Stresses in thick-walled FGM cylinders with exponentially-varying properties

Engineering Structures 29 (2007) 2032–2035 www.elsevier.com/locate/engstruct

Short communication

Stresses in thick-walled FGM cylinders with exponentially-varying properties Naki Tutuncu ∗ C ¸ ukurova University, Department of Mechanical Engineering, 01330 Adana, Turkey Received 11 July 2006; received in revised form 30 November 2006; accepted 4 December 2006 Available online 3 January 2007

Abstract Power series solutions for stresses and displacements in functionally-graded cylindrical vessels subjected to internal pressure alone are obtained using the infinitesimal theory of elasticity. The material is assumed to be isotropic with constant Poisson’s ratio and exponentially-varying elastic modulus through the thickness. Stress distributions depending on an inhomogeneity constant are calculated and presented in the form of graphs. The inhomogeneity constant which includes continuously varying volume fraction of the constituents is empirically determined. The values used in this study are arbitrarily chosen to demonstrate the effect of inhomogeneity on stress distribution. c 2006 Elsevier Ltd. All rights reserved.

Keywords: FGM; Pressure vessel; Elasticity; Axisymmetry

1. Introduction Functionally graded materials (FGMs) have attracted much interest primarily as heat-shielding materials. The possibility of tailoring the desired thermomechanical properties holds enormous application potential for FGMs. Aside from the thermal barrier coatings, some of the potential applications of FGMs include their use as interfacial zones to improve the bonding strength and to reduce residual stresses in bonded dissimilar materials and as wear-resistant layers such as gears, cams, ball and roller bearings and machine tools (Erdogan [1]). Most of the studies conducted on FGMs are confined to the analysis of thermal stress and deformation (see, e.g., Wetherhold et al. [2], Takezono et al. [3], Zhang et al. [4], Obata and Noda [5]). The works concerning the stress analysis of cylindrical and spherical structural elements involve finite elements and other numerical techniques due to the nature of functions chosen to describe the inhomogeneous properties (Fukui and Yamanaka, [6] Loy et al. [7], Salzar [8]). Developing sufficiently general methods for solving specific boundary value problems in solid mechanics involving

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E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.12.003

inhomogeneous media has always been difficult. Because of this difficulty, all existing treatments dealing with the mechanics of inhomogeneous solids are based on a simple function representing material inhomogeneity. For example, in the half-plane elasticity problems considered by Kassir and Chauprasert [9] and Kassir [10] it is assumed that the shear modulus is a power function of the depth coordinate of the form µ(y) = µ0 y m and the Poisson’s ratio ν is constant. Modeling of density and stiffness by the same power-law are proposed by Bert and Niedenfuhr [11], Reddy and Srinath [12] and Gurushankar [13]. The functionally gradient material considered by Loy et al. [7] is composed of stainless steel and nickel where the volume fractions follow a power-law distribution. Closed-form solutions are obtained by Tutuncu and Ozturk [14] for cylindrical and spherical vessels with variable elastic properties obeying a simple power law through the wall thickness which resulted in simple Euler–Cauchy equations whose solutions were readily available. A similar work was also published by Horgan and Chan [15] where it was noted that increasing the positive exponent of the radial coordinate provided a stress shielding effect whereas decreasing it created stress amplification. Three-dimensional solutions for FGM plates are obtained numerically by Reddy and Cheng [16] using the transfer matrix method. The overall material properties were calculated from the constituent

N. Tutuncu / Engineering Structures 29 (2007) 2032–2035

properties by the well-known Mori–Tanaka method. On the stress analysis of piezoelectric plates where the piezoelectric properties are functionally graded the work by Lim and He [17] presents exact solutions. Similar works on piezoelectric FGM plates are also presented by Liew et al. [18,19]. Free vibration analysis of such plates is performed by Lim et al. [20] where the transfer matrix method combined with the asymptotic expansion method is used. The parametric resonance of FGM rectangular plates is studied by Ng et al. [21] where Hamilton’s principle and Bolotin’s method are used to determine the instability regions. The present paper aims to present stress and displacement solutions in thick-walled cylinders subjected to internal pressure only. The material is assumed to be isotropic with exponentially-varying elastic modulus through the thickness as E(r ) = E 0 eβr yielding governing equations solutions of which are not readily available and can only be obtained in the form of power series by employing the lengthy process of Frobenius method. When the functional dependence is assumed for both the elastic modulus and Poisson’s ratio a simple tractable solution cannot be obtained necessitating the employment of numerical and perturbation techniques. For the sake of simplicity the insignificant influence of the variation in Poisson’s ratio on stresses is neglected and a constant Poisson’s ratio ν is assumed throughout the thickness as it is done in numerous works in the literature such as those by Erdogan [1], Horgan and Chan [15], Chen and Erdogan [22] and Jabbari et al. [23]. Various β values are used to demonstrate the effect of inhomogeneity on the stress distribution. The arbitrary values used in this study for the inhomogeneity constant β do not necessarily represent a certain material.

Using Eqs. (1)–(3), the governing equation of radial displacement becomes r 2 u 00 + r (1 + rβ)u 0 + (νβr − 1)u = 0 C12 C11

(5)

ν0 1−ν0 .

where ν = = Eq. (5) can be solved by Frobenius Method with the solution in the form ∞ X

u(r ) =

ak r k+s .

(6)

k=0

Substituting in Eq. (5) gives the recurrence formula ak = −

(k + s − 1) + ν βak−1 (k + s + 1)(k + s − 1)

(7)

and the indicial equation (s − 1)(s + 1) = 0. Since the roots of the indicial equation differ by an integer (s1 = 1, s2 = −1) only one of the solutions is in the form of Eq. (6). Expansion of the recurrence formula for k = 1, 2, 3, . . . gives the coefficients ak in terms of a0 and Gamma functions as: ak =

(−1)k 0(2 + s)2 0(k + s + ν) β k a0 . 0(s + ν)0(k + s)0(k + 2 + s)(1 + s)s

(8)

For the first root s = 1, taking the nonzero arbitrary constant a0 = 1 the recurrence relation takes the following form: ak =

2(−1)k 0(k + 1 + ν) k β . k!(k + 2)! 0(1 + ν)

(9)

Here, for an integer k, the property 0(k + 1) = k! has been used. The first solution is given as u1 =

2. Analysis

2033

∞ X

ak r k+1 .

(10)

k=0

The stress distribution in thick-walled cylindrical pressure vessels will be calculated. Elastic modulus for the isotropic material is assumed to vary as E(r ) = E 0 eβr .

The second solution for s = −1 will be of the form u2 =

du , dr

εθ =

u , r

+

 ∞  X d [(s + 1)ak (s)] r k−1 . ds s=−1 k=0

(11)

The multiplier of the logarithmic term is expanded first as γrθ = 0

σr = C11 εr + C12 εθ σθ = C12 εr + C11 εθ

(2)

∞ X

{(s + 1)ak (s)r k+s }s=−1

k=0

(3a,b)

where, with ν0 being the Poisson’s ratio,   E 0 (1 − ν0 ) 0 βr eβr ve C12 C11 = C11 e = (1 + ν0 )(1 − 2ν0 )   E 0 ν0 0 βr = C12 e = eβr . (1 + ν0 )(1 − 2ν0 )

# (s + 1)(−1)k 0(2 + s)2 0(k + s + ν) k k+s β r = a0 0(s + ν)0(k + s)0(k + 2 + s)(1 + s)s k=0 s=−1 ∞ X (−1)k−1 0(k − 1 + ν) = a0 β k r k−1 0(ν − 1)0(k − 1)0(k + 1) k=0 ∞ X

= a0 β 2

The only nontrivial equilibrium equation is dσr σr − σθ + = 0. dr r

{(s + 1)ak (s)r k+s }s=−1 log r

k=0

(1)

Employing the plain-strain assumption and axisymmetry, the strain–displacement and constitutive equations are εr =

∞ X

(4)

"

∞ X (−1)k+1 0(k + 1 + ν) k k+1 β r . 0(ν − 1)k!(k + 2)! k=0

(12)

It should be noted that since 0(0) = ∞ and 0(−1) = ∞, the summation should start from k = 2. Subsequently, the indices are changed as k → k + 2 to obtain the final form.

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N. Tutuncu / Engineering Structures 29 (2007) 2032–2035

To expand the second term in Eq. (11) the following relations are noted first:   (−1)k 0(2 + s)2 0(k + s + ν) k log[(s + 1)ak ] = log β a0 0(s + ν)0(k + s)0(k + 2 + s)s = log[(−1)k β k a0 ] + 2 log 0(2 + s) + log 0(k + s + ν) − log 0(s + ν) − log 0(k + s) − log 0(k + 2 + s) − log s. To simplify the differentiation, the logarithm of the term is treated. Taking the derivative now and introducing the 0 (z) d Psi(Digamma) function ψ(z) = dz log 0(z) = 00(z) the following is obtained in view of Eq. (8): {(s + 1)ak (s)}s=−1 = a0

(−1)k−1 0(k − 1 + ν) βk . 0(ν − 1)0(k − 1)0(k + 1)

Fig. 1. Radial displacement through the wall thickness.

Here, too, since 0(0) = ∞ and 0(−1) = ∞ the indices are adjusted as in Eq. (12). The final form of the second term in Eq. (11) is  ∞  X d [(s + 1)ak (s)] r k−1 ds s=−1 k=0 = a0 β 2

∞ X (−1)k+1 0(k + 1 + ν) k=0

0(ν − 1)k!(k + 2)!

β k r k+1 (1 + 2ψ(1)

− ψ(ν − 1) + ψ(k + 1 + ν) − ψ(k + 1) − ψ(k + 3)). (13) The second solution is thus obtained and is given concisely as follows: u2 =

∞ X

bk r k+1 log r +

k=0

∞ X

where,

bk ck r k+1 .

(14)

k=0

A1 =

In the above expression, since a0 is an arbitrary constant, the term a0 β 2 is set equal to 1. The coefficients are explicitly given as: (−1)k+1 0(k + 1 + ν) k β k!(k + 2)! 0(k − 1) ck = 1 + 2ψ(1) − ψ(ν − 1) + ψ(k + 1 + ν) − ψ(k + 1) − ψ(k + 3).

bk =

B2 =

bk r k+1 log r +

k=0

∞ X

A2 RHS A2 B1 − A1 B2 A1 RHS C2 = −A2 B1 + A1 B2

(k + 1 + ν)ak Ro

k=0 ∞ X

bk (1 + (k + 1 + ν)(ck + log Ro ))Ro

k=0 ∞ X

(k + 1 + ν)ak Ri

k=0 ∞ X

RHS = −

bk (1 + (k + 1 + ν)(ck + log Ri ))Ri P 0 e β Ri C11

0 and C11 =

E 0 (1 − ν0 ) . (1 + ν0 )(1 − 2ν0 )

Stress expressions take the following form !

bk ck r k+1 .

(15)

k=0

Here, the constants C1 and C2 are found using the boundary conditions σr = −P at (r = Ri ) and σr = 0 at (r = Ro ) to be C1 =

∞ X

k=0

k=0 ∞ X

A2 = B1 =

Finally, the complete solution of Eq. (5) is ! ∞ X k+1 ak r u(r ) = C1 u 1 + C2 u 2 = C1

+ C2

Fig. 2. Radial stress distribution through the wall thickness.

σr = eβr (εr + νεθ ) σθ = eβr (νεr + εθ ). 0 u , σ , σ are presented in the form The results of U ∗ = C11 r r θ of graphs in Figs. 1–3 for unit inside pressure and the values ν = 0.3, Ri = 0.6, Ro = 1.0 and β = 1, 2, 3.

(16)

3. Results and conclusions

(17)

The main objective of this work is to obtain tractable solutions rather than numerical results to allow for further

N. Tutuncu / Engineering Structures 29 (2007) 2032–2035

Fig. 3. Circumferential stress distribution through the wall thickness.

parametric studies. Stress and displacement solutions in the form of power series are presented in FGM thick-walled cylinders with exponentially-varying elastic modulus in the radial direction. A five-digit accuracy was obtained by taking twenty terms in the power series. Although the case of FGM cylinders with variation of elastic properties obeying a simple power law is extensively studied, the results for exponentiallyvarying properties are scarcely available in the literature. A positive inhomogeneity constant refers to increasing stiffness in the radial direction. Figs. 1–3 show, respectively, the radial displacement distribution, the radial stress distribution and the hoop stress distribution through the thickness. The results show that a positive inhomogeneity constant provides a stress shielding effect. It is obvious that a negative inhomogeneity constant would create a stress amplification effect. Increasing the positive inhomogeneity constant causes an increase in the stresses. It should be noted that, although no Poisson’s ratio is present in the solutions for homogeneous isotropic cylinders it is clearly noticed in the FGM case along with the inhomogeneity constant. The inhomogeneity constant, which includes continuously varying volume fraction of the constituents, is empirically determined and is a useful parameter from a design point of view in that it can be tailored for specific applications to control the stress distribution. Acknowledgements This work was supported by the research grant MMF.2000. 14 from C ¸ ukurova University. The author also gratefully acknowledges the contribution of Dr. Murat Ozturk of Lehigh University, USA. References [1] Erdogan F. Fracture mechanics of functionally graded materials. Composites Engineering 1995;5(7):753–70. [2] Wetherhold RC, Seelman S, Wang JZ. Use of functionally graded materials to eliminate or control thermal deformation. Composites Science and Technology 1996;56(4):1099–104.

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