25 Thick Cylinders Stresses Due to Internal and External Pressures

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T. Singh, V. K. Gupta: Modeling creep in a thick composite cylinder subjected to internal and external pressures

Tejeet Singha, Vinay K. Guptab a b

Department of Mechanical Engg., SBS College of Engg. and Tech., Ferozepur, India Mech. Engg., UCoE, Punjabi Univ., Patiala, India

International Journal of Materials Research downloaded from www.hanser-elibrary.com by McMaster University on October 22, 2014 For personal use only.

Modeling creep in a thick composite cylinder subjected to internal and external pressures A mathematical model to describe secondary creep in a thick composite cylinder made of Al–SiCp and subjected to both internal and external pressures has been developed. The creep behavior of the composite has been described by a threshold stress based creep law with a stress exponent of 5. The model developed has been used to investigate the effect of varying size and content of the SiCp dispersoid on the stresses and strain rates in a composite cylinder. It is observed that the stress distributions in the cylinder do not have significant variation with varying size and content of the SiCp. Unlike stresses, the strain rates in the cylinder are reduced to a significant extent by decreasing the size of SiCp and increasing the content of SiCp. Keywords: Modeling; Second stage creep; Composite; Cylinder; Pressure

1. Introduction The cylinder is a commonly used component design in various applications such as hydraulic cylinders, gun barrels, nuclear reactors, boilers, fuel tanks, accumulator shells, emergency breathing cylinders, [1 – 5]. In some of these applications viz. piping of nuclear reactors, pressure vessels for industrial gases or transportation of highly-pressurized fluids, the material of the cylinder is exposed to high temperature and severe mechanical loadings, leading to significant creep and thereby reducing the service life [3, 6, 7]. Prediction of creep life in many axisymmetric problems, including pressure vessels, is a very important and complex task. Even the most elaborate finite element procedure yields results that are very time consuming and are not always reliable. Creep analysis of thick-walled cylinder made of an isotropic monolithic material and subjected to internal pressure has been presented by Weir [8], and King and Mackie [9]. Pai [10] used a piecewise linear model to obtain the solutions for creep stresses and creep rates in a thick walled orthotropic cylinder subjected to internal pressure. Rimrott [11] solved the problem of a closed end, thick-walled, hollow circular cylinder subjected to internal pressure and estimated the creep stresses, creep strains and creep rates in the cylinder. Bhatnagar and Gupta [12] have solved the problem of creep in thick walled cylinder made of an orthotropic material and subjected to internal pressure. Sim and Penny [13] analyzed a range of thick-walled tubes operating under creep conditions for different loadings, which include internal pressure, external surface loading and inertia loading. Int. J. Mat. Res. (formerly Z. Metallkd.) 101 (2010) 2

Bhatnagar and Arya [14] investigated large strain creep deformation in a thick-walled cylinder made of an anisotropic material subjected to internal pressure. Chen et al. [15] analyzed creep behavior of a functionally graded cylinder under both internal and external pressures. They derived an asymptotic solution on the basis of Taylor expansion series by assuming axisymmetric properties of the graded material, dependent on the radial coordinate. The comparison of approximate solutions corresponding to different higher-order terms with the results of finite element analysis indicates that the use of fifth-order form is sufficiently accurate to calculate the creep stress distribution with satisfactory approximation. You et al. [16] analyzed secondary creep in thick-walled cylindrical vessels made of functionally graded materials subjected to internal pressure. They assumed material parameters involved in Norton’s law to be the functions of radial coordinate. The effect of material parameters, varying radially, on the stresses was investigated. Under severe mechanical and thermal loadings cylinders made of monolithic materials may not perform well. The metal matrix composites (MMCs) such as aluminum or aluminum alloy matrix reinforced with silicon carbide (SiC) particles or whiskers offer excellent mechanical properties, viz. high specific strength, high specific stiffness and high temperature stability. Therefore, these MMCs could be successfully employed for cylinders operating under high pressure and exposed to high service temperature [17 – 21]. Since the primary creep stage is transient and occurs over a short interval of time, most of the useful life of the components, such as cylindrical pressure vessels, operating under high temperature is subject to secondary creep effects. The present work is aimed at developing a mathematical model to describe secondary creep in a thick walled composite cylinder made of Al–SiCp (subscript p refers to the particulate form of SiC) operating under simultaneous action of internal and external pressures and exposed to elevated temperature. The model developed is used to evolve an understanding of the role of material parameters, viz. size and content of the reinforcement (SiCp) on the creep stresses and strain rates in the composite cylinder. The study presented may help designers to select the optimum size and content of the costly reinforcement in a composite cylindrical vessel operating under a given set of operating pressures and temperature, in order to control the resulting creep deformations.

2. Creep law In this study, the cylinder is assumed to be composed of Al– SiCp composites. Several investigators have used the fol279

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Basic T. Singh, V. K. Gupta: Modeling creep in a thick composite cylinder subjected to internal and external pressures


lowing form of threshold stress based law to relate effective creep rate, e_ e , to the effective applied stress, re , in aluminum based composites [22, 23], r r n Q e 0 e_ e ¼ A0 ð1Þ exp RT E where the symbols A’, r0 , n, Q, E, R and T denote respectively the structure dependent parameter, threshold stress, true stress exponent, true activation energy, temperaturedependent Young’s modulus, gas constant and operating temperature. The values of true stress exponent n appearing in Eq. (1) are chosen as 3, 5 or 8, respectively corresponding to creep controlled by viscous glide processes of dislocation, creep controlled by high temperature dislocation climb (lattice diffusion), and lattice diffusion-controlled creep with a constant structure [22]. Some investigators [24 – 27] have used a true stress exponent of 8 to describe secondary creep in Al–SiCp,w (subscript “p” for particle and “w” for whisker) composites, however, a large number of other investigators [28 – 35] have noticed that a stress exponent of either 3 or 5, rather than 8, provides a better description of creep data observed for discontinuously reinforced Al–SiC composites. In the light of the abovementioned, the value of stress exponent (n) appearing in Eq. (1) is chosen as 5 in the present work.

tematic error, if any, in the available experimental results, the creep results from a single source have been used. Figure 1 reveals that e_ 1=5 versus r plots corresponding to the observed experimental data points of Al–SiCp composites for various combinations of particle size and particle content exhibit an excellent linearity. The coefficient of correlation for these plots has been reported in excess to 0.945 as evident from Table 1. Therefore, the choice of stress exponent (n) = 5 in the present work, to describe second stage creep behavior of Al–SiCp composite, appears to be appropriate.

3. Assessment of creep parameters (a)

The creep law given by Eq. (1) may be alternatively expressed as, n e_ e ¼ Mðre r0 Þ ð2Þ 1=n where M ¼ E1 A0 exp Q . RT The creep parameters M and r0 appearing in Eq. (2) are dependent on the type of material and in addition they are also affected by the operating temperature (T). In a composite, the reinforcement size (P) and the content (V) are the primary material variables affecting these parameters. In the present study, the values of M and r0 have been extracted from the experimental creep results reported for Al–SiCp composite under uniaxial loading [25], by plotting the variation of e_ 1=5 with r on linear scales as shown in Fig. 1. Following a linear extrapolation technique [36], the values of creep parameters M and r0 have been obtained from the slope and intercepts of these graphs and are reported in Table 1. In order to avoid variation due to sys(b)

Table 1. Estimated values of creep parameters for Al–SiCp composites P (lm)

T V M (8C) (vol.%) (s – 1/5/MPa)

r0 (MPa)

Coefficient of correlation

1.7 14.5 45.9

350

10

4.35 · 10 – 3 8.72 · 10 – 3 9.39 · 10 – 3

19.83 16.50 16.29

0.945 0.999 0.998

1.7

350

10 20 30

4.35 · 10 – 3 2.63 · 10 – 3 2.27 · 10 – 3

19.83 32.02 42.56

0.945 0.995 0.945

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Fig. 1. Variation of e_ 1=5 with r in Al–SiCp composite for different (a) size and (b) content (vol.%) of SiCp [25].

The accuracy of creep stresses and creep rates estimated in a composite (Al–SiCp) cylinder will depend on the accuracy involved with the prediction of values of creep parameters M and r0 for various combinations of size and content of the SiCp reinforcement. To verify the accuracy of the creep parameters given in Table 1, the estimated values of parameters M and r0 have been back substituted in the creep law given by Eq. (2), to estimate the strain rates corresponding to the observed experimental stress values reInt. J. Mat. Res. (formerly Z. Metallkd.) 101 (2010) 2

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ported by Pandey et al. [25] for various combinations of particle size and particle content in Al–SiCp composite (refer to Table 1). The strain rates thus estimated have been compared with the experimental strain rates observed by Pandey et al. [25]. An excellent agreement between the experimental and the estimated strain rates is noticed in Fig. 2, which inspires confidence in the creep parameters used in this study.

are respectively taken along radial, tangential and axial direction of the cylinder. For the purpose of analysis following assumptions are made in the present work. (i) Creep deformation is infinitesimally small. (ii) Steady state condition of stress is assumed. (iii) Material of the cylinder is incompressible, isotropic and possesses uniform distribution of SiCp in aluminum matrix. (iv) The pressures are applied gradually and held constant during the loading history. The geometric relationships between radial (_er ) and tangential (_eh ) strain rates are, e_ r ¼

du_ r dr

ð3Þ

e_ h ¼

u_ r r

ð4Þ

where u_ r ¼ du dt is the radial displacement rate and u is the radial displacement. Eliminating u_ r from Eqs. (3) and (4), we get the deformation compatibility equation given by, r

d_eh ¼ e_ r e_ h dr

ð5Þ

The boundary conditions for a cylinder subjected to both internal and external pressures are, (a)

ðiÞ

At

r ¼ a;

ðiiÞ

At r ¼ b;

rr ¼ pi

ð6Þ

rr ¼ po

ð7Þ

The negative sign of rr in Eqs. (6) and (7) implies the compressive nature of radial stress. The equilibrium equation for a thick-walled cylindrical vessel subjected to uniform internal and external pressures is [16], r

drr ¼ rh rr dr

ð8Þ

Due to the assumed condition of incompressibility, one may write, e_ r þ e_ h þ e_ z ¼ 0

ð9Þ

where e_ Z is the strain rate in the axial (z) direction. The generalized constitutive equations for creep, when reference frame is along the principal directions of r, h and z, are given by [16], (b)

e_ r ¼

e_ e ½2rr rh rz 2re

ð10Þ

e_ h ¼

e_ e ½2rh rz rr 2re

ð11Þ

e_ z ¼

e_ e ½2rz rr rh 2re

ð12Þ

Fig. 2. Comparison of experimental [25] and estimated strain rates in Al–SiCp composite for different (a) size and (b) content (vol.%) of SiCp

4. Mathematical modeling of creep behavior Let us consider a long thick-walled cylinder made of Al– SiCp composite, with closed end and having inner and outer radii a and b respectively. The cylinder is subjected to internal pressure pi and external pressure po. The axes r, h and z Int. J. Mat. Res. (formerly Z. Metallkd.) 101 (2010) 2

where e_ e and re are the effective strain rate and the effective stress respectively. 281

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The effective stress for isotropic material may be expressed using von Mises yield criterion [37] as, i12 1 h re ¼ pffiffiffi ðrh rz Þ2 þ ðrz rr Þ2 þ ðrr rh Þ2 2

ð13Þ

For a cylinder made of incompressible material, the closed end condition reduces to the Plane Strain condition i. e. e_ z ¼ 0 [14, 38]. Therefore, by virtue of Plane Strain condition (_ez ¼ 0) and Eqs. (3), (4) and (9), the radial displacement rate is given by,


K u_ r ¼ r

e_ h ¼

K r2

K r2

ð14Þ

ð15Þ

Substituting Eqs. (22) and (23) into Eq. (17), the axial stress is expressed as,

1 2 b 2n n r 0:5 po ð24Þ I2 ln rz ¼ X b 1 n r

ð16Þ

The Eqs. (17) and (18) may be substituted into Eqs. (10) and (11) to obtain the radial and tangential strain rates in the composite cylinder in terms of effective strain rate as,

ð17Þ

e_ r ¼ _eh ¼ 0:87 e_ e

Under plane strain condition, Eq. (12) becomes, rz ¼

rr þ rh 2

Therefore, the axial stress at any point in the cylinder is arithmetic mean of radial and tangential stresses at that location. Substituting Eq. (17) into Eq. (13), the effective stress is obtained as, pffiffiffi 3 re ¼ ðr h r r Þ ð18Þ 2 Equations (15) and (17) are substituted in Eq. (10) to get, 4 re K rh rr ¼ ð19Þ 3 e_ e r2 Substituting e_ e and re respectively from Eqs. (2) and (18) into above equation and simplifying we get, rh rr ¼

I1

2 þ I2 rn

nþ1 4 2n where, I1 ¼ 3

ð20Þ 1

Kn M

! and

2 I2 ¼ pffiffiffi r0 . 3

Substituting Eq. (20) into equilibrium Eq. (8) and integrating the resulting equation, n I1 rr ¼ 2 þ I2 ln r þ B 2 rn

ð21Þ

where B is another constant of integration. Using boundary conditions given by Eqs. (6) and (7) into Eq. (21), the values of constants B and I1 are obtained as, n 2 B ¼ I1 bn I2 ln b po 2 I1 ¼ 282

2 X n

The constants B and I1 obtained above are substituted in Eq. (21) to get the radial stress,

b 2 2 þ po ð22Þ rr ¼ X rn bn þ I2 ln r Using Eq. (22) in Eq. (20), the tangential stress is obtained,

2 b 2 2 rn þ I2 1 ln po ð23Þ rh ¼ X b n 1 n r

where K is the constant of integration. Using Eq. (14) into Eqs. (3) and (4) we get, e_ r ¼

where, pi po I2 ln ba X¼ 2 2 ðan bn Þ

ð25Þ

Therefore, the radial and tangential strain rates in the composite cylinder are 87 % of the effective strain rate at the corresponding radial location.

5. Numerical calculations Based on the mathematical model developed in the previous section, computations have been carried out to obtain the second stage creep response of the composite cylinder for various combinations of size and content of the reinforcement (SiCp). The inner and outer radii of the cylinder are taken 25.4 mm and 50.8 mm respectively while the internal and external pressures are assumed to be 85.25 MPa and 42.62 MPa respectively. The external pressure is kept half of the internal pressure in line with the earlier reported work of Bhatnagar and Arya [14]. The dimensions of the cylinder used and the value of internal pressure chosen in this study are similar to those adopted by Johnson et al. [39] in their work on thick walled cylinder made of aluminum alloy (RR59). The cylinder is assumed to operate at a temperature of 350 8C. The creep parameters M and r0 for the composite material, required during the computation process, have been taken from Table 1 for the desired combination of size and content of the SiCp reinforcement.

6. Results and discussions Before presenting the results computed, it is necessary to check the validity of the mathematical formulation carried out. To achieve this goal, following the current analysis, we have computed the tangential, radial and axial stresses in a cylinder made of monolithic material for which the results are reported in an earlier investigation by Bhatnagar and Arya [14]. The dimensions of the cylinder, operating pressures and creep parameters used for validation are summarized in Table 2. In order to obtain the creep parameters Int. J. Mat. Res. (formerly Z. Metallkd.) 101 (2010) 2

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Table 2. Data used for validation Cylinder dimensions: a = 20 mm, b =100 mm Internal Pressure (pi) = 281.23MPa, External Pressure (po) = 0

6 Values of constants in power law e_ i ¼ Bc rrc , Bc = 5.787 · 10 – 17 s – 1, rc ¼ 7:03 MPa


Creep parameters estimated: M = 1.7723 · 10 – 4 s – 1/5/MPa, r0 = 24.38 MPa

M and r0 for the arbitrary monolithic material reported in the study of Bhatnagar and Arya [14], the effective strain rates ð_ee Þ have been computed at the inner and outer radii of the cylinder by substituting the values of effective stresses at the corresponding locations in the power law creep reported in Table 2. The values of effective stress and effective strain rates at the inner and outer radii: (i) at inner radius, re = 189.93 MPa and e_ e = 2.168 · 10 – 8 s – 1, and (ii) at outer radius, re = 116 MPa and e_ e = 1.128 · 10 – 9 s – 1, have been substituted in Eq. (2) to estimate the creep parameters M and r0 (see Table 2) for the monolithic material used by Bhatnagar and Arya [14]. The creep parameters thus obtained have been used to compute the distribution of tangential, radial and axial stresses in the monolithic cylinder. A good agreement between the stresses calculated from the current analysis procedure and those reported by Bhatnagar and Arya [14] is observed in Fig. 3, therefore, validating the current analysis.

Fig. 4. Variation of creep stresses for varying particle size of SiC (V = 10 vol.%, T = 350 8C).

Fig. 3. Comparison of stresses in monolithic cylinder estimated from current analysis and reported by Bhatnagar and Arya [14].

6.1. Effect of particle size Figure 4 shows the distribution of radial, tangential, axial and effective stresses in the composite cylinder for varying size of SiCp reinforcement from 1.7 lm to 45.9 lm. The radial stress remains compressive over the entire radius with maximum value at the inner radius and minimum value at the outer raInt. J. Mat. Res. (formerly Z. Metallkd.) 101 (2010) 2

dius, due to the imposed boundary conditions given by Eqs. (6) and (7). The tangential stress varies from maximum compressive value at the inner radius to reach a maximum tensile value at the outer radius of the cylinder. The axial stress, which is the average of radial and tangential stresses, see Eq. (17), exhibits a variation similar to that of the tangential stress. The variation in size of reinforcement (SiCp) does not exhibit a sizable effect on the values of stresses in the cylinder, except for some marginal variation observed in tangential stress near the inner and outer radii. The maximum variation observed in tangential stress is about 4 % at the inner as well as at the outer radius whereas for axial stress it is less than 1 % at inner radius and about 2 % at outer radius of the cylinder having coarser SiCp of size 45.9 lm as compared to those observed for cylinder having finer SiCp of 1.7 lm. The tangential stresses, both compressive (near inner radius) as well as tensile (near the outer radius), in the cylinder having finer sized (1.7 lm) SiCp are marginally higher than those observed in cylinders with relatively coarser SiCp (i. e. 283

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Fig. 5. Variation of stress difference for varying particle size of SiC (V = 10 vol.%, T = 350 8C).

14.5 lm and 45.9 lm). The effective stress decreases on moving from the inner towards the outer radius of the cylinder. With increase in SiCp size from 1.7 lm to 45.9 lm, the effective stress exhibits marginal increase near the inner radius whereas it decreases marginally towards the outer radius. The strain rates, given by Eqs. (10) and (11), are related to the effective strain rate, e_ e , which ultimately depends upon ðre r0 Þ, the difference between effective stress (re ) and threshold stress (r0 ), as evident from creep law given by Eq. (2). Therefore, to investigate the effect of reinforcement (SiCp) size on the creep rates, the variation of stress difference ðre r0 Þ, with radial distance is depicted in Fig. 5. It is noticed that the stress difference ðre r0 Þ decreases significantly with decreasing SiCp size from 45.9 lm to 1.7 lm over the entire radius. The decrease observed may be attributed to the increase in threshold stress with decrease in particle size as evident from Table 1. As a consequence, the effective strain rate (_ee ) also reduces significantly over the entire radius with decreasing particle (SiCp) size, Fig. 6. The decrease observed is about two or-

Fig. 6. Variation of strain rates for varying particle size of SiC (V = 10 vol.%, T = 350 8C).

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ders of magnitude with decrease in SiCp size from 45.9 lm to 1.7 lm. It is quite evident from Eq. (2) that the decrease observed in effective strain rate is due to the decrease in parameter M and increase in threshold stress r0 , with decreasing size of SiCp, as revealed in Table 1. The radial and tangential strain rates are equal in magnitude but with opposite nature due to the condition of incompressibility (Eq. (9)) and the assumed plane strain condition ð_eZ ¼ 0Þ. The radial (compressive) and tangential (tensile) strain rates in the cylinder, for a given size of reinforcement (SiCp), are 13 % lower than the corresponding effective strain rates (Eq. (25)) as is evident from Fig. 6. The effect of particle size on strain rates is similar to those observed for effective strain rate. Therefore, it may be concluded that the second stage creep rates in the cylinder could be reduced significantly by employing finer reinforcement (SiCp) in the aluminum matrix. For the same volume fraction of reinforcement, the smaller size particles will be larger in number, and therefore, lead to more load transfer to the reinforcement with a corresponding reduction in the level of effective stress shared by the matrix material, which enhances the substructure strength [40 – 43] and ultimately helps in restraining the creep flow of cylinder. 6.2. Effect of particle content Figure 7 shows the variation of stresses in composite cylinder containing different content (volume) of SiCp i. e. 10 %,

Fig. 7. Variation of creep stresses for varying particle content of SiC (P = 1.7 lm, T = 350 8C).

Int. J. Mat. Res. (formerly Z. Metallkd.) 101 (2010) 2

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20 % and 30 %. The radial stress does not exhibit sizable variation on modifying the content of SiCp except for a small increase observed somewhere in the middle region of the cylinder with increase in particle content from 10 % to 30 %. Unlike particle size, Fig. 4, the increase in particle content induces some sizable variation on tangential and axial stresses as observed in Fig. 7. With increase in content of SiCp from 10 % to 30 %, the tangential stress (compressive near inner and tensile near outer radius) increases over the entire radius of the cylinder. The maximum increase observed in tangential stress is about 25 % at the outer radius. The axial stress (compressive) increases near the inner radius but decreases towards the outer radius, on increasing the content of SiCp from 10 % to 30 %. At the inner radius, the axial stress increases by about 4 % but at the outer radius it decreases by about 12 % with increase in particle content from 10 % to 30 %. Unlike axial stress, the effective stress decreases near the inner radius but increases near the outer radius with increase in content of SiCp from 10 % to 30 %. The maximum decrease (at inner radius) and increase (at outer radius) observed are respectively about 5 % and 6 %. The stress difference, ðre r0 Þ, shown in Fig. 8, decreases significantly over the entire radial distance with increasing SiCp content from 10 % to 30 %. The decrease observed is relatively more towards the inner radius. As expected, the

Fig. 9. Variation of strain rates for varying particle content of SiC (P = 1.7 lm, T = 350 8C).

which causes the increase in threshold stress [44] but decrease in creep parameter M (Table 1), both these factors contribute in reducing the strain rates significantly. Mishra and Pandey [24] in their review of uniaxial creep data of Nieh [17] also noticed that creep rate in SiC (whisker) reinforced aluminum alloy (6061Al) composite could be significantly reduced by increasing the content of reinforcement. A similar effect of increasing SiC (particle) content on strain rate has been observed by Pandey et al. [25] for Al– SiCp composite under uniaxial creep. 6.3. Selection of material parameters

Fig. 8. Variation of stress difference for varying particle content of SiC (P = 1.7 lm, T =350 8C).

effective strain rate, Fig. 9, decreases significantly with increase in content of SiCp. The effective strain rate decreases by about four orders of magnitude throughout the cylinder on increasing the content of SiCp from 10 % to 30 %. The decrease observed in effective strain rate may be attributed to decrease in parameter M and increase in threshold stress r0 with increasing content of SiCp, as evident from Table 1. The impact of varying particle content on tangential and radial creep rates is similar to those noticed for effective strain rate in Fig. 9. By increasing the content of SiCp in the composite cylinder the inter-particle spacing decreases, Int. J. Mat. Res. (formerly Z. Metallkd.) 101 (2010) 2

It is evident from the above discussion that the creep stresses in a thick composite cylinder do not vary significantly by varying size and content of reinforcement (SiCp) as compared to the variation observed in strain rates. From the point of view of designing a composite pressure vessel, operating under elevated temperature, the strain rates are considered to be primary design parameters. In order to reduce the secondary creep rates in a composite cylinder operating under a given set of operating conditions (i. e. operating pressures and temperature) any of the following three options could be selected: (i) employing finer reinforcement (SiCp) without varying its content (Fig. 6), (ii) incorporating higher quantities of dispersoids (SiCp) without altering its size (Fig. 9), and (iii) simultaneously decreasing the size and increasing the content of SiCp reinforcement. The selection of optimum size and content of reinforcement in the composite pressure vessel for a given set of operating conditions can be decided by simultaneously optimizing the cost of composite and the value of maximum strain rate in the composite cylinder for different combinations of size and content of reinforcement within the specified range. 285

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7. Conclusions The stress distributions in thick walled composite cylinder do not vary significantly by varying size and content of reinforcement (SiCp), except for some sizable variation observed in tangential and axial stresses with varying content of SiCp. However, the strain rates (radial and tangential) induced in the cylinder could be reduced significantly by incorporating finer reinforcement and increasing the content of reinforcement. The optimum size and content of the reinforcement in a composite cylinder under a given set of operating conditions (i. e. operating pressures and temperature) can be obtained by performing simultaneous optimization of the cost of composite and the value of maximum strain rate in the composite cylinder for different combinations of size and content of the reinforcement within the desired range. References [1] V.K. Arya, N.S. Bhatnagar: J. Mech. Eng. Sci. 18 (1976) 1. DOI:10.1243/JMES_JOUR_1976_018_003_02 [2] C. Becht-IV, Y. Chen: J. Pressure Vessel Technol. 122 (2000) 121. DOI:10.1115/1.556161 [3] S.K. Gupta, S. Pathak: Indian J. Pure Appl. Math. 32 (2001) 237. [4] J. Perry, J. Aboudi: J. Pressure Vessel Technol. 125 (2003) 248. DOI:10.1115/1.1593078 [5] J.M. Bergheau, J. Devaux, G. Mottet, P. Gilles: J. Pressure Vessel Technol. 126 (2004) 163. DOI:10.1115/1.1687799 [6] Y. Tachibana, T. Iyoku: Nuclear Eng. Des. 233 (2004) 261. DOI:10.1016/j.nucengdes.2004.08.013 [7] S. Hagihara, N. Miyazaki: Nuclear Eng. Des. 238 (2008) 33. DOI:10.1016/j.nucengdes.2007.04.009 [8] C.D. Weir: J. Appl. Mech. 24 (1957) 464. [9] R.H. King, W.W. Mackie: J. Basic Eng. 89 (1967) 877. [10] D.H. Pai: Int. J. Mech. Sci. 9 (1967) 335. DOI:10.1016/0020-7403(67)90039-2 [11] F.P.J. Rimrott: J. Appl. Mech. 26 (1959) 271. [12] N.S. Bhatnagar, S.K. Gupta: J. Physical Soc. Japan 27 (1969) 1655. DOI:10.1143/JPSJ.27.1655 [13] R.G. Sim, R.K. Penny: Int. J. Mech. Sci. 13 (1971) 987. DOI:10.1016/0020-7403(71)90023-3 [14] N.S. Bhatnagar, V.K. Arya: Int. J. Non Linear Mechanics. 9 (1974) 127. DOI:10.1016/0020-7462(74)90004-3 [15] J.J. Chen, S.T. Tu, F.Z. Xuan, Z.D. Wang: J. Strain Analysis 42 (2007) 62. DOI:10.1243/03093247JSA237 [16] L.H. You, H. Ou, Z.Y. Zheng: Composite Structures 78 (2007) 285. DOI:10.1016/j.compstruct.2005.10.002 [17] T.G. Nieh: Metall. Trans. A 15 (1984) 139. DOI:10.1007/BF02644396 [18] A.K. Roy, S.W. Tsai: J. Pressure Vessel Technol. 110 (1988) 255. DOI:10.1115/1.3265597 [19] Y. Fukui, N. Yamanaka, K. Wakashima: JSME A 36 (1993) 156. [20] R.S. Salzar, M.J. Pindera, F.W. Barton: J. Pressure Vessel Technol. 118 (1996) 13. DOI:10.1115/1.2842155 [21] V.K. Gupta, S.B. Singh, H.N. Chandrawat, S. Ray: Metall. Mater. Trans. A 35 (2004) 1381. DOI:10.1007/s11661-004-0313-3 [22] S.C. Tjong, Z.Y. Ma: Mater. Sci. Eng. R 29 (2000) 49. DOI:10.1016/S0927-796X(00)00024-3 [23] Z.Y. Ma, S.C. Tjong: Composites Sci. Technol. 61 (2001) 771. DOI:10.1016/S0266-3538(01)00018-5 [24] R.S. Mishra, A.B. Pandey: Metall. Trans. 21A (1990) 2089. [25] A.B. Pandey, R.S. Mishra, Y.R. Mahajan: Acta Metall. Mater. 40 (1992) 2045. DOI:10.1016/0956-7151(92)90190-P [26] D.G. Gonzalez, O.D. Sherby: Acta Metall. Mater. 41 (1993) 2797. DOI:10.1016/0956-7151(93)90094-9

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(Received September 6, 2008; accepted April 16, 2009) Bibliography DOI 10.3139/146.110273 Int. J. Mat. Res. (formerly Z. Metallkd.) 101 (2010) 2; page 279 – 286 # Carl Hanser Verlag GmbH & Co. KG ISSN 1862-5282

Correspondence address Dr. Vinay K. Gupta Reader (Mechanical Eng.) University College of Eng., Punjabi University, Patiala-147002 (Punjab), India Tel.: +91 175 2280 166 Fax: +91 175 3046 333 E-mail: [email protected]

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Int. J. Mat. Res. (formerly Z. Metallkd.) 101 (2010) 2

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