The application of 2-dimensional elasticity for the elastic analysis of solid sphere made of exponential functionally graded material/Dvimates tamprumo teorijos taikymai sferos pagamintos is eksponentiskai funkcionaliai sluoksniuotos medziagos tamprumo analizei.

Nejad, Mohammad Zamani ; Abedi, Majid ; Lotfian, Mohammad Hassan 等

1. Introduction

Hollow and Solid spherical shells are a common type of structure in engineering mechanics. This problem is studied by several researchers in the past. Among them, Srinath [1] obtained the analytical expressions of stresses and displacement in a solid sphere subjected to external pressure. Xiao-Ming and Zong-Da [2] using the method of weighted residuals, obtained the general solutions in forms of Legendre series for thick spherical shell and solid sphere.

Functionally graded materials (FGMs) are a class of new advanced composite materials with continuously varying material properties in one or multi spatial directions and consist of two or more constituents by changing their volume fraction for the goal of optimizing their performance. Closed-form solutions are obtained by Tutuncu and Ozturk [3] 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. Elastic analysis of internally pressurized thick-walled spherical pressure vessels of functionally graded materials was studied by You et al. [4].

Based on the assumption that Poisson's ratio is constant and modulus of elasticity is an exponential function of radius, Chen and Lin [5] have analyzed stresses and displacements in FG cylindrical and spherical pressure vessels. Singh et al. [6], making use of the particular forms of heterogeneity, solved the equation of equilibrium for torsional vibrations of a solid sphere made of functionally graded materials. A hollow sphere made of FGMs subjected to radial pressure was analyzed by Li et al. [7]. Using plane elasticity theory and Complementary Functions method, Tutuncu and Temel [8] are obtained axisymmetric displacements and stresses in functionally-graded hollow cylinders, disks and spheres subjected to uniform internal pressure. Zamani Nejad et al. [9] developed 3D set of field equations of FGM thick shells of revolution in curvilinear coordinate system by tensor calculus. An analytical solution is obtained by Wei [10] for inhomogeneous strain and stress distributions within solid spheres of [Si.sub.1-x][Ge.sub.x] alloy under diametrical compression.

Deformations and stresses inside multilayered thick-walled spheres are investigated by Borisov [11]. In the paper, each sphere is characterized by its elastic modules. Assuming the volume fractions of two phases of a functionally graded (FG) material (FGM) vary only with the radius, Nie et al. [12] obtained a technique to tailor materials for FG linear elastic hollow cylinders and spheres to attain through-the-thickness either a constant circumferential (or hoop) stress or a constant in-plane shear stress. Ghannad and Zamani Nejad [13] presented a complete analytical solution for FGM thick-walled spherical shells subjected to internal and/or external pressures. In another work, Zamani Nejad et al. [14] obtained an exact analytical solution and a numerical solution for stresses and displacements of pressurized thick spheres made of functionally graded material with exponentially-varying properties. On the basis of plane elasticity theory (PET), the displacement and stress components in a thick-walled spherical pressure vessels made of heterogeneous materials subjected to internal and external pressure is developed [15].

In this study, an elastic solution and a numerical solution for pressurized solid sphere made of functionally graded material is presented.

2. Analysis

An axisymmetric solid sphere with radius b is shown in Fig. 1 with the properties changing continuously along radial direction. The sphere is subjected axisymmetric constant pressure [P.sub.o] on its outer surface.

The problem can be studied in the spherical coordinates (r, [theta], [phi]). In this paper, it is assumed that the Poison's ratio [upsilon], takes a constant value and the modulus of elasticity E, is assumed to vary radially according to exponential form as follows [16],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [E.sub.0] and [E.sub.out] are modulus of elasticity in center and outer surface, respectively. n and [eta] are material parameters. The displacement in the r-direction is denoted by u. Three strain components can be expressed as:

[[epsilon].sub.r] = du/dr; (2)

[[epsilon].sub.[theta]] = [[epsilon].sub.[phi]] = u/r, (3)

where [[epsilon].sub.r] and [[epsilon].sub.[theta]] = [[epsilon].sub.[phi]] are radial and circumferential strains.

The Hooke's law are given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

where [[sigma].sub.r] and [[sigma].sub.[theta]] = [[sigma].sub.[phi]] are radial and circumferential stresses.

Substituting Eqs. (2) and (3) into Eq. (4) yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

The equilibrium equation of the FGM solid sphere, in the absence of body forces, is expressed as:

[d/dR]([R.sup.2][[sigma].sub.r]) - 2R[[sigma].sub.[theta]] = 0. (7)

Substituting Eq. (5), into Eq. (7), the equilibrium equation is expressed as:

[R.sup.2][[d.sup.2]u/d[R.sup.2]] + R(2 + [RE'/E])[du/dR] - 2(1 - [[upsilon].sup.*][RE'/E])u = 0, (8)

here, prime denotes differentiation with respect to R.

The general solution of Eq. (8) is as follows:

u(R) = [C.sub.1]G(R) + [C.sub.2]H(R), (9)

where [C.sub.1] and [C.sub.2] are arbitrary integration constants, and G(R) and H(R) are homogeneous solutions.

Substituting Eq. (9) into Eq. (5), yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

[FIGURE 1 OMITTED]

The forms of G(R) and H(R) will be determined next.

Substituting Eq. (1) into Eq. (8), the governing differential equation is as follows:

[R.sup.2][[d.sup.2]u/d[R.sup.2]] + R(2 - [eta]n[R.sup.[eta]])[du/dR] - 2(1 + [[upsilon].sup.*][eta]n[R.sup.[eta]])u = 0. (11)

Eq. (11) is a homogeneous hypergeometric differential equation. Using a new variable x = n[R.sup.[eta]] and applying the transformation u(R) = Ry(x), the result Eq. (11) is:

x[[d.sup.2]y/d[x.sup.2]] + (1 + [3/[eta]] - x)[dy/dx] - [[1 + 2[[upsilon].sup.*]]/[eta]]y = 0. (12)

The solution of Eq. (12) is given as:

y(x) = [C.sub.1][F.sub.C]([alpha], [beta];x) +

+ [[bar.C].sub.2][x.sup.-3/[eta]][F.sub.C]([alpha] - [beta] + 1,2 - [beta]; x). (13)

In Eq. (13), [F.sub.C]([alpha], [beta];x) is the confluent hypergeometric function defined by the series [17]:

[F.sub.C]([alpha], [beta];x) = 1 + [[infinity].summation over (k=1)][[([alpha]).sub.k]/[([beta]).sub.k]][[x.sup.k]/k!]. (14)

where

[([alpha]).sub.k] = [alpha]([alpha] + 1)([alpha] + 2) ... ([alpha] + k - 1). (15)

Thus

[F.sub.C]([alpha],[beta];x) = 1 + [[alpha]/[beta]][x/1!] + [[alpha]([alpha] + 1)/[beta]([beta] + 1)][[x.sup.2]/2!] +

+ [[alpha]([alpha] + 1)([alpha] + 2)/[beta]([beta] + 1)([beta] + 2)][[x.sup.3]/3!] + ... (16)

The arguments [alpha], [beta] of [F.sub.C] in Eq. (16) are determined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

From u(R) = Ry(n[R.sup.[eta]]), the homogeneous solutions G(R) and H(R) are found in the form:

G(R)= R[F.sub.C]([alpha],[beta];n[R.sup.[eta]]); (18)

H(R) = [1/[R.sup.2]][F.sub.C]([alpha] - [beta] + 1,2 - [beta];n[R.sup.[eta]]). (19)

The Eqs. (9) and (10) may be rewritten with nondimensional paparmeters as:

U(R) = [C.sub.3]G(R) + [C.sub.4]H(R); (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

Integration constants [C.sub.1] and [C.sub.2] are determined by using the following boundary conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

Hence, non-dimensional radial displacement, radial stress and circumferential stress are found as follows:

U = -[e.sup.n]G(R)/[AG'(1) + 2BG(1)]; (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

4. Numerical analysis

The finite element method is a powerful numerical method in solid mechanics. In this study in order to numerical analysis of problem, a geometry specimen was modeled using a commercial FE code, ANSYS 12, for a comparative study. The numerical solution is done by using of PLANE82 element, and the number of elements and nodes are considered 1371 and 2846, respectively. An axisymmetric element has been applied for modeling and meshing. For modeling of FGM solid sphere, the variation in material properties was implemented by having 20 layers, with each layer having a constant value of material properties.

5. Results and discussion

Consider a solid sphere with an arbitrary radius of b, subjected to an arbitrary constant uniform pressure [P.sub.o]. It is assumed that the Poisson's ratio [upsilon], has a constant value of 0.3.

For the presentation of the results, use the following dimensionless and normalized variables.

In Fig. 2, for different values of n and [eta], dimensionless modulus of elasticity along through the radial direction is plotted. It is apparent from the curve that at the same position (0 < R < 1), for n = -0.5, dimensionless modulus of elasticity increases as [eta] decreases, while for n = +0.5, the reverse holds true.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Distribution of the radial displacement and the radial stress along the radial direction for different values of n and constant value of [eta] = 0.9 are shown in Figs. 3 and 4. According to these figures, at the same position (0 < R < 1), for higher values of n, radial displacement and radial stress increase.

The circumferential stress along the radial direction for different values of n and constant value of [eta] = 0.9 is plotted in Fig. 5. It must be noted from this figure that at the same position, almost for R < 0.65, there is an increase in the value of the circumferential stress as n increases, whereas for R > 0.65 this situation was reversed. Besides, along the radial direction for the positive magnitudes of n the circumferential stress decreases, while for negative magnitude of n, the circumferential stress increases.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Using values n = -0.5 and [eta] = 1.5 for the material parameters, the stresses and displacement in an FGM solid sphere is calculated and compared to those in a homogeneous solid sphere (n = 0) in Figs. 6 and 7. The effect of material parameter n on the deformation behavior of the solid sphere is also evaluated.

6. Conclusions

In this work, elastic and numerical solutions for stresses, and displacement in pressurized FGM solid sphere are obtained. The material properties except Poisson's ratio are assumed to be exponential-varying in the radial direction.

To show the effect of inhomogeneity on the stress distributions, different values were considered for material parameter n. Numerical results showed that the inhomogeneity parameter n has great effect on the distributions of elastic fields. For example, the maximum of radial and circumferential stresses for negative values of material parameter n, occur on the external surface, whereas for positive values of n, this situation was reversed. Thus by selecting a proper value of n, it is possible for engineers to design a solid sphere that can meet some special requirements.

References

[1.] Srinath, L.S. 2003. Advanced Mechanics of Solids. 3nd Edition, New York: McGraw-Hill, 285p.

[2.] Xiao-Ming, B.; Zong-Da, Y. 1992. A general solution of axisymmetric problem of arbitrary thick spherical shell and solid sphere, Applied Mathematics and Mechanics-English Edition 13(6): 515-521. http://dx.doi.org/ 10.1007/BF02451514.

[3.] Tutuncu, N.; Ozturk, M. 2001. Exact solutions for stresses in functionally graded pressure vessels, Composites Part B-Engineering 32(8): 683-686. http://dx.doi.org/10.1016/S1359-8368(01)00041-5.

[4.] You, L.H.; Zhang, J.J.; You, X.Y. 2005. Elastic analysis of internally pressurized thick-walled spherical pressure vessels of functionally graded materials, International Journal of Pressure Vessels and Piping 82(5): 347-354. http://dx.doi.org/10.1016/j.ijpvp.2004.11.001.

[5.] Chen, Y.Z.; Lin, X.Y. 2008. Elastic analysis for thick cylinders and spherical pressure vessels made of functionally graded materials, Computational Materials Science 44(2): 581-587. http://dx.doi.org/10.1016/j.commatsci.2008.04.018.

[6.] Singh, B.M.; Rokne, J.; Dhaliwal, R.S. 2008. Vibrations of a solid sphere or shell of Functionally Graded Materials, European Journal of Mechanics A-Solids 27(3): 460-468. http://dx.doi.org/10.1016/j.euromechsol.2007.08.006.

[7.] Li, X-F.; Peng, X-L.; Kang, Y-A. 2009. Pressurized hollow spherical vessels with arbitrary radial nonhomogeneity, AIAA Journal 47(9): 2262-2265. http://dx.doi.org/10.2514/L41995.

[8.] Tutuncu, N.; Temel, B. 2009. A novel approach to stress analysis of pressurized FGM cylinders, disks and spheres, Composite Structures 91(3): 385-390. http://dx.doi.org/10.1016/j.compstruct.2009.06.009.

[9.] Nejad, M.Z.; Rahimi, G.H.; Ghannad, M. 2009. Set of field equations for thick shell of revolution made of functionally graded materials in curvilinear coordinate system, Mechanika 77(3): 18-26.

[10.] Wei, X.X. 2009. Analytical solutions for solid spheres of [Si.sub.1-x][Ge.sub.x] alloy under diametrical compression, Mechanics Research Communications 36(6): 682-689. http://dx.doi.org/10.1016/j.mechrescom.2009.03.002.

[11.] Borisov, A.V. 2010. Elastic analysis of multilayered thick-walled spheres under external load, Mechanika 84(4): 28-32.

[12.] Nie, G.J.; Zhong, Z.; Batra, R.C. 2011. Material tailoring for functionally graded hollow cylinders and spheres, Composites Science and Technology 71(5): 666-673. http://dx.doi.org/10.1016/j.compscitech.2011.01.009.

[13.] Ghannad, M.; Nejad, M.Z. 2012. Complete closed-form solution for pressurized heterogeneous thick spherical shells, Mechanika 18(5): 508-516. http://dx.doi.org/10.5755/j01.mech.18.5.2702.

[14.] Nejad, M.Z.; Abedi, M.; Lotfian, M.H.; Ghannad, M. 2012. An exact solution for stresses and displacements of pressurized FGM thick-walled spherical shells with exponential-varying properties, Journal of Mechanical Science and Technology 26(12): 4081-4087. http://dx.doi.org/10.1007/s12206-012-0908-3.

[15.] Karami, K.; Abedi, M.; Nejad, M.Z.; Lotfian, M.H. 2012. Elastic analysis of heterogeneous thick-walled spherical pressure vessels with parabolic varying properties, Frontiers of Mechanical Engineering 7(4): 433-438. http://dx.doi.org/10.1007/s11465-012-0336-1.

[16.] Eraslan, A.N.; Akis, T. 2006. On the plane strain and plane stress solutions of functionally graded rotating solid shaft and solid disk problems, Acta Mechanica 181(1-2): 43-63. http://dx.doi.org/10.1007/s00707-005-0276-5.

[17.] Abramowitz, M.; Stegun, A.I. (eds.). 1972. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 10th Edition, Washington, D.C.: US Government Printing Office, 556p.

Received September 07, 2012

Accepted May 09, 2014

Mohammad Zamani Nejad *, Majid Abedi **, Mohammad Hassan Lotfian ***, Mehdi Ghannad ****

* Mechanical Engineering Department, Yasouj University, P. O. Box: 75914-353, Yasouj, Iran, E-mails: m.zamani.n@gmail.com, m_zamani@jnail.yu.ac.ir

** Mechanical Engineering Department, Yasouj University, P. O. Box: 75914-353, Yasouj, Iran

*** Mechanical Engineering Department, Yasouj University, Yasouj P. O. Box: 75914-353 Iran

**** Mechanical Engineering Faculty, Shahrood University of Technology, Shahrood, Iran

cross ref http://dx.doi.org/10.5755/j01.mech.20.3.7395