Influence of coupling effects on analytical solutions of functionally graded (FG) spherical shells of revolution

Due to the lack of commercially available finite elements packages allowing us to analyse the behaviour of porous functionally graded (FG) structures in this paper, axisymmetric deformations of coupled FG spherical shells are studied. The analytical solution is presented by using complex hypergeometric polynomial series. The results presented agree closely with the reference results for isotropic spherical shells of revolution. The influence of the effects of material properties is characterized by a multiplier characterizing an unsymmetric shell wall construction (stiffness coupling). The results can be easily adopted in design procedures. The present results can be treated as the benchmark for finite element investigations.


Introduction
The analytical description of axisymmetric isotropic shells of revolution is demonstrated in monographs [1][2][3]. Galletly and Muc [4,5] and Muc [6] studied buckling and deformations of laminated shells of revolution considering only laminated unidirectional symmetric configurations. Due to coupling effects between the membrane and bending stress resultants, the analysis of arbitrarily laminated shells was usually carried out using numerical methods, e.g. Czebyshev expansions [7].
Considering the structural deformations in only the elastic range, the analysis of composite constructions can be divided into three categories: -Unidirectional laminatesthe potential abrupt variation of mechanical properties between laminae. -Porous functionally graded materials (FGM)a smooth variation of the properties from the bottom to the top surface. -Nanostructures reinforcing isotropic matrixthey are made of a polymer matrix reinforced with nanoplatelets or carbon nanotubes; the material properties are derived with the use of homogenization theories, e.g. the Mori-Tanaka method.
Due to the possible variations of material characteristics along the spatial coordinates (x, y, zthe thickness direction), it is rather difficult or impossible to find analytical solutions characterizing structural deformations even for the simplest shells as, e.g. cylindrical, conical, spherical, etc. The variations along the thickness direction z can be eliminated by the 2D approximations (power series, Legendre, Czebyshev polynomials -Muc [8], Fantuzzi et al. [9]). Considering two geometrical variables (x = Rφ, y = Rθ, where R is the radius of the sphere, φ the meridional coordinate, and θ the circumferential coordinate) defining the variations along the shell reference midsurface for axisymmetric shell constructions, it is possible to formulate and solve three independent problems: (1) shells closed at the vertex, (2) shells opened at the vertex and (3) shell panels (Figure 1).
For the 2D Love-Kirchhoff kinematical hypothesis, the stress-strain relations can be expressed in the following way: where N denotes the membrane forces, M the bending moments, ε the membrane strains and κ the changes in curvatures. Each of the above values has three components. A, B and D denotes the matrices [3 × 3], and their explicit form will be presented in the next section.
For anisotropic shell structures (unidirectional laminates, nanoplates and nanotubes embedded in a matrix), the terms of the stiffness matrices A, B and D are functions of two global coordinates x and y and define the structural configuration along the thickness coordinate z. Considering porous FGM, the variability of the terms of the matrices A, B and D depends on the definition of the thickness distributions t(x, y, z) (assuming that the Young modulus E is a function of the z coordinate). If the shell thickness t is the function of the z coordinate only, the B terms are not equal to zero for unsymmetric configurations of pores ( Figure 2).
Note that in the literature a variety of papers exists solving different problems for 2D axisymmetric shell constructions (anisotropic or porous FGM); see Table 1. In general, these problems can be divided into two classes: 1. Solutions are functions of two geometrical variables x and y for shells with unsymmetric loading and boundary conditions and shell panels ( Figure 1) and shells having unsymmetric wall configurations; see Muc and Muc-Wierzgoń [10]. This class deals also with buckling and free vibrations problems where the expansion along the circumferential direction is required.
2. Solutions are functions of one meridional variable only for symmetric shell wall configurations, axisymmetric boundary and loading conditions for constructions plotted in Figures 1a and b; the above condition is also satisfied for unsymmetric shell wall constructions made of FG materials (Figure 1c).
Although a great number of studies have been done for static, buckling and free vibration analysis of axisymmetric shells using isotropic and laminated materials, as pointed out in Table 1, a general lack of information for axisymmetric structures made of FGM is observed since the problems were solved with the use of numerical methods only (finite element [11,12] or finite difference methods [13][14][15] including also the generalized differential quadrature method GDQ [16][17][18][19]). Note that the applied method of solutions is directly connected with the used numerical methodfor instance, the problems with singular points are solved with the use of finite difference methods [13][14][15], whereas the GDQ method was adopted to the discussion of problems without singular points only (see e.g. Sayyad and Ghugal [20]).  The aim of the present article is to extend the analytical solutions for axisymmetric spherical shells made of FGM and subject to the normal uniform pressure. The solutions are obtained with the use of hypergeometric functions and compared with the available published results for isotropic shells. Since such a class of problems has not been considered and discussed in the literature, to the best of our knowledge, the demonstrated herein solutions are the novelty in the existing literature and in addition they can constitute and can be easily adopted to the design codes of composite spherical shells. It has been proved analytically that for FGM shells the influence of the unsymmetric shell wall construction (the non-zero terms of the stiffness matrix B, Figure 1c) can be described by only one parameter (functionmultiplier) and is independent of the form of boundary value problems.

Formulation of the problem
The shells of revolution can be defined in the curvilinear orthogonal coordinates φ and θ of a point on the shell midsurface. It is convenient to take the spherical coordinates, where the angle φ defines the location of a point along the meridian, and θ describes the location of a point along the parallel circlethe circumferential coordinate (see Figure 3). R 1 = R 1 (φ) is the principal radius of the meridian and R 2 = R 2 (φ)is the principal radius of the parallel circle. Note that r = R 2 sin(φ) and the Lame parameters are defined as follows: When an arbitrary shell of revolution is subjected to rotationally symmetrical loads, its deformations and stress resultants and couples do not depend upon the circumferential variable θ. The axisymmetric deformations of shells of revolution are described by two displacement components of the shell midsurface u and w in the meridional and normal directions to the shell cross-section, respectively ( Figure 3).
For shells of revolution under axisymmetrical loads, the strain displacement relations take the following form: where ε denotes the membrane strain, κ the change in curvature, and 1, 2 correspond to the meridional φ and circumferential directions θ, respectively. The constitutive equations are written in the following way:   21 11 where N r are the stress resultants and M r the moment resultants. The elastic modulus E variation characterizes the distribution of porosity along the thickness direction  Figure 3: Cross-section of an axisymmetric shell of revolution.
Coupling effects in FG spherical shells  763 z and is defined as where t and b refer to the material properties on top and bottom surfaces, n is the power index and ν is Poisson's ratio.
To complete the set of fundamental relations, we add the equilibrium equations derived using the principle of virtual work [1][2][3]: where T denotes the transverse shear force and q is the normal pressure to the shell midsurface. Note that the present formulation (the Love-Kirchhoff hypothesis) is valid for thin FG plates or shells, i.e. for t/R < 0.05. For thicker structures, it is necessary to adopt more complicated forms of kinematical approximations taking into account possible deformations of the normal to a shell/plate midsurface, e.g. in the form proposed by Vlasov [21]. A broader discussion of these problems is presented by Muc and Flis [22]. In the buckling, free vibration analysis the use of geometrically nonlinear relations is required, e.g. in the form proposed by Muc [8,15].
Both for complex kinematical hypotheses and nonlinear relations, the stress-strain relations (3) become more complex as defined in refs [8,9,15].

Solution of governing relation for axisymmetric spherical shells
Let us examine deformations of spherical shells (R 1 = R 2 = R) having constant thickness, i.e. t(φ) = const. Using the classical approach [1][2][3] for isotropic shells, the system of equations is reduced to the following form: 11 11 φ e is a coordinate of the shell edge and the constant C characterizes an axial coordinate of the external load applied at the top edge. By inserting B = 0, relations (7) and (8) are reduced to the classical Meissner equations for isotropic shells. Elimination of each variable (T or β) in equations (7) and (8) leads to fourth-order differential equations, i.e.
Each of equations (11) and (12) can be solved for one of the dependent variables. The solutions can be represented as Each of the equations has complex solutions (AD > B 2 ). The hypergeometric equation (14) has three regular singular points: 0 (φ = 0), 1 (φ = π/2) and ∞. Usually, for spherical shells, the analytical solutions of complementary equations are formulated for two regions corresponding to singularity. Using the Mathematica package, the solutions of the fourth-order differential equations for complementary variables can be derived analytically with the aid of one command DSolve; see the general formulation presented by Polyanin and Zaitsev [78]. The general solution is represented by the complex hypergeometric functions and have the following form:  (15c) where C 1 …C 4 are the complex constants of integration. Since Re(Z 1 ) = Re(Z 2 ), Im(Z 1 ) = −Im(Z 2 ), Re(Z 3 ) = Re(Z 4 ), and Im(Z 3 ) = −Im(Z 4 ), the solutions of equation (15a) can be represented as real variables with real constants of integration C C , ...,1 4 . For FGM spherical shells, the deformation functions are controlled by the value of the parameter μ 2 (equation (14)). For isotropic shells, this value is the function of the ratio R/t: but for FGM shells, it is also the function of the coupling effects expressed by the term B (equation (14)). Figure 4 illustrates the influence of the coupling effects on the variations of the parameter μ 2 . It is represented by two values: the ratio E t /E b and the porosity index n (equation (5)).
The evaluation of T c and β c variables is identical and therefore the value of the parameter μ affects the solutions of the complementary equation (14) in a similar way. The results are presented in Figure 5 for a spherical segment where 0 < φ 1 ≤ φ ≤ φ 2 < π/2. In such a case, the solutions (equation (15b)) have no singular points and demonstrate only the coupling effects. For isotropic and porous FG structures, the difference in the complementary solution Re[Z 1 ], Im[Z 1 ] (equation (15b)) can reach even almost 10%; however, for the solution, Im[Z 1 ] is almost negligible. The identical plots can be drawn for equation (15c), which are the solutions for φ 1 ≤ φ ≤ π/2 < φ 2 (the singular point at π/2; Figure 6).
Comparing the complementary solutions for isotropic and FGM shells (Figures 5 and 6), one can observe the effects of the / μ μ 2 isotr 2 ratio. Both the changes in the index parameter n and of the E b /E t ratio affect the solutions expressed by the solution functions. In general, the results illustrate the localized effects at the lower boundary (i.e. φ = φ 2 , φ = π/2).
From the equilibrium equation (17), one can obtain the analytical form of the stress resultants N φ c , N θ c :  Coupling effects in FG spherical shells  765 A u denotes a real constant of integration. The computations can be made easily applying the symbolic packages, e.g. Mathematica.
It is convenient and necessary to distinguish two classes of problems that should be introduced independently, referring directly to the convenience and simplicity of analytical solutions given for two types of singular points (equation (15b) and (15c)).

Spherical shells under a uniform internal pressure
Let us consider the case of the spherical shell loaded by the uniform external pressure q. For particular solutions, the appropriate differential equations (11) and (12) are homogeneous and Now, we consider two solutions.

Clamped spherical segment
The clamped edge conditions are the following: Conditions (20) correspond to β c = 0 at the edges (equation (2)).
Using the equilibrium equation (6) and taking into account the form of particular solutions (20), one can find that For clamped spherical shells, it is convenient to express the boundary conditions (20) in the equivalent form The solution of equation (11) has four real constants of integration C C , ...,1 4 . They are evaluated from equation (23). T c is represented by β c in equation (8), and ε 2 is computed from equations (3), (21), and (22), where all unknown functions are expressed by functions β c . Figure 7 illustrates the distributions of circumferential membrane forces N 2 for spherical segments. These results are similar to those presented by Muc [70] for shallow shell segments. The maximal difference between isotropic and porous FG spherical shell segments occurs at the lower φ 1 boundary and corresponds to the results of complementary solutions plotted in Figure 5.

Clamped hemispherical shell 0 ≤ φ ≤ π/2
For the structure, the boundary conditions can be written in the following way:  and clamped at the edge φ = π/2: The value of the particular solutions: Since equations (11) and (12) are homogeneous. However, the effects of the distributed load q are brought by the equilibrium equation (6), i.e.
The stress resultants can be derived from the equilibrium conditions (6) and the relations (24) and (26)-(28): Using the above relations and the definitions (2) and (3) and the relation (7), the stress couples can be written in the following way: Inserting the coupling stiffness B = 0, one can find the relations (29)-(32) for isotropic shells [2,3]. The symbols P1, P2 are integration constants and, for the boundary conditions (26), they are equal to Figure 8 demonstrates the distributions of the dimensionless circumferential forces N 2 /(qt) along the shell meridian. The results demonstrate that the localized effects at the clamped edge increase with the decrease of the thickness ratio. The coupling effects expressed by the controlling parameter μ 2 (14) have a significant influence on the values of the circumferential forces at the clamped edgethe growth of this value results in the increase of bending effects.   The distributions of the stress resultants N 2 show that these values are almost equal to 0.5qR (the membrane state).

Concluding remarks
The analytical solutions presented herein demonstrate evidently the significant influence of the coupling terms B illustrating the unsymmetry in the construction of shell walls of porous FGM. These effects are characterized by the controlling parameter μ. It is worth mentioning that a similar controlling parameter is proposed and introduced in the description of flutter problems for structures made of porous FGM or plates reinforced by nanostructures (nanoplates or nanotubes)(see Muc et al. [10,22]).
It is very important to formulate precisely the assumptions of the analysis, particularly in view of the used approaches. In our opinion, the classification presented in Table 1 can be useful in understanding the limitations (or generality) of the results and not only for spherical shells.
Let us note that the distributions of the stress resultants and bending moments are sensitive to the variations of the thickness ratio t/R. In the present work, the classical Love-Kirchhoff hypothesis is employed but, for thicker structures (t/R > 0.1), the higher-order 2D shell theories should be used, e.g. in the form proposed in the Appendix of ref. [22].
The present results are derived for spherical shells; however, the identical analysis can be easily extended to the analysis of shells or pressure vessels having various forms: paraboloidal, hyperbolical, elliptical, and torispherical discussed from the numerical and optimization point of view, e.g. in ref. [79,80].
The results presented demonstrate that the analytical results can be treated as a benchmark for finite element analysis.
Funding information: The paper publication has received funding from Cracow University of Technology.