Effect of Material Parameters on Steady State Creep in a Thick Composite Cylinder Subjected to Internal Pressure

The steady state creep in AlSiCP composite cylinder subjected to internal pressure was investigated. The creep behavior of the material were described by threshold stress based creep law by assuming a stress exponent of 5. The effect of size and content of the reinforcement (SiCP), and operating temperature on the stresses and strain rates in the composite cylinder were investigated. The stresses in the cylinder did not have significant variation with varying size and content of the reinforcement, and operating temperature. However, the tangential as well as radial strain rates in the cylinder could be reduced to a significant extent by decreasing size of SiCP, increasing the content of SiCP and decreasing operating temperature.


Introduction
Cylinder made of monolithic material such as metal and concrete (eg.asphalt concrete), is a common component employed in numerous applications such as pressure vessels (eg.hydraulic cylinders, gun barrels, pipes, boilers and fuel tanks), accumulator shells, emergency breathing cylinders, cylinders for aerospace industries, nuclear reactors, military applications and civil structures etc. (Arya, Bhatnagar, 1976;Bhatnagar, et al. 1980;Becht, et al. 2000;Gupta, et al. 2001;Perry and Aboudi, 2003;Buttlar, et al. 2004 andYou andButtlar, 2005).In some of these applications such as pressure vessel for industrial gases or a media transportation of high-pressurized fluids and piping of nuclear reactors, the cylinder has to operate under severe mechanical and thermal loads, causing significant creep hence reduced service life (Gupta and Pathak, 2001;Tachibana andIyoku, 2004 andHagihara andMiyazaki, 2008).As an example, in the high temperature engineering test reactor, the temperature reaches of the order of 900 0 C (Tachibana and Iyoku, 2004).The piping of reactor cooling system is subjected to high _______________________________________________ *Corresponding author's e-mail: guptavk_70@yahoo.co.in temperature and pressure and may be damaged due to high heat generated from the reactor core (Hagihara and Miyazaki, 2008).
Creep analysis of thick-walled cylinder made of isotropic monolithic material and subjected to internal pressure has been presented by (Weir, 1957;King andMackie 1967 andPai 1967) has solved the problem for orthotropic cylinders.In all these analyses it was assumed that the strains are infinitesimal and the deformation is referred with respect to original dimensions of the cylinder.Rimrott, 1959, used generally accepted assumptions of constant density, zero axial strain and distortion energy theory to derive equations for creep rate, creep strains and creep stresses in a, closed end, thick-walled hollow cylinder subjected to internal pressure.Bhatnagar and Gupta, 1969 obtained the solution for thick walled cylinder made of an orthotropic material and subjected to internal pressure.In recent years, the problem of creep in composite cylinders made of Functionally Graded Materials (FGMs) operating at high pressure and temperature has attracted the interest of many researchers.Fukui and Yamanaka, 1992 investigated the effect of gradation of components on strength and deformation of thick-walled Functionally Graded (FG) tubes subjected to internal pressure under plain strain conditions.Fukui et al. 1993 extended the work to investigate the effect of graded components on residual stresses in a thick walled FG tube under uniform thermal loading.Chen et al. 2007 studied the creep behavior of a thick walled FGM cylinder subjected to internal and external pressures.The asymptotic solutions were obtained on the basis of Taylor expansion series and compared with the results obtained by finite element analysis.You et al. 2007 investigated steady state creep in a thick walled FGM cylinder subjected to internal pressure.The effect of variation in material parameters on the stresses induced in cylinder was investigated.Abrinia et al. 2008 obtained the analytical solution for radial and circumferential stresses in a thick FGM cylindrical vessel under the influence of internal pressure and temperature.The study indicated that stresses in FGM cylinder could be lowered by tailoring the material properties along radial direction of the cylinder.
Under severe thermo-mechanical loads cylinder made of monolithic materials may not perform well.The metal matrix composites (MMCs) such as aluminium/aluminium alloy matrix reinforced with silicon carbide offer excellent mechanical properties like high specific strength and stiffness along with high temperature stability.Therefore, these are suitable for cylinder applications exposed to high pressure and high operating temperature (Nieh, 1984;Roy and Tsai, 1988;Fukui et al. 1993;Salzar et al. 1996 andGupta et al. 2004.With these forethoughts, it is decided to investigate the steady state creep in a cylinder made of Al-SiC P composite and subjected to high pressure and high temperature.A mathematical model has been developed to describe the steady creep behavior of the composite cylinder.The model developed is used to investigate the effect of material parameters viz particle size and particle content, and operating temperatures on the steady state creep response of the composite cylinder.

Selection of Creep Law
(1) where the symbols A', n, Q, E, R and T denote respectively structure dependent parameter, true stress exponent, true activation energy, temperature-dependent Young's modulus, gas constant and operating temperature.
The true stress exponent n appearing in Eq. ( 1) is usually selected as 3, 5 and 8 which correspond to three welldocumented creep cases for metals and alloys: (i) n = 3 for creep controlled by viscous glide processes of dislocation, (ii) n = 5 for creep controlled by high temperature dislocation climb (lattice diffusion), and (iii) n = 8 for lattice diffusion-controlled creep with a constant structure (Tjong and Ma, 2000).Though, some of the research groups (Mishra and Pandey, 1990;Pandey et al. 1992;Gonzalex-Doncel andSherby, 1993 andPandey et al. 1994) have used a true stress exponent of 8 to describe steady state creep in Al-SiC P , W (subscript 'p' for particle and 'w' for whisker) composites but a number of other research groups (Park et al. 1990;Mohamed et al. 1992;Park and Mohamed, 1995;Cadek et al. 1995;Yoshioka et al. 1998;Li and Mohamed, 1997;Li andLangdon, 1997 andLi andLangdon, 1999) have observed that a stress exponent of either ~3 or ~5, rather than 8, provides a better description of steady state creep data observed for discontinuously reinforced Al-SiC composites.Keeping this in view, a stress exponent of 5 is used to describe steady state creep behavior of the composite cylinders in this study.The justification regarding choice of stress exponent is further elaborated in the next section.
The creep parameters M and so appearing in Eq .( 2) are dependent on the type of material and are also affected by the temperature (T) of application .In a composite, the dispersoid size (P) and the content of dispersoid (V) are the primary material variables determining these parameters .In the present study , the values of M and σ 0 have been extracted from the experimental creep results reported for Al -SiCP composite under uniaxial loading (Pandey et al. 1992).Though, Pandey et al. 1992 suggested a stress exponent of 8 to describe steady state creep in these composites .But due to the objections pointed by several research groups (Park et al. 1990;Mohamed et al. 1992;Park and Mohamed, 1995;Cedek et al. 1995;Yoshioka et al. 1998;Li and Mohamed , 1997;Li andLongdon , 1997 andLi andLangdon , 1999), we have used a stress exponent of 5 to describe steady state creep in composite (Al-SiCP) taken in the study of (Pandey et al. 1992).The individual set of creep data of (Pandey et al. 1992) have been plotted as 1 5 / ε versus σ on linear scales as shown in Figs.1(a)-(c).From the slope and intercepts of these graphs, the values of creep parameters M and σ ο have been obtained and are reported in Table-1.This approach of determining the threshold stress σ ο , is known as linear extrapolation technique (Lagneborg and Bergman, 1976).To avoid variation due to systematic error, if any, in experimental results, the creep results from a single source have been used.
The ε 1/5 versus σ plots corresponding to the observed experimental data points of Al-SiC P composites (Pandey et al. 1992) for various combinations of particle size, particle content and operating temperature exhibit an excellent linearity as evident from Figs. 1(a)-(c).The coefficient of correlation for these plots has been reported in excess to 0.916 as given in Table-1.In the light of these results, the choice of stress exponent n = 5, to describe the steady state creep behavior of Al-SiC P composite, is justified.
The accuracy of the creep response of the composite cylinder, to be estimated in subsequent sections, will depend on the accuracy associated with prediction of creep parameters M and σ ο for various combinations of material parameters and operating temperature.To accomplish this task, the creep parameters given in Table -1 have been substituted in the constitutive creep model, Eq. ( 2), to estimate the strain rates corresponding to the experimental stress values reported by (Pandey et al. 1992) for Al-SiC P composite for the various combinations of material parameters and temperature as given in Table-1.The estimated strain rates have been compared with the strain rates observed experimentally by (Pandey et al. 1992).Figs.2(a)-(c) show an excellent agreement between the strain rates estimated from Eq. ( 2) and those observed experimentally, to inspire confidence in the creep parameters estimated in this study.

Mathematical Formulation
Consider a long, closed end, thick-walled hollow cylinder made of Al-SiC P composite having inner and outer radii of a and b respectively and subjected to internal pressure p.The coordinates axes r, θ and z are taken respectively along radial, tangential and axial directions of the cylinder.The present analysis is based on the following assumptions: i. Material of the cylinder is incompressible, isotropic and has uniform distribution of SiC P in aluminium matrix.ii.Pressure is applied gradually and held constant during the loading history.
iii.Stresses at any point in the cylinder remain constant with time ie.steady state condition of stress is assumed.iv.Elastic deformations are small and are neglected as compared to creep deformations.
(3) (4) (5) Considering the equilibrium of forces on an element of the cylinder in the radial direction, we may write, (Gupta and Pathak, 2001 where σ θ is the tangential stress. Since material of the cylinder is assumed to be incompressible, therefore, (7) The generalized constitutive equations for creep in an isotropic composite (Gupta et al. 2005), when reference frame is along the principal directions of r, θ and z are given by, (8) (9) (10) where σ r , σ θ , σ z are respectively the radial, tangential and axial stresses.
Following Von-Mises yield criterion (Dieter, 1988), the effective stress is given by,  The radial ( r ε ) and tangenti al ( θ ε ) strain rates in the cylinder are respectively given by: is the radial displacement rate and u is the radial displacement.
where Z ε is the strain rate in the axial ( z) direction.
2 2 In a cylinder made of incompressible material with closed end, the plane strain condition exist i.e. the axial strain rate ) ( z ε is zero (Popov 2001).Therefore, Eqs.
The boundary conditions for a cylinder subjected to only internal pressure are given as, (i) At r = a, σ r = -P (negative sign implies compressive stress) (20) Applying boundary conditions stated above, in Eq. ( 19), we get, The values C 1 and I 1 , thus obtained, are substituted in Eq. ( 19) to get the radial stress, σ r , (22) where, Using Eq. ( 22) in Eq. ( 18), the tangential stress, σ θ , is obtained, (23) Substituting Eqs. ( 22) and ( 23) in Eq. ( 15), we get the axial stress, σ z , ( Based on the analysis presented, a computer program has been developed to calculate the steady state creep response of the composite cylinder for various combinations of size and content of the reinforcement (SiC P ), and operating temperature.For the purpose of numerical computation, the inner and outer radii of the cylinder are taken 25.4 mm and 50.8 mm respectively, and the internal pressure is assumed to be 85.25 MPa.The dimensions of cylinder and the operating pressure chosen in this study are similar to those used in earlier work (Johnson et al. 1961) on thick walled aluminum alloy (RR59) cylinder.The radial, tangential and axial stresses at different radial locations of the cylinder are calculated respectively from Eqs. ( 22), ( 23) and ( 24).The distributions of radial and tangential strain rates are computed from Eqs. ( 8) and ( 9) respectively.The creep parameters M and σ o required during the computation process are taken from Table-1 for the desired combination of particle size, particle content and operating temperature.

Results and Discussions
Before discussing the results obtained in the present study, it is necessary to check the accuracy of analysis carried out and the computer program developed.To accomplish this task, following present analysis, the tangential, radial and axial stresses have been computed for a copper cylinder, for which the results are available in the literature (Johnson et al. 1961).The dimensions of the cylin- ) Putting e ε and e σ respectively from Eqs. ( 2) and ( 16) in the above equation and simplifying, one gets, 1 2 2 der, operating pressure and temperature, and the values of creep parameters used for the purpose of validation are summarized in Table-2.To estimate the parameters M and σ o for copper cylinder, firstly the values of σ e have been calculated at inner and outer radii of cylinder by substituting in Eq. ( 11) the values of σ e , σ r , σ θ and σ z at these locations as reported in the study of (Johnson et al. 1961).
The values of stresses σ r , σ θ and σ z and the tangential strain rates (ε θ ) reported in reference (Johnson et al. (1961) at inner and outer radii are substituted in Eqn. ( 9) to estimate the effective strain rates (ε e ) at the corresponding radial locations.The effective stresses and effective strain rates estimated at the inner (ε e = 189.83MPa and ε e = 2.168 x 10 -8 s -1 ) and the outer (ε e = 116 MPa and ε e = 1.128 x 10 -9 s -1 ) radii of the copper cylinder are substituted in creep law, Eq. ( 2), to estimate the creep parameters M and σ o for copper, Table-2.The creep parameters have been used in the developed software to compute the distribution of tangential strain rate in the copper cylinder.The tangential strain rates thus obtained have been compared with those reported by (Johnson et al. 1961).A nice agreement is observed in Fig. 3, therefore verifying the accuracy of analysis presented and software developed in the current study.

Effect of Particle Size and Particle Content
Figs. 4(a)-(d) show the variation of creep stresses in composite cylinder for varying SiC P size from 1.7 µm to 45.9 µm.The trend of stresses obtained is similar to those reported by (Bhatnagar and Arya 1974).The radial stress, Fig. 4(a), remains compressive over the entire cylinder radius with maximum at the inner radius and zero at the outer radius, due to the imposed boundary conditions given in Eqs. ( 20) and ( 21).The tangential stress shown in, Fig. 4(b), remains always tensile and increases on moving from the inner to the outer radius of cylinder.Unlike radial and tangential stresses, the axial stress, Figs.4(c), decreases from maximum compressive, observed at the inner radius, to reach a maximum tensile value at the outer radius of the cylinder.It is clearly evident from Figs. 4(a)-(c) that the variation in particle size does not have a sizable effect on the stresses.The maximum variation observed in radial and tangential stresses are less than 1% and for axial stress it is around 5% at a radius of 30.5 mm, in cylinders having relatively coarser SiC P of size 14.5 and 45.9 µm.when compared to cylinder having finer SiC P of 1.7 µm size.The variation of effective stress,σ e , given by Eq. ( 11), for varying particle size of SiC is shown in Fig. 4(d marginally with increasing particle size.The maximum decrease of about 1% is observed at a radius 40.6 mm in cylinders containing relatively coarser SiC P (14.5µm and 45.9 µm) when compared to cylinder having finer SiC P of 1.7µm size.The strain rates given by Eqs. ( 8) and ( 9) are related to the effective strain rate, ε e , which ultimately depends upon (σ e -σ o ), as evident from creep law given by Eq. (2).Therefore, to investigate the effect of particle (SiC p ) size on the creep rates, the distribution of (σ e -σ o ), is plotted in Fig. 5. Though, the effective stress varies a little with particle size but the threshold, σ o , decreases from 19.83 MPa to 16.29 MPa with the increase in particle size from 1.7µm to 45.9 µm, Table 1.Under external load, as the deformation of material reaches to threshold value, the interfacial debonding occurs between particles (SiC p ) and the matrix (Al) (Chen et al. 2008).Due to which the stress triaxiality in the matrix near the reinforcement will decrease and the debonded particles are no longer be able to transfer the stress.The damage dissipation caused by interfacial debonding increases with increase in particle (SiC p ) size (Chen et al. 2007), thereby decreasing the threshold stress, Table 1.As a result of this, the stress difference (σ e -σ o ) exhibits sizable variation throughout the cylinder with decreasing SiC P size from 45.9 µm to 1.7µm.The decrease observed towards the inner radius are relatively more than those observed near the outer radius.
As a consequence, the effective strain rate (ε e ) also decreases significantly with decreasing SiC P size as shown in Fig. 6(a).The decrease observed is about two orders of magnitude on decreasing SiC P size from 45.9µm to 1.7 µm.The decrease in effective strain rate may be attributed to decrease in creep parameter M and increase in threshold stress σ o with decreasing size of SiC P reinforcement (Table-1) as evident from creep law given in Eq. ( 2).The radial (compressive) and tangential (tensile) strain rates, Fig. 6(b), decrease on moving from the inner to the outer radius of the cylinder.Throughout the cylinder, the radial and tangential strain rates remain equal in magnitude but have opposite nature due to the condition of incompressibility (Eq.7) and of plane strain condition (ε z = 0).The effect of particle size on both the strain rates is similar to those observed for effective strain rate in Fig. 6(a).The tangential and radial creep rates in the cylinder may be reduced to a significant extent by reinforcing finer SiC P particles.The smaller size particles will be larger in number for the same volume fraction of reinforcement, thereby, leading to: (i) more transfer of load to the reinforcement with a corresponding reduction in the level of effective stress acting on the matrix, and (ii) enhancement of substructure strength, which help in restraining the creep flow (Li and Langdon, 1998;Peng et al. 1998;Peng et al. 1999;Han and Langdon, 2002).Further, the threshold stress observed for finer particles is higher than those observed for coarser one (Table 1) due to lesser degree of debonding between particles and matrix (Chen et al. 2007).As a result, the load is effectively transferred from the matrix to reinforcement hence reducing the creep rates in cylinder having finer sized SiC p .8), decreases significantly over the entire radii with increasing SiC P content.The decrease observed is relatively more prominent towards the inner radius compared to those observed around the outer radius.As expected, the effective strain rate decreases significantly with increasing content of SiC P , Fig. 9 (a).The effective strain rate reduces by about two orders of magnitude on increasing the content of SiC P from 10% to 30%.The reduction in effective strain rate, given by Eq. ( 2), may be attributed to decrease in creep parameter M and increase in threshold stress σ o with increasing content of SiC P (Table-1).The impact of particle content on tangential and radial creep rates, Fig. 9(b), is similar to those observed for effective strain in Fig. 9(a).
By increasing the content of SiC P in the composite cylinder, the inter-particle spacing decreases which cause the increase in threshold stress (Li and Langdon, 1999) but decrease in creep parameter M (Table-1), both these factors contribute in reducing the strain rates significantly.Mishra and Pandey (1990), in their review of uniaxial creep data of Nieh (1984), have also observed that the creep rate in SiC (whisker) reinforced aluminum alloy (6061Al) composite could be reduced significantly by increasing the content of reinforcement.Similar effect of increasing SiC (particle) content on strain rate has been observed by (Pandey et al. 1992) for Al-SiC P composite under uniaxial test.

Effect of Temperature
The creep in any material is significantly affected by operating temperature.Therefore, this section discusses the effect of varying operating temperature on the stresses and strain rates in a thick cylinder made of aluminium matrix composite containing 20 vol% of SiC P .Figs. 10(a Sherby, 1993;Cadek et al. 1998) and the creep parameter M increases (Table -1), as a result of which the strain rates in the composite cylinder increase to a significant extent.The effect of temperature on the creep rate observed in this study are similar to those reported by (Pandey et al. 1992) for Al-SiC P composites under uniaxial creep test.

Conclusions
The present study has led to the following conclusions: . The steady state radial, tangential and axial stresses increases on moving from the inner to the outer radius of a thick walled composite cylinder.The stress distributions do not vary significantly for various combinations of particle size, particle content and operating temperature, except for some sizable variation observed in tangential and axial stresses with varying content of SiC P .
. The tangential as well as radial strain rates in an isotropic thick-wall internally pressurized Al-SiC P cylinder decreases on moving from the inner to the outer radius.
The strain rates induced in the composite cylinder could be reduced significantly by incorporating finer size of reinforcement (SiC P ), increasing the content of reinforcement and decreasing the operating temperature.

Figure 1 .
Figure 1.Variation of ε 1/5 versus σ in Al-SiC p composite for different (a) particle sizes of SiC, (b) vol% of SiC and (c) temperature

Figure 2 .
Figure 2. Comparison of experimental and estimated strain rates in Al-SiC p composite for different (a) particle sizes of SiC, (b) vol% of SiC and (c) temperature

Figure 3 .
Fig. 4(a), remains compressive over the entire cylinder radius with maximum at the inner radius and zero at the outer radius, due to the imposed boundary conditions given in Eqs.(20) and (21).The tangential stress shown in, Fig. 4(b), remains always tensile and increases on moving from the inner to the outer radius of cylinder.Unlike radial and tangential stresses, the axial stress, Figs.4(c), decreases from maximum compressive, observed at the inner radius, to reach a maximum tensile value at the outer radius of the cylinder.It is clearly evident from Figs. 4(a)-(c) that the variation in particle size does not have a sizable effect on the stresses.The maximum variation observed in radial and tangential stresses are less than 1% and for axial stress it is around 5% at a radius of 30.5 mm, in cylinders having relatively coarser SiC P of size 14.5 and 45.9 µm.when compared to cylinder having finer SiC P of 1.7 µm size.The variation of effective stress,σ e , given by Eq. (11), for varying particle size of SiC is shown in Fig.4(d).The effective stress shown does not change on varying the particle size near the inner radius; however, towards the outer radius it decreases Figure 4. Variation of creep stresses in composite cylinder for varying particle size of SiC (V = 10 vol%, T = 350 o C) Figure 6.Variation of strain rates in composite cylinder for varying particle size of SiC (V = 10 vol%, T = 350 o C).

Figure 7 .Figure 8 .
Figure 7. Variation of creep stresses in composite cylinder for varying particle content of SiC (P = 1.7 µm, T = 350 o C) )-(c) show respectively the variation of radial, tangential, axial stresses in a composite cylinder for three different operating temperatures ie.350 0 C, 400 0 C and 450 0 C. The effect of temperature on the stresses is not significant, except for a slight variation observed in tangential stress, Fig.10(b).On increasing the operating temperature from 350 0 C to 450 0 C, the tangential stress increases a little (around 1%) near the inner radius but towards the outer radius of the cylinder it does not change.The effective stress, Fig.10(d), at the inner radius does not change with varying temperature; however, it exhibits a marginal decrease at the outer radius with increase in temperature.The stress difference, (σ e -σ o ), decreases throughout the cylinder with decreasing temperature, Fig.11.The effective as well as radial and tangential strain rates observed in Figs.12(a) and 12(b) respectively increases by about two orders of magnitude on increasing the temperature from 350 0 C to 450 0 C.With increase in operating temperature, the threshold stress σ o decreases (Gonzalez and Figure 9. Variation of strain rates in composite cylinder for varying particle content of SiC (P = 1.7 µm, T = 350 o C)