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Time-dependent thermo-creep analysis of rotating FGM thick-walled cylindrical pressure vessels under heat flux
International Journal of Engineering Science 82 (2014) 222–237
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Time-dependent thermo-creep analysis of rotating FGM thick-walled cylindrical pressure vessels under heat flux Mohammad Zamani Nejad ⇑, Mosayeb Davoudi Kashkoli Mechanical Engineering Department, Yasouj University, P. O. Box: 75914-353, Yasouj, Iran
a r t i c l e
i n f o
Article history: Received 15 May 2014 Received in revised form 13 June 2014 Accepted 14 June 2014
Keywords: Thick cylindrical pressure vessel Thermo-creep Time-dependent Functionally graded material (FGM) Heat flux
a b s t r a c t In the present study, time-dependent thermoelastic creep response for isotropic rotating thick-walled cylindrical pressure vessels made of functionally graded material (FGM) has been investigated, taking into account the creep behavior of the FGM pressure vessels, as described in Norton’s model. For the purpose of stress analysis in an FGM pressure vessel, material creep behavior and the solutions of the stresses at a time equal to zero (i.e. the initial stress state) are needed. This corresponds to the solution of materials with linear elastic behavior. Therefore, using equations of equilibrium, stress–strain and strain– displacement, a differential equation for displacement is obtained and subsequently the stresses at a time equal to zero are calculated. Using Norton’s law in the multi-axial form in conjunction with the above-mentioned equations in the rate form, the radial displacement rate is obtained and then the radial, circumferential and axial creep stress rates are calculated for the conditions of plane strain and plane stress. When the stress rates are known, the stresses at any time are calculated iteratively. Assuming that the inner surface is exposed to a uniform flux, and that the outer surface is exposed to an airstream, the heat conduction equation for the one-dimensional problem in polar coordinates is used to obtain temperature distribution in the cylinder. Assuming that material properties are a function of the radius of the cylinder and that the Poisson’s ratio is constant, creep stresses, creep strains and radial displacement are plotted against dimensionless radius and time for different values of the powers of the material properties. It has been found that inhomogeneity constants have significant influence on the distributions of the creep stresses and radial displacement. Ó 2014 Published by Elsevier Ltd.
1. Introduction Functionally graded materials (FGMs) are inhomogeneous composites which are usually made from a mixture of metals and ceramics, providing the specific benefits of both the constituents. The continuously compositional variation of the constituents in FGMs from one surface to another offers a graceful solution to the problem of appearing high magnitude shear stresses that may be induced in laminated composites, where two materials with large differences in properties are bonded (Kahrobaiyan, Rahaeifard, Tajalli, & Ahmadian, 2012; Rahaeifard, Kahrobaiyan, Ahmadian, & Firoozbakhsh, 2013). Today, structures made of FGMs have attracted wide and increased attention from scientists and engineers (Birman, 2014; Chun & Ernian, 2013). Rotating discs are important structural components. Their significance lies in their extensive use in steam ⇑ Corresponding author. Tel./fax: +98 741 2221711. E-mail addresses: [email protected], [email protected] (M.Z. Nejad). http://dx.doi.org/10.1016/j.ijengsci.2014.06.006 0020-7225/Ó 2014 Published by Elsevier Ltd.
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and gas turbine rotors, ship propellers, aircraft engines, high speed gears, flywheels, pumps, centrifugal compressors, turbo generators, automotive braking systems, rotational air cleaners, computer disc drives and many other applications. In some of these applications, the disc has to operate at high temperatures while subjected to severe thermo-mechanical loading (Garg, Salaria, & Gupta, 2013; Hassani, Hojjati, Mahdavi, Alashti, & Farrahi, 2012). As a result of severe thermo-mechanical loading, the material of disc undergoes creep deformations, thereby affecting performance of the system (Deepak, Gupta, & Dham, 2010). Over the last years, stresses in FGM rotating thick-walled cylindrical pressure vessels under thermal loading have been analyzed extensively with regard to the elastic material behavior. Thermoelastic analysis of FGM rotating discs was carried out by Kordkheili and Naghdabadi (2007) without taking into account the creep analysis of the disc. Attia, Fitzgeorge, and Pope (1954) investigated the residual stresses produced in cast iron cylinders by the creep-relaxation of thermal stresses. In this study, a series of thick-hollow cylinders were subjected to a radial flow of heat by heating the bore and water-cooling the outer diameter for a chosen period of time, during which the thermal stresses were relaxed by creep. Weir (1957) investigated the creep stresses in pressurized thick walled tubes. Assuming the simply supported and fixed-end boundary conditions for the cylinders, Wah (1961) developed a theory for the collapse of cylindrical shells under steady-state creep and under external radial pressure and high temperature (300–500 F). Considering large strains, Rimrott and Luke (1961) obtained the creep stresses of a rotating hollow circular cylinder made of isotropic and homogeneous materials. Simonian (1979) investigated the thermal stresses in a long thick-walled cylinder in a temperature field, under a centrifugal force, axial force and internal–external pressure. In this paper, it is suggested that the material of the cylinder deforms according to the nonlinear theory of heredity. Besseling (1962) investigated the feasibility of the numerical analysis of nonstationary creep problems for thick-walled tubes under axially symmetric loading. Pai (1967) studied the steady-state creep of a thick-walled orthotropic cylinder subjected to internal pressure. It was observed that the creep anisotropy has a significant effect on the cylinder behavior particularly in terms of creep rates, which may differ by an order of magnitude compared with an isotropic analysis. Sankaranarayanan (1969) studied the steady creep behavior of thin circular cylindrical shells subjected to combined lateral and axial pressures. The analysis was based on the Tresca criterion and the associated flow rule. Assuming that the total strain consists of elastic and creep components, Murakami and Iwatsuki (1969) investigated the transient creep analysis of circular cylindrical shells on the basis of the strain-hardening and timehardening theories. Bhatnagar and Gupta (1969) obtained the solution for an orthotropic thick-walled internally pressurized cylinder, using constitutive equations of anisotropy creep and Norton’s creep law. Murakami and Iwatsuki (1971) developed a numerical analysis of the steady state creep of a pressurized circular cylindrical shell on the basis of Mises’ criterion and the power law of creep. Sim and Penny (1971) studied the deformation behavior of thick-walled tubes subjected to a variety of loadings during stress redistribution caused by creep. Murakami and Suzuki (1971) investigated the steady state creep of simply supported circular cylindrical shells with open ends under internal pressure by using Nortons’s law. Using finite-strain theory Bhatnagar and Arya (1974) studied the creep behavior of a thick-walled cylinder under large strains. Murakami and Tanaka (1976) investigated the creep buckling of clamped circular cylindrical shells subjected to axial compression combined with internal pressure with special emphasis on the concept of creep stability and the accuracy of the analysis. Jones and Sullivan (1976) studied the advantages and limitations of a perturbation method of analysis for the creep buckling of shells by examining the particular case of a long cylindrical shell subjected to a uniform external pressure. Arya, Debnath, and Bhatnagar (1983) investigated the creep problem in a thin circular cylindrical shell made of a homogeneous, incompressible and orthotropic material, using a non-steady creep law. Assuming the plane strain condition, Bhatnagar, Kulkarni, and Arya (1984) obtained the analysis of an internally pressurized, homogeneous, orthotropic rotating cylinder subjected to a steady state creep condition. In another study, considering the effect of anisotropy on stress and strain, Bhatnagar, Kulkarni, and Arya (1986) investigated the creep analysis of thick-walled orthotropic rotating cylinders. Creep damage simulation of thick-walled tubes, using the theta projection concept was investigated by Loghman and Wahab (1996). They obtained a closed-form solution for steady state creep stresses in FGM cylinders. Yang (2000) presented an analytical solution for the calculation of stresses in FGMs for the elastic and creep behavior of the materials. This solution can be used to study the dependence of stress on temperature and time for FG structures. Finally, the analytical results were compared with FEM. Using Seth’s transition theory, Gupta, Sharma, and Pathak (2000) obtained the creep stresses and strain rates for a nonhomogeneous thick-walled rotating cylinder. Gupta and Pathak (2001) studied thermo creep analysis in a pressurized thick hollow cylinder. Jahed and Bidabadi (2003) presented a general axisymmetric method for an inhomogeneous body for a disc with varying thickness. An approximation was employed during their solution algorithm. It means that they avoided considering the differentiation constitutive terms of governing equations for creep analysis. Gupta, Singh, Chandrawat, and Ray (2004) investigated the creep behavior of a rotating FG disc made of silicon carbide particles in a matrix of pure aluminum by using Cherby’s law. The FG disc that they considered had a constant thickness and they assumed linear particles distribution. In addition, they assigned a fixed function to radius direction temperature distribution. Chen, Tu, Xuan, and Wang (2007) studied the creep behavior of a functionally graded cylinder under both internal and external pressures. They observed that an asymptotic solution can be derived on the basis of a Taylor series expansion if the properties of the graded material are axisymmetric and dependent on radial coordinate. In order to investigate creep performance of thick-walled cylindrical vessels or cylinders made of functionally graded materials, You, Ou, and Zheng (2007) proposed a simple and accurate method to determine stresses and creep strain rates in thick-walled cylindrical vessels subjected to internal pressure. Based on the power law constitutive equation, Altenbach, Gorash, and Naumenko (2008) presented the classical solution of the steadystate creep problem for a pressurized thick-walled cylinder. In this paper, they applied an extended constitutive equation which includes both the linear and the power law stress dependencies. Sharma and Sahni (2008) obtained the creep stresses
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for a transversely isotropic thick-walled rotating cylinder under internal pressure by using Seth’s transition theory. Singh (2008) studied the steady state creep in a rotating disc of anisotropic aluminum silicon carbide whisker composite by using Norton’s law. In this study stress and strain rate distributions for anisotropic discs were calculated and then compared with those obtained for an isotropic disc. Sharma (2009) obtained the creep stresses for a non-homogeneous thick-walled rotating cylinder by using transition theory, which is based on the concept of ‘generalized principal strain measure’. Assuming that the creep response of the material is governed by threshold stress-based creep law, Deepak, Gupta, and Dham (2009) studied the effect of the stress exponent on the steady state creep in a rotating disc made of isotropic aluminum–silicon carbide particulate (Al–SiCp) composite. Singh and Gupta (2009, 2010) developed a mathematical model to describe the steady-creep behavior of functionally graded composite cylinders containing linearly varying silicon carbide particles in a matrix of pure aluminum involving threshold stress-based creep law. The model developed was used to investigate the effect of gradient on the distribution of SiCp on the steady-state creep response of the composite cylinder. Assuming that the creep response of the material is governed by Sherby’s law, Rattan, Chamoli, and Singh (2010) investigated the creep response of isotropic axisymmetric rotating discs made of a particle-reinforced FGM. Chamoli, Rattan, and Singh (2010) investigated the creep behavior of an anisotropic rotating disc made of Al–SiCp composite, using Hill’s yield criteria. The creep behavior in this case was assumed to follow Sherby’s constitutive model. Sharma, Sahni, and Kumar (2010) obtained the thermal creep stresses and strain rates for a transversely isotropic thick-walled rotating cylinder under internal pressure by using Seth’s transition theory. They observed that rotating circular cylinder under internal pressure made of transversely isotropic material is on the safer side of the design as compared with rotating circular cylinders under internal pressure made of isotropic material. Deepak et al. (2010) further extended their work to investigate creep behavior of a rotating discs made of functionally graded materials with linearly varying thickness. The discs under investigation were made of composite containing silicon carbide particles in a matrix of pure aluminum. The effect of imposing linear particle gradient on the distribution of stresses and strain rates in the composite disc was also investigated. Assuming total strains to be the sum of elastic, thermal and creep strains, Loghman, Ghorbanpour Arani, Amir, and Vajedi (2010) studied the time-dependent creep stress redistribution analysis of a thick-walled FGM cylinder placed in uniform magnetic and temperature fields and subjected to an internal pressure. Following Norton’s law for material creep behavior, using equations of equilibrium, strain displacement and stress–strain relations in the rate form and considering Prandtl–Reuss relations for creep strain rate-stress equation, they obtained a differential equation for the displacement rate and then calculated the radial and circumferential creep stress rates. In another study Loghman, Ghorbanpour Arani, and Aleayoub (2011) studied the time-dependent creep stress redistribution analysis of thick-walled spheres made of functionally graded material (FGM) subjected to an internal pressure and a uniform temperature field, using the method of successive elastic solution. Singh and Gupta (2011) investigated the steady state creep in transversely isotropic functionally graded cylinder, operating under internal and external pressures. In this paper, the effect of anisotropy on creep stresses and creep rates in the FGM cylinder was analyzed and then compared with an isotropic FGM cylinder. Hoseini, Nejad, Niknejad, and Ghannad (2011) presented a new analytical solution for the steady state creep in rotating thick cylindrical shells subjected to internal and external pressure by using Norton’s law. Sharma, Sahay, and Kumar (2012) investigated the creep stresses in thick-walled circular cylinders under internal and external pressure, using transition theory, which is based on the concept of ‘generalized principal strain measure’. Gupta and Singh (2012) investigated the steady state creep in an anisotropic rotating disc made of Al–SiCp composite having hyperbolically varying thickness, using Hill’s yield criterion followed by the Sherby’s law. Chamoli and Singh (2012) investigated the steady-state creep response of an isotropic FG rotating disc of aluminum silicon carbide particulate composite by taking into account the residual stress present in the disc. Loghman and Atabakhshian (2012) studied the time-dependent creep behavior of rotating cylinders made from exponentially graded material, using Bailey–Norton creep constitutive model. They observed that using exponentially graded material significantly decreases creep strains, stresses and deformations of the EGM rotating cylinder. Jamian, Sato, Tsukamoto, and Watanabe (2013) investigated the creep analysis of a thick-walled cylinder made of functionally graded materials (FGMs) subjected to thermal and internal pressure. Garg et al. (2013) investigated the effect of varying disc thickness gradient on the creep stresses and creep rates in a rotating functionally graded composite disc containing non-linearly varying radial distribution of silicon carbide particles in a matrix of pure aluminum. Singh and Gupta (2014) studied the steady state creep behavior in a functionally graded thick composite cylinder subjected to internal pressure in the presence of residual stress. Hoffman’s yield criterion was used to describe the yielding of the cylinder material in order to account for residual stress. The present study aims to investigate the time-dependent thermoelastic creep response of isotropic rotating thick-walled cylindrical pressure vessels made of functionally graded material subjected to a steady-state thermal loading and the effect of material properties on the distributions of stress and radial displacement through the dimensionless radius and time. For the creep material behavior, the solution is asymptotic. Subsequent to creeping for a long time, the iterative procedure is necessary for the stress analysis. 2. Geometry and loading condition, material properties, and creep constitutive model 2.1. Geometry and loading condition Consider a thick-walled FGM cylindrical vessel with an inner radius a, and outer radius b. Loading is an internal pressure Pi and an external pressure Po, which are axisymmetric, and also an inertia body force due to the rotation of the cylindrical
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Fig. 1. Geometry and boundary conditions of the cylinder.
vessel with a constant angular velocity of x = 1200 rpm and a distributed temperature field due to a steady-state heat conduction from inner surface to outer surface of the vessel. The coordinate axes r, h and z are taken, respectively, along the radial, circumferential, and axial directions of the cylinder (see Fig. 1). 2.2. Material properties In this study, Poisson’s ratio, m, is considered to be a constant and E, q, a and k are assumed to obey the power-law variation as
EðrÞ ¼ Ei ðr=aÞn1
qðrÞ ¼ qi ðr=aÞn2 aðrÞ ¼ ai ðr=aÞn3
ð1Þ
kðrÞ ¼ ki ðr=aÞn4 here Ei, qi, ai and ki are the modulus of elasticity, density, linear expansion and thermal conductivity at the inner surface r = a and n1, n2, n3 and n4 are the in-homogeneity constants determined empirically. 2.3. Creep constitutive model The material creep behavior is described by Norton’s constitutive model in the multi-axial form as follows (Finnie & Heller, 1959)
1þm m 3 r_ ij r_ kk dij þ DrðeffN1Þ Sij 2 E E 1 Sij ¼ rij rkk dij 3 ffi rffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 reff ¼ Sij Sij ¼ pffiffiffi ðrrr rhh Þ2 þ ðrrr rzz Þ2 þ ðrzz rhh Þ2 2 2
e_ ij ¼
ð2Þ ð3Þ ð4Þ
The parameters D and N are the creep coefficient and the creep exponent, respectively. reff is the effective stress, Sij is the deviator stress tensor and rrr, r hh and rzz are, respectively, the radial, circumferential and axial stresses. 3. Heat conduction formulation In the steady-state case, the heat conduction equation for the one-dimensional problem in polar coordinates simplifies to
@ @T ¼0 rk @r @r
ð5Þ
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We may determine the temperature distribution in the cylindrical vessel by solving Eq. (5) and applying appropriate boundary conditions. Eq. (5) may be integrated twice to obtain the general solution
TðrÞ ¼ A1 r n4 þ A2
ð6Þ
It is assumed that the inner surface is exposed to a uniform heat flux, and that the outer surface is exposed to an airstream. To obtain the constants of integration A1 and A2, the following boundary conditions are introduced:
8 0 r¼a > < kT ¼ qa ; 0 kT ¼ h1 ðT T 1 Þ; r ¼ b > : 0 dT T ¼ dr
ð7Þ
Applying these conditions to the general solution, we then obtain
an4 þ1 qa ; n4 ki
A1 ¼
A2 ¼ T 1 þ
aqa aq a K n4 ; bh1 n4 ki
K¼
b a
ð8Þ
Substituting the constants of integration A1 and A2 into the general solution, we then obtain the temperature distribution
TðrÞ ¼ T 1 þ
aqa r n4 n4 ki þ K n4 n4 ki a bh1
ð9Þ
4. Theoretical analysis 4.1. Solution for linear elastic behavior of FGM rotating thick cylindrical pressure vessels For the stress analysis in an FGM thick cylindrical pressure vessel, material creep behavior and the solutions of the stresses at a time equal to zero (i.e. the initial stress state) are needed, which corresponds to the solution of materials with linear elastic behavior. In this section, equations needed to calculate such linear stresses in FGM rotating thick cylindrical pressure vessels analytically will be given briefly for two conditions: (a) plane strain; (b) plane stress. 4.1.1. Plane strain condition For the case of plane strain (ezz = 0) the elastic stress–strain relations in each material are
Eð1 mÞ m 1þm err þ aT ehh ð1 þ mÞð1 2mÞ 1m 1m Eð1 mÞ m 1þm ¼ ehh þ aT err ð1 þ mÞð1 2mÞ 1m 1m
rrr ¼
ð10Þ
rhh
ð11Þ
rzz ¼ mðrhh þ rrr Þ EaT
ð12Þ
where rrr, rhh and rzz are radial, circumferential and axial stresses, respectively. Here E, m and a are the Young’s modulus, Poisson’s ratio and thermal expansion coefficient, respectively, and T is the temperature distribution in the cylindrical vessel T = T(r). The strains are related to the displacement, as follows
dur dr ur ¼ r
err ¼
ð13Þ
ehh
ð14Þ
where err and ehh are radial and circumferential strains and ur is the displacement in the r-direction. In determining the stress distribution in a rotating cylindrical vessel, the following equation is used, based on the equilibrium condition of an element of the vessel, subjected to radial, circumferential and axial stresses, rrr, rhh and rzz,
drrr rrr rhh þ ¼ qrx2 dr r
ð15Þ
Substituting strains from Eqs. (13) and (14) into Eqs. (10) and (11) and then substituting radial and circumferential stresses from Eqs. (10) and (11) into equilibrium Eq. (15), the following differential equation for displacement is obtained 2
d ur dr
2
þ
i q x2 ð1 þ mÞð1 2mÞan1 n2 ðn1 þ 1Þ dur mðn1 þ 1Þ 1 ur ð1 þ mÞ h n2 1 ¼ Ar þ Br n2 n3 1 i rn2 n1 r ð1 mÞ dr r 2 ð1 mÞ Ei ð1 mÞ
ð16Þ
where
T 1 ai aqa ai n4 aqa ai A ¼ ðn1 þ n3 Þ K þ n n n a3 n4 ki a 3 bh1 a 3 aqa ai aqa ai n n B ¼ ðn1 þ n3 Þ n4 ki an3 n4 ki a 3 4
ð17Þ
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The general solution of the displacement ur is
"
Ar n3 þ1 Br n3 n4 þ1 þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ 2 q x ð1 þ mÞð1 2mÞan1 n2 rn2 n1 þ2 i Ei ð1 mÞ ðn2 n1 x1 þ 1Þðn2 n1 x2 þ 1Þ
227
#
ur ðrÞ ¼ C 1 r x1 þ C 2 r x2 þ m00
ð18Þ
where
x1;2 ¼
m00
n1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n21 þ 4ð1 n1 m0 Þ 2
1þm ¼ ; 1m
m0 ¼
m 1m
ð19Þ ð20Þ
The corresponding stresses for the case of plane strain are n E r 1 ð1 mÞ Aðn3 þ m0 þ 1Þm00 r n3 Bðn3 n4 þ m0 þ 1Þm00 r n3 n4 rrr ¼ i a C 1 r x1 1 ðm0 þ x1 Þ þ C 2 rx2 1 ðm0 þ x2 Þ þ þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ ð1 þ mÞð1 2mÞ r n3 aq a r n3 r n4 n k q x2 ð1 þ mÞð1 2mÞan1 n2 ðn2 n1 þ m0 þ 2Þr n2 n1 þ2 4 i n4 a i 00 m T 1 ai i þ þ K a a n4 ki a bh1 Ei ð1 mÞ ðn2 n1 x1 þ 1Þðn2 n1 x2 þ 1Þ
ð21Þ n Ei ar 1 ð1 mÞ Að1 þ m0 ðn3 þ 1ÞÞm00 r n3 Bð1 þ m0 ðn3 n4 þ 1ÞÞm00 r n3 n4 rhh ¼ C 1 rx1 ð1 þ m0 x1 Þ þ C 2 rx2 ð1 þ m0 x2 Þ þ þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ ð1 þ mÞð1 2mÞ r n3 aq a r n3 r n4 n k q x2 ð1 þ mÞð1 2mÞan1 n2 ð1 þ m0 ðn2 n1 þ 2ÞÞr n2 n1 þ2 i 4 i n a m00 T 1 ai i þ þ K 4 ðn2 n1 x1 þ 1Þðn2 n1 x2 þ 1Þ a a n 4 ki a bh1 Ei ð1 mÞ ð22Þ
rzz ¼ mðrhh þ rrr Þ EaT
ð23Þ
To determine the unknown constants C1 and C2 in each material, boundary conditions have to be used, which are as follows:
rrr ¼ Pi ; r ¼ a rrr ¼ Po ; r ¼ b
ð24Þ
The unknown constants C1 and C2 are given in the Appendix A. 4.1.2. Plane stress condition For the case of plane stress the stress and strain relation is:
E ½err þ mehh ð1 þ mÞaT 1 m2 E rhh ¼ ½ehh þ merr ð1 þ mÞaT 1 m2 rzz ¼ 0
rrr ¼
ð25Þ ð26Þ ð27Þ
The relations between the displacement and strains, and the equilibrium equation for stresses are the same as those for the case of plane strain. The differential equation for displacement ur is
h i q x2 1 m2 an1 n2 ðn1 þ 1Þ dur ur n3 1 n3 n4 1 þ ðmn1 þ 1Þ 2 ¼ ð1 þ mÞ Ar þ Br r n1 n2 þ1 i 2 r dr r Ei dr 2
d ur
The solution of Eq. (28) is
# Ar n3 þ1 Br n3 n4 þ1 þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ
2 qi x 1 m2 an1 n2 r n2 n1 þ3 Ei ðn2 n1 x1 þ 3Þðn2 n1 x2 þ 3Þ
ð28Þ
"
uðrÞ ¼ C 01 r x1 þ C 02 r x2 þ m
ð29Þ
where
x1;2 ¼
n1
m ¼ 1 þ m
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n21 þ 4ð1 þ n1 mÞ 2
ð30Þ ð31Þ
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The corresponding stresses for the case of plane stress are
n r n3 r n3 n4 Ei ar 1 Aðn3 þ m þ 1Þm Bðn3 n4 þ m þ 1Þm C 01 r x1 1 ðm þ x1 Þ þ C 02 rx2 1 ðm þ x2 Þ þ þ 2 ð1 m Þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ
r n3 aq a r n3 r n4 n k q x2 1 m2 ðn2 n1 þ m þ 1Þan1 n2 n2 n1 þ2 i 4 i a T 1 ai i þ þ K n4 r ð32Þ m a a n 4 ki a bh1 Ei ðn2 n1 x1 þ 3Þðn2 n1 x2 þ 3Þ
rrr ¼
n r n3 r n3 n4 Ei ar 1 Að1 þ mðn3 þ 1ÞÞm Bð1 þ mðn3 n4 þ 1ÞÞm C 01 r x1 1 ð1 þ mx1 Þ þ C 02 r x2 1 ð1 þ mx2 Þ þ þ ð1 m2 Þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ
r n3 aq a r n3 r n4 n k q x2 1 m2 ð1 þ mðn2 n1 þ 1ÞÞan1 n2 n2 n1 þ2 4 i i T 1 ai i þ a þ K n4 r ð33Þ m a a n 4 ki a bh1 Ei ðn2 n1 x1 þ 3Þðn2 n1 x2 þ 3Þ
rhh ¼
To determine the constants C 01 and C 02 , boundary conditions have to be used, which are the same as those for the case of plane strain. The unknown constants C 01 and C 02 are given in the Appendix A. 4.2. Solution for creep behavior of FGM rotating thick cylindrical pressure vessel The geometric relationships between radial and circumferential strain rates and the radial displacement rates are
du_ r dr u_ r ¼ r
e_ rr ¼
ð34Þ
e_ hh
ð35Þ
where e_ rr ; e_ hh and u_ r are, respectively, radial strain rate, circumferential strain rate, and the radial displacement rate. The equilibrium equation of the stress rate is:
dr_ rr r_ rr r_ hh þ ¼0 dr r
ð36Þ
For the case of plane strain ðe_ zz ¼ 0Þ, the relations between the rates of stress and strain are:
Eð1 mÞ m _ 3 m e_ rr þ ehh DrðeffN1Þ S0rr þ S0hh ð1 þ mÞð1 2mÞ 2 ð1 mÞ ð1 mÞ Eð1 mÞ m 3 m ¼ e_ hh þ e_ rr DrðeffN1Þ S0hh þ S0rr ð1 þ mÞð1 2mÞ 2 ð1 mÞ ð1 mÞ
r_ rr ¼
ð37Þ
r_ hh
ð38Þ
where
S0rr ¼ Srr þ mSzz ;
S0hh ¼ Shh þ mSzz
ð39Þ
For the case of plane stress ðr_ zz ¼ 0Þ, the relations between the rates of stress and strain are
E 3 _ rr þ me_ hh DrðeffN1Þ S00rr e 2 ð1 m2 Þ E 3 _ hh þ me_ rr DrðeffN1Þ S00hh ¼ e ð1 m2 Þ 2
r_ rr ¼
ð40Þ
r_ hh
ð41Þ
where
S00rr ¼ Srr þ mShh ;
S00hh ¼ Shh þ mSrr
ð42Þ
4.2.1. The case of e_ zz being zero Substituting Eqs. (34) and (35) into Eqs. (37) and (38) and then into Eq. (36) gives the differential equation for u_ r in an FGM cylindrical vessel 2 d u_ r
dr
2
þ
d 3
du_ r 1 dðln EÞ dðln EÞ 3 u_ r 0 dðln EÞ 1 ðN1Þ 0 ðN1Þ 0 þ ¼ þ m Dreff Srr þ m0 S0hh þ Dreff Srr þ m0 S0hh dr dr dr 2 r dr 2 dr r r 0 0 S S 3 ðN1Þ rr hh þ Dreff ð1 m0 Þ 2 r
ð43Þ
where
m0 ¼
m 1m
ð44Þ
M.Z. Nejad, M.D. Kashkoli / International Journal of Engineering Science 82 (2014) 222–237
229
In general, the quantities reff ; S0rr and S0hh are very complicated functions of the coordinate r, even in an implicit function form. Therefore, it is almost impossible to find an exact analytical solution for Eq. (43). In this study, an attempt is made to find a semi-analytical solution for Eq. (43). We can find an asymptotical solution for Eq. (43). At first, we assume that, reff ; S0rr and S0hh are constant, i.e. they are independent of the coordinate r. 2 d u_ r
dr
2
þ
0 0
0
3 ð1 þ n1 Þ du_ r mn1 1 3 Dn1 ðN1Þ 0 ðN1Þ ð1 m Þ Srr Shh 0 0 _r ¼ r S þ m S D r u þ rr hh eff eff r r2 2 r 2 r dr
ð45Þ
Then, the solution of Eq. (45) is
u_ r ðrÞ ¼ D1 rx1 þ D2 r x2 þ
ðN1Þ 0 Srr ð1
3 Drreff 2
þ n1 m0 Þ þ S0hh ðn1 m0 þ m0 1Þ
n 1 ð1 þ m 0 Þ
ð46Þ
where the unknown constants D1 and D2 can be determined from the boundary conditions. The corresponding stress rates are
n
Ei ð1 mÞ ar 1 3 1 m0 ðN1Þ 0 D1 rx1 1 ðm0 þ x1 Þ þ D2 r x2 1 ðm0 þ x2 Þ þ Dreff Srr S0hh 2 n1 ð1 2mÞð1 þ mÞ r n 1
Ei ð1 mÞ a 3 ð1 m0 Þð1 þ n1 Þ ðN1Þ 0 ¼ D1 rx1 1 ð1 þ m0 x1 Þ þ D2 r x2 1 ð1 þ m0 x2 Þ þ Dreff Srr S0hh 2 n1 ð1 2mÞð1 þ mÞ
r_ rr ¼
ð47Þ
r_ hh
ð48Þ
3 2
r_ zz ¼ mðr_ rr þ r_ hh Þ DrðeffN1Þ Szz
ð49Þ
Since internal and external pressures do not change with time, the boundary conditions for stress rates on the inner and outer surfaces may be written as:
r_ rr ¼ 0; r ¼ a r_ rr ¼ 0; r ¼ b
ð50Þ
Substituting the above-mentioned boundary condition into Eq. (47), the constants D1 and D2 are obtained.
D1 ¼
3 2
i
h x 1 S0rr S0hh b 2 ax2 1 h i x 1 x 1 ðm0 þ x1 Þ ax2 1 b 1 b 2 ax1 1
m0 1 DrðN1Þ n1
eff
ð51Þ
ðN1Þ 0 0 Srr S0hh D1 ðm0 þ x1 Þax1 1 3 ð1 m ÞDreff þ D2 ¼ 2 ðm0 þ x2 Þax2 1 n1 ðm0 þ x2 Þax2 1
ð52Þ
When the stress rate is known, the calculation of stresses at any time ti should be performed iteratively
rijðiÞ ðr; ti Þ ¼ rðiji1Þ ðr; ti1 Þ þ r_ ijðiÞ ðr; ti ÞdtðiÞ
ð53Þ
where
ti ¼
i X ðkÞ dt
ð54Þ
k¼0
To obtain a generally useful solution, a higher order approximation of
d
reff ðrÞ
2
d dr 2
reff ðrÞ
3
r¼r
d dr 3
reff ðrÞ r¼r
ðr r Þ þ ðr r Þ þ ðr r Þ3 þ . . . 1! 2! 3!
0
0
0 d2 d3 d S ðrÞ r¼r 2 Srr ðrÞ r¼ 3 Srr ðrÞ r¼ r r ðr r Þ þ dr ðr rÞ2 þ dr ðr rÞ3 þ . . . S0rr ðrÞ ¼ S0rr ðr Þ þ dr rr 1! 2! 3!
0
0
0 d2 d3 d Shh ðrÞ r¼r 2 Shh ðrÞ r¼ 3 Shh ðrÞ r¼ 2 r r 0 0 dr dr dr ðr r Þ þ ðr r Þ þ ðr r Þ3 þ . . . Shh ðrÞ ¼ Shh ðr Þ þ 1! 2! 3!
reff ðrÞ ¼ reff ðrÞ þ
dr
r¼r
2
reff ; S0rr and S0hh should be made.
ð55Þ
ð56Þ
ð57Þ
where r is the center point of the wall thickness in the following analysis. 4.2.2. The case of r_ zz being zero For the case of plane stress the differential equation for u_ r is 2 d u_ r
dr
2
þ
ð1 þ n1 Þ du_ r ðmn1 1Þ 3 D ðN1Þ 00 r Srr S00hh u_ r ¼ r r2 2 r eff dr
ð58Þ
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The solution of Eq. (58) is
_ uðrÞ ¼ D01 r x1 þ D02 rx2 þ
ðN1Þ 00 Srr þ S00hh 3 Drreff n1 ð1 þ mÞ 2
ð59Þ
where the unknown constants D01 and D02 can be determined from the boundary conditions. Using the previous boundary conditions, the constants D01 and D02 are obtained
D01
3 D 2
h
rðeffN1Þ S00rr
1n1 n1
S00hh
ih 1 n1
x2 1
b
ax2 1 i
i
h x 1 x 1 ðm þ x1 Þ ax2 1 b 1 b 2 ax1 1 h i ðN1Þ 1 x 1 S00hh n11 S00rr 1n n1 D1 ðm þ x1 Þb 1 3 Dreff 0 D2 ¼ x 1 x 1 2 ðm þ x2 Þb 2 ðm þ x2 Þb 2 ¼
ð60Þ
ð61Þ
The corresponding stress rates for the case of plane stress are
n Ei ar 1 3 1 ðN1Þ 00 1 n1 00 0 0 x1 1 x2 1 S D ð m þ x Þr þ D ð m þ x Þr þ D r S 1 2 rr hh 1 2 eff 2 n1 ð1 m2 Þ n1 n Ei ar 1 3 1 1 þ n1 ðN1Þ 00 00 0 x1 1 0 x2 1 S ¼ D r ð 1 þ m x Þ þ D r ð 1 þ m x Þ þ D r S 1 2 1 2 rr hh eff 2 n1 ð1 m2 Þ n1
r_ rr ¼
ð62Þ
r_ hh
ð63Þ
5. Numerical results and discussion In the previous section, the analytical solution of creep stresses for an FGM rotating thick hollow cylinder subjected to uniform pressures on the inner and outer surfaces was obtained. In this section, some profiles are plotted for the radial displacement, radial, circumferential and axial stresses and radial and circumferential strains as a function of radial direction and time. An FGM cylinder with creep behavior under internal and external pressure is considered. Radii of the cylinder are a = 20 mm and b = 40 mm. Mechanical properties of the cylinder, such as modulus of elasticity, density, linear expansion and thermal conductivity are assumed to be varying through the radius. In-homogeneity constants n1 = n2 = n3 = n4 = n, and n range from 2 to +2. The following data for loading and material properties are used in this investigation.
m ¼ 0:292; ai ¼ 10:8 106 K1 ; Pi ¼ 80 MPa; Po ¼ 0 MPa; qa ¼ 3000 W=m2 ; ki ¼ 43 W=m: C; h1 ¼ 6:5 W=m2 C; T 1 ¼ 25 C; D ¼ 1:4 108 ; N ¼ 2:25; x ¼ 1200 rpm; qi ¼ 7798 Kg=m3
Ei ¼ 207 GPa;
5.1. Plane strain condition The distributions of creep stress components rrr, rhh and rzz after 10 h of creeping for values of n = ± 0.1, ± 0.3, are plotted in Fig. 2. It can be seen from Fig. 2(a) that radial stress remains compressive throughout the cylinder for all values of n, with maximum values at the outer radius for material n = 0.3 and minimum values at the inner radius for all values of n. It is also clear from Fig. 2(a) that the minimum changes in radial stresses take place for material n = 0.1, +0.1, +0.3 and maximum changes occur for n = 0.3. It must be noted from Fig. 2(b) that circumferential stress at the inner radius decreases as n decreases. The absolute maximums of radial and circumferential stresses occur at the outer edge. It means the maximum shear stress for n = 0.3 which is smax ¼ rhh rrr =2 will be very high on the outer surface of the vessel. The axial stress shown in Fig. 2(c) remains compressive throughout and increases with an increase in n at the inner radius. Time dependent creep stress redistributions at point r = 30 mm, for different values of n, are shown in Fig. 3. It can be seen that after 3 h of creeping, all stresses are reduced to a constant. The radial displacement along the radius for the condition of plane strain is plotted in Fig. 4(a). There is an increase in the value of the radial displacement as n decreases. Fig. 4(b) shows the time-dependent radial displacement at the point r = 30 mm. It can be seen that after 3 h of creeping, radial displacement is reduced to almost zero for n = 0.1 and reaches a constant for n = 0.3, + 0.1, + 0.3. Fig. 5 shows the radial and circumferential creep strains along the thickness of FGM cylinder, for different values of n. It can be seen from Fig. 5 that the absolute maximums of radial and circumferential strains occur at the outer edge for n = 0.3. Fig. 6 shows the effect of adding external pressure on radial, circumferential and axial stresses. It can be seen from Fig. 6 that radial, circumferential and axial stresses at the outer radius decrease as external pressure increases while at the inner radius circumferential and axial stresses increase as external pressure increases. It must be noted that in these figures the internal pressure is Pi = 80 MPa and n = + 0.3. Temperature distribution of four different values of n is shown in Fig. 7. It can be seen from Fig. 7 that the maximums of temperature occur at the inner radius for n = 0.3 whereas the minimums of temperature occur at the outer radius for all values of n under the imposed boundary conditions.
M.Z. Nejad, M.D. Kashkoli / International Journal of Engineering Science 82 (2014) 222–237
Fig. 2. Variation of normalized radial, circumferential and axial stresses along the dimensionless radius after 10 h of creeping.
Fig. 3. Variation of normalized time-dependent radial, circumferential and axial stresses at the point r = 30 mm.
231
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Fig. 4. (a) Variation of normalized radial displacement along the dimensionless radius after 10 h of creeping (b) Variation of normalized time-dependent radial displacement at the point r = 30 mm.
Fig. 5. Variation of radial and circumferential strains along the dimensionless radius after 10 h of creeping.
Fig. 6. The effect of adding external pressure on radial, circumferential and axial stresses.
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233
Fig. 7. Temperature distribution of an FGM thick-walled cylindrical vessel for values of n = ± 0.1, ± 0.3.
Fig. 8. Variation of normalized radial and circumferential stresses along the dimensionless radius after 10 h of creeping.
Fig. 9. Variation of normalized time-dependent radial and circumferential stresses at the point r = 30 mm.
5.2. Plane stress condition The distributions of creep stress components rrr and rhh after 10 h of creeping for values of n = ± 1, ± 2, are plotted in Fig. 8. It must be noted from Fig. 8(a) that radial stress increases as n increases, and the radial stress for different values of n is compressive. According to Fig. 8(b), for n = 1, 2, circumferential stress is tensile whereas for n = 1, 2, it is compressive. The absolute maximums of radial and circumferential stresses occur at the outer edge. It means the maximum shear stress for n = 1, 2 will be very high on the outer surface of the vessel.
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Fig. 10. (a) Variation of normalized radial displacement along the dimensionless radius after 10 h of creeping (b) Variation of normalized time-dependent radial displacement at the point r = 30 mm.
Fig. 11. Variation of radial and circumferential strains along the dimensionless radius after 10 h of creeping.
Fig. 12. The effect of adding external pressure on radial and circumferential stresses.
The time dependent stresses at point r = 30 mm are plotted in Fig. 9. It can be seen that after 3 h of creeping, all stresses are reduced to a constant. The radial displacement along the radius for the case of plane stress is plotted in Fig. 10(a). There is an increase in the value of the radial displacement as n increases. Fig. 10(b) shows the time-dependent radial displacement at the point r = 30 mm. It can be seen that after 3 h of creeping, radial displacement is reduced to a constant. Fig. 11 shows the distribution of radial and circumferential strains in the radial direction. From Fig. 11(a), it could be observed that the minimums of radial and circumferential strains occur at the inner radius for all values of n, whereas the maximums of radial and circumferential strains occur at the outer radius for n = + 2.
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235
Fig. 12 shows the effect of adding external pressure to radial and circumferential stresses. It can be seen from Fig. 12 that radial and circumferential stresses at the outer radius decrease as external pressure increases. It is also necessary to point out that in these figures the internal pressure is Pi = 80 MPa and n = + 1. 6. Conclusions Time-dependent thermoelastic creep response of isotropic rotating thick-walled cylindrical pressure vessels made of functionally graded material (FGM) has been investigated in the present study by taking into account the creep behavior of the FGM pressure vessels, as described in Norton’s model. For the stress analysis in a cylinder, material creep behavior and the solutions of the stresses at a time equal to zero (i.e. the initial stress state) are needed, which corresponds to the solution of materials with linear elastic behavior. The analytical solution is obtained for the conditions of plane strain and plane stress. It is assumed that the material properties change as graded in radial direction to a power law function. To show the effect of in-homogeneity on the stress distributions, different values are considered for in-homogeneity constants. The pressure, inner radius and outer radius are considered constant. The heat conduction equation for the one-dimensional problem in polar coordinates is used to obtain temperature distribution in the cylinder. For the creep material behavior, the solution is asymptotic. After creeping for a long time, for the purpose of stress analysis, the iterative procedure is necessary. It could be seen that the in-homogeneity constants have significant influence on the distributions of the creep stresses and radial displacement. For the case of plane strain, the absolute maximums of radial and circumferential stresses occur at the outer edge for n = 0.3 and by increasing grading parameter n, the normalized radial and circumferential stresses decrease. The absolute minimum of axial stress occurs at the inner edge for n = 0.3 and the absolute maximum occurs at almost r/a = 1.8 for n = 0.3. It must be noted that for the case of plane strain, the radial, circumferential and axial stresses at the point r = 30 mm for different values of n, are compressive and that after 3 h of creeping, all stresses are reduced to a constant. The absolute maximum of normalized radial displacement occurs at the outer surface for n = 0.3 and by decreasing n, the values increase. For the case of plane stress, by increasing grading parameter n, the normalized radial stress increases, and also the absolute maximum of radial stress occurs at the outer edge for n = 2 whereas the absolute maximum of circumferential stress occurs at the outer edge for n = 1. It must be noted that for n = 1, 2, circumferential stress is tensile whereas for n = 1, 2, it is compressive. The maximum values of radial and circumferential stresses at the point r = 30 mm, occur at a time equal to zero (i.e. the initial stress state). As could be seen, after 3 h of creeping, radial and circumferential stresses are reduced to a constant. The maximum value of the normalized radial displacement takes place on the outer surface for n = 2 and by increasing n, the values increase. For both cases of plane strain and plane stress, by increasing external pressure, the normalized radial, circumferential and axial stresses decrease at the outer radius. Appendix A The unknown constants in Eqs. (21) and (22) are
C1 ¼
Pi ð1 þ mÞð1 2mÞ C 2 ðm0 þ x2 Þax2 1 Aðn3 þ m0 þ 1Þm00 an3 x 1 x 1 0 0 Ei ð1 þ mÞðm þ x1 Þa 1 ðm þ x1 Þa 1 ðn3 x2 þ 1Þðn3 x1 þ 1Þðm0 þ x1 Þax1 1 0 00 n3 n4 Bðn3 n4 þ m þ 1Þm a m00 aqa ai n 4 ki n4 T a þ 1 þ K 1 i ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þðm0 þ x1 Þax1 1 ðm0 þ x1 Þax1 1 n4 ki bh1 2 n1 n2 0 n2 n1 þ1 qi ð1 þ mÞð1 2mÞx a ðn2 n1 þ m þ 2Þa þ Ei ð1 mÞðm0 þ x1 Þax1 1 ðn2 n1 x1 þ 2Þðn2 n1 x2 þ 2Þ
ð1Þ
h i n þx 1 n x 1 Po a b1n1 1 ð1 þ mÞð1 2mÞ Am00 ðn3 þ m0 þ 1Þ b 3 ax1 1 b 1 an3 h i h i C2 ¼ x 1 x 1 x 1 x 1 ðn3 x2 þ 1Þðn3 x1 þ 1Þðm0 þ x2 Þ b 2 ax1 1 b 1 ax2 1 Ei ð1 mÞðm0 þ x2 Þ b 2 ax1 1 b 1 ax2 1 h i n n x 1 Bm00 ðn3 n4 þ m0 þ 1Þ b 3 4 ax1 1 b 1 an3 n4 h i x 1 x 1 ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þðm0 þ x2 Þ b 2 ax1 1 b 1 ax2 1 n3 n n4 m00 b aq ai n4 k i aqa ai x1 1 b 3 b n4 k i h i T 1 ai bx1 1 ax1 1 1þ K n4 a þ K n4 þ a x 1 x 1 n4 ki bh1 n4 k i bh1 a a a ðm0 þ x2 Þ b 2 ax1 1 b 1 ax2 1 h i qi ð1 þ mÞð1 2mÞðn2 n1 þ m0 þ 2Þx2 an1 n2 an2 n1 þ1 bx1 1 ax1 1 bn2 n1 þ1 h i x 1 x 1 Ei ð1 mÞðm0 þ x2 Þðn2 n1 x1 þ 2Þðn2 n1 x2 þ 2Þ b 2 ax1 1 b 1 ax2 1 Pi b
x1 1
ð2Þ
The unknown constants in Eqs. (32) and (33) are
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C 01 ¼
C 02
an3 Pi 1 m2 C 0 ðm þ x2 Þax2 1 Aðn3 þ m þ 1Þm 2 x 1 x 1 1 1 ðm þ x1 Þa ðn3 x2 þ 1Þðn3 x1 þ 1Þðm þ x1 Þax1 1 Ei ðm þ x1 Þa n a 3 n4 Bðn3 n4 þ m þ 1Þm m aqa ai n4 ki n4 þ T a þ 1 þ K 1 i ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þðm þ x1 Þax1 1 ðm þ x1 Þax1 1 n4 ki bh1
2 2 2 qi x a 1 m ðn2 n1 þ m þ 3Þ þ Ei ðn2 n1 x1 þ 3Þðn2 n1 x2 þ 3Þðm þ x1 Þax1 1
ð3Þ
h i
n þx 1 x 1 ðn3 þ m þ 1Þ bn3 ax1 1 bx1 1 an3 1 m2 P i b 1 Po a b1n1 1 Am h i h i ¼ x 1 x 1 x 1 x 1 ðn3 x2 þ 1Þðn3 x1 þ 1Þðm þ x2 Þ b 2 ax1 1 b 1 ax2 1 Ei ðm þ x2 Þ b 2 ax1 1 b 1 ax2 1 h i ðn3 n4 þ m þ 1Þ bn3 n4 ax1 1 bx1 1 an3 n4 Bm h i x 1 x 1 ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þðm þ x2 Þ b 2 ax1 1 b 1 ax2 1 n3 n n4 m b aqa ai n4 ki aqa ai x1 1 b 3 b n4 ki n4 n4 h i T 1 ai bx1 1 ax1 1 1 þ K a þ K þ x 1 x 1 a a a n4 k i bh1 n4 k i bh1 ðm þ x2 Þ b 2 ax1 1 b 1 ax2 1 h i
qi x2 1 m2 ðn2 n1 þ m þ 3Þ a2 bx1 1 bn2 n1 þ2 an1 n2 þx1 1 h i x 1 x 1 Ei ðn2 n1 x1 þ 3Þðn2 n1 x2 þ 3Þðm þ x2 Þ b 2 ax1 1 b 1 ax2 1
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Loghman, A., & Wahab, M. A. (1996). Creep damage simulation of thick-walled tubes using the theta projection concept. International Journal of Pressure Vessels and Piping, 67, 105–111. Murakami, S., & Iwatsuki, Sh. (1969). Transient creep of circular cylindrical shells. International Journal of Mechanical Sciences, 11, 897–912. Murakami, S., & Iwatsuki, Sh. (1971). Steady-state creep of circular cylindrical shells. Bulletin of the JSME, 73, 615–623. Murakami, S., & Suzuki, K. (1971). On the creep analysis of pressurized circular cylindrical shells. International Journal of Non-Linear Mechanics, 6, 377–392. Murakami, S., & Tanaka, E. (1976). On the creep buckling of circular cylindrical shells. International Journal of Mechanical Sciences, 18, 185–194. Pai, D. H. (1967). Steady-state creep analysis of thick-walled orthotropic cylinders. International Journal of Mechanical Sciences, 9, 335–348. Rahaeifard, M., Kahrobaiyan, M. H., Ahmadian, M. T., & Firoozbakhsh, K. (2013). Strain gradient formulation of functionally graded nonlinear beams. International Journal of Engineering Science, 65, 49–63. Rattan, M., Chamoli, N., & Singh, S. B. (2010). Creep analysis of an isotropic functionally graded rotating disc. International Journal of Contemporary Mathematical Sciences, 5, 419–431. Rimrott, F. P. J., & Luke, J. R. (1961). Large strain creep of rotating cylinders. Journal of Applied Mathematics and Mechanics, 41, 485–500. Sankaranarayanan, R. (1969). Steady creep of circular cylindrical shells under combined lateral and axial pressures. International Journal of Solids and Structures, 5, 17–32. Sharma, S. (2009). Thermo creep transition in non-homogeneous thick-walled rotating cylinders. Defence Science Journal, 59, 30–36. Sharma, S., Sahay, I., & Kumar, R. (2012). Creep transition in non homogeneous thick-walled circular cylinder under internal and external pressure. Applied Mathematical Sciences, 122, 6075–6080. Sharma, S., & Sahni, M. (2008). Creep transition of transversely isotropic thick-walled rotating cylinder. Advances in Theoretical and Applied Mechanics, 1, 315–325. Sharma, S., Sahni, M., & Kumar, R. (2010). Thermo creep transition of transversely isotropic thick-walled rotating cylinder under internal pressure. International Journal of Contemporary Mathematical Sciences, 5, 517–527. Simonian, A. M. (1979). Calculation of thermal stresses in thick-walled cylinders taking account of non-linear creep. International Journal of Engineering Science, 17, 513–519. Sim, R. G., & Penny, R. K. (1971). Plane strain creep behaviour of thick-walled cylinders. International Journal of Mechanical Sciences, 12, 987–1009. Singh, S. B. (2008). One parameter model for creep in a whisker reinforced anisotropic rotating disc of Al–SiCw composite. European Journal of Mechanics A/ Solids, 27, 680–690. Singh, T., & Gupta, V. K. (2009a). Creep analysis of an internally pressurized thick cylinder made of a functionally graded composite. Journal of Strain Analysis, 44, 583–594. Singh, T., & Gupta, V. K. (2009b). Effect of material parameters on steady state creep in a thick composite cylinder subjected to internal pressure. The Journal of Engineering Research, 6, 20–32. Singh, T., & Gupta, V. K. (2010b). Modeling of creep in a thick composite cylinder subjected to internal and external pressures. International Journal of Materials Research, 2, 279–286. Singh, T., & Gupta, V. K. (2010c). Modeling steady state creep in functionally graded thick cylinder subjected to internal pressure. Journal of Composite Materials, 44, 1317–1333. Singh, T., & Gupta, V. K. (2011). Effect of anisotropy on steady state creep in functionally graded cylinder. Composite Structures, 93, 747–758. Singh, T., & Gupta, V. K. (2014). Analysis of steady state creep in whisker reinforced functionally graded thick cylinder subjected to internal pressure by considering residual stress. Mechanics of Advanced Materials and Structures, 21, 384–392. Wah, T. (1961). Creep collapse of cylindrical shells. Journal of the Franklin Institute, 272, 45–60. Weir, C. D. (1957). The creep of thick walled tube under internal pressure. Journal of Applied Mechanics, 24, 464–466. Yang, Y. Y. (2000). Time-dependent stress analysis in functionally graded materials. International Journal of Solids and Structures, 37, 7593–7608. You, L. H., Ou, H., & Zheng, Z. Y. (2007). Creep deformations and stresses in thick-walled cylindrical vessels of functionally graded materials subjected to internal pressure. Composite Structures, 78, 285–291.
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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Time-dependent thermo-creep analysis of rotating FGM thick-walled cylindrical pressure vessels under heat flux Mohammad Zamani Nejad ⇑, Mosayeb Davoudi Kashkoli Mechanical Engineering Department, Yasouj University, P. O. Box: 75914-353, Yasouj, Iran
a r t i c l e
i n f o
Article history: Received 15 May 2014 Received in revised form 13 June 2014 Accepted 14 June 2014
Keywords: Thick cylindrical pressure vessel Thermo-creep Time-dependent Functionally graded material (FGM) Heat flux
a b s t r a c t In the present study, time-dependent thermoelastic creep response for isotropic rotating thick-walled cylindrical pressure vessels made of functionally graded material (FGM) has been investigated, taking into account the creep behavior of the FGM pressure vessels, as described in Norton’s model. For the purpose of stress analysis in an FGM pressure vessel, material creep behavior and the solutions of the stresses at a time equal to zero (i.e. the initial stress state) are needed. This corresponds to the solution of materials with linear elastic behavior. Therefore, using equations of equilibrium, stress–strain and strain– displacement, a differential equation for displacement is obtained and subsequently the stresses at a time equal to zero are calculated. Using Norton’s law in the multi-axial form in conjunction with the above-mentioned equations in the rate form, the radial displacement rate is obtained and then the radial, circumferential and axial creep stress rates are calculated for the conditions of plane strain and plane stress. When the stress rates are known, the stresses at any time are calculated iteratively. Assuming that the inner surface is exposed to a uniform flux, and that the outer surface is exposed to an airstream, the heat conduction equation for the one-dimensional problem in polar coordinates is used to obtain temperature distribution in the cylinder. Assuming that material properties are a function of the radius of the cylinder and that the Poisson’s ratio is constant, creep stresses, creep strains and radial displacement are plotted against dimensionless radius and time for different values of the powers of the material properties. It has been found that inhomogeneity constants have significant influence on the distributions of the creep stresses and radial displacement. Ó 2014 Published by Elsevier Ltd.
1. Introduction Functionally graded materials (FGMs) are inhomogeneous composites which are usually made from a mixture of metals and ceramics, providing the specific benefits of both the constituents. The continuously compositional variation of the constituents in FGMs from one surface to another offers a graceful solution to the problem of appearing high magnitude shear stresses that may be induced in laminated composites, where two materials with large differences in properties are bonded (Kahrobaiyan, Rahaeifard, Tajalli, & Ahmadian, 2012; Rahaeifard, Kahrobaiyan, Ahmadian, & Firoozbakhsh, 2013). Today, structures made of FGMs have attracted wide and increased attention from scientists and engineers (Birman, 2014; Chun & Ernian, 2013). Rotating discs are important structural components. Their significance lies in their extensive use in steam ⇑ Corresponding author. Tel./fax: +98 741 2221711. E-mail addresses: [email protected], [email protected] (M.Z. Nejad). http://dx.doi.org/10.1016/j.ijengsci.2014.06.006 0020-7225/Ó 2014 Published by Elsevier Ltd.
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and gas turbine rotors, ship propellers, aircraft engines, high speed gears, flywheels, pumps, centrifugal compressors, turbo generators, automotive braking systems, rotational air cleaners, computer disc drives and many other applications. In some of these applications, the disc has to operate at high temperatures while subjected to severe thermo-mechanical loading (Garg, Salaria, & Gupta, 2013; Hassani, Hojjati, Mahdavi, Alashti, & Farrahi, 2012). As a result of severe thermo-mechanical loading, the material of disc undergoes creep deformations, thereby affecting performance of the system (Deepak, Gupta, & Dham, 2010). Over the last years, stresses in FGM rotating thick-walled cylindrical pressure vessels under thermal loading have been analyzed extensively with regard to the elastic material behavior. Thermoelastic analysis of FGM rotating discs was carried out by Kordkheili and Naghdabadi (2007) without taking into account the creep analysis of the disc. Attia, Fitzgeorge, and Pope (1954) investigated the residual stresses produced in cast iron cylinders by the creep-relaxation of thermal stresses. In this study, a series of thick-hollow cylinders were subjected to a radial flow of heat by heating the bore and water-cooling the outer diameter for a chosen period of time, during which the thermal stresses were relaxed by creep. Weir (1957) investigated the creep stresses in pressurized thick walled tubes. Assuming the simply supported and fixed-end boundary conditions for the cylinders, Wah (1961) developed a theory for the collapse of cylindrical shells under steady-state creep and under external radial pressure and high temperature (300–500 F). Considering large strains, Rimrott and Luke (1961) obtained the creep stresses of a rotating hollow circular cylinder made of isotropic and homogeneous materials. Simonian (1979) investigated the thermal stresses in a long thick-walled cylinder in a temperature field, under a centrifugal force, axial force and internal–external pressure. In this paper, it is suggested that the material of the cylinder deforms according to the nonlinear theory of heredity. Besseling (1962) investigated the feasibility of the numerical analysis of nonstationary creep problems for thick-walled tubes under axially symmetric loading. Pai (1967) studied the steady-state creep of a thick-walled orthotropic cylinder subjected to internal pressure. It was observed that the creep anisotropy has a significant effect on the cylinder behavior particularly in terms of creep rates, which may differ by an order of magnitude compared with an isotropic analysis. Sankaranarayanan (1969) studied the steady creep behavior of thin circular cylindrical shells subjected to combined lateral and axial pressures. The analysis was based on the Tresca criterion and the associated flow rule. Assuming that the total strain consists of elastic and creep components, Murakami and Iwatsuki (1969) investigated the transient creep analysis of circular cylindrical shells on the basis of the strain-hardening and timehardening theories. Bhatnagar and Gupta (1969) obtained the solution for an orthotropic thick-walled internally pressurized cylinder, using constitutive equations of anisotropy creep and Norton’s creep law. Murakami and Iwatsuki (1971) developed a numerical analysis of the steady state creep of a pressurized circular cylindrical shell on the basis of Mises’ criterion and the power law of creep. Sim and Penny (1971) studied the deformation behavior of thick-walled tubes subjected to a variety of loadings during stress redistribution caused by creep. Murakami and Suzuki (1971) investigated the steady state creep of simply supported circular cylindrical shells with open ends under internal pressure by using Nortons’s law. Using finite-strain theory Bhatnagar and Arya (1974) studied the creep behavior of a thick-walled cylinder under large strains. Murakami and Tanaka (1976) investigated the creep buckling of clamped circular cylindrical shells subjected to axial compression combined with internal pressure with special emphasis on the concept of creep stability and the accuracy of the analysis. Jones and Sullivan (1976) studied the advantages and limitations of a perturbation method of analysis for the creep buckling of shells by examining the particular case of a long cylindrical shell subjected to a uniform external pressure. Arya, Debnath, and Bhatnagar (1983) investigated the creep problem in a thin circular cylindrical shell made of a homogeneous, incompressible and orthotropic material, using a non-steady creep law. Assuming the plane strain condition, Bhatnagar, Kulkarni, and Arya (1984) obtained the analysis of an internally pressurized, homogeneous, orthotropic rotating cylinder subjected to a steady state creep condition. In another study, considering the effect of anisotropy on stress and strain, Bhatnagar, Kulkarni, and Arya (1986) investigated the creep analysis of thick-walled orthotropic rotating cylinders. Creep damage simulation of thick-walled tubes, using the theta projection concept was investigated by Loghman and Wahab (1996). They obtained a closed-form solution for steady state creep stresses in FGM cylinders. Yang (2000) presented an analytical solution for the calculation of stresses in FGMs for the elastic and creep behavior of the materials. This solution can be used to study the dependence of stress on temperature and time for FG structures. Finally, the analytical results were compared with FEM. Using Seth’s transition theory, Gupta, Sharma, and Pathak (2000) obtained the creep stresses and strain rates for a nonhomogeneous thick-walled rotating cylinder. Gupta and Pathak (2001) studied thermo creep analysis in a pressurized thick hollow cylinder. Jahed and Bidabadi (2003) presented a general axisymmetric method for an inhomogeneous body for a disc with varying thickness. An approximation was employed during their solution algorithm. It means that they avoided considering the differentiation constitutive terms of governing equations for creep analysis. Gupta, Singh, Chandrawat, and Ray (2004) investigated the creep behavior of a rotating FG disc made of silicon carbide particles in a matrix of pure aluminum by using Cherby’s law. The FG disc that they considered had a constant thickness and they assumed linear particles distribution. In addition, they assigned a fixed function to radius direction temperature distribution. Chen, Tu, Xuan, and Wang (2007) studied the creep behavior of a functionally graded cylinder under both internal and external pressures. They observed that an asymptotic solution can be derived on the basis of a Taylor series expansion if the properties of the graded material are axisymmetric and dependent on radial coordinate. In order to investigate creep performance of thick-walled cylindrical vessels or cylinders made of functionally graded materials, You, Ou, and Zheng (2007) proposed a simple and accurate method to determine stresses and creep strain rates in thick-walled cylindrical vessels subjected to internal pressure. Based on the power law constitutive equation, Altenbach, Gorash, and Naumenko (2008) presented the classical solution of the steadystate creep problem for a pressurized thick-walled cylinder. In this paper, they applied an extended constitutive equation which includes both the linear and the power law stress dependencies. Sharma and Sahni (2008) obtained the creep stresses
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for a transversely isotropic thick-walled rotating cylinder under internal pressure by using Seth’s transition theory. Singh (2008) studied the steady state creep in a rotating disc of anisotropic aluminum silicon carbide whisker composite by using Norton’s law. In this study stress and strain rate distributions for anisotropic discs were calculated and then compared with those obtained for an isotropic disc. Sharma (2009) obtained the creep stresses for a non-homogeneous thick-walled rotating cylinder by using transition theory, which is based on the concept of ‘generalized principal strain measure’. Assuming that the creep response of the material is governed by threshold stress-based creep law, Deepak, Gupta, and Dham (2009) studied the effect of the stress exponent on the steady state creep in a rotating disc made of isotropic aluminum–silicon carbide particulate (Al–SiCp) composite. Singh and Gupta (2009, 2010) developed a mathematical model to describe the steady-creep behavior of functionally graded composite cylinders containing linearly varying silicon carbide particles in a matrix of pure aluminum involving threshold stress-based creep law. The model developed was used to investigate the effect of gradient on the distribution of SiCp on the steady-state creep response of the composite cylinder. Assuming that the creep response of the material is governed by Sherby’s law, Rattan, Chamoli, and Singh (2010) investigated the creep response of isotropic axisymmetric rotating discs made of a particle-reinforced FGM. Chamoli, Rattan, and Singh (2010) investigated the creep behavior of an anisotropic rotating disc made of Al–SiCp composite, using Hill’s yield criteria. The creep behavior in this case was assumed to follow Sherby’s constitutive model. Sharma, Sahni, and Kumar (2010) obtained the thermal creep stresses and strain rates for a transversely isotropic thick-walled rotating cylinder under internal pressure by using Seth’s transition theory. They observed that rotating circular cylinder under internal pressure made of transversely isotropic material is on the safer side of the design as compared with rotating circular cylinders under internal pressure made of isotropic material. Deepak et al. (2010) further extended their work to investigate creep behavior of a rotating discs made of functionally graded materials with linearly varying thickness. The discs under investigation were made of composite containing silicon carbide particles in a matrix of pure aluminum. The effect of imposing linear particle gradient on the distribution of stresses and strain rates in the composite disc was also investigated. Assuming total strains to be the sum of elastic, thermal and creep strains, Loghman, Ghorbanpour Arani, Amir, and Vajedi (2010) studied the time-dependent creep stress redistribution analysis of a thick-walled FGM cylinder placed in uniform magnetic and temperature fields and subjected to an internal pressure. Following Norton’s law for material creep behavior, using equations of equilibrium, strain displacement and stress–strain relations in the rate form and considering Prandtl–Reuss relations for creep strain rate-stress equation, they obtained a differential equation for the displacement rate and then calculated the radial and circumferential creep stress rates. In another study Loghman, Ghorbanpour Arani, and Aleayoub (2011) studied the time-dependent creep stress redistribution analysis of thick-walled spheres made of functionally graded material (FGM) subjected to an internal pressure and a uniform temperature field, using the method of successive elastic solution. Singh and Gupta (2011) investigated the steady state creep in transversely isotropic functionally graded cylinder, operating under internal and external pressures. In this paper, the effect of anisotropy on creep stresses and creep rates in the FGM cylinder was analyzed and then compared with an isotropic FGM cylinder. Hoseini, Nejad, Niknejad, and Ghannad (2011) presented a new analytical solution for the steady state creep in rotating thick cylindrical shells subjected to internal and external pressure by using Norton’s law. Sharma, Sahay, and Kumar (2012) investigated the creep stresses in thick-walled circular cylinders under internal and external pressure, using transition theory, which is based on the concept of ‘generalized principal strain measure’. Gupta and Singh (2012) investigated the steady state creep in an anisotropic rotating disc made of Al–SiCp composite having hyperbolically varying thickness, using Hill’s yield criterion followed by the Sherby’s law. Chamoli and Singh (2012) investigated the steady-state creep response of an isotropic FG rotating disc of aluminum silicon carbide particulate composite by taking into account the residual stress present in the disc. Loghman and Atabakhshian (2012) studied the time-dependent creep behavior of rotating cylinders made from exponentially graded material, using Bailey–Norton creep constitutive model. They observed that using exponentially graded material significantly decreases creep strains, stresses and deformations of the EGM rotating cylinder. Jamian, Sato, Tsukamoto, and Watanabe (2013) investigated the creep analysis of a thick-walled cylinder made of functionally graded materials (FGMs) subjected to thermal and internal pressure. Garg et al. (2013) investigated the effect of varying disc thickness gradient on the creep stresses and creep rates in a rotating functionally graded composite disc containing non-linearly varying radial distribution of silicon carbide particles in a matrix of pure aluminum. Singh and Gupta (2014) studied the steady state creep behavior in a functionally graded thick composite cylinder subjected to internal pressure in the presence of residual stress. Hoffman’s yield criterion was used to describe the yielding of the cylinder material in order to account for residual stress. The present study aims to investigate the time-dependent thermoelastic creep response of isotropic rotating thick-walled cylindrical pressure vessels made of functionally graded material subjected to a steady-state thermal loading and the effect of material properties on the distributions of stress and radial displacement through the dimensionless radius and time. For the creep material behavior, the solution is asymptotic. Subsequent to creeping for a long time, the iterative procedure is necessary for the stress analysis. 2. Geometry and loading condition, material properties, and creep constitutive model 2.1. Geometry and loading condition Consider a thick-walled FGM cylindrical vessel with an inner radius a, and outer radius b. Loading is an internal pressure Pi and an external pressure Po, which are axisymmetric, and also an inertia body force due to the rotation of the cylindrical
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Fig. 1. Geometry and boundary conditions of the cylinder.
vessel with a constant angular velocity of x = 1200 rpm and a distributed temperature field due to a steady-state heat conduction from inner surface to outer surface of the vessel. The coordinate axes r, h and z are taken, respectively, along the radial, circumferential, and axial directions of the cylinder (see Fig. 1). 2.2. Material properties In this study, Poisson’s ratio, m, is considered to be a constant and E, q, a and k are assumed to obey the power-law variation as
EðrÞ ¼ Ei ðr=aÞn1
qðrÞ ¼ qi ðr=aÞn2 aðrÞ ¼ ai ðr=aÞn3
ð1Þ
kðrÞ ¼ ki ðr=aÞn4 here Ei, qi, ai and ki are the modulus of elasticity, density, linear expansion and thermal conductivity at the inner surface r = a and n1, n2, n3 and n4 are the in-homogeneity constants determined empirically. 2.3. Creep constitutive model The material creep behavior is described by Norton’s constitutive model in the multi-axial form as follows (Finnie & Heller, 1959)
1þm m 3 r_ ij r_ kk dij þ DrðeffN1Þ Sij 2 E E 1 Sij ¼ rij rkk dij 3 ffi rffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 reff ¼ Sij Sij ¼ pffiffiffi ðrrr rhh Þ2 þ ðrrr rzz Þ2 þ ðrzz rhh Þ2 2 2
e_ ij ¼
ð2Þ ð3Þ ð4Þ
The parameters D and N are the creep coefficient and the creep exponent, respectively. reff is the effective stress, Sij is the deviator stress tensor and rrr, r hh and rzz are, respectively, the radial, circumferential and axial stresses. 3. Heat conduction formulation In the steady-state case, the heat conduction equation for the one-dimensional problem in polar coordinates simplifies to
@ @T ¼0 rk @r @r
ð5Þ
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We may determine the temperature distribution in the cylindrical vessel by solving Eq. (5) and applying appropriate boundary conditions. Eq. (5) may be integrated twice to obtain the general solution
TðrÞ ¼ A1 r n4 þ A2
ð6Þ
It is assumed that the inner surface is exposed to a uniform heat flux, and that the outer surface is exposed to an airstream. To obtain the constants of integration A1 and A2, the following boundary conditions are introduced:
8 0 r¼a > < kT ¼ qa ; 0 kT ¼ h1 ðT T 1 Þ; r ¼ b > : 0 dT T ¼ dr
ð7Þ
Applying these conditions to the general solution, we then obtain
an4 þ1 qa ; n4 ki
A1 ¼
A2 ¼ T 1 þ
aqa aq a K n4 ; bh1 n4 ki
K¼
b a
ð8Þ
Substituting the constants of integration A1 and A2 into the general solution, we then obtain the temperature distribution
TðrÞ ¼ T 1 þ
aqa r n4 n4 ki þ K n4 n4 ki a bh1
ð9Þ
4. Theoretical analysis 4.1. Solution for linear elastic behavior of FGM rotating thick cylindrical pressure vessels For the stress analysis in an FGM thick cylindrical pressure vessel, material creep behavior and the solutions of the stresses at a time equal to zero (i.e. the initial stress state) are needed, which corresponds to the solution of materials with linear elastic behavior. In this section, equations needed to calculate such linear stresses in FGM rotating thick cylindrical pressure vessels analytically will be given briefly for two conditions: (a) plane strain; (b) plane stress. 4.1.1. Plane strain condition For the case of plane strain (ezz = 0) the elastic stress–strain relations in each material are
Eð1 mÞ m 1þm err þ aT ehh ð1 þ mÞð1 2mÞ 1m 1m Eð1 mÞ m 1þm ¼ ehh þ aT err ð1 þ mÞð1 2mÞ 1m 1m
rrr ¼
ð10Þ
rhh
ð11Þ
rzz ¼ mðrhh þ rrr Þ EaT
ð12Þ
where rrr, rhh and rzz are radial, circumferential and axial stresses, respectively. Here E, m and a are the Young’s modulus, Poisson’s ratio and thermal expansion coefficient, respectively, and T is the temperature distribution in the cylindrical vessel T = T(r). The strains are related to the displacement, as follows
dur dr ur ¼ r
err ¼
ð13Þ
ehh
ð14Þ
where err and ehh are radial and circumferential strains and ur is the displacement in the r-direction. In determining the stress distribution in a rotating cylindrical vessel, the following equation is used, based on the equilibrium condition of an element of the vessel, subjected to radial, circumferential and axial stresses, rrr, rhh and rzz,
drrr rrr rhh þ ¼ qrx2 dr r
ð15Þ
Substituting strains from Eqs. (13) and (14) into Eqs. (10) and (11) and then substituting radial and circumferential stresses from Eqs. (10) and (11) into equilibrium Eq. (15), the following differential equation for displacement is obtained 2
d ur dr
2
þ
i q x2 ð1 þ mÞð1 2mÞan1 n2 ðn1 þ 1Þ dur mðn1 þ 1Þ 1 ur ð1 þ mÞ h n2 1 ¼ Ar þ Br n2 n3 1 i rn2 n1 r ð1 mÞ dr r 2 ð1 mÞ Ei ð1 mÞ
ð16Þ
where
T 1 ai aqa ai n4 aqa ai A ¼ ðn1 þ n3 Þ K þ n n n a3 n4 ki a 3 bh1 a 3 aqa ai aqa ai n n B ¼ ðn1 þ n3 Þ n4 ki an3 n4 ki a 3 4
ð17Þ
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The general solution of the displacement ur is
"
Ar n3 þ1 Br n3 n4 þ1 þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ 2 q x ð1 þ mÞð1 2mÞan1 n2 rn2 n1 þ2 i Ei ð1 mÞ ðn2 n1 x1 þ 1Þðn2 n1 x2 þ 1Þ
227
#
ur ðrÞ ¼ C 1 r x1 þ C 2 r x2 þ m00
ð18Þ
where
x1;2 ¼
m00
n1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n21 þ 4ð1 n1 m0 Þ 2
1þm ¼ ; 1m
m0 ¼
m 1m
ð19Þ ð20Þ
The corresponding stresses for the case of plane strain are n E r 1 ð1 mÞ Aðn3 þ m0 þ 1Þm00 r n3 Bðn3 n4 þ m0 þ 1Þm00 r n3 n4 rrr ¼ i a C 1 r x1 1 ðm0 þ x1 Þ þ C 2 rx2 1 ðm0 þ x2 Þ þ þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ ð1 þ mÞð1 2mÞ r n3 aq a r n3 r n4 n k q x2 ð1 þ mÞð1 2mÞan1 n2 ðn2 n1 þ m0 þ 2Þr n2 n1 þ2 4 i n4 a i 00 m T 1 ai i þ þ K a a n4 ki a bh1 Ei ð1 mÞ ðn2 n1 x1 þ 1Þðn2 n1 x2 þ 1Þ
ð21Þ n Ei ar 1 ð1 mÞ Að1 þ m0 ðn3 þ 1ÞÞm00 r n3 Bð1 þ m0 ðn3 n4 þ 1ÞÞm00 r n3 n4 rhh ¼ C 1 rx1 ð1 þ m0 x1 Þ þ C 2 rx2 ð1 þ m0 x2 Þ þ þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ ð1 þ mÞð1 2mÞ r n3 aq a r n3 r n4 n k q x2 ð1 þ mÞð1 2mÞan1 n2 ð1 þ m0 ðn2 n1 þ 2ÞÞr n2 n1 þ2 i 4 i n a m00 T 1 ai i þ þ K 4 ðn2 n1 x1 þ 1Þðn2 n1 x2 þ 1Þ a a n 4 ki a bh1 Ei ð1 mÞ ð22Þ
rzz ¼ mðrhh þ rrr Þ EaT
ð23Þ
To determine the unknown constants C1 and C2 in each material, boundary conditions have to be used, which are as follows:
rrr ¼ Pi ; r ¼ a rrr ¼ Po ; r ¼ b
ð24Þ
The unknown constants C1 and C2 are given in the Appendix A. 4.1.2. Plane stress condition For the case of plane stress the stress and strain relation is:
E ½err þ mehh ð1 þ mÞaT 1 m2 E rhh ¼ ½ehh þ merr ð1 þ mÞaT 1 m2 rzz ¼ 0
rrr ¼
ð25Þ ð26Þ ð27Þ
The relations between the displacement and strains, and the equilibrium equation for stresses are the same as those for the case of plane strain. The differential equation for displacement ur is
h i q x2 1 m2 an1 n2 ðn1 þ 1Þ dur ur n3 1 n3 n4 1 þ ðmn1 þ 1Þ 2 ¼ ð1 þ mÞ Ar þ Br r n1 n2 þ1 i 2 r dr r Ei dr 2
d ur
The solution of Eq. (28) is
# Ar n3 þ1 Br n3 n4 þ1 þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ
2 qi x 1 m2 an1 n2 r n2 n1 þ3 Ei ðn2 n1 x1 þ 3Þðn2 n1 x2 þ 3Þ
ð28Þ
"
uðrÞ ¼ C 01 r x1 þ C 02 r x2 þ m
ð29Þ
where
x1;2 ¼
n1
m ¼ 1 þ m
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n21 þ 4ð1 þ n1 mÞ 2
ð30Þ ð31Þ
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The corresponding stresses for the case of plane stress are
n r n3 r n3 n4 Ei ar 1 Aðn3 þ m þ 1Þm Bðn3 n4 þ m þ 1Þm C 01 r x1 1 ðm þ x1 Þ þ C 02 rx2 1 ðm þ x2 Þ þ þ 2 ð1 m Þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ
r n3 aq a r n3 r n4 n k q x2 1 m2 ðn2 n1 þ m þ 1Þan1 n2 n2 n1 þ2 i 4 i a T 1 ai i þ þ K n4 r ð32Þ m a a n 4 ki a bh1 Ei ðn2 n1 x1 þ 3Þðn2 n1 x2 þ 3Þ
rrr ¼
n r n3 r n3 n4 Ei ar 1 Að1 þ mðn3 þ 1ÞÞm Bð1 þ mðn3 n4 þ 1ÞÞm C 01 r x1 1 ð1 þ mx1 Þ þ C 02 r x2 1 ð1 þ mx2 Þ þ þ ð1 m2 Þ ðn3 x2 þ 1Þðn3 x1 þ 1Þ ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þ
r n3 aq a r n3 r n4 n k q x2 1 m2 ð1 þ mðn2 n1 þ 1ÞÞan1 n2 n2 n1 þ2 4 i i T 1 ai i þ a þ K n4 r ð33Þ m a a n 4 ki a bh1 Ei ðn2 n1 x1 þ 3Þðn2 n1 x2 þ 3Þ
rhh ¼
To determine the constants C 01 and C 02 , boundary conditions have to be used, which are the same as those for the case of plane strain. The unknown constants C 01 and C 02 are given in the Appendix A. 4.2. Solution for creep behavior of FGM rotating thick cylindrical pressure vessel The geometric relationships between radial and circumferential strain rates and the radial displacement rates are
du_ r dr u_ r ¼ r
e_ rr ¼
ð34Þ
e_ hh
ð35Þ
where e_ rr ; e_ hh and u_ r are, respectively, radial strain rate, circumferential strain rate, and the radial displacement rate. The equilibrium equation of the stress rate is:
dr_ rr r_ rr r_ hh þ ¼0 dr r
ð36Þ
For the case of plane strain ðe_ zz ¼ 0Þ, the relations between the rates of stress and strain are:
Eð1 mÞ m _ 3 m e_ rr þ ehh DrðeffN1Þ S0rr þ S0hh ð1 þ mÞð1 2mÞ 2 ð1 mÞ ð1 mÞ Eð1 mÞ m 3 m ¼ e_ hh þ e_ rr DrðeffN1Þ S0hh þ S0rr ð1 þ mÞð1 2mÞ 2 ð1 mÞ ð1 mÞ
r_ rr ¼
ð37Þ
r_ hh
ð38Þ
where
S0rr ¼ Srr þ mSzz ;
S0hh ¼ Shh þ mSzz
ð39Þ
For the case of plane stress ðr_ zz ¼ 0Þ, the relations between the rates of stress and strain are
E 3 _ rr þ me_ hh DrðeffN1Þ S00rr e 2 ð1 m2 Þ E 3 _ hh þ me_ rr DrðeffN1Þ S00hh ¼ e ð1 m2 Þ 2
r_ rr ¼
ð40Þ
r_ hh
ð41Þ
where
S00rr ¼ Srr þ mShh ;
S00hh ¼ Shh þ mSrr
ð42Þ
4.2.1. The case of e_ zz being zero Substituting Eqs. (34) and (35) into Eqs. (37) and (38) and then into Eq. (36) gives the differential equation for u_ r in an FGM cylindrical vessel 2 d u_ r
dr
2
þ
d 3
du_ r 1 dðln EÞ dðln EÞ 3 u_ r 0 dðln EÞ 1 ðN1Þ 0 ðN1Þ 0 þ ¼ þ m Dreff Srr þ m0 S0hh þ Dreff Srr þ m0 S0hh dr dr dr 2 r dr 2 dr r r 0 0 S S 3 ðN1Þ rr hh þ Dreff ð1 m0 Þ 2 r
ð43Þ
where
m0 ¼
m 1m
ð44Þ
M.Z. Nejad, M.D. Kashkoli / International Journal of Engineering Science 82 (2014) 222–237
229
In general, the quantities reff ; S0rr and S0hh are very complicated functions of the coordinate r, even in an implicit function form. Therefore, it is almost impossible to find an exact analytical solution for Eq. (43). In this study, an attempt is made to find a semi-analytical solution for Eq. (43). We can find an asymptotical solution for Eq. (43). At first, we assume that, reff ; S0rr and S0hh are constant, i.e. they are independent of the coordinate r. 2 d u_ r
dr
2
þ
0 0
0
3 ð1 þ n1 Þ du_ r mn1 1 3 Dn1 ðN1Þ 0 ðN1Þ ð1 m Þ Srr Shh 0 0 _r ¼ r S þ m S D r u þ rr hh eff eff r r2 2 r 2 r dr
ð45Þ
Then, the solution of Eq. (45) is
u_ r ðrÞ ¼ D1 rx1 þ D2 r x2 þ
ðN1Þ 0 Srr ð1
3 Drreff 2
þ n1 m0 Þ þ S0hh ðn1 m0 þ m0 1Þ
n 1 ð1 þ m 0 Þ
ð46Þ
where the unknown constants D1 and D2 can be determined from the boundary conditions. The corresponding stress rates are
n
Ei ð1 mÞ ar 1 3 1 m0 ðN1Þ 0 D1 rx1 1 ðm0 þ x1 Þ þ D2 r x2 1 ðm0 þ x2 Þ þ Dreff Srr S0hh 2 n1 ð1 2mÞð1 þ mÞ r n 1
Ei ð1 mÞ a 3 ð1 m0 Þð1 þ n1 Þ ðN1Þ 0 ¼ D1 rx1 1 ð1 þ m0 x1 Þ þ D2 r x2 1 ð1 þ m0 x2 Þ þ Dreff Srr S0hh 2 n1 ð1 2mÞð1 þ mÞ
r_ rr ¼
ð47Þ
r_ hh
ð48Þ
3 2
r_ zz ¼ mðr_ rr þ r_ hh Þ DrðeffN1Þ Szz
ð49Þ
Since internal and external pressures do not change with time, the boundary conditions for stress rates on the inner and outer surfaces may be written as:
r_ rr ¼ 0; r ¼ a r_ rr ¼ 0; r ¼ b
ð50Þ
Substituting the above-mentioned boundary condition into Eq. (47), the constants D1 and D2 are obtained.
D1 ¼
3 2
i
h x 1 S0rr S0hh b 2 ax2 1 h i x 1 x 1 ðm0 þ x1 Þ ax2 1 b 1 b 2 ax1 1
m0 1 DrðN1Þ n1
eff
ð51Þ
ðN1Þ 0 0 Srr S0hh D1 ðm0 þ x1 Þax1 1 3 ð1 m ÞDreff þ D2 ¼ 2 ðm0 þ x2 Þax2 1 n1 ðm0 þ x2 Þax2 1
ð52Þ
When the stress rate is known, the calculation of stresses at any time ti should be performed iteratively
rijðiÞ ðr; ti Þ ¼ rðiji1Þ ðr; ti1 Þ þ r_ ijðiÞ ðr; ti ÞdtðiÞ
ð53Þ
where
ti ¼
i X ðkÞ dt
ð54Þ
k¼0
To obtain a generally useful solution, a higher order approximation of
d
reff ðrÞ
2
d dr 2
reff ðrÞ
3
r¼r
d dr 3
reff ðrÞ r¼r
ðr r Þ þ ðr r Þ þ ðr r Þ3 þ . . . 1! 2! 3!
0
0
0 d2 d3 d S ðrÞ r¼r 2 Srr ðrÞ r¼ 3 Srr ðrÞ r¼ r r ðr r Þ þ dr ðr rÞ2 þ dr ðr rÞ3 þ . . . S0rr ðrÞ ¼ S0rr ðr Þ þ dr rr 1! 2! 3!
0
0
0 d2 d3 d Shh ðrÞ r¼r 2 Shh ðrÞ r¼ 3 Shh ðrÞ r¼ 2 r r 0 0 dr dr dr ðr r Þ þ ðr r Þ þ ðr r Þ3 þ . . . Shh ðrÞ ¼ Shh ðr Þ þ 1! 2! 3!
reff ðrÞ ¼ reff ðrÞ þ
dr
r¼r
2
reff ; S0rr and S0hh should be made.
ð55Þ
ð56Þ
ð57Þ
where r is the center point of the wall thickness in the following analysis. 4.2.2. The case of r_ zz being zero For the case of plane stress the differential equation for u_ r is 2 d u_ r
dr
2
þ
ð1 þ n1 Þ du_ r ðmn1 1Þ 3 D ðN1Þ 00 r Srr S00hh u_ r ¼ r r2 2 r eff dr
ð58Þ
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The solution of Eq. (58) is
_ uðrÞ ¼ D01 r x1 þ D02 rx2 þ
ðN1Þ 00 Srr þ S00hh 3 Drreff n1 ð1 þ mÞ 2
ð59Þ
where the unknown constants D01 and D02 can be determined from the boundary conditions. Using the previous boundary conditions, the constants D01 and D02 are obtained
D01
3 D 2
h
rðeffN1Þ S00rr
1n1 n1
S00hh
ih 1 n1
x2 1
b
ax2 1 i
i
h x 1 x 1 ðm þ x1 Þ ax2 1 b 1 b 2 ax1 1 h i ðN1Þ 1 x 1 S00hh n11 S00rr 1n n1 D1 ðm þ x1 Þb 1 3 Dreff 0 D2 ¼ x 1 x 1 2 ðm þ x2 Þb 2 ðm þ x2 Þb 2 ¼
ð60Þ
ð61Þ
The corresponding stress rates for the case of plane stress are
n Ei ar 1 3 1 ðN1Þ 00 1 n1 00 0 0 x1 1 x2 1 S D ð m þ x Þr þ D ð m þ x Þr þ D r S 1 2 rr hh 1 2 eff 2 n1 ð1 m2 Þ n1 n Ei ar 1 3 1 1 þ n1 ðN1Þ 00 00 0 x1 1 0 x2 1 S ¼ D r ð 1 þ m x Þ þ D r ð 1 þ m x Þ þ D r S 1 2 1 2 rr hh eff 2 n1 ð1 m2 Þ n1
r_ rr ¼
ð62Þ
r_ hh
ð63Þ
5. Numerical results and discussion In the previous section, the analytical solution of creep stresses for an FGM rotating thick hollow cylinder subjected to uniform pressures on the inner and outer surfaces was obtained. In this section, some profiles are plotted for the radial displacement, radial, circumferential and axial stresses and radial and circumferential strains as a function of radial direction and time. An FGM cylinder with creep behavior under internal and external pressure is considered. Radii of the cylinder are a = 20 mm and b = 40 mm. Mechanical properties of the cylinder, such as modulus of elasticity, density, linear expansion and thermal conductivity are assumed to be varying through the radius. In-homogeneity constants n1 = n2 = n3 = n4 = n, and n range from 2 to +2. The following data for loading and material properties are used in this investigation.
m ¼ 0:292; ai ¼ 10:8 106 K1 ; Pi ¼ 80 MPa; Po ¼ 0 MPa; qa ¼ 3000 W=m2 ; ki ¼ 43 W=m: C; h1 ¼ 6:5 W=m2 C; T 1 ¼ 25 C; D ¼ 1:4 108 ; N ¼ 2:25; x ¼ 1200 rpm; qi ¼ 7798 Kg=m3
Ei ¼ 207 GPa;
5.1. Plane strain condition The distributions of creep stress components rrr, rhh and rzz after 10 h of creeping for values of n = ± 0.1, ± 0.3, are plotted in Fig. 2. It can be seen from Fig. 2(a) that radial stress remains compressive throughout the cylinder for all values of n, with maximum values at the outer radius for material n = 0.3 and minimum values at the inner radius for all values of n. It is also clear from Fig. 2(a) that the minimum changes in radial stresses take place for material n = 0.1, +0.1, +0.3 and maximum changes occur for n = 0.3. It must be noted from Fig. 2(b) that circumferential stress at the inner radius decreases as n decreases. The absolute maximums of radial and circumferential stresses occur at the outer edge. It means the maximum shear stress for n = 0.3 which is smax ¼ rhh rrr =2 will be very high on the outer surface of the vessel. The axial stress shown in Fig. 2(c) remains compressive throughout and increases with an increase in n at the inner radius. Time dependent creep stress redistributions at point r = 30 mm, for different values of n, are shown in Fig. 3. It can be seen that after 3 h of creeping, all stresses are reduced to a constant. The radial displacement along the radius for the condition of plane strain is plotted in Fig. 4(a). There is an increase in the value of the radial displacement as n decreases. Fig. 4(b) shows the time-dependent radial displacement at the point r = 30 mm. It can be seen that after 3 h of creeping, radial displacement is reduced to almost zero for n = 0.1 and reaches a constant for n = 0.3, + 0.1, + 0.3. Fig. 5 shows the radial and circumferential creep strains along the thickness of FGM cylinder, for different values of n. It can be seen from Fig. 5 that the absolute maximums of radial and circumferential strains occur at the outer edge for n = 0.3. Fig. 6 shows the effect of adding external pressure on radial, circumferential and axial stresses. It can be seen from Fig. 6 that radial, circumferential and axial stresses at the outer radius decrease as external pressure increases while at the inner radius circumferential and axial stresses increase as external pressure increases. It must be noted that in these figures the internal pressure is Pi = 80 MPa and n = + 0.3. Temperature distribution of four different values of n is shown in Fig. 7. It can be seen from Fig. 7 that the maximums of temperature occur at the inner radius for n = 0.3 whereas the minimums of temperature occur at the outer radius for all values of n under the imposed boundary conditions.
M.Z. Nejad, M.D. Kashkoli / International Journal of Engineering Science 82 (2014) 222–237
Fig. 2. Variation of normalized radial, circumferential and axial stresses along the dimensionless radius after 10 h of creeping.
Fig. 3. Variation of normalized time-dependent radial, circumferential and axial stresses at the point r = 30 mm.
231
232
M.Z. Nejad, M.D. Kashkoli / International Journal of Engineering Science 82 (2014) 222–237
Fig. 4. (a) Variation of normalized radial displacement along the dimensionless radius after 10 h of creeping (b) Variation of normalized time-dependent radial displacement at the point r = 30 mm.
Fig. 5. Variation of radial and circumferential strains along the dimensionless radius after 10 h of creeping.
Fig. 6. The effect of adding external pressure on radial, circumferential and axial stresses.
M.Z. Nejad, M.D. Kashkoli / International Journal of Engineering Science 82 (2014) 222–237
233
Fig. 7. Temperature distribution of an FGM thick-walled cylindrical vessel for values of n = ± 0.1, ± 0.3.
Fig. 8. Variation of normalized radial and circumferential stresses along the dimensionless radius after 10 h of creeping.
Fig. 9. Variation of normalized time-dependent radial and circumferential stresses at the point r = 30 mm.
5.2. Plane stress condition The distributions of creep stress components rrr and rhh after 10 h of creeping for values of n = ± 1, ± 2, are plotted in Fig. 8. It must be noted from Fig. 8(a) that radial stress increases as n increases, and the radial stress for different values of n is compressive. According to Fig. 8(b), for n = 1, 2, circumferential stress is tensile whereas for n = 1, 2, it is compressive. The absolute maximums of radial and circumferential stresses occur at the outer edge. It means the maximum shear stress for n = 1, 2 will be very high on the outer surface of the vessel.
234
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Fig. 10. (a) Variation of normalized radial displacement along the dimensionless radius after 10 h of creeping (b) Variation of normalized time-dependent radial displacement at the point r = 30 mm.
Fig. 11. Variation of radial and circumferential strains along the dimensionless radius after 10 h of creeping.
Fig. 12. The effect of adding external pressure on radial and circumferential stresses.
The time dependent stresses at point r = 30 mm are plotted in Fig. 9. It can be seen that after 3 h of creeping, all stresses are reduced to a constant. The radial displacement along the radius for the case of plane stress is plotted in Fig. 10(a). There is an increase in the value of the radial displacement as n increases. Fig. 10(b) shows the time-dependent radial displacement at the point r = 30 mm. It can be seen that after 3 h of creeping, radial displacement is reduced to a constant. Fig. 11 shows the distribution of radial and circumferential strains in the radial direction. From Fig. 11(a), it could be observed that the minimums of radial and circumferential strains occur at the inner radius for all values of n, whereas the maximums of radial and circumferential strains occur at the outer radius for n = + 2.
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235
Fig. 12 shows the effect of adding external pressure to radial and circumferential stresses. It can be seen from Fig. 12 that radial and circumferential stresses at the outer radius decrease as external pressure increases. It is also necessary to point out that in these figures the internal pressure is Pi = 80 MPa and n = + 1. 6. Conclusions Time-dependent thermoelastic creep response of isotropic rotating thick-walled cylindrical pressure vessels made of functionally graded material (FGM) has been investigated in the present study by taking into account the creep behavior of the FGM pressure vessels, as described in Norton’s model. For the stress analysis in a cylinder, material creep behavior and the solutions of the stresses at a time equal to zero (i.e. the initial stress state) are needed, which corresponds to the solution of materials with linear elastic behavior. The analytical solution is obtained for the conditions of plane strain and plane stress. It is assumed that the material properties change as graded in radial direction to a power law function. To show the effect of in-homogeneity on the stress distributions, different values are considered for in-homogeneity constants. The pressure, inner radius and outer radius are considered constant. The heat conduction equation for the one-dimensional problem in polar coordinates is used to obtain temperature distribution in the cylinder. For the creep material behavior, the solution is asymptotic. After creeping for a long time, for the purpose of stress analysis, the iterative procedure is necessary. It could be seen that the in-homogeneity constants have significant influence on the distributions of the creep stresses and radial displacement. For the case of plane strain, the absolute maximums of radial and circumferential stresses occur at the outer edge for n = 0.3 and by increasing grading parameter n, the normalized radial and circumferential stresses decrease. The absolute minimum of axial stress occurs at the inner edge for n = 0.3 and the absolute maximum occurs at almost r/a = 1.8 for n = 0.3. It must be noted that for the case of plane strain, the radial, circumferential and axial stresses at the point r = 30 mm for different values of n, are compressive and that after 3 h of creeping, all stresses are reduced to a constant. The absolute maximum of normalized radial displacement occurs at the outer surface for n = 0.3 and by decreasing n, the values increase. For the case of plane stress, by increasing grading parameter n, the normalized radial stress increases, and also the absolute maximum of radial stress occurs at the outer edge for n = 2 whereas the absolute maximum of circumferential stress occurs at the outer edge for n = 1. It must be noted that for n = 1, 2, circumferential stress is tensile whereas for n = 1, 2, it is compressive. The maximum values of radial and circumferential stresses at the point r = 30 mm, occur at a time equal to zero (i.e. the initial stress state). As could be seen, after 3 h of creeping, radial and circumferential stresses are reduced to a constant. The maximum value of the normalized radial displacement takes place on the outer surface for n = 2 and by increasing n, the values increase. For both cases of plane strain and plane stress, by increasing external pressure, the normalized radial, circumferential and axial stresses decrease at the outer radius. Appendix A The unknown constants in Eqs. (21) and (22) are
C1 ¼
Pi ð1 þ mÞð1 2mÞ C 2 ðm0 þ x2 Þax2 1 Aðn3 þ m0 þ 1Þm00 an3 x 1 x 1 0 0 Ei ð1 þ mÞðm þ x1 Þa 1 ðm þ x1 Þa 1 ðn3 x2 þ 1Þðn3 x1 þ 1Þðm0 þ x1 Þax1 1 0 00 n3 n4 Bðn3 n4 þ m þ 1Þm a m00 aqa ai n 4 ki n4 T a þ 1 þ K 1 i ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þðm0 þ x1 Þax1 1 ðm0 þ x1 Þax1 1 n4 ki bh1 2 n1 n2 0 n2 n1 þ1 qi ð1 þ mÞð1 2mÞx a ðn2 n1 þ m þ 2Þa þ Ei ð1 mÞðm0 þ x1 Þax1 1 ðn2 n1 x1 þ 2Þðn2 n1 x2 þ 2Þ
ð1Þ
h i n þx 1 n x 1 Po a b1n1 1 ð1 þ mÞð1 2mÞ Am00 ðn3 þ m0 þ 1Þ b 3 ax1 1 b 1 an3 h i h i C2 ¼ x 1 x 1 x 1 x 1 ðn3 x2 þ 1Þðn3 x1 þ 1Þðm0 þ x2 Þ b 2 ax1 1 b 1 ax2 1 Ei ð1 mÞðm0 þ x2 Þ b 2 ax1 1 b 1 ax2 1 h i n n x 1 Bm00 ðn3 n4 þ m0 þ 1Þ b 3 4 ax1 1 b 1 an3 n4 h i x 1 x 1 ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þðm0 þ x2 Þ b 2 ax1 1 b 1 ax2 1 n3 n n4 m00 b aq ai n4 k i aqa ai x1 1 b 3 b n4 k i h i T 1 ai bx1 1 ax1 1 1þ K n4 a þ K n4 þ a x 1 x 1 n4 ki bh1 n4 k i bh1 a a a ðm0 þ x2 Þ b 2 ax1 1 b 1 ax2 1 h i qi ð1 þ mÞð1 2mÞðn2 n1 þ m0 þ 2Þx2 an1 n2 an2 n1 þ1 bx1 1 ax1 1 bn2 n1 þ1 h i x 1 x 1 Ei ð1 mÞðm0 þ x2 Þðn2 n1 x1 þ 2Þðn2 n1 x2 þ 2Þ b 2 ax1 1 b 1 ax2 1 Pi b
x1 1
ð2Þ
The unknown constants in Eqs. (32) and (33) are
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C 01 ¼
C 02
an3 Pi 1 m2 C 0 ðm þ x2 Þax2 1 Aðn3 þ m þ 1Þm 2 x 1 x 1 1 1 ðm þ x1 Þa ðn3 x2 þ 1Þðn3 x1 þ 1Þðm þ x1 Þax1 1 Ei ðm þ x1 Þa n a 3 n4 Bðn3 n4 þ m þ 1Þm m aqa ai n4 ki n4 þ T a þ 1 þ K 1 i ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þðm þ x1 Þax1 1 ðm þ x1 Þax1 1 n4 ki bh1
2 2 2 qi x a 1 m ðn2 n1 þ m þ 3Þ þ Ei ðn2 n1 x1 þ 3Þðn2 n1 x2 þ 3Þðm þ x1 Þax1 1
ð3Þ
h i
n þx 1 x 1 ðn3 þ m þ 1Þ bn3 ax1 1 bx1 1 an3 1 m2 P i b 1 Po a b1n1 1 Am h i h i ¼ x 1 x 1 x 1 x 1 ðn3 x2 þ 1Þðn3 x1 þ 1Þðm þ x2 Þ b 2 ax1 1 b 1 ax2 1 Ei ðm þ x2 Þ b 2 ax1 1 b 1 ax2 1 h i ðn3 n4 þ m þ 1Þ bn3 n4 ax1 1 bx1 1 an3 n4 Bm h i x 1 x 1 ðn3 n4 x2 þ 1Þðn3 n4 x1 þ 1Þðm þ x2 Þ b 2 ax1 1 b 1 ax2 1 n3 n n4 m b aqa ai n4 ki aqa ai x1 1 b 3 b n4 ki n4 n4 h i T 1 ai bx1 1 ax1 1 1 þ K a þ K þ x 1 x 1 a a a n4 k i bh1 n4 k i bh1 ðm þ x2 Þ b 2 ax1 1 b 1 ax2 1 h i
qi x2 1 m2 ðn2 n1 þ m þ 3Þ a2 bx1 1 bn2 n1 þ2 an1 n2 þx1 1 h i x 1 x 1 Ei ðn2 n1 x1 þ 3Þðn2 n1 x2 þ 3Þðm þ x2 Þ b 2 ax1 1 b 1 ax2 1
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