341
Fig. 4. Creep response of composite disc for varying particle content of dispersoids (SiCP ). RPM of disc = 15 000, T = 623 K, P = 1.7 µm. − − 10 vol.%, · · · 20 vol.%, − · − 30 vol.%.
the tangential strain rate, the radial strain rate is also significantly affected by the particle size of the reinforcement. The radial strain rate is also reduced by almost three orders of magnitude when the particle size decreases from 45.9 µm to 1.7 µm. Fig. 4(a) shows the variation of tangential stress distribution in a particle-reinforced composite rotating disc containing 1.7 µm SiC particles for three values of particle contents. Tangential stress for each particle content varies as discussed in Fig. 3(a). When the particle content changes from 10 vol.% to 30 vol.%, the density of the composite increases by about 3 percent. The tangential stress also, generally, varies by a similar order. The maximum variation observed in tangential stress is about 5%. The radial stresses are not significantly affected by the particle content over the entire disc radius, Fig. 4(b). The maximum variation observed for radial stress with varying particle content is of the order of 1%. The effect of changing SiC particle content on the tangential strain rate is significant as shown in Fig. 4(c). The variation observed is similar to that observed for varying particle size as shown in Fig. 3(c). When the particle content increases from 10 vol.% to 30 vol.% in a particle reinforced composite containing 1.7 µm SiC particles the strain rate decreases by about four orders of magnitude. The variation of radial strain rate with radial distance for varying particle content is shown in Fig. 4(d). The effect of increasing particle content from 10 vol.% to 30 vol.% on radial strain rate is similar to that observed previously in Fig. 3(d) on tangential strain rate. Mishra and coworker (Mishra and Pandey, 1990) in their review of creep data of (Nieh, 1984) and (Morimoto et al., 1988) also noticed that creep rate in 6061 alloy-SiC whisker composites could be significantly reduced by increasing the whisker content.
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(a)
(b)
(c)
(d)
Fig. 5. Creep response of composite disc for varying operating temperature. RPM of disc = 15 000, P = 1.7 µm, V = 20 vol.%. − − 623 K, · · · 673 K, − · − 723 K.
4.2. Effect of temperature Temperature is one of the key factors influencing creep flow of any material. In this subsection, the effect of variation of temperature on stress distribution and creep deformation is reported for a rotating disc made of aluminium base particle reinforced composite containing 20 vol.% SiC particles of size 1.7 µm. Fig. 5(a) shows the tangential stress distribution with radial distance for three values of temperatures, i.e., 623 K, 673 K and 723 K. It is observed that the tangential stress varies little when the temperature is increased from 623 K to 723 K. The temperature also has little effect on radial stress distribution, Fig. 5(b). The maximum variation observed in tangential as well as radial stress is only about 0.15%. Inspite of very little change in the tangential and radial stresses with temperature, the tangential strain rates at different temperatures are quite different, Fig. 5(c). The tangential strain rate increases by about three orders of magnitude as the temperature increases from 623 K to 723 K. Likewise the temperature shows similar effect on the radial strain rate, Fig. 5(d). The effect of temperature on strain rates observed in this study are similar to the earlier experimental results (Pandey et al., 1992) obtained under uniaxial creep tests on Al–SiCP composites.
V.K. Gupta et al. / European Journal of Mechanics A/Solids 23 (2004) 335–344
(a)
343
(b)
Fig. 6. Effect of varying particle size, particle content and operating temperature on maximum strain rates in the disc. − − V = 10 vol.%, T = 623 K; · · · V = 20 vol.%, T = 623 K; − · − V = 20 vol.%, T = 623 K.
4.3. Selection of material parameters It is evident from the above discussion that the creep stresses do not vary significantly with particle size, particle content and operating temperature as compared to variation in strain rates. Further, from the point of view of designing a disc, operating at elevated temperature, the strain rates are considered to be primary design parameters. Therefore, a study that includes cumulative effect of the above mentioned parameters on strain rates would provide appropriate data for selecting optimum material parameters for the disc design. Fig. 6(a) shows the variation of maximum tangential strain rate, observed at the inner radius of the disc, with respect to particle size for different particle content and operating temperature. It is interesting to note that in a disc containing coarse particle and operating at specified temperature (623 K) the tangential strain rate can be reduced to the desired level by increasing the particle content of the reinforcement (i.e., from 10 vol.% to 20 vol.%). Further, for higher operating temperature (723 K), the increased strain rate can be controlled by selecting fine particles of the order of 5 µm, Fig. 6(a). The maximum radial strain rate, observed at the inner radius of the disc, can also be controlled to the desired level by selecting appropriate particle size and/or particle content of the reinforcement as shown in Fig. 6(b).
5. Conclusions Based on the results and discussions presented the following conclusions may be drawn: The tangential as well as radial stresses increases as one move from the inner towards the outer radius of the disc, reaches maximum before decreasing near the outer radius. Further, the stress distribution in the disc does not vary significantly for various combinations of material parameters and operating temperatures. The tangential as well as radial strain rates decreases on moving from the inner towards the outer radius of the disc but radial strain rate reaches a minimum somewhere near the middle region of the disc. The tangential as well as radial strain rates in the disc reduces significantly with reducing particle size, with increasing particle content and with decreasing operating temperature. The maximum strain rates in the disc for given operating conditions can be controlled by selecting optimum particle content and/or particle size of the reinforcement.
References Arya, V.K., Bhatnagar, N.S., 1979. Creep analysis of rotating orthotropic discs. Nucl. Eng. Des. 55, 326–330. Mishra, R.S., Pandey, A.B., 1990. Some observations on the high-temperature creep behaviour of 6061 Al–SiC composites. Metall. Trans. A 21, 2089–2090.
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Morimoto, T., Yamaoka, T., Lilholt, H., Taya, M., 1988. Second stage creep of SiC whiskers/6061 aluminum composite at 573 K. J. Eng. Mater. Tech. 110, 70–76. Nieh, T.G., 1984. Creep rupture of a silicon carbide reinforced aluminum composites. Metall. Trans. A 15, 139–146. Pandey, A.B., Mishra, R.S., Mahajan, Y.R., 1992. Steady state creep behaviour of silicon carbide particulate reinforced aluminium composites. Acta Met. Mater. 40 (8), 2045–2052. Sherby, O.D., Klundt, R.H., Miller, A.K., 1977. Flow stress, subgrain size and subgrain stability at elevated temperatures. Metall. Trans. A 8, 843–850. Singh, S.B., Ray, S., 2001. Steady state creep behaviour of an isotropic functionally graded rotating disc of Al–SiC composites. Metall. Trans. A 32, 1679–1685. Singh, S.B., Ray, S., 2002. Modeling the anisotropy and creep in orthotropic aluminum–silicon carbide composite rotating disc. Mech. Mater. 34, 363–372. Wahl, A.M., Shankey, G.O., Manjoine, M.J., Shoemaker, E., 1954. Creep tests of rotating discs at elevated temperature and comparison with theory. J. Appl. Mech. 21 (3), 225–235.
Steady state creep and material parameters in a rotating disc of Al–SiCP composite V.K. Gupta a,∗ , S.B. Singh b , H.N. Chandrawat a , S. Ray c a Department of Mechanical Engineering, T.I.E.T., Patiala-147004, India b School of Basic & Applied Sciences, T.I.E.T., Patiala-147004, India c Department of Metallurgical and Materials Engineering, I.I.T., Roorkee-247667, India
Received 1 April 2003; accepted 26 November 2003
Abstract The steady state creep in a rotating disc made of isotropic aluminium–silicon carbide particulate composite has been investigated in the present study. The creep behaviour of the composite has been described by Sherby’s constitutive model. The creep parameters in the law have been determined using regression equations developed on the basis of available experimental results. The radial and tangential stresses and steady state creep rates in the disc have been calculated and presented for various combinations of material parameters (like particle size and particle content) and temperatures. The study revealed that for given operating conditions, the strain rates in the disc can be controlled by selecting optimum particle content and/or particle size of the reinforcement. 2003 Elsevier SAS. All rights reserved. Keywords: Creep; Particulate composite; Rotating disc
1. Introduction A rotating disc at elevated temperature is a common component in aero-engines, automobiles, turbines, pumps, compressors and in a number of other dynamic applications (Singh and Ray, 2002). A reduced weight of such components, resulting from the use of aluminium/aluminium base alloys, is expected to save power and fuel due to a reduction in the payload. However, the enhanced creep of aluminium and its alloys may be a big hindrance in such applications. Particle- and whisker-reinforced metal matrix composites have shown superior high temperature properties and are finding increasing application in components exposed to higher temperature (Singh and Ray, 2001). Earlier study (Nieh, 1984) reveals that aluminium matrix based composite has better creep resistance compared to the base aluminium alloy. The study pertaining to steady state creep behaviour of Al–SiCP composites under uniaxial condition (Pandey et al., 1992) in the temperature range between 623–723 K for different particle sizes and with varying volume fraction of reinforcement indicated that the composite with finer particle size has better creep resistance than that containing coarser ones. Wahl et al. (1954) theoretically studied steady state creep deformation in a rotating turbine disc made of 12-pct chromium steel using von Mises and Tresca yield criteria and validated his results experimentally. They described creep behaviour by a power function relation of the form ε˙ = kσ n f (t). Arya and Bhatnagar (1979) investigated the effect of orthotropicity in a rotating disc made of wood on stress and strain distributions using Hill yield criterion. Singh et al. (2002) analysed steady * Corresponding author.
E-mail address: [email protected] (V.K. Gupta). 0997-7538/$ – see front matter 2003 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2003.11.005
336
V.K. Gupta et al. / European Journal of Mechanics A/Solids 23 (2004) 335–344
Nomenclature a A b |br | DL E f (t) k M n P r t T V Vavg ε˙ ε˙¯ ε˙ r ε˙ θ
inner radius of disc (= 31.75 mm) constant outer radius of disc (= 152.4 mm) magnitude of burgers vector lattice diffusivity Young’s modulus function of time constant creep parameter, (1/E)(ADL λ3 /|br |5 )1/8 constant size of dispersoid, SiCP (µm) radius of disc (a r b) thickness of the disc (= 5 mm) temperature (K) dispersoid (SiCP ) content (vol.%) average particle content in disc (vol.%) strain rate (s−1 ) effective creep rate under multiaxial stress (s−1 ) strain rate (compressive) along radial (r) direction (s−1 ), (¯ε/(2σ¯ ))(2σr − σθ ) strain rate along tangential (θ) direction (s−1 ), (ε˙¯ /(2σ¯ ))(2σθ − σr )
ε˙ z ε˙ rmax ε˙ θmax λ ρ σ σ¯
σr , σθ σθavg σ0 u˙ r u˙ a ω
strain rate along axial (z) direction, (ε˙¯ /(2σ¯ ))(−σr − σθ ) radial strain rate (compressive) at inner radius (s−1 ) tangential strain rate at inner radius (s−1 ) subgrain size density of the composite (kg/m3 ) stress (MPa) effective stress under biaxial state of stress (MPa), = √1 [σθ2 + σr2 + (σr − σθ )2 ]1/2 2 stresses along radial and tangential directions (MPa) average tangential stress over cross section of disc (MPa) threshold stress (MPa) radial strain rate at any radius ‘r’ of disc (s−1 ) rdial strain rate at inner radius ‘a’ of the disc (s−1 ) angular velocity of disc rotating at 15 000 rpm (radian/sec)
state creep using Norton’s creep model in a rotating disc made of aluminium–silicon carbide whiskers using Hill yield criterion. However, it has been observed that application of Norton’s creep model to composite results in very high value of stress exponent as compared to Sherby’s model (Mishra and Pandey, 1990). The survey also revealed that, the effect of size and content of the reinforcement and operating temperature on the steady state creep behaviour of the rotating disc made of composite has not been investigated. In the present study, the analysis of steady state creep in a rotating disc made of isotropic composites containing SiC particles in an aluminium matrix has been carried out. Sherby’s constitutive creep model (Sherby et al., 1977), which is claimed to work better than Norton’s creep model (Mishra and Pandey, 1990) has been used to describe the creep behaviour of the composite. The radial and tangential stress distributions have been determined and the steady state creep rates have been calculated for various combinations of material parameters (like particle size and particle content) and temperatures.
2. Mathematical formulation of creep behaviour A particle reinforced composite disc (Al–SiCP ) of constant thickness with inner and outer radii of a and b respectively, is considered rotating with angular velocity ω (radian/sec). For the purpose of analysis the following assumptions are made: (i) (ii) (iii) (iv)
Steady state condition of stress is assumed. Elastic deformations are small and are neglected as compared to creep deformations. Biaxial state of stress (i.e., stress along axial (z) direction, σz = 0) exist at any point in the disc. The composite shows a steady state creep behaviour, which may be described by following Sherby’s constitutive model of the form (Sherby et al., 1977), ADL λ3 σ¯ − σ0 8 ε˙¯ = E |br |5
or alternatively 8 ε˙¯ = M(σ¯ − σ0 ) .
(1)
(2)
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Table 1 Values of material parameters from experimental study Particle size P (µm)
Temperature T (K)
Particle content V (vol.%)
M (s−1/8 /MPa)
σ0 (MPa)
1.7 14.5 45.9
623
10
0.00963 0.01444 0.01897
15.24 11.46 13.65
1.7
623
10 20 30
0.00963 0.00594 0.00518
15.24 24.83 34.32
1.7
623 673 723
20
0.00594 0.00897 0.01295
24.83 24.74 25.72
In a particle-reinforced composite, the values of creep parameters M and σ0 depend on the size and the percentage of dispersed particles (SiCP ) and the temperature of application. For the present work, the values of M and σ0 have been extracted from the creep results experimentally obtained for Al–SiCP composites under uniaxial loading (Pandey et al., 1992) and are shown in Table 1. The following regression equations describe the values of M and σ0 as functions of T , P and V . ln(M) = −35.38 + 0.2077 ln(P ) + 4.98 ln(T ) − 0.622 ln(V ),
(3)
σ0 = −0.03507P + 0.01057T + 1.00536V − 2.11916.
(4)
Taking the reference frame along the axial directions r, θ and z of the disc, the relations for ε˙ r , ε˙ θ and ε˙ z for biaxial state of stress are obtained as (Singh and Ray, 2001), 8 du˙ r (2x(r) − 1) M(σ¯ − σ0 ) , = 2 1/2 dr 2((x(r)) − x(r) + 1) 8 u˙ r (2 − x(r)) M(σ¯ − σ0 ) , = ε˙ θ = r 2((x(r))2 − x(r) + 1)1/2 ε˙ z = −(˙εr + ε˙ θ ),
ε˙ r =
(5) (6) (7)
where x = σr /σθ is the ratio of radial and tangential stresses at any radius r. Eqs. (5) and (6) can be solved to obtain σθ (r) as given below: 1/8
u˙ σθ (r) = a ψ1 (r) + ψ2 (r), M
(8)
where M((b − a)σθavg − ab ψ2 (r) dr) 1/8 , u˙ a = b a ψ1 (r) dr ρω2 (b3 − a 3 ) , σθavg = 3(b − a) ψ(r) ψ1 (r) = , (x(r)2 − x(r) + 1)1/2
(9)
(10)
ψ2 (r) =
σ0 (r) [(x(r))2 − x(r) + 1]1/2
(11)
and
2 {(x(r))2 − x(r) + 1}1/2 ψ(r) = exp r {2 − x(r)}
r a
1 2x(r) − 1 dr . r 2 − x(r)
(12)
In determining the stress distribution in a rotating disc of constant thickness the following equation based on the equilibrium condition of an element of the disc, subjected to radial and tangential stresses, σθ (r) and σr (r), is used (Singh and Ray, 2001), d rσr (r) − σθ + ρω2 r 2 = 0. dr
(13)
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The boundary conditions are σr (a) = σr (b) = 0 at r = a and r = b.
(14)
Using above boundary conditions in equilibrium equation (13), one can obtain the radial stress, σr (r), as given below: σr (r) =
1 r
r σθ (r) dr − a
ρω2 (r 3 − a 3 ) = 0. 3r
(15)
Knowing the radial distribution of σθ (r), values of σr (r) can be obtained from the above equation and thereafter, the strain rates ε˙ r , ε˙ θ and ε˙ z are calculated using Eqs. (5), (6) and (7), respectively.
3. Numerical computations The stress distribution is obtained from the above analysis by an iterative numerical scheme as shown in Fig. 1.To find the first approximation of x(r), [x(r)]1 , to be used in Eq. (12) in the first iteration, it is assumed that σθ (r) = σθavg in Eq. (13)
Fig. 1. Numerical scheme of computation.
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and on integration one gets the first approximation of σr (r), i.e., [σr (r)]1 . The subscript on the first bracket has been used to indicate the iteration number. Then dividing [σr (r)]1 by σθavg one gets [x(r)]1 . Substituting this value of [x(r)]1 for x(r) in Eq. (12), [ψ(r)]1 , to be used in the first iteration, is calculated. Using this [ψ(r)]1 in Eq. (11), [ψ1 (r)]1 and [ψ2 (r)]1 are found which are used in Eq. (9) to find (u˙ a )1 . Using [ψ1 (r)]1 , [ψ2 (r)]1 and (u˙ a )1 thus obtained in Eq. (8), [σθ (r)]1 is obtained. On substituting [σθ (r)]1 for [σθ (r)] in Eq. (15), the second approximation of σr , i.e., [σr (r)]2 is found by numerical integration and from this value, the second approximation of x, i.e., (x)2 is found. The iteration is continued till the process converges yielding the values of stresses at different points of the radius grid. For rapid convergence σθ to be used in the next iteration has been obtained using the following relation; σθNext = 0.25σθPrevious + 0.75σθCurrent . Since the problem has cylindrical symmetry the state of stress is now completely known under condition of creep. The strain rates are calculated now from Eqs. (4), (5) and (6).
4. Results and discussion Numerical calculations based on the analysis presented in this paper have been carried out to obtain the steady-state creep response of the composite disc. The results have been obtained for various combinations of material parameters and temperatures. Before discussing these results, however, it is considered necessary to validate the analysis and the developed computer program. To accomplish this task, the results for tangential and radial creep strains were computed for a rotating steel disc following the same analysis and compared with the available experimental results (Wahl et al., 1954) for the same type of disc and operating conditions as mentioned in Table 2. A good agreement between the theoretical and experimental results is observed, Fig. 2.
Fig. 2. Comparison of theoretical (present study) and experimental strains in a rotating steel disc. Table 2 Values of Wahl’s experimental data Parameters for steel disc: Density of disc material (ρ) = 7823.18 kg/m3 Disc radii: a = 31.75 mm, b = 152.4 mm Creep parameters: M = 4.72 × 10−4 s−1/8 /MPa, σ0 = −54.05 MPa Operating conditions: Disc rpm = 15 000 rpm Operating temperature = 810.78 K Creep duration = 180 hr
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(a)
(b)
(c)
(d)
Fig. 3. Creep response of composite disc for varying particle size of dispersoids (SiCP ). RPM of disc = 15 000, T = 623 K, V = 30 vol%. − − 1.7 µm, · · · 14.5 µm, − · − 45.9 µm.
4.1. Effect of material parameters Fig. 3(a)–(d) shows the creep response of the composite disc for varying particle size of the reinforcement. The variations obtained are similar to that obtained by Wahl and coworkers (Wahl et al., 1954). It is observed that in the region near the inner radius of the disc σθ increases rapidly and reaches a maximum value before decreasing near the outer radius, Fig. 3(a). The effect of particle size on the tangential stress is, however small. The radial stress, Fig. 3(b), also varies in a similar way with the maximum value somewhere near the middle of the disc. The particle size, in this case too, has very little effect. The maximum variation observed in tangential as well as radial stress is about 1%. Fig. 3(c) shows the variation of tangential strain rate with radial distance. The change in particle size of the reinforcement affects the tangential strain rate quite significantly. It is expected that the smaller size particles will be larger in number for the same volume fraction and are able to restrain creep flow more effectively. This is clearly evident from the figure as the creep rate at any radius is reduced by almost three orders of magnitude when the particle size decreases from 45.9 µm to 1.7 µm. Similar effects of particle size have been observed but under uniaxial creep by earlier workers (Pandey et al., 1992). Fig. 3(d) shows the variation of compressive radial strain rate with radial distance. The strain rates decreases with radius up to a certain radius where it becomes minimum. Beyond this value of the radius, the compressive radial strain rate increases slightly. Like
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(a)
(b)
(c)
(d)
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Fig. 4. Creep response of composite disc for varying particle content of dispersoids (SiCP ). RPM of disc = 15 000, T = 623 K, P = 1.7 µm. − − 10 vol.%, · · · 20 vol.%, − · − 30 vol.%.
the tangential strain rate, the radial strain rate is also significantly affected by the particle size of the reinforcement. The radial strain rate is also reduced by almost three orders of magnitude when the particle size decreases from 45.9 µm to 1.7 µm. Fig. 4(a) shows the variation of tangential stress distribution in a particle-reinforced composite rotating disc containing 1.7 µm SiC particles for three values of particle contents. Tangential stress for each particle content varies as discussed in Fig. 3(a). When the particle content changes from 10 vol.% to 30 vol.%, the density of the composite increases by about 3 percent. The tangential stress also, generally, varies by a similar order. The maximum variation observed in tangential stress is about 5%. The radial stresses are not significantly affected by the particle content over the entire disc radius, Fig. 4(b). The maximum variation observed for radial stress with varying particle content is of the order of 1%. The effect of changing SiC particle content on the tangential strain rate is significant as shown in Fig. 4(c). The variation observed is similar to that observed for varying particle size as shown in Fig. 3(c). When the particle content increases from 10 vol.% to 30 vol.% in a particle reinforced composite containing 1.7 µm SiC particles the strain rate decreases by about four orders of magnitude. The variation of radial strain rate with radial distance for varying particle content is shown in Fig. 4(d). The effect of increasing particle content from 10 vol.% to 30 vol.% on radial strain rate is similar to that observed previously in Fig. 3(d) on tangential strain rate. Mishra and coworker (Mishra and Pandey, 1990) in their review of creep data of (Nieh, 1984) and (Morimoto et al., 1988) also noticed that creep rate in 6061 alloy-SiC whisker composites could be significantly reduced by increasing the whisker content.
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(a)
(b)
(c)
(d)
Fig. 5. Creep response of composite disc for varying operating temperature. RPM of disc = 15 000, P = 1.7 µm, V = 20 vol.%. − − 623 K, · · · 673 K, − · − 723 K.
4.2. Effect of temperature Temperature is one of the key factors influencing creep flow of any material. In this subsection, the effect of variation of temperature on stress distribution and creep deformation is reported for a rotating disc made of aluminium base particle reinforced composite containing 20 vol.% SiC particles of size 1.7 µm. Fig. 5(a) shows the tangential stress distribution with radial distance for three values of temperatures, i.e., 623 K, 673 K and 723 K. It is observed that the tangential stress varies little when the temperature is increased from 623 K to 723 K. The temperature also has little effect on radial stress distribution, Fig. 5(b). The maximum variation observed in tangential as well as radial stress is only about 0.15%. Inspite of very little change in the tangential and radial stresses with temperature, the tangential strain rates at different temperatures are quite different, Fig. 5(c). The tangential strain rate increases by about three orders of magnitude as the temperature increases from 623 K to 723 K. Likewise the temperature shows similar effect on the radial strain rate, Fig. 5(d). The effect of temperature on strain rates observed in this study are similar to the earlier experimental results (Pandey et al., 1992) obtained under uniaxial creep tests on Al–SiCP composites.
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(b)
Fig. 6. Effect of varying particle size, particle content and operating temperature on maximum strain rates in the disc. − − V = 10 vol.%, T = 623 K; · · · V = 20 vol.%, T = 623 K; − · − V = 20 vol.%, T = 623 K.
4.3. Selection of material parameters It is evident from the above discussion that the creep stresses do not vary significantly with particle size, particle content and operating temperature as compared to variation in strain rates. Further, from the point of view of designing a disc, operating at elevated temperature, the strain rates are considered to be primary design parameters. Therefore, a study that includes cumulative effect of the above mentioned parameters on strain rates would provide appropriate data for selecting optimum material parameters for the disc design. Fig. 6(a) shows the variation of maximum tangential strain rate, observed at the inner radius of the disc, with respect to particle size for different particle content and operating temperature. It is interesting to note that in a disc containing coarse particle and operating at specified temperature (623 K) the tangential strain rate can be reduced to the desired level by increasing the particle content of the reinforcement (i.e., from 10 vol.% to 20 vol.%). Further, for higher operating temperature (723 K), the increased strain rate can be controlled by selecting fine particles of the order of 5 µm, Fig. 6(a). The maximum radial strain rate, observed at the inner radius of the disc, can also be controlled to the desired level by selecting appropriate particle size and/or particle content of the reinforcement as shown in Fig. 6(b).
5. Conclusions Based on the results and discussions presented the following conclusions may be drawn: The tangential as well as radial stresses increases as one move from the inner towards the outer radius of the disc, reaches maximum before decreasing near the outer radius. Further, the stress distribution in the disc does not vary significantly for various combinations of material parameters and operating temperatures. The tangential as well as radial strain rates decreases on moving from the inner towards the outer radius of the disc but radial strain rate reaches a minimum somewhere near the middle region of the disc. The tangential as well as radial strain rates in the disc reduces significantly with reducing particle size, with increasing particle content and with decreasing operating temperature. The maximum strain rates in the disc for given operating conditions can be controlled by selecting optimum particle content and/or particle size of the reinforcement.
References Arya, V.K., Bhatnagar, N.S., 1979. Creep analysis of rotating orthotropic discs. Nucl. Eng. Des. 55, 326–330. Mishra, R.S., Pandey, A.B., 1990. Some observations on the high-temperature creep behaviour of 6061 Al–SiC composites. Metall. Trans. A 21, 2089–2090.
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Morimoto, T., Yamaoka, T., Lilholt, H., Taya, M., 1988. Second stage creep of SiC whiskers/6061 aluminum composite at 573 K. J. Eng. Mater. Tech. 110, 70–76. Nieh, T.G., 1984. Creep rupture of a silicon carbide reinforced aluminum composites. Metall. Trans. A 15, 139–146. Pandey, A.B., Mishra, R.S., Mahajan, Y.R., 1992. Steady state creep behaviour of silicon carbide particulate reinforced aluminium composites. Acta Met. Mater. 40 (8), 2045–2052. Sherby, O.D., Klundt, R.H., Miller, A.K., 1977. Flow stress, subgrain size and subgrain stability at elevated temperatures. Metall. Trans. A 8, 843–850. Singh, S.B., Ray, S., 2001. Steady state creep behaviour of an isotropic functionally graded rotating disc of Al–SiC composites. Metall. Trans. A 32, 1679–1685. Singh, S.B., Ray, S., 2002. Modeling the anisotropy and creep in orthotropic aluminum–silicon carbide composite rotating disc. Mech. Mater. 34, 363–372. Wahl, A.M., Shankey, G.O., Manjoine, M.J., Shoemaker, E., 1954. Creep tests of rotating discs at elevated temperature and comparison with theory. J. Appl. Mech. 21 (3), 225–235.