The application of 2-dimensional elasticity for the elastic analysis of solid sphere made of exponential functionally graded material

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@mail.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


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 1-x Ge 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.

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 o P on its outer surface.
The problem can be studied in the spherical coordinates   r, ,  .In this paper, it is assumed that the Poi- son's ratio  , takes a constant value and the modulus of elasticity E , is assumed to vary radially according to ex- ponential form as follows [16], where 0 E and out E are modulus of elasticity in center and outer surface, respectively.n and  are material parame- ters.The displacement in the r -direction is denoted by u .
Three strain components can be expressed as: ; where r  and    are radial and circumferential strains.The Hooke's law are given by: 12 1 1 where r  and    are radial and circumferential stresses.Substituting Eqs. ( 2) and (3) into Eq.( 4) yields: where Substituting Eq. ( 5), into Eq.( 7), the equilibrium equation is expressed as: here, prime denotes differentiation with respect to R.
The general solution of Eq. ( 8) is as follows: where 1 C and 2 C are arbitrary integration constants, and   GR and   HR are homogeneous solutions.Substituting Eq. ( 9) into Eq.( 5), yields: The forms of

 
GR and   HR will be deter- mined next.
Substituting Eq. ( 1) into Eq.( 8), the governing differential equation is as follows: Eq. ( 11) is a homogeneous hypergeometric differential equation.Using a new variable x nR   and applying the transformation     u R Ry x  , the result Eq. ( 11) is: The solution of Eq. ( 12) is given as: In Eq. ( 13), x  is the confluent hypergeometric function defined by the series [17]: where The arguments  ,  of C F in Eq. ( 16) are de- termined as: From     u R Ry nR   , the homogeneous solutions   GR and   HR are found in the form: The Eqs. ( 9) and ( 10) may be rewritten with nondimensional paparmeters as: Hence, non-dimensional radial displacement, radial stress and circumferential stress are found as follows:

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.

Results and discussion
Consider a solid sphere with an arbitrary radius of b , subjected to an arbitrary constant uniform pressure  , there is an increase in the value of the circumferential stress as n increases, whereas for 0 65 R.  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.  for the material parameters, the stresses and displacement in an FGM solid sphere is calculated and compared to those in a homogeneous solid sphere ( 0 n  ) in Figs. 6 and 7.The ef- fect of material parameter n on the deformation behavior of the solid sphere is also evaluated.

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 inhomo- geneity 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.

Fig. 1
Fig. 1 Geometry of the FGM solid sphere subjected to constant external pressureThe equilibrium equation of the FGM solid sphere, in the absence of body forces, is expressed as: that the Poisson's ratio , has a constant value of 03 . .For the presentation of the results, use the following dimensionless and normalized variables.In Fig. 2, for different values of n and  , dimen- sionless modulus of elasticity along through the radial direction is plotted.It is apparent from the curve that at the same position   01 R , for 05 n.  , dimensionless modulus of elasticity increases as  decreases, while for 05 n.  , the reverse holds true.

Fig. 2
Fig. 2 Radial distribution of Modulus of elasticity