1 Introduction

Foams made up of hollow spheres are becoming more and more attractive for engineering because of their light weight and long plateau stress range, which leads to their excellent energy-absorption capacity [1]. Hollow spheres can make up a kind of very important category of materials that is used in many fields, such as aerospace, chemistry, biotechnology and material science [2]. Various kinds of new technologies have been developed to manufacture hollow spheres and bulk foams with a wide range of diameter and wall thickness, which may exhibit very different mechanical property [3–5]. For example, Fig. 1 shows the structures of K405 alloy hollow sphere foams [6]. In order to understand the failure mechanism of these foams, it is vital to know the mechanical properties of the individual sphere under diametrical loads [7], because each hollow sphere can be considered as an element of the foams, and the present solution can be directly used for the failure analysis for the spheres shown in Fig. 1. In addition, hollow spheres with a wide range of sphere diameter and wall thickness have been manufactured and used in fields of medical science, chemistry, biotechnology, electronic materials and civil engineering [8, 9].

Fig. 1
figure 1

K405 alloy hollow sphere foams [6]

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Extensive experimental studies have been reported on the mechanical behavior of single hollow sphere. Carlisle et al. [10] investigated microstructure and compressive properties of carbon microballoons under uniaxial compression. Dalla et al. [11] and Ebrahimi et al. [12] studied mechanical properties of nanocrystalline nickel. Gasser et al. [13] investigated uniaxial tensile elastic properties of brazed hollow spheres, and Gupta et al. [14] and Gupta and Venkatesh [15] studied the mechanical behavior of metallic spherical shells under axial compression between rigid plates. Karagiozova et al. [7] analyzed stress–strain relationship for metal hollow sphere materials. Koopman et al. [16] studied the mechanical properties of hollow microspheres by using compression test. Li et al. [17] used high-speed photography and studied strain rate-dependent compressive properties of glass microballoon and quantification of impact energy dissipation capacity in metallic thin-walled hollow sphere foams. Dong et al. [18] investigated the effect of the impact velocity on the dynamic deformation and crushing of celluloid ping pong balls. Vesenjak et al. [19], Zeng et al. [20] and Lim et al. [21] studied dynamic behaviors of metallic hollow sphere structures and forms.

Numerical methods, such as finite element method (FEM), have long been employed to study the behavior of hollow spheres and spherical structures. For example, Carlisle et al. [10] used FEM and studied the uniaxial compression behavior of carbon microballoons. Gupta and Venkatesh [15] studied the dynamic axial compression of thin-walled spherical shells. Sanders and Gibson [22] investigated the mechanical properties of hollow sphere foams. Shorter et al. [23] analyzed the elastic buckling deformation of thin-walled hollow elastic spheres compressed between two parallel rigid surfaces.

However, few analytical solutions are obtained for hollow spheres under external loads. Fok and Allwright [24] obtained an elastic solution for free vibrations of an eccentric hollow sphere and studied the buckling of a spherical shell embedded in an elastic medium. Koiter [25, 26] did nonlinear analysis for the buckling problem of a complete spherical shell under uniform external pressure. Kobayashi et al. [27] and Kobayashi and Ishimaru [28] derived an elastodynamic solution for an anisotropic hollow sphere subject to a uniform sudden load. By using the ray theory and the Laplace transforms with rational approximations for inverse Laplace transforms, Pao and Ceranoglu [29] and Bickford and Warren [30] obtained exact solutions for an isotropic and transversely isotropic hollow sphere under uniform external stress, respectively. For non-axisymmetric deformation of hollow spheres, only approximate solutions for thin hollow spheres based on shell theory [31–33], Gregory et al. [34] obtained an asymptotic solution for a thick hollow sphere compressed by equal and opposite concentrated axial loads based on a refined theory. Although some other analytical solutions for solid spheres under point loads have been obtained [35–37], there is no exact analytical solution for the elastic hollow spheres subjected to the diametrical point loads.

Therefore, in this paper, based on the system of fifteen partial differential equations of the governing equations for elastic hollow spheres, which is composed of six physical equations, three equilibrium equations and six geometrical equations, new variables are introduced in order to uncouple the system of fifteen partial differential equations so that explicit differential equations are obtained for displacement components, which are further expressed in terms of Legendre’s polynomial of the first kind. Similar to the classical solution by Hiramatsu and Oka [38] for isotropic solid spheres and the analytical solutions by Chau and Wei [35] for anisotropic solid spheres under the point loads, the diametrical point loads acting on the hollow sphere are also modeled by uniform pressure acting on two small areas on the surface of the sphere, which is further expressed in terms of Fourier–Legendre series in order to determine the unknown coefficient in the analytical solutions for hollow spheres. The present exact analytical solution is suitable for thin and thick hollow spheres and can be considered as an extension of the classical solution by Hiramatsu and Oka [38] for solid spheres under the point loads, which provided theoretical basis for the point load strength test (PLST) and has been recognized as an extremely useful index test in engineering for rock classification and strength estimation. The present solution, however, can be used as a basic solution as well as a benchmark for numerical solutions for studying the failure mechanism of bulk foams made up of hollow spheres.

Note that the present analytical solution is not suitable for analyzing the stress distributions within eccentric hollow spheres or solid spheres with ellipsoidal defects, or hollow spheres under biaxial or even triaxial loads, or other more complicated cases. Since FEM has been extensively used to study the stresses under complicated loading situations, FEM can be conveniently used to study the stress distributions within hollow spheres in these complicated cases.

2 Governing equations

Consider a spherical polar coordinate system \((r,\theta ,\varphi )\) with the origin locating at the center of a hollow sphere subjected to point loads, as shown in Fig. 2. The generalized Hooke’s law for an elastic isotropic hollow sphere can be written as

$$\begin{aligned} \sigma _{rr}&= \frac{E}{1+\nu }\left( {\frac{\nu }{1{-}2\nu }\varepsilon _{_H} +\varepsilon _{rr} } \right) ,\quad \sigma _{\theta \theta } \!=\!\frac{E}{1+\nu }\left( {\frac{\nu }{1{-}2\nu }\varepsilon _{_H} +\varepsilon _{\theta \theta } } \right) ,\quad \sigma _{\varphi \varphi } =\frac{E}{1+\nu }\left( {\frac{\nu }{1{-}2\nu }\varepsilon _{_H} +\varepsilon _{\varphi \varphi } } \right) , \nonumber \\ \tau _{\theta \varphi }&= \frac{E}{\left( {1+\nu } \right) }\varepsilon _{\theta \varphi },\quad \tau _{r\varphi } =\frac{E}{\left( {1+\nu } \right) }\varepsilon _{r\varphi },\quad \tau _{r\theta } =\frac{E}{\left( {1+\nu } \right) }\varepsilon _{r\theta } \end{aligned}$$
(1)

where \(\varepsilon _{_H} =\varepsilon _{rr} +\varepsilon _{\varphi \varphi } +\varepsilon _{\theta \theta }\). The Cauchy stress tensor is denoted by \(\sigma \), and the associated strain tensor is denoted by \(\varepsilon \). Symbols \(E\) and \(\nu \) are Young’s modulus and Poisson’s ratio, respectively.

Fig. 2
figure 2

A hollow sphere under the point loads is modeled by uniform radial stress \(p\) applied over two small spherical areas subtending an angle of \(2\theta _0\), the radius of the outer and the inner surface is denoted by \(R\) and \(R_0\), and the pressure acting on the inner surface is \(p_0\)

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For the present problem of elastic hollow spheres under the diametrical point loads, body forces can be neglected. The equations of equilibrium are

$$\begin{aligned}&\frac{\partial \sigma _{rr} }{\partial r}+\frac{1}{r}\frac{\partial \tau _{\theta r} }{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial \tau _{r\varphi } }{\partial \varphi }+\frac{1}{r}\left( {2\sigma _{rr} -\sigma _{\theta \theta } -\sigma _{\varphi \varphi } +\tau _{r\theta } \cot \theta } \right) =0 \nonumber \\&\frac{\partial \tau _{r\theta } }{\partial r}+\frac{1}{r}\frac{\partial \sigma _{\theta \theta } }{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial \tau _{\theta \varphi } }{\partial \varphi }+\frac{1}{r}\left[ {\left( {\sigma _{\theta \theta } -\sigma _{\varphi \varphi } } \right) \cot \theta +3\tau _{r\theta } } \right] =0 \nonumber \\&\frac{\partial \tau _{r\varphi } }{\partial r}+\frac{1}{r}\frac{\partial \tau _{\theta \varphi } }{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial \sigma _{\varphi \varphi } }{\partial \varphi }+\frac{1}{r}\left( {3\tau _{r\varphi } +2\tau _{\theta \varphi } \cot \theta } \right) =0 \end{aligned}$$
(2)

The relations between the strain and displacement components in a spherical polar coordinate are

$$\begin{aligned} \varepsilon _{rr}&= \frac{\partial u_r }{\partial r},\quad \varepsilon _{\theta \theta } =\frac{1}{r}\frac{\partial u_\theta }{\partial \theta }+\frac{u_r }{r},\quad \varepsilon _{\varphi \varphi } =\frac{1}{r\sin \theta }\frac{\partial u_\varphi }{\partial \varphi }+\frac{u_r }{r}+\frac{u_\theta }{r}\cot \theta ,\nonumber \\ \varepsilon _{r\varphi }&= \frac{1}{2}\left( \frac{1}{r\sin \theta }\frac{\partial u_r }{\partial \varphi }-\frac{u_\varphi }{r}+\frac{\partial u_\varphi }{\partial r}\right) ,\quad \varepsilon _{r\theta } =\frac{1}{2}\left( \frac{1}{r}\frac{\partial u_r }{\partial \theta }-\frac{u_\theta }{r}+\frac{\partial u_\theta }{\partial r}\right) ,\nonumber \\ \varepsilon _{\theta \varphi }&= \frac{1}{2}\left( \frac{1}{r}\frac{\partial u_\varphi }{\partial \theta }-\frac{u_\varphi }{r}\cot \theta +\frac{1}{r\sin \theta }\frac{\partial u_\theta }{\partial \varphi }\right) \end{aligned}$$
(3)

where \(u_\theta , u_\varphi \) and \(u_r\) are displacements in \(\theta ,\varphi \) and \(r\) directions, respectively.

3 Boundary conditions for the hollow spheres under the diametrical point loads

For the diametrical compression of hollow spheres of radius \(R\), the pair of point forces of magnitude \(F\) are modeled by uniform radial stress \(p\) applied over two opposite spherical areas on the surface \(r=R\) of the hollow sphere, which subtend an overall angle of \(2\theta _0 \) from the origin symmetrically with respect to the \(z\)-axis, as shown in Fig. 2. All other tractions are zero on the surface \(r=R\) of the hollow sphere. Mathematically, this boundary condition can be expressed as

$$\begin{aligned} \sigma _{rr}&= \left\{ \begin{array}{lll} -p&{}\quad \hbox {for}&{} \quad 0\le \theta \le \theta _0\,\hbox {and}\,\pi -\theta _0 \le \theta \le \pi \\ 0&{}\quad \hbox {for}&{}\quad \theta _0 <\theta <\pi -\theta _0 \end{array}\right. \end{aligned}$$
(4)
$$\begin{aligned} \tau _{r\varphi }&= \tau _{r\theta } =0 \end{aligned}$$
(5)

on \(r=R\), where \(p\) can be expressed in terms of \(F, R\) and \(\theta _0 \) as [38]

$$\begin{aligned} p=F/\left[ 2\pi R^{2}(1-\cos \theta _0 )\right] \end{aligned}$$
(6)

Moreover, the boundary condition for the inner surface of the hollow sphere is

$$\begin{aligned} \sigma _{rr} =-p_0, \quad \tau _{r\varphi } =\tau _{r\theta } =0 \end{aligned}$$
(7)

on \(r=R_0 \), and \(p_0 \) is the magnitude of a constant pressure acting on the inner surface of the hollow sphere.

Now, our problem is to solve fifteen governing equations expressed in Eqs. (1–3) under the boundary conditions (4–7). The techniques for the analytical method of solutions are considered next.

4 Method of solution

The key step for deriving the exact analytical solution for the hollow sphere under the diametrical point loads is to uncouple fifteen governing equations expressed in Eqs. (1–3). Considering that the present problem is axisymmetric with respect to \(\varphi \), all terms related to the partial differential with respect to \(\varphi \) in (2) and (3) are reasonably set to zero.

Substituting (3) into (1) yields

$$\begin{aligned} \sigma _{rr}&= \frac{\lambda }{r^{2}\sin \theta }\left\{ {\frac{\partial }{\partial r}\left( {r^{2}\cdot u_r \sin \theta } \right) +\frac{\partial }{\partial \theta }\left( {ru_\theta \cdot \sin \theta } \right) } \right\} +2\mu \frac{\partial u_r }{\partial r}\end{aligned}$$
(8)
$$\begin{aligned} \sigma _{\theta \theta }&= \frac{\lambda }{r^{2}\sin \theta }\left\{ {\frac{\partial }{\partial r}\left( {r^{2}\cdot u_r \sin \theta } \right) +\frac{\partial }{\partial \theta }\left( {ru_\theta \cdot \sin \theta } \right) } \right\} +2\mu \left( {\frac{1}{r}\frac{\partial u_\theta }{\partial \theta }+\frac{u_r }{r}} \right) \end{aligned}$$
(9)
$$\begin{aligned} \sigma _{\varphi \varphi }&= \frac{\lambda }{r^{2}\sin \theta }\left\{ {\frac{\partial }{\partial r}\left( {r^{2}u_r \sin \theta } \right) +\frac{\partial }{\partial \theta }\left( {ru_\theta \sin \theta } \right) } \right\} +2\mu \left( {\frac{u_r }{r}+\frac{u_\theta }{r}\cot \theta } \right) \end{aligned}$$
(10)
$$\begin{aligned} \tau _{r\theta }&= \mu \left( {\frac{1}{r}\frac{\partial u_r }{\partial \theta }+\frac{\partial u_\theta }{\partial r}-\frac{u_\theta }{r}} \right) \end{aligned}$$
(11)
$$\begin{aligned} \tau _{\theta \varphi }&= \mu \left( {\frac{1}{r}\frac{\partial u_\varphi }{\partial \theta }-\frac{u_\varphi }{r}\cot \theta } \right) \end{aligned}$$
(12)
$$\begin{aligned} \tau _{r\varphi }&= \mu \left( {\frac{\partial u_\varphi }{\partial r}-\frac{u_\varphi }{r}} \right) \end{aligned}$$
(13)

where \(\lambda =\frac{E}{1+\nu }\frac{\nu }{1-2\nu }, \mu =\frac{E}{2\left( {1+\nu } \right) }\).

The substitution of (8–13) into the first and second equations of (2) leads to

$$\begin{aligned}&\left( {\lambda +2\mu } \right) \frac{\partial \varDelta }{\partial r}-\frac{2\mu }{r}\frac{\partial \varOmega }{\partial \theta }-\frac{2\mu }{r}\varOmega \cdot \cot \theta =0\end{aligned}$$
(14)
$$\begin{aligned}&\left( {\lambda +2\mu } \right) \frac{1}{r}\frac{\partial \varDelta }{\partial \theta }+2\mu \frac{\partial \varOmega }{\partial r}+2\mu \frac{\varOmega }{r}=0 \end{aligned}$$
(15)

where

$$\begin{aligned} \varDelta&= \frac{1}{r^{2}\sin \theta }\left[ {\frac{\partial }{\partial r}\left( {r^{2}u_r \sin \theta } \right) +\frac{\partial }{\partial \theta }\left( {ru_\theta \sin \theta } \right) } \right] \end{aligned}$$
(16)
$$\begin{aligned} \varOmega&= \frac{1}{2}\left( {\frac{\partial u_\theta }{\partial r}-\frac{1}{r}\frac{\partial u_r }{\partial \theta }+\frac{u_\theta }{r}} \right) \end{aligned}$$
(17)

Eliminating \(\varDelta \) from (14) and (15), and after detailed calculation, the following equation is obtained

$$\begin{aligned} \frac{\partial ^{2}\varOmega }{\partial r^{2}}+\frac{2}{r}\frac{\partial \varOmega }{\partial r}+\frac{1}{r^{2}}\frac{\partial ^{2}\varOmega }{\partial \theta ^{2}}+\frac{\cot \theta }{r^{2}}\frac{\partial \varOmega }{\partial \theta }-\frac{1}{\sin ^{2}\theta }\frac{\varOmega }{r^{2}}=0 \end{aligned}$$
(18)

The general solution for (18) can be obtained as

$$\begin{aligned} \varOmega =-\sum _{n=0}^\infty {\left[ {A_n r^{n}+A^{\prime }_n r^{-(n+1)}} \right] \frac{\partial P_n (\cos \theta )}{\partial \theta }} \end{aligned}$$
(19)

where \(P_n (\cos \theta )\) is the Legendre’s polynomial of the first kind, and \(A_n\) and \(A^{\prime }_n\) are unknown coefficients to be determined by the boundary condition.

Substituting of (19) into (14) or (15) leads to the expression of \(\varDelta \) as

$$\begin{aligned} \varDelta =\frac{2\mu }{\lambda +2\mu }\sum _{n=0}^\infty {\left[ {(n+1)A_n r^{n}-nA^{\prime }_n r^{-(n+1)}} \right] P_n \left( {\cos \theta } \right) } \end{aligned}$$
(20)

The general expressions for \(u_r\) and \(u_\theta \) can be further obtained by substitution of (19) and (20) into (16) and (17), i.e.,

$$\begin{aligned} u_r&= \sum _{n=0}^\infty {\left[ {\lambda _1 (n)\cdot A_n r^{n+1}+\lambda _2 (n)\cdot A^{\prime }_n r^{-n}+n\cdot C_n r^{n-1}+(n+1)\cdot C^{\prime }_n r^{-(n+2)}} \right] P_n (\cos \theta )}\end{aligned}$$
(21)
$$\begin{aligned} u_\theta&= \sum _{n=0}^\infty {\left[ {\lambda _3 (n)\cdot A_n r^{n+1}+\lambda _4 (n)\cdot A^{\prime }_n r^{-n}+C_n r^{n-1}-C^{\prime }_n r^{-(n+2)}} \right] } \frac{\partial P_n (\cos \theta )}{\partial \theta } \end{aligned}$$
(22)

where \(C_n\) and \(C^{\prime }_n\) are other two unknown coefficients to be determined by the boundary condition, and

$$\begin{aligned} \lambda _1 (n)&= -(n+1)\frac{n\lambda +(n-2)\mu }{(2n+3)(\lambda +2\mu )},\quad \lambda _2 (n)=n\frac{(n+1)\lambda +(n+3)\mu }{(2n-1)(\lambda +2\mu )}, \nonumber \\ \lambda _3 (n)&= -\frac{(n+3)\lambda +(n+5)\mu }{(2n+3)(\lambda +2\mu )},\quad \lambda _4 (n)=-\frac{(n-2)\lambda +(n-4)\mu }{(2n-1)(\lambda +2\mu )} \end{aligned}$$
(23)

Considering that the hollow sphere under the diametrical point loads is an axisymmetric problem with respect to \(\varphi \), the displacement component \(u_\varphi \) should be zero. That is,

$$\begin{aligned} u_\varphi =0 \end{aligned}$$
(24)

Substitution of (24) into (12) and (13) yields

$$\begin{aligned} \tau _{\theta \varphi }&= 0\end{aligned}$$
(25)
$$\begin{aligned} \tau _{r\varphi }&= 0 \end{aligned}$$
(26)

Therefore, the third of equilibrium equation (2) is satisfied automatically.

Substitution of (21) and (22) into (8–11) leads to the expressions of the other stress components as

$$\begin{aligned} \sigma _{rr}&= 2\mu \sum _{n=0}^\infty \left[ \alpha _1 (n)\cdot A_n r^{n}+\alpha _2 (n)\cdot A^{\prime }_n r^{-(n+1)}\right. \nonumber \\&\qquad \quad \left. +\,n(n-1)\cdot C_n r^{n-2}-(n+1)(n+2)\cdot C^{\prime }_n r^{-(n+3)} \right] P_n (\cos \theta )\end{aligned}$$
(27)
$$\begin{aligned} \sigma _{\theta \theta }&= 2\mu \sum _{n=0}^\infty {\left[ {\alpha _3 (n)\cdot A_n r^{n}+\alpha _4 (n)\cdot A^{\prime }_n r^{-(n+1)}+n\cdot C_n r^{n-2}+(n+1)\cdot C^{\prime }_n r^{-(n+3)}} \right] } P_n (\cos \theta ) \nonumber \\&\quad +\,2\mu \sum _{n=0}^\infty {\left[ {\lambda _3 (n)\cdot A_n r^{n}+\lambda _4 (n)\cdot A^{\prime }_n r^{-(n+1)}+C_n r^{n-2}-C^{\prime }_n r^{-(n+3)}} \right] } \frac{\partial ^{2}P_n (\cos \theta )}{\partial \theta ^{2}} \end{aligned}$$
(28)
$$\begin{aligned} \sigma _{\varphi \varphi }&= 2\mu \sum _{n=0}^\infty {\left[ {\alpha _3 (n)\cdot A_n r^{n}+\alpha _4 (n)\cdot A^{\prime }_n r^{-(n+1)}+n\cdot C_n r^{n-2}+(n+1)\cdot C^{\prime }_n r^{-(n+3)}} \right] } P_n (\cos \theta ) \nonumber \\&\quad +\,2\mu \sum _{n=0}^\infty {\left[ {\lambda _3 (n)\cdot A_n r^{n}+\lambda _4 (n)\cdot A^{\prime }_n r^{-(n+1)}+C_n r^{n-2}-C^{\prime }_n r^{-(n+3)}} \right] } \cot \theta \frac{\partial P_n (\cos \theta )}{\partial \theta } \end{aligned}$$
(29)
$$\begin{aligned} \tau _{r\theta }&= 2\mu \sum _{n=0}^\infty {\left[ {\alpha _5 (n)\cdot A_n r^{n}+\alpha _6 (n)\cdot A^{\prime }_n r^{-(n+1)}+(n-1)\cdot C_n r^{n-2}+(n+2)\cdot C^{\prime }_n r^{-(n+3)}} \right] \frac{\partial P_n (\cos \theta )}{\partial \theta }}\nonumber \\ \end{aligned}$$
(30)

where

$$\begin{aligned} \alpha _1 (n)&= (n+1)\left[ \frac{\lambda }{\lambda +2\mu }+\lambda _1 (n)\right] ,\quad \alpha _2 (n)=-n\left[ \frac{\lambda }{\lambda +2\mu }+\lambda _2 (n)\right] \nonumber \\ \alpha _3 (n)&= \frac{(n+1)\lambda }{\lambda +2\mu }+\lambda _1 (n),\quad \alpha _4 (n)=-\frac{n\lambda }{\lambda +2\mu }+\lambda _2 (n) \nonumber \\ \alpha _5 (n)&= \frac{1}{2}\left[ \lambda _1 (n)+n\lambda _3 (n)\right] ,\quad \alpha _6 (n)=\frac{1}{2}\left[ \lambda _2 (n)-(n+1)\lambda _4(n)\right] \end{aligned}$$
(31)

5 Determination of unknown coefficients

In order to determine the unknown coefficients \(A_n,A^{\prime }_n ,C_n \) and \(C^{\prime }_n \) in the expressions for the stress components (27–30), the Fourier–Legendre expansion technique [39] adopted by Hiramatsu and Oka [38] is employed here to rewrite boundary condition (4) as

$$\begin{aligned} \sigma _{rr} =(\cos \theta _0 -1)p+\sum _{n=1}^\infty {E_{2n} P_{2n} (\cos \theta )} \end{aligned}$$
(32)

where

$$\begin{aligned} E_{2n} =\frac{4n+1}{2n+1}\left[ \cos \theta _0 P_{2n} (\cos \theta _0 )-P_{2n-1} (\cos \theta _0 )\right] p \end{aligned}$$
(33)

In addition, since the point loads acting on the outer surface of the hollow sphere are symmetric, the expressions for \(\sigma _{rr} ,\sigma _{\theta \theta }\) and \(\sigma _{\varphi \varphi } \) should be even functions with respect to \(\theta \) while \(\tau _{r\theta } \) should be an odd function with respect to \(\theta \). Consequently, \(n\) should be an even number. Thus, the integer \(n\) in all of the expressions for stress components in (27–30) should be replaced by \(2n\).

Considering the specific values of the corresponding expressions (27–30) of the stress components on the boundary conditions (4–7) together with (32), the following equations for the unknown coefficients are obtained as

$$\begin{aligned}&2\mu \left[ \alpha _1 (0)\cdot A_0 -2R_0^{-3} C^{\prime }_0 \right] =-p_0\end{aligned}$$
(34)
$$\begin{aligned}&2\mu \left[ \alpha _1 (0)\cdot A_0 -2R^{-3}C^{\prime }_0 \right] =(\cos \theta _0 -1)p\end{aligned}$$
(35)
$$\begin{aligned}&\alpha _5 (0)\cdot A_0 +\alpha _6 (0)\cdot R_0^{-1} A^{\prime }_0 -R_0^{-2} C_0 +2R_0^{-3} C^{\prime }_0 =0\end{aligned}$$
(36)
$$\begin{aligned}&\alpha _5 (0)\cdot A_0 +\alpha _6 (0)\cdot R^{-1}A^{\prime }_0 -R^{-2}C_0 +2R^{-3}C^{\prime }_0 =0 \end{aligned}$$
(37)

for \(n=0\), while

$$\begin{aligned}&\alpha _1 (2n)\cdot R_0^{2n} A_{2n} +\alpha _2 (2n)\cdot R_0^{-(2n+1)} A^{\prime }_{2n} \nonumber \\&\quad +\,2n(2n-1)R_0^{2n-2} C_{2n} -(2n+1)(2n+2)R_0^{-(2n+3)} C^{\prime }_{2n} =0 \end{aligned}$$
(38)
$$\begin{aligned}&2\mu \left[ \alpha _1 (2n)\cdot R^{2n}A_{2n} +\alpha _2 (2n)\cdot R^{-(2n+1)}A^{\prime }_{2n} \right. \nonumber \\&\left. \qquad +\,2n(2n-1)R^{2n-2}C_{2n} -(2n+1)(2n+2)R^{-(2n+3)}C^{\prime }_{2n} \right] =E_{2n} \end{aligned}$$
(39)
$$\begin{aligned}&\alpha _5 (2n)\cdot R_0^{2n} A_{2n} +\alpha _6 (2n)\cdot R_0^{-(2n+1)} A^{\prime }_{2n} +(2n-1)R_0^{2n-2} C_{2n} +(2n+2)R_0^{-(2n+3)} C^{\prime }_{2n} =0\end{aligned}$$
(40)
$$\begin{aligned}&\alpha _5 (2n)\cdot R^{2n}A_{2n} +\alpha _6 (2n)\cdot R^{-(2n+1)}A^{\prime }_{2n} +(2n-1)R^{2n-2}C_{2n} +(2n+2)R^{-(2n+3)}C^{\prime }_{2n} =0 \end{aligned}$$
(41)

for \(n\ne 0\).

By now, all boundary conditions have been satisfied, and all unknown coefficients can be uniquely determined. More specifically, the unknown coefficients \(A_0,A^{\prime }_0,C_0\) and \(C^{\prime }_0\) can be simply obtained by solving four Eqs. (34–37), while the other unknown coefficients \(A_{2n},A^{\prime }_{2n} ,C_{2n}\) and \(C^{\prime }_{2n}\) for \(n>0\) can be derived simultaneously by solving the four coupled algebraic system Eqs. (38–41).

As described in boundary conditions in Sect. 3, the present solution is only limited for analyzing the stress distributions within hollow spheres under diametrical point loads. However, in engineering, biaxial or triaxial point loads, even more pair of point loads, may simultaneously act on the surface of the hollow sphere. In such cases, all of the stress and strain components are, except for dependent with coordinate \(\theta \), also dependent with coordinate \(\varphi \). Consequently, much more complicated calculations are needed. In particular, all terms related to coordinate \(\varphi \) should also be considered and included since Eq. (8) and all of the following equations, and spherical harmonics \(S_n (\theta ,\varphi )\), instead of Legendre’s polynomial \(P_n (\cos \theta )\), should be used in Eq. (19). The obtained analytical expressions for stress components are much more complicated than the present case, which will be considered in a separate paper.

Numerical results for the present analytical solutions are given and discussed in the next section.

6 Numerical results and discussions

6.1 Check the convergence of the analytical solution

The analytical solution given in the previous section involves the solution of coupled algebraic equations for the coefficients of the infinite series. In actual computation, we have to truncate the infinite series and retain only finite number of terms in Eqs. (38–41).

Similar to the analysis for isotropic and anisotropic solid spheres under diametrical compression [35–38], the present paper mainly investigates the stress distributions within hollow spheres along the axis of loading of the point loads (i.e., \(\theta =0\)). In order to check the convergence of the present series analytical solution, Fig. 3 plots the normalized stress components \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) versus the normalized distance \(r/R\) for \(\theta =0, \nu =0.3, R_0/R=0.7, p_0 =0\) and \(\theta _0 =5^{\circ }\) for various values of \(n\). Note that \(\sigma _0 =F/2\pi R^{2}\). Figure 3 indicates that all lines obtained by taking 20 terms are the same as those by taking 100 terms or more, which shows that 20 terms are sufficient to achieve a satisfactory convergence. To ensure the accuracy of our numerical results, all subsequent calculations were done using 100 terms for summation indices \(n\).

Fig. 3
figure 3

The normalized stress components \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) versus the normalized distance \(r/R\) for \(\theta =0, \nu =0.3, R_0 /R=0.7, p_0 =0\) and \(\theta _0 =5^{\circ }\) for various values of \(n\)

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6.2 The comparison of the present solution with the asymptotic solution and numerical results by FEM

Gregory et al. [34] employed a refined theory and obtained an asymptotic solution for a thick hollow sphere compressed by equal and opposite concentrated axial loads. Figure 4 shows the comparison of the normalized stress \(\sigma _{\varphi \varphi } (R^{2}-R_0^2 )/(4F)\) on the outer surface of the hollow sphere obtained from the present solution with the asymptotic solution by the refined theory and the approximate solution by the thin shell theory for \(\nu =1/3\) and \(R_0 /R=9/11\). In addition, Fig. 5 shows the comparison of the present solution with the numerical results by using FEM for \(\nu =0.3, \theta _0 =5^{\circ }, R_0 /R=0.7, p_0 =0\) and \(\theta =0\), and it is obvious that the two kinds of solutions are almost the same except near the point loads. Figures 4 and 5 show that the present solution agrees well with the asymptotic solution and FEM numerical results.

Fig. 4
figure 4

The comparison of the present solution with the asymptotic solution by the refined theory and the approximate solution by the thin shell theory for \(\nu =1/3, R_0 /R=9/11\), and \(r/R=1\)

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Fig. 5
figure 5

The comparison of the present solution with the FEM numerical results for \(\nu =0.3, \theta _0 =5^{\circ }, R_0 /R=0.7, p_0 =0\) and \(\theta =0\)

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6.3 The stress distributions of the normal stresses along the axis of loading

Since few discussions have been done on the stress distributions of thick hollow sphere under the point loads, Fig. 6 plots the normalized stress components \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) versus the normalized distance \(r/R\) for \(\theta =0, \nu =0.3, p_0 =0\) and \(\theta _0 =5^{\circ }\) for various values of \(R_0 /R\) (for \(R_0 /R\le 0.6\)), which shows that the tensile stress concentrations are developed at the inner surface \(r=R_0 \) as well as near the loading area with \(r/R\approx 0.85\) for a relatively thick hollow sphere (say \(R_0 /R<0.3\)), while the tensile stress concentration is only developed at the inner surface \(r=R_0 \) for a relatively thin hollow sphere (say \(R_0 /R>0.5\)), and it was found that the maximum normalized tensile stress \(\sigma _{\theta \theta } /\sigma _0 \) is always developed at the inner surface. This indicates that the hollow sphere under the point loads always starts to fail at the inner surface except for a hollow sphere with a very small Poisson’s ratio (i.e., \(\nu =0.1\)) and a very large thickness (i.e., \(R_0 /R\le 0.23)\). This conclusion is very different from the well-known classical solution by Hiramatsu and Oka [38] for the isotropic solid spheres and the analytical solutions by Chau and Wei [35] for anisotropic solid spheres under the point loads, which shows that the tensile stress concentration is always developed near \(r/R\approx 0.9\), and the maximum tensile stress at \(r/R\approx 0.9\) is even ten times larger than that at the central part of the sphere [35]. In addition, Figs. 6 and 7 show that the normalized compressive stress \(\sigma _{rr} /\sigma _0\) always decreases rapidly with the increase in \(r/R\) for hollow spheres with any thickness.

Fig. 6
figure 6

The normalized stress components \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) versus the normalized distance \(r/R\) for \(\theta =0, \nu =0.3, p_0 =0\) and \(\theta _0 =5^{\circ }\) for various values of \(R_0 /R\) (for \(R_0 /R\le 0.6\))

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Fig. 7
figure 7

The normalized stress components \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) versus the normalized distance \(r/R\) for \(\theta =0, \nu =0.3, p_0 =0\) and \(\theta _0 =5^{\circ }\)for various values of \(R_0 /R\) (for \(R_0 /R\ge 0.7\))

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In addition, the stress distribution within a thin shell under the point loads is a very interesting problem in engineering. Figure 7 plots the normalized stress components \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) versus the normalized distance \(r/R\) for \(\theta =0, \nu =0.3, p_0 =0\) and \(\theta _0=5^{\circ }\) for various values of \(R_0 /R\) (for \(R_0 /R\ge 0.7\)). Figure 7 shows that the normalized tensile stress \(\sigma _{\theta \theta } /\sigma _0 \) at the inner surface is always the largest tensile stress for thin shell, and the thinner the thickness of the hollow sphere, the larger the stress concentrations developed at the inner surface. But the normalized tensile stress \(\sigma _{\theta \theta } /\sigma _0 \) decreases drastically with the increase in \(r/R\) and even become compressive when it approaches the point loads.

If the thickness of the hollow sphere is relatively small, say \(R_0 /R=0.7\), Fig. 8 further plots the normalized stress components \(\sigma _{rr} /\sigma _0\) and \(\sigma _{\theta \theta } /\sigma _0 \) versus the normalized distance \(r/R\) for \(\theta =0, R_0 /R=0.7, p_0 =0\) and \(\theta _0 =5^{\circ }\) for various values of Poisson’s ratio \(\nu \), Fig. 8 shows that the normalized tensile stress \(\sigma _{\theta \theta } /\sigma _0\) always increases with the increase in Poisson’s ratio \(\nu \) for \(0.7\le r/R\le 0.81\), and it decreases with the increase in Poisson’s ratio \(\nu \) for \(0.81<r/R\le 1.0\), while the normalized compressive stress \(\sigma _{rr} /\sigma _0\) decreases slightly with the increase in Poisson’s ratio \(\nu \).

Fig. 8
figure 8

The normalized stress components \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0\) versus the normalized distance \(r/R\) for \(\theta =0, R_0 /R=0.7, p_0 =0\) and \(\theta _0 =5^{\circ }\) for various value of Poisson’s ratio \(\nu \)

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Figure 9 illustrates the effect of the parameter \(\theta _0\), which indicates the size of the loading area, on the normalized stress components \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) versus the normalized distance \(r/R\) for \(\theta =0, R_0 /R=0.7, p_0 =0\) and \(\nu =0.3\). Figure 9 shows that the stress distributions within the hollow sphere are very sensitive to the value of \(\theta _0 \), and the smaller the size of the loading area, the larger the tensile stress concentrations developed at the inner surface, but the normalized compressive stress \(\sigma _{rr} /\sigma _0 \) increases with the increase in \(\theta _0 \).

Fig. 9
figure 9

The normalized stress components \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) versus the normalized distance \(r/R\) for \(\theta =0, R_0 /R=0.7, p_0 =0\) and \(\nu =0.3\) for various value of \(\theta _0 \)

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All of the above discussions are for internal pressure \(p_0 =0\). If internal pressure \(p_0 \ne 0\), Fig. 10 plots the normalized stress components \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) versus the normalized distance \(r/R\) for \(\theta =0, R_0 /R=0.7, \theta _0 =5^{\circ }\) and \(\nu =0.3\) for various internal pressure \(p_0 \), which shows that the normalized tensile stress \(\sigma _{\theta \theta } /\sigma _0\) increases with the increase in the internal pressure \(p_0 \), but the normalized compressive stress \(\sigma _{rr} /\sigma _0 \) decreases with the increase in the internal pressure \(p_0 \). However, \(p_0 \) has little effect on the magnitude and distribution of stresses near the point loads, which agrees with Saint-Venant’s Principle.

Fig. 10
figure 10

The normalized stress components \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) versus the normalized distance \(r/R\) for \(\theta =0, R_0 /R=0.7, \theta _0 =5^{\circ }\) and \(\nu =0.3\) for various internal pressure \(p_0 \)

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6.4 The stress distributions of the normal stresses away from the axis of loading

All of the above discussions are for stress distribution along the axis of loading with \(\theta =0\). If \(\theta \ne 0\), Figs. 11 and 12 plot the normalized stresses \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0\) versus coordinate \(\theta \) for \(R_0 /R=0.7, \theta _0 =5^{\circ }, p_0 =0\) and \(\nu =0.3\) for various value of \(r/R\). It seems that the maximum compressive stresses of both \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) are always developed near the loading area on the outer surface of the hollow sphere (i.e., \(r/R=1.0\)). Both of the normalized stresses \(\sigma _{rr} /\sigma _0 \) and \(\sigma _{\theta \theta } /\sigma _0 \) increase rapidly with the decrease in \(r/R\) when \(\theta <6^{\circ }\) and eventually become tensile stresses, but the maximum tensile stress of \(\sigma _{\theta \theta } /\sigma _0 \) is very large, and it is almost 13 times larger than that of \(\sigma _{rr} /\sigma _0 \). Figure 13 plots the normalized stress \(\sigma _{\varphi \varphi } /\sigma _0\) versus coordinate \(\theta \) for \(R_0 /R=0.7, \theta _0 =5^{\circ }, p_0 =0\) and \(\nu =0.3\) for various value of \(r/R\). Figure 13 shows that the normalized stress \(\sigma _{\varphi \varphi } /\sigma _0 \) is tensile at the inner surface (that is \(R_0 /R=0.7)\) of the hollow sphere, but it becomes compressive for about \(r /R>0.9\) and \(\theta <\pi /5\), and the maximum compressive stress of \(\sigma _{\varphi \varphi } /\sigma _0 \) is developed at the outer surface near the loading area.

Fig. 11
figure 11

The normalized stress \(\sigma _{rr} /\sigma _0\) versus coordinate \(\theta \) for \(R_0 /R=0.7, \theta _0 =5^{\circ }, p_0 =0\) and \(\nu =0.3\) for various value of \(r/R\)

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Fig. 12
figure 12

The normalized stress \(\sigma _{\theta \theta } /\sigma _0 \) versus coordinate \(\theta \) for \(R_0 /R=0.7, \theta _0 =5^{\circ }, p_0 =0\) and \(\nu =0.3\) for various value of \(r/R\)

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Fig. 13
figure 13

The normalized stress \(\sigma _{\varphi \varphi } /\sigma _0 \) versus coordinate \(\theta \) for \(R_0 /R=0.7, \theta _0 =5^{\circ }, p_0 =0\) and \(\nu =0.3\) for various value of \(r/R\)

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6.5 The stress distributions of the shear stresses

All of the above discussions are for the stress distribution of the normal stresses within the hollow sphere under the point loads. Figure 14, however, plots the normalized shear stress \(\tau _{r\theta } /\sigma _0 \) versus coordinate \(\theta \) for \(R_0 /R=0.7, \theta _0 =5^{\circ }, p_0 =0\) and \(\nu =0.3\) for various value of \(r/R\). It seems that a significant value of 83 of the normalized shear stress \(\tau _{r\theta }/\sigma _0\) is developed near the loading area at \(r/R\approx 0.99\), and it decreases drastically with the decrease in \(r/R\).

Fig. 14
figure 14

The normalized shear stress \(\tau _{r\theta } /\sigma _0 \) versus coordinate \(\theta \) for \(R_0 /R=0.7, \theta _0 =5^{\circ }, p_0 =0\) and \(\nu =0.3\) for various value of \(r/R\)

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Since buckling may easily occur for very thin hollow spheres under the diametrical point loads, the related analysis is necessary and useful in engineering. However, this issue is out of the scope of the present paper and is not discussed here.

7 Conclusion

An exact analytical solution for the stress distributions within a hollow sphere under a pair of diametrical point loads is obtained by uncoupling the governing equations analytically together with a Fourier–Legendre expansion for the boundary applied loads. The present solution for hollow spheres can be considered as an extension of the classical solution by Hiramatsu and Oka [38] for solid spheres, which provided the famous theoretical basis for the PLST. Note that PLST is an extremely convenient and extensively used index test in engineering for rock classification and strength estimation. Unlike in solid spheres, the stress concentrations within the hollow spheres under the point loads are usually developed at the joint point of the inner surface and the loading axis, and the thinner the thickness of the hollow sphere, the larger the stress concentrations developed at the inner surface. This numerical result indicates that the failure crack of the hollow spheres usually starts at the inner surface, and the peak of the normalized tensile stress at the inner surface increases with the increase in Poisson’s ratio and internal pressure, but decreases with the increase in the size of the loading area. The normalized compressive stress seems insensitive to Poisson’s ratio. Moreover, significant shear stress zone is usually developed in areas immediately inside the outer surface, and the maximum shear stress is often developed at the point immediately inside the outer surface jointing the edge of the loading area and the center of the hollow sphere.