Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials

Abstract

Novel unified analytical displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions for three-dimensional, generally anisotropic and linear elastic materials are presented in this paper. Adequate integral expressions for the displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions are evaluated analytically by using the Cauchy residue theorem. The resulting explicit displacement fundamental solutions and their first and second derivatives are recast into convenient analytical forms which are valid for non-degenerate, partially degenerate, fully degenerate and nearly degenerate cases. The correctness and the accuracy of the novel unified and closed-form three-dimensional anisotropic fundamental solutions are verified by using some available analytical expressions for both transversely isotropic (non-degenerate or partially degenerate) and isotropic (fully degenerate) linear elastic materials.

1. Introduction

Fundamental solutions or Green’s functions have many applications in elasticity theory, for example inclusion problems, dislocation problems, crack problems and so on [1,2]. In particular, they are indispensable in the boundary element method (BEM) or boundary integral equation method (BIEM) [3–8]. The efficiency and the accuracy of the BEM or BIEM are significantly dependent on the mathematical structure of the corresponding fundamental solutions. Therefore, explicit and closed-form analytical fundamental solutions are preferable and play an important role in BEM or BIEM. In displacement-based BEM, both the displacement and the stress or traction fundamental solutions are required. In stress- or traction-based BEM, both the stress fundamental solutions and the higher order derivatives of the displacement fundamental solutions are required. In both displacement- and stress-based BEM formulations, the displacement evaluation at interior points requires both the displacement and the stress or traction fundamental solutions, while the stress computation at interior points necessitates also the stress or traction fundamental solutions and the higher order derivatives of the displacement fundamental solutions.

For isotropic and linear elastic materials, three-dimensional, simple and closed-form elastostatic fundamental solutions are available. In contrast, the corresponding three-dimensional fundamental solutions for anisotropic materials are much more complex, and they have closed-form analytical expressions only in some special cases such as for transversely isotropic materials [9–15]. By using either the Fourier or the Radon transform, contour or infinite line integral expressions for the three-dimensional displacement fundamental solutions have been presented by Fredholm [16], Lifshitz & Rozenzweig [17], Barnett [18], Mura [1], Wang [19] and Lee [20], among others. By applying the residue theorem to the integral expressions of the displacement fundamental solutions, Ting & Lee [21] presented explicit expressions for three-dimensional anisotropic displacement fundamental solutions in terms of the Stroh eigenvalues p_i (i=1,2,3). The most important and distinct feature of the work by Ting & Lee [21] is the fact that no factors like p_i−p_j appear in the denominators of the displacement fundamental solutions, and hence they are generally valid for non-degenerate (p₁≠p₂≠p₃), partially degenerate (p₁=p₂≠p₃ or p₁≠p₂=p₃) and fully degenerate (p₁=p₂=p₃) cases. It should be noted here that the partially degenerate cases may appear in transversely isotropic materials, whereas the fully degenerate cases correspond to isotropic materials. Malén [22] and Nakamura & Tanuma [23] expressed the displacement fundamental solutions in terms of the Stroh eigenvectors instead of the eigenvalues. While the three-dimensional anisotropic displacement fundamental solutions can be written in relatively simple explicit forms in terms of the Stroh eigenvalues, the stress fundamental solutions and the higher order derivatives of the displacement fundamental solutions are very complicated due to the additional derivatives of the displacement fundamental solutions. By using the Cauchy residue theorem, Phan et al. [24,25] presented the displacement fundamental solutions and their first derivatives needed for the stress or traction fundamental solutions, but their results were given separately for non-degenerate, partially degenerate and fully degenerate cases. Based on the explicit displacement fundamental solutions of Ting & Lee [21] and using the spherical coordinate system, Lee [26] and Shiah et al. [27] derived explicit expressions for the first and second derivatives of the displacement fundamental solutions by performing the spatial differentiations with respect to the spherical coordinates. Unified and explicit expressions for the displacement fundamental solutions and their derivatives based on the displacement fundamental solutions of Ting & Lee [21] have been recently presented by Buroni & Sáez [28], who also performed the spatial derivatives in the spherical coordinate system and argued that their expressions are valid for both non-degenerate and degenerate cases. In any case, care should be taken by using fundamental solutions containing terms like p_i−p_j in the denominators, which may either destroy the validity of the fundamental solutions or cause considerable numerical errors and instability in nearly degenerate cases, where p_i is very close to p_j [24,25,29,30]. A prevalent way to overcome this difficulty is to disturb the material constants by a sufficiently small number to avoid the degeneracy problem.

Another route is the numerical computation of the three-dimensional anisotropic fundamental solutions. The series solution was presented by Mura & Kinoshita [31], which is differentiable term by term and converges uniformly. A perturbation expansion technique was suggested by Gray et al. [32], who used isotropic fundamental solutions as the zeroth-order term, but the method is only good for a weak anisotropy. Based on pre-calculated values of the fundamental solutions, an interpolation scheme using cubic splines was first suggested by Wilson & Cruse [33] and then followed by many researchers [34,35]. The finite difference method for computing the derivatives of the displacement fundamental solutions was used by Pan & Yuan [36] and Tonon et al. [37]. Recently, an efficient and elegant numerical procedure for computing the three-dimensional anisotropic fundamental solutions was developed by Shiah et al. [38] and Tan et al. [39], who used a double Fourier-series expansion technique and performed the partial derivatives in the spherical coordinate system.

A comprehensive review and a detailed description of several different methods for deriving the anisotropic fundamental solutions or Green’s functions and their derivatives can be found in the excellent monograph by Pan & Chen [40] and references therein.

The main objective of this paper is to present novel explicit displacement and stress fundamental solutions as well as the derivatives of the stress fundamental solutions or the higher order derivatives of the displacement fundamental solutions in terms of the Stroh eigenvalues for generally anisotropic elastic materials. For this purpose, the contour-integral expressions of the three-dimensional anisotropic fundamental solutions given by Mura [1] are rewritten into their equivalent integral forms, which can be reduced to a special fundamental integral. Then, the Cauchy residue theorem is applied to evaluate this special fundamental integral. The resulting explicit fundamental solutions are then recast into their equivalent but more convenient forms, which are unified, in closed-form and valid for non-degenerate, partially degenerate, fully degenerate and nearly degenerate cases. In contrast to some previous works [26–28], in this analysis the partial derivatives of the displacement fundamental solutions are performed with respect to the Cartesian coordinates instead of the spherical coordinates, which is easier, especially for the second derivatives of the displacement fundamental solutions. To the best of the authors’ knowledge, the present unified and closed-form analytical fundamental solutions for three-dimensional anisotropic elasticity are novel and have not yet been reported elsewhere. To check and verify the present three-dimensional anisotropic fundamental solutions, some available analytical fundamental solutions for transversely isotropic and isotropic elastic materials are taken.

2. Governing equations, representation formulae and boundary integral equations

(a). Governing equations

We consider a three-dimensional, infinite, homogeneous, anisotropic and linear elastic solid. The equilibrium equations of the anisotropic elastic solid are given by

$${\sigma }_{ij,j}+f_i=0,$$ 2.1

where σ_ij is the stress tensor, f_i is the body force vector, a comma after a quantity denotes partial derivatives with respect to spatial variables, and the conventional summation rule over repeated indices is applied throughout the analysis. Hooke’s law is given by

$${\sigma }_{ij}=C_{ijkl}{ϵ}_{kl},$$ 2.2

where ϵ_kl is the strain tensor, which is related to the displacement gradients by the following kinematic relation:

$${ϵ}_{kl}=\frac{1}{2}(\frac{\mathrm{\partial }{u}_k}{\mathrm{\partial }{x}_l}+\frac{\mathrm{\partial }{u}_l}{\mathrm{\partial }{x}_k}),$$ 2.3

with u_k being the displacement vector. The fourth-order elasticity tensor C_ijkl has the following symmetry properties:

$$C_{ijkl}=C_{jikl}=C_{ijlk}=C_{klij}.$$ 2.4

By substituting equations (2.2) and (2.3) into equation (2.1), the equilibrium equations can be written in terms of the displacement components as

$$C_{ijkl}{u}_{k,lj}+f_i=0.$$ 2.5

(b). Representation formulae and boundary integral equations

Let us consider a three-dimensional, anisotropic and linear elastic solid with the volume Ω bounded by the surface S=∂Ω. By using the Betti reciprocal theorem, the displacement components at an arbitrary internal point of the domain can be obtained by using the following representation formula:

$$u_i(\mathit{\text{x}})={\int }_S{u}_{ij}^G(\mathit{\text{x}},\mathit{\text{y}})t_j(\mathit{\text{y}})\hspace{0.17em}\mathrm{d}S-{\int }_S{t}_{ij}^G(\mathit{\text{x}},\mathit{\text{y}})u_j(\mathit{\text{y}})\hspace{0.17em}\mathrm{d}S,\mathit{\text{x}}\in \mathit{\Omega },$$ 2.6

where $u_{ij}^G(\mathit{\text{ x}},\mathit{\text{ y}})$ denotes the displacement fundamental solutions, $t_{ij}^G(\mathit{\text{ x}},\mathit{\text{ y}})={\sigma }_{ikj}^G(\mathit{\text{ x}},\mathit{\text{ y}})n_k(\mathit{\text{ y}})$ represents the traction fundamental solutions with ${\sigma }_{ikj}^G(\mathit{\text{ x}},\mathit{\text{ y}})$ being the stress fundamental solutions and n_k(y) the outward unit normal vector on the boundary S, t_j=σ_jkn_k stands for the traction vector, x is the position vector of the observation point, and y is the position vector of the source point. Physically, the displacement fundamental solutions $u_{ij}^G(\mathit{\text{ x}},\mathit{\text{ y}})$ denote the displacement components at the observation point x in the x_i-direction due to a unit point force applied at the source point y in the y_j-direction. By taking the limit process x→S, one obtains the following displacement boundary integral equations:

$$c_{ij}(\mathit{\text{x}})u_j(\mathit{\text{x}})={\int }_S{u}_{ij}^G(\mathit{\text{x}},\mathit{\text{y}})t_j(\mathit{\text{y}})\hspace{0.17em}\mathrm{d}S-{\int }_S{t}_{ij}^G(\mathit{\text{x}},\mathit{\text{y}})u_j(\mathit{\text{y}})\hspace{0.17em}\mathrm{d}S,\mathit{\text{x}}\in S,$$ 2.7

where the free-term coefficients c_ij depend on the smoothness, shape and orientation of the surface at x.

Once the unknown boundary values have been obtained by solving the above boundary integral equations, the displacement field in the interior domain of interest can be computed by using the displacement representation formula (2.6). The corresponding stress components at an arbitrary internal point of the domain can be obtained by substituting the displacement representation formula (2.6) into Hooke’s law (2.2), which results in the following representation formula for the stress components:

$${\sigma }_{kl}(\mathit{\text{x}})=-{\int }_S{\sigma }_{klj}^G(\mathit{\text{x}},\mathit{\text{y}})t_j(\mathit{\text{y}})\hspace{0.17em}\mathrm{d}S-{\int }_S{s}_{klj}^G(\mathit{\text{x}},\mathit{\text{y}})u_j(\mathit{\text{y}})\hspace{0.17em}\mathrm{d}S,\mathit{\text{x}}\in \mathit{\Omega },$$ 2.8

where the stress fundamental solutions ${\sigma }_{klj}^G(\mathit{\text{ x}},\mathit{\text{ y}})$ and the higher order derivatives of the displacement fundamental solutions are given by

$$\begin{array}{rl}{\sigma }_{klj}^G& =C_{klim}\frac{\mathrm{\partial }{u}_{ij}^G}{\mathrm{\partial }{y}_m}=-C_{klim}\frac{\mathrm{\partial }{u}_{ij}^G}{\mathrm{\partial }{x}_m}\\ \text{and}{s}_{klj}^G& =C_{klim}\frac{\mathrm{\partial }{t}_{ij}^G}{\mathrm{\partial }{x}_m}=C_{klim}{n}_p(\mathit{\text{y}})\frac{\mathrm{\partial }{\sigma }_{ipj}^G}{\mathrm{\partial }{x}_m}=-C_{klim}{n}_p(\mathit{\text{y}})\frac{\mathrm{\partial }{\sigma }_{ipj}^G}{\mathrm{\partial }{y}_m}=-C_{klim}{C}_{ipnq}{n}_p(\mathit{\text{y}})\frac{{\mathrm{\partial }}^2{u}_{nj}^G}{\mathrm{\partial }{y}_q\mathrm{\partial }{y}_m}.\end{array}\}$$ 2.9

It should be noted here that the relation ∂(⋅)/∂x_i=−∂(⋅)/∂y_i has been used in deriving equation (2.9). By letting x tend to S, the corresponding stress or traction boundary integral equations can be obtained, which are of special interests in crack analysis.

From equation (2.9), it can be seen that the stress or traction fundamental solutions involve the first derivatives of the displacement fundamental solutions, while the derivatives of the stress or traction fundamental solutions contain the second derivatives of the displacement fundamental solutions.

The representation formulae and the boundary integral equations are presented here just to demonstrate the important applications of the fundamental solutions in the BEM. However, the main objective of this paper is devoted to the derivation of explicit fundamental solutions rather than the numerical solution of the corresponding boundary integral equations. In the following sections, the displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions will be derived.

3. Displacement fundamental solutions

In the following sections, the source point is chosen as the origin of the coordinate system, i.e. y=0. In this case, the displacement fundamental solutions $u_{ij}^G(\mathit{\text{ x}},\mathit{\text{ y}})$ can be simply denoted by $u_{ij}^G(\mathit{\text{ x}})$, which satisfy the following partial differential equations:

$$C_{ijkl}{u}_{km,lj}^G(\mathit{\text{x}})+{\delta }_{im}\delta (\mathit{\text{x}})=0,$$ 3.1

where δ_im is the Kronecker delta, and δ(x) is the three-dimensional Dirac delta function which is zero everywhere except at point x=0. Physically, the displacement fundamental solutions $u_{ij}^G(\mathit{\text{ x}})$ correspond to the displacement component at x (field or observation point) in the x_i-direction due to a unit point-force in the x_j-direction at the origin x=0 of the Cartesian coordinate system (source point).

By applying either the Fourier transform or the Radon transform to equation (3.1) and after some mathematical manipulations, the displacement fundamental solutions $u_{ij}^G(\mathit{\text{ x}})$ can be obtained as [1,2]

$$u_{ij}^G(\mathit{\text{x}})=\frac{1}{8{\pi }^2}{\int }_{\mathit{\Omega }}\delta (r\mathit{\xi }\cdot \overline{\mathit{\text{x}}}){\mathit{\Gamma }}_{ij}^{-1}(\mathit{\xi })\hspace{0.17em}\mathrm{d}S(\mathit{\xi }),$$ 3.2

where Γ_ij(ξ)=C_ikjlξ_kξ_l, Ω is a unit sphere whose centre is the origin of the ξ₁–ξ₂–ξ₃-coordinate system, r=|x| is the distance between the field point x and the source point, $\overline{\mathit{\text{ x}}}$ is the unit vector of x defined by $\overline{\mathit{\text{ x}}}=\mathit{\text{ x}}/r$ and ξ is a parameter vector. From equation (3.2), the following integral expression for the displacement fundamental solutions can be obtained [1,2]:

$$u_{ij}^G(\mathit{\text{x}})=\frac{1}{8{\pi }^{2}r}{\oint }_s{\mathit{\Gamma }}_{ij}^{-1}(\mathit{\xi })\hspace{0.17em}\mathrm{d}\varphi ,$$ 3.3

where s is the unit circle on the oblique plane perpendicular to x. It should be noted here that the displacement fundamental solutions given by equation (3.3) are not explicit in the sense that they still contain an integral expression.

By choosing any two mutually orthogonal unit vectors n and m on the oblique plane as the bases, the parameter vector ξ can be written as

$$\mathit{\xi }=\mathit{\text{n}}\cos \varphi +\mathit{\text{m}}\sin \varphi .$$ 3.4

By setting $p=\tan \varphi$ and ξ′=n+pm, we have

$$\mathit{\xi }=\cos \varphi {\mathit{\xi }}^{\prime },\hspace{0.17em}\mathrm{d}p=\frac{1}{{\cos }^2\varphi }\hspace{0.17em}\mathrm{d}\varphi .$$ 3.5

By substituting equation (3.5) into equation (3.3), it follows that

$$u_{ij}^G(\mathit{\text{x}})=\frac{1}{4{\pi }^{2}r}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\mathit{\Gamma }}_{ij}^{-1}(p)\hspace{0.17em}\mathrm{d}p,$$ 3.6

where Γ_ij(p)=C_ikjlξ′_kξ′_l. To obtain the above equation, the periodicity of the integrand in equation (3.3) and the symmetry properties of C_ijkl and Γ_ij are used.

Equation (3.6) can be rewritten as

$$u_{ij}^G(\mathit{\text{x}})=\frac{1}{4{\pi }^{2}r}{H}_{ij}(\overline{\mathit{\text{x}}}),$$ 3.7

where

$$H_{ij}(\overline{\mathit{\text{x}}})={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{{\hat{\mathit{\Gamma }}}_{ij}(p)}{D(p)}\hspace{0.17em}\mathrm{d}p,$$ 3.8

in which ${\hat{\mathit{\Gamma }}}_{ij}(p)$ is the adjoint of the matrix Γ_ij(p) defined by

$${\hat{\mathit{\Gamma }}}_{ij}(p){\mathit{\Gamma }}_{jk}(p)=D(p){\delta }_{ik},$$ 3.9

and D(p) is the determinant of the matrix Γ_ij(p) given by

$$D(p)=\text{det}[{\mathit{\Gamma }}_{ij}(p)].$$ 3.10

It should be noted that the matrix $H_{ij}(\overline{\mathit{\text{ x}}})$ is independent of the distance of the observation point x to the origin, and only dependent on the direction of the observation point. Apart from the factor 1/π, the matrix $H_{ij}(\overline{\mathit{\text{ x}}})$ is identical to one of the three Barnett–Lothe tensors [2].

The determinant D(p) is a sixth-order polynomial in p. Since the elasticity tensor C_ijkl is positive definite, the roots of the determinant D(p) are complex and there are three pairs of complex conjugates [2]. So D(p) can be written as

$$\begin{array}{rl}D(p)& =\alpha (p-p_1)(p-p_2)(p-p_3)(p-{\overline{p}}_1)(p-{\overline{p}}_2)(p-{\overline{p}}_3)\\ & =\alpha \prod _{i=1}^3(p-p_i)(p-{\overline{p}}_i),\end{array}$$ 3.11

where α is the coefficient of p⁶ and equal to the determinant of the matrix T_ij=C_ijklm_km_l, i.e.

$$\alpha =\text{det}[T_{ij}(p)]$$ 3.12

and

$$p_k={\eta }_k+i{\beta }_k,{\beta }_k>0(k=1,2,3)$$ 3.13

are well known as the Stroh eigenvalues (see appendix A) and an over-bar denotes the complex conjugate. It can be concluded from the expression of $D(p)$ that the Stroh eigenvalues depend on the material constants, the direction from the coordinate origin to the observation point and also the chosen coordinates in the oblique plane.

Since ${\hat{\mathit{\Gamma }}}_{ij}(p)$ is a fourth-order polynomial in p, we define [21]

$${\hat{\mathit{\Gamma }}}_{ij}(p)=\sum _{n=0}^4{\hat{\mathit{\Gamma }}}_{ij}^n{p}^n.$$ 3.14

Substituting equations (3.11) and (3.14) into equation (3.8), we obtain

$$H_{ij}(\overline{\mathit{\text{x}}})=\frac{1}{\alpha }\sum _{n=0}^4{\hat{\mathit{\Gamma }}}_{ij}^n{I}_n,$$ 3.15

where

$$I_n={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{p^n}{f(p)}\hspace{0.17em}\mathrm{d}p$$ 3.16

and

$$\begin{array}{rl}f(p)& =(p-p_1)(p-p_2)(p-p_3)(p-{\overline{p}}_1)(p-{\overline{p}}_2)(p-{\overline{p}}_3)=\prod _{i=1}^3(p-p_i)(p-{\overline{p}}_i).\end{array}$$ 3.17

Although the explicit expressions of the coefficients ${\hat{\mathit{\Gamma }}}_{ij}^n$ are very difficult to obtain, they can be numerically evaluated almost exactly by a polynomial algorithm. It can also be found that both the coefficients and the integrals I_n are real.

Substituting equation (3.17) into equation (3.16), we obtain

$$I_n={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{p^n}{\prod _{i=1}^3(p-p_i)(p-{\overline{p}}_i)}\hspace{0.17em}\mathrm{d}p,n=0,1,\dots ,4.$$ 3.18

It can easily be shown that I_n can be expressed by the following special fundamental integral:

$$X_m^{(\alpha )}={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{p^{\alpha }}{\prod _{i=1}^m(p-p_i)(p-{\overline{p}}_i)}\hspace{0.17em}\mathrm{d}p,\alpha =0,1,$$ 3.19

where m≥1 if α=0 while m≥2 when α=1. After some mathematical manipulations based on rewriting the numerator pⁿ in equation (3.18), for example when n=3, the numerator can be rewritten as $p^3=p(p-p_3)(p-{\overline{p}}_3)+(p_3+{\overline{p}}_3)p^2-p_3{\overline{p}}_{3}p$, the higher order integrals I_n can be expressed in terms of the lower order ones and $X_m^{(\alpha )}\hspace{0.17em}(\alpha =0,1)$ as follows:

$$\begin{array}{rl}{I}_0& =X_3^{(0)},\\ I_1& =X_3^{(1)},\\ I_2& =X_2^{(0)}+2\mathrm{Re}(p_3)I_1-|p_3{|}^2{I}_0,\\ I_3& =X_2^{(1)}+2\mathrm{Re}(p_3)I_2-|p_3{|}^2{I}_1\\ \text{and}{I}_4& =X_1^{(0)}+2\mathrm{Re}(p_2+p_3)I_3-[|p_2{|}^2+|p_3{|}^2+4\mathrm{Re}(p_2)\mathrm{Re}(p_3)]I_2\\ & +2[\mathrm{Re}(p_2)|p_3{|}^2+\mathrm{Re}(p_3)|p_2{|}^2]I_1-|p_2{|}^2|p_3{|}^2{I}_0.\end{array}\}$$ 3.20

By applying the Cauchy residue theorem to equation (3.19) with the assumption of distinct Stroh eigenvalues (p₁≠p₂≠p₃), we arrive at the following explicit algebraic expression in terms of p_i:

$$X_m^{(\alpha )}=2\pi i\sum _{i=1}^m\frac{p_i^{\alpha }}{(p_i-{\overline{p}}_i)\prod _{1\le j\ne i\le m}(p_i-p_j)(p_i-{\overline{p}}_j)}.$$ 3.21

However, the denominator in this expression has the factor p_i−p_j, which makes the expression invalid when p_i=p_j (i≠j) and may also cause remarkable errors in the numerical computation when p_i and p_j are very close to each other. To overcome this difficulty, a suitable procedure for the rearrangement of $X_m^{(\alpha )}$ is required to remove the factor p_i−p_j in the denominator. The key idea of our procedure is to group and rearrange some adequate terms in an appropriate manner such that the term p_i−p_j in the denominator can be eliminated. This procedure is made possible by using the performance of the computer software Mathematica. The rearranged $X_m^{(\alpha )}$ can be finally written as

$$\begin{array}{rl}{X}_1^{(0)}& =\frac{\pi }{{\beta }_1},\\ X_2^{(\alpha )}& =-\frac{\pi }{{\beta }_1{\beta }_2}\mathrm{Im}\left(\frac{p_1^{\alpha }}{p_1-{\overline{p}}_2}\right)\\ \text{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{X}_3^{(\alpha )}& =-\frac{\pi }{2{\beta }_1{\beta }_2{\beta }_3}\mathrm{Re}[\frac{p_1^{\alpha }}{(p_1-{\overline{p}}_2)(p_1-{\overline{p}}_3)}+\frac{p_2^{\alpha }}{(p_2-{\overline{p}}_1)(p_2-{\overline{p}}_3)}+\frac{p_3^{\alpha }}{(p_3-{\overline{p}}_1)(p_3-{\overline{p}}_2)}],\end{array}\}$$ 3.22

where β_i is the imaginary part of p_i. It should be mentioned here that $X_m^{(\alpha )}$ are real-valued.

So far, the displacement fundamental solutions are expressed by $X_m^{(\alpha )}$. Since the obtained explicit expressions for the displacement fundamental solutions have no factors like p_i−p_j in the denominators, they are valid for both degenerate and non-degenerate cases. It can be proved that the present displacement fundamental solutions are equivalent to that presented by Ting & Lee [21]. It should be mentioned here that the present formulation for obtaining the displacement fundamental solutions is very similar to the seminal work by Ting & Lee [21] in the sense that both formulations use the Cauchy residue theorem applied to infinite integrals. Ting & Lee [21] applied the residue theorem to equation (3.8), while it is applied to equation (3.16) in the present formulation. If only the displacement fundamental solutions are concerned, it is recommended to use the neat expressions of Ting & Lee [21], which are applicable for both non-degenerate and degenerate cases, because there is no real advantage to using the results given by equations (3.20)–(3.22). However, the main target of this paper is also to derive the stress fundamental solutions and their derivatives rather than the displacement fundamental solutions alone. This goal can be realized systematically by using our formulation, which should be considered as an overall concept and will be shown in the following sections.

4. Stress fundamental solutions

Stress fundamental solutions can be obtained by substituting the displacement fundamental solutions into Hooke’s law (2.2) as

$${\sigma }_{ijk}^G(\mathit{\text{x}})=C_{ijlm}{u}_{lk,m}^G(\mathit{\text{x}}).$$ 4.1

To obtain the stress fundamental solutions, the first derivatives of the displacement fundamental solutions are required. To this end, we use the following integral expression [1]:

$$\begin{array}{r}{u}_{ij,k}^G(\mathit{\text{x}})=\frac{1}{8{\pi }^2{r}^2}{\oint }_s[-{\overline{x}}_k{\mathit{\Gamma }}_{ij}^{-1}(\mathit{\xi })+{\xi }_k{C}_{lpmq}({\overline{x}}_p{\xi }_q+{\xi }_p{\overline{x}}_q){\mathit{\Gamma }}_{li}^{-1}(\mathit{\xi }){\mathit{\Gamma }}_{mj}^{-1}(\mathit{\xi })]\hspace{0.17em}\mathrm{d}\varphi .\end{array}$$ 4.2

By using equations (3.4) and (3.5), we obtain from equation (4.2)

$$\begin{array}{rl}{u}_{ij,k}^G(\mathit{\text{x}})& =\frac{1}{4{\pi }^2{r}^2}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[-{\overline{x}}_k{\mathit{\Gamma }}_{ij}^{-1}(p)+{\xi }_k^{\prime }{C}_{lpmq}({\overline{x}}_p{\xi }_q^{\prime }+{\xi }_p^{\prime }{\overline{x}}_q){\mathit{\Gamma }}_{li}^{-1}(p){\mathit{\Gamma }}_{mj}^{-1}(p)]\hspace{0.17em}\mathrm{d}p,\end{array}$$ 4.3

where use is made of the periodicity of the integrand in equation (4.2) and the symmetry of C_ijkl and Γ_ij. Equation (4.3) can be rewritten as

$$u_{ij,k}^G(\mathit{\text{x}})=\frac{1}{4{\pi }^2{r}^2}[-{\overline{x}}_k{H}_{ij}(\overline{\mathit{\text{x}}})+P_{ijk}(\overline{\mathit{\text{x}}})],$$ 4.4

where $H_{ij}(\overline{\mathit{\text{ x}}})$ is defined by equation (3.15), and $P_{ijk}(\overline{\mathit{\text{ x}}})$ is given by

$$P_{ijk}(\overline{\mathit{\text{x}}})={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{N_{ijk}(p)}{D^2(p)}\hspace{0.17em}\mathrm{d}p.$$ 4.5

In equation (4.5), N_ijk(p) is defined by

$$\begin{array}{rl}{N}_{ijk}(p)& =F_{im}(p){\hat{\mathit{\Gamma }}}_{jm}(p){\xi }_k^{\prime },\\ F_{im}(p)& =E_{hm}(p){\hat{\mathit{\Gamma }}}_{ih}(p)\\ \text{and}{E}_{hm}(p)& =C_{hpmq}({\overline{x}}_p{\xi }_q^{\prime }+{\overline{x}}_q{\xi }_p^{\prime }).\end{array}\}$$ 4.6

Since N_ijk is a polynomial of order 10, we define

$$N_{ijk}(p)=\sum _{n=0}^{10}{\hat{N}}_{ijk}^n{p}^n,$$ 4.7

where the coefficients ${\hat{N}}_{ijk}^n$ are independent of p. Substitution of equations (4.7) and (3.11) into equation (4.5), after the term-by-term integration, leads to

$$P_{ijk}(\overline{\mathit{\text{x}}})=\frac{1}{{\alpha }^2}\sum _{n=0}^{10}{\hat{N}}_{ijk}^n{J}_n,$$ 4.8

where

$$J_n={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{p^n}{f^2(p)}\hspace{0.17em}\mathrm{d}p,n=0,1,\dots ,10.$$ 4.9

If p₁, p₂ and p₃ are distinct, the order of the poles in equation (4.9) is 2, which increases the complexity of the expressions resulting from the Cauchy residue theorem. Therefore, instead of equation (4.9) we consider

$$J_n={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{p^n}{\prod _{i=1}^6(p-p_i)(p-{\overline{p}}_i)}\hspace{0.17em}\mathrm{d}p,n=0,1,\dots ,10,$$ 4.10

which is identical to equation (4.9) when p₄=p₁, p₅=p₂ and p₆=p₃ are taken. Similar to I_n, the integral J_n can be expressed by $X_m^{(\alpha )}$ as

$$\begin{array}{rl}{J}_{2k}& =X_{6-k}^{(0)}-\sum _{i=1}^{2k}{(-1)}^i{E}_i^{(6(7-k))}{J}_{2k-i},(k=0,1,\dots ,5)\\ \text{and}{J}_{2k+1}& =X_{6-k}^{(1)}-\sum _{i=1}^{2k}{(-1)}^i{E}_i^{(6(7-k))}{J}_{2k+1-i},(k=0,1,\dots ,4),\end{array}\}$$ 4.11

in which the sum terms in equation (4.11) should be taken as zero when k=0, and

$$E_i^{(kl)}=\{\begin{array}{ll}{e}_i(p_k,{\overline{p}}_k,\dots ,p_l,{\overline{p}}_l),& l<k,\\ e_i(p_k,{\overline{p}}_k),& l=k,\end{array}$$ 4.12

where e_i(z₁,…,z_n) is the elementary symmetric polynomial, which is the sum of all products of the i distinct variables out of z₁,…,z_n. That is,

$$\begin{array}{rl}{e}_1(z_1,\dots ,z_n)& =\sum _{i=1}^n{z}_i,\\ e_2(z_1,\dots ,z_n)& =\sum _{1\le i_1<i_2\le n}{z}_{i_1}{z}_{i_2},\\ & ⋮\\ e_m(z_1,\dots ,z_n)& =\sum _{1\le i_1<\cdots <i_m\le n}{z}_{i_1}\dots z_{i_m}\\ & ⋮\\ \text{and}{e}_n(z_1,\dots ,z_n)& =z_1{z}_2\dots z_n.\end{array}\}$$ 4.13

By taking p₄=p₁, p₅=p₂ and p₆=p₃, and following the same procedure as for the rearrangement of $X_m^{(\alpha )}(m=1,2,3;\alpha =0,1)$ appearing in the displacement fundamental solutions, then $X_m^{(\alpha )}$ (m=4,5,6;α=0,1) having no factors like p_i−p_j in any denominators are given by

$$\begin{array}{rl}{X}_4^{(\alpha )}& =\frac{\pi }{4{\beta }_1^2{\beta }_2{\beta }_3}\mathrm{Im}[\frac{-i{p}_1^{\alpha }}{{\beta }_1(p_1-{\overline{p}}_2)(p_1-{\overline{p}}_3)}+\frac{p_2^{\alpha }}{{(p_2-{\overline{p}}_1)}^2(p_2-{\overline{p}}_3)}\\ & +\frac{p_3^{\alpha }}{{(p_3-{\overline{p}}_1)}^2(p_3-{\overline{p}}_2)}+2F_0^{(\alpha )}(1,2,\overline{1},\overline{3})+F_0^{(\alpha )}(1,1,\overline{2},\overline{3})],\\ X_5^{(\alpha )}& =\frac{\pi }{8{\beta }_1^2{\beta }_2^2{\beta }_3}\mathrm{Re}[\frac{-p_1^{\alpha }i}{{\beta }_1{(p_1-{\overline{p}}_2)}^2(p_1-{\overline{p}}_3)}+\frac{-p_2^{\alpha }i}{{\beta }_2{(p_2-{\overline{p}}_1)}^2(p_2-{\overline{p}}_3)}\\ & +\frac{p_3^{\alpha }}{{(p_3-{\overline{p}}_1)}^2{(p_3-{\overline{p}}_2)}^2}+4F_1^{(\alpha )}(1,2,\overline{1},\overline{2},\overline{3})+2F_1^{(\alpha )}(1,3,\overline{1},\overline{2},\overline{2})\\ & \phantom{\frac{\pi }{8{\beta }_1^2{\beta }_2^2{\beta }_3}}+2F_1^{(\alpha )}(2,3,\overline{1},\overline{1},\overline{2})+F_1^{(\alpha )}(1,1,\overline{2},\overline{2},\overline{3})+F_1^{(\alpha )}(2,2,\overline{1},\overline{1},\overline{3})]\\ \text{and}{X}_6^{(\alpha )}& =\frac{-\pi }{16{\beta }_1^2{\beta }_2^2{\beta }_3^2}\mathrm{Im}[\frac{-p_1^{\alpha }i}{{\beta }_1{(p_1-{\overline{p}}_2)}^2{(p_1-{\overline{p}}_3)}^2}+\frac{-p_2^{\alpha }i}{{\beta }_2{(p_2-{\overline{p}}_1)}^2{(p_2-{\overline{p}}_3)}^2}\\ & +\frac{-p_3^{\alpha }i}{{\beta }_3{(p_3-{\overline{p}}_1)}^2{(p_3-{\overline{p}}_2)}^2}+4F_2^{(\alpha )}(1,2,\overline{1},\overline{2},\overline{3},\overline{3})+4F_2^{(\alpha )}(1,3,\overline{1},\overline{2},\overline{2},\overline{3})\\ & +4F_2^{(\alpha )}(2,3,\overline{1},\overline{1},\overline{2},\overline{3})+F_2^{(\alpha )}(1,1,\overline{2},\overline{2},\overline{3},\overline{3})+F_2^{(\alpha )}(2,2,\overline{1},\overline{1},\overline{3},\overline{3})\\ & +F_2^{(\alpha )}(3,3,\overline{1},\overline{1},\overline{2},\overline{2})+4F_3^{(\alpha )}(1,2,3,\overline{1},\overline{2},\overline{3})+2F_3^{(\alpha )}(1,1,2,\overline{2},\overline{3},\overline{3})\\ & \phantom{\frac{\pi }{8{\beta }_1^2{\beta }_2^2{\beta }_3}}+2F_3^{(\alpha )}(1,2,2,\overline{1},\overline{3},\overline{3})+2F_3^{(\alpha )}(1,1,3,\overline{2},\overline{2},\overline{3})].\end{array}\}$$ 4.14

In equation (4.14), the abbreviations p_k=k and ${\overline{p}}_k=\overline{k}$ for the variables of the functions $F_m^{(\alpha )}(\cdots )$ (m=1,2,3;α=0,1) are introduced for convenience, and

$$\begin{array}{rl}{F}_0^{(0)}(z_1,\dots ,z_4)& ={[\prod _{i=3}^4(z_1-z_i)(z_2-z_i)]}^{-1}\times (z_1+z_2-z_3-z_4),\\ F_1^{(0)}(z_1,\dots ,z_5)& ={[\prod _{i=3}^5(z_1-z_i)(z_2-z_i)]}^{-1}\\ & \times [(z_1-z_3)(z_1-z_4)+(z_1-z_3)(z_2-z_5)+(z_2-z_4)(z_2-z_5)],\\ F_2^{(0)}(z_1,\dots ,z_6)& ={[\prod _{i=3}^6(z_1-z_i)(z_2-z_i)]}^{-1}\\ & \times [(z_1-z_3)(z_1-z_4)(z_1-z_5)+(z_1-z_3)(z_1-z_4)(z_2-z_6)\\ & +(z_1-z_3)(z_2-z_5)(z_2-z_6)+(z_2-z_4)(z_2-z_5)(z_2-z_6)],\\ F_3^{(0)}(z_1,\dots ,z_6)& ={[\prod _{i=4}^6(z_1-z_i)(z_2-z_i)(z_3-z_i)]}^{-1}\\ & \times [v_2^2-v_1{v}_3+v_3{v}_4+v_2(-v_1{v}_4+v_4^2-2v_5)\\ & +(v_1-v_4)v_6+v_5(v_1^2-v_1{v}_4+v_5)],\\ F_0^{(1)}(z_1,\dots ,z_4)& ={[(z_1-z_3)(z_2-z_3)]}^{-1}+z_4{F}_0^{(0)}(z_1,\dots ,z_4),\\ F_1^{(1)}(z_1,\dots ,z_5)& =F_0^{(0)}(z_1,\dots ,z_4)+z_5{F}_1^{(0)}(z_1,\dots ,z_5),\\ F_2^{(1)}(z_1,\dots ,z_6)& =F_1^{(0)}(z_1,\dots ,z_5)+z_6{F}_2^{(0)}(z_1,\dots ,z_6)\\ \text{and}{F}_3^{(1)}(z_1,\dots ,z_6)& =F_1^{(0)}(z_4,z_5,z_1,z_2,z_3)+z_6{F}_3^{(0)}(z_1,\dots ,z_6).\end{array}\}$$ 4.15

In equation (4.15), v_i are elementary symmetric polynomials defined by

$$v_i=\{\begin{array}{ll}{e}_i(z_1,z_2,z_3),& i=1,2,3,\\ e_{i-3}(z_4,z_5,z_6),& i=4,5,6.\end{array}$$ 4.16

Substitution of equation (4.11) into equation (4.8), and then equations (3.15) and (4.8) into equation (4.4), yields explicit expressions for the first derivatives of the displacement fundamental solutions. Subsequently, the stress fundamental solutions can be obtained by using equation (4.1).

Similar to the displacement fundamental solutions, the final explicit expressions of the stress fundamental solutions have no factors like p_i−p_j in the denominators, which implies that the expressions remain valid even in degenerate and nearly degenerate cases. Compared with the expressions derived by Lee [20,26], the new expressions may be more suitable for the applications since the expressions of Lee [20] are not valid in degenerate and nearly degenerate cases while the expressions of Lee [26] may cause numerical errors in the nearly degenerate cases. Buroni & Sáez [28] also presented unified explicit expressions by taking the derivatives of the expressions given by Ting & Lee [21] in the spherical coordinate system. However, it seems that our expressions are more explicit and clear than those presented by Buroni & Sáez [28] in the sense that our expressions without terms like p_i−p_j in the denominators can be more evidently identified.

5. Higher order derivatives of the displacement fundamental solutions

To obtain the derivatives of the stress fundamental solutions, the second derivatives of the displacement fundamental solutions are needed. For this purpose, the following integral expression can be used [1]:

$$\begin{array}{rl}{u}_{ij,kl}^G(\mathit{\text{x}})& =\frac{1}{8{\pi }^2{r}^3}{\oint }_s\{2{\overline{x}}_k{\overline{x}}_l{\mathit{\Gamma }}_{ij}^{-1}-2[({\overline{x}}_k{\xi }_l+{\xi }_k{\overline{x}}_l)({\overline{x}}_p{\xi }_q+{\xi }_p{\overline{x}}_q)+{\xi }_k{\xi }_l{\overline{x}}_p{\overline{x}}_q]C_{hpmq}{\mathit{\Gamma }}_{ih}^{-1}{\mathit{\Gamma }}_{jm}^{-1}\\ & +{\xi }_k{\xi }_l{C}_{hpmq}({\overline{x}}_p{\xi }_q+{\xi }_p{\overline{x}}_q)C_{satb}({\overline{x}}_a{\xi }_b+{\xi }_a{\overline{x}}_b)({\mathit{\Gamma }}_{jm}^{-1}{\mathit{\Gamma }}_{is}^{-1}{\mathit{\Gamma }}_{ht}^{-1}+{\mathit{\Gamma }}_{ih}^{-1}{\mathit{\Gamma }}_{js}^{-1}{\mathit{\Gamma }}_{mt}^{-1})\}\hspace{0.17em}\mathrm{d}\varphi .\end{array}$$ 5.1

By using equations (3.4) and (3.5), equation (5.1) can be rewritten as

$$\begin{array}{rl}{u}_{ij,kl}^G(\mathit{\text{x}})& =\frac{1}{2{\pi }^2{r}^3}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\{{\overline{x}}_k{\overline{x}}_l{\mathit{\Gamma }}_{ij}^{-1}-[({\overline{x}}_k{\xi }_l^{\prime }+{\xi }_k^{\prime }{\overline{x}}_l)({\overline{x}}_p{\xi }_q^{\prime }+{\xi }_p^{\prime }{\overline{x}}_q)+{\xi }_k^{\prime }{\xi }_l^{\prime }{\overline{x}}_p{\overline{x}}_q]C_{hpmq}{\mathit{\Gamma }}_{ih}^{-1}{\mathit{\Gamma }}_{jm}^{-1}\\ & +{\xi }_k^{\prime }{\xi }_l^{\prime }{C}_{hpmq}({\overline{x}}_p{\xi }_q^{\prime }+{\xi }_p^{\prime }{\overline{x}}_q)C_{satb}({\overline{x}}_a{\xi }_b^{\prime }+{\xi }_a^{\prime }{\overline{x}}_b){\mathit{\Gamma }}_{jm}^{-1}{\mathit{\Gamma }}_{is}^{-1}{\mathit{\Gamma }}_{ht}^{-1}\}\mathrm{d}\varphi ,\end{array}$$ 5.2

where use is made of the periodicity of the integrand in (5.1) and the symmetry of C_ijkl and Γ_ij. Equation (5.2) can be recast into

$$u_{ij,kl}^G(\mathit{\text{x}})=\frac{1}{2{\pi }^2{r}^3}\{H_{ij}(\overline{\mathit{\text{x}}}){\overline{x}}_k{\overline{x}}_l-[P_{ijk}(\overline{\mathit{\text{x}}}){\overline{x}}_l+P_{ijl}(\overline{\mathit{\text{x}}}){\overline{x}}_k]+Q_{ijkl}(\overline{\mathit{\text{x}}})\},$$ 5.3

where $H_{ij}(\overline{\mathit{\text{ x}}})$ and $P_{ijk}(\overline{\mathit{\text{ x}}})$ are given in equations (3.15) and (4.8), and $Q_{ijkl}(\overline{\mathit{\text{ x}}})$ is defined by

$$Q_{ijkl}(\overline{\mathit{\text{x}}})={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{S_{ijkl}(p)}{D^3(p)}\hspace{0.17em}\mathrm{d}p.$$ 5.4

In equation (5.4), S_ijkl(p) is given by

$$\begin{array}{rl}{S}_{ijkl}(p)& =[L_{ij}(p)-R_{ij}(p)D(p)]{\xi }_k^{\prime }{\xi }_l^{\prime },\\ L_{ij}(p)& =F_{jh}(p)F_{im}(p){\hat{\mathit{\Gamma }}}_{hm}(p)\\ \text{and}{R}_{ij}(p)& ={\overline{x}}_p{\overline{x}}_q{C}_{hpmq}{\hat{\mathit{\Gamma }}}_{ih}(p){\hat{\mathit{\Gamma }}}_{jm}(p).\end{array}\}$$ 5.5

Since S_ijkl is a polynomial of order 16, we define

$$S_{ijkl}(p)=\sum _{n=0}^{16}{\hat{S}}_{ijkl}{p}^n.$$ 5.6

Substitution of equation (5.6) into equation (5.4) yields

$$Q_{ijkl}(\overline{\mathit{\text{x}}})=\frac{1}{{\alpha }^3}\sum _{n=0}^{16}{\hat{S}}_{ijkl}{K}_n,$$ 5.7

where

$$K_n={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{p^n}{f^3(p)}\hspace{0.17em}\mathrm{d}p,n=0,1,\dots ,16.$$ 5.8

If p₁, p₂ and p₃ are distinct, the order of the poles in equation (5.8) is 3, which increases the complexity of the expressions resulting from the Cauchy residue theorem. Therefore, instead of equation (5.8) we consider

$$K_n={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{p^n}{\prod _{i=1}^9(p-p_i)(p-{\overline{p}}_i)}\hspace{0.17em}\mathrm{d}p,n=0,1,\dots ,16,$$ 5.9

which is equivalent to equation (5.8) when p₄=p₇=p₁, p₅=p₈=p₂ and p₆=p₉=p₃, respectively. Similar to I_n and J_n, the integral K_n can be expressed in terms of $X_m^{(\alpha )}$ as

$$\begin{array}{lll}& K_{2k}=X_{9-k}^{(0)}-\sum _{i=1}^{2k}{(-1)}^i{E}_i^{(9(10-k))}{K}_{2k-i},& (k=0,1,\dots ,8)\\ \text{and}& K_{2k+1}=X_{9-k}^{(1)}-\sum _{i=1}^{2k}{(-1)}^i{E}_i^{(9(10-k))}{K}_{2k+1-i},& (k=0,1,\dots ,7),\end{array}\}$$ 5.10

where the sum terms in equation (5.10) should be taken as zero for k=0, and

$$\begin{array}{rl}{X}_7^{(\alpha )}& =\frac{-\pi }{32{\beta }_1^3{\beta }_2^2{\beta }_3^2}\mathrm{Re}[\frac{-3p_1^{\alpha }}{4{\beta }_1^2{(p_1-{\overline{p}}_2)}^2{(p_1-{\overline{p}}_3)}^2}+\frac{-p_2^{\alpha }i}{{\beta }_2{(p_2-{\overline{p}}_1)}^3{(p_2-{\overline{p}}_3)}^2}\\ & +\frac{-p_3^{\alpha }i}{{\beta }_3{(p_3-{\overline{p}}_1)}^3{(p_3-{\overline{p}}_2)}^2}+6F_4^{(\alpha )}(1,2,\overline{1},\overline{1},\overline{2},\overline{3},\overline{3})+6F_4^{(\alpha )}(1,3,\overline{1},\overline{1},\overline{2},\overline{2},\overline{3})\\ & +4F_4^{(\alpha )}(2,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{3})+3F_4^{(\alpha )}(1,1,\overline{1},\overline{2},\overline{2},\overline{3},\overline{3})+F_4^{(\alpha )}(2,2,\overline{1},\overline{1},\overline{1},\overline{3},\overline{2})\\ & +F_4^{(\alpha )}(3,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{2})+12F_5^{(\alpha )}(1,2,3,\overline{1},\overline{1},\overline{2},\overline{3})+6F_5^{(\alpha )}(1,1,2,\overline{1},\overline{2},\overline{3},\overline{3})\\ & +6F_5^{(\alpha )}(1,1,3,\overline{1},\overline{2},\overline{2},\overline{3})+3F_5^{(\alpha )}(1,2,2,\overline{1},\overline{1},\overline{3},\overline{3})+3F_5^{(\alpha )}(1,3,3,\overline{1},\overline{1},\overline{2},\overline{2})\\ & +2F_5^{(\alpha )}(2,2,3,\overline{1},\overline{1},\overline{1},\overline{3})+2F_5^{(\alpha )}(2,3,3,\overline{1},\overline{1},\overline{1},\overline{2})\phantom{\frac{-\pi }{32{\beta }_1^3{\beta }_2^2{\beta }_3^2}}+F_5^{(\alpha )}(1,1,1,\overline{2},\overline{2},\overline{3},\overline{3})],\end{array}$$ 5.11a

$$\begin{array}{rl}{X}_8^{(\alpha )}& =\frac{\pi }{64{\beta }_1^3{\beta }_2^3{\beta }_3^2}\mathrm{Im}[\frac{-3p_1^{\alpha }}{4{\beta }_1^2{(p_1-{\overline{p}}_2)}^3{(p_1-{\overline{p}}_3)}^2}+\frac{-3p_2^{\alpha }}{4{\beta }_2^2{(p_2-{\overline{p}}_1)}^3{(p_2-{\overline{p}}_3)}^2}\\ & +\frac{-p_3^{\alpha }i}{{\beta }_3{(p_3-{\overline{p}}_1)}^3{(p_3-{\overline{p}}_2)}^3}+9F_6^{(\alpha )}(1,2,\overline{1},\overline{1},\overline{2},\overline{2},\overline{3},\overline{3})+6F_6^{(\alpha )}(1,3,\overline{1},\overline{1},\overline{2},\overline{2},\overline{2},\overline{3})\\ & +6F_6^{(\alpha )}(2,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{2},\overline{3})+3F_6^{(\alpha )}(1,1,\overline{1},\overline{2},\overline{2},\overline{2},\overline{3},\overline{3})+3F_6^{(\alpha )}(2,2,\overline{1},\overline{1},\overline{1},\overline{2},\overline{3},\overline{3})\\ & +F_6^{(\alpha )}(3,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{2},\overline{2})+18F_7^{(\alpha )}(1,2,3,\overline{1},\overline{1},\overline{2},\overline{2},\overline{3})+9F_7^{(\alpha )}(1,1,2,\overline{1},\overline{2},\overline{2},\overline{3},\overline{3})\\ & +9F_7^{(\alpha )}(1,2,2,\overline{1},\overline{1},\overline{2},\overline{3},\overline{3})+6F_7^{(\alpha )}(1,1,3,\overline{1},\overline{2},\overline{2},\overline{2},\overline{3})+6F_7^{(\alpha )}(2,2,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{3})\\ & +3F_7^{(\alpha )}(2,3,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{2})+3F_7^{(\alpha )}(1,3,3,\overline{1},\overline{1},\overline{2},\overline{2},\overline{3})+F_7^{(\alpha )}(1,1,1,\overline{2},\overline{2},\overline{2},\overline{3},\overline{3})\\ & +F_7^{(\alpha )}(2,2,2,\overline{1},\overline{1},\overline{1},\overline{3},\overline{3})+18F_8^{(\alpha )}(1,1,2,3,\overline{1},\overline{2},\overline{2},\overline{3})+9F_8^{(\alpha )}(1,1,2,2,\overline{1},\overline{2},\overline{3},\overline{3})\\ & +3F_8^{(\alpha )}(1,1,3,3,\overline{1},\overline{2},\overline{2},\overline{2})+3F_8^{(\alpha )}(1,1,1,2,\overline{2},\overline{2},\overline{3},\overline{3})\phantom{\frac{-\pi }{32{\beta }_1^3{\beta }_2^2{\beta }_3^2}}+2F_8^{(\alpha )}(1,1,1,3,\overline{2},\overline{2},\overline{2},\overline{3})]\end{array}$$ 5.11b

$$\begin{array}{rl}\text{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{X}_9^{(\alpha )}& =\frac{\pi }{128{\beta }_1^3{\beta }_2^3{\beta }_3^3}\mathrm{Re}[\frac{-3p_1^{\alpha }}{4{\beta }_1^2{(p_1-{\overline{p}}_2)}^3{(p_1-{\overline{p}}_3)}^3}+\frac{-3p_2^{\alpha }}{4{\beta }_2^2{(p_2-{\overline{p}}_1)}^3{(p_2-{\overline{p}}_3)}^3}\\ & \frac{-3p_3^{\alpha }}{4{\beta }_3^2{(p_3-{\overline{p}}_1)}^3{(p_3-{\overline{p}}_2)}^3}+9F_9^{(\alpha )}(1,2,\overline{1},\overline{1},\overline{2},\overline{2},\overline{3},\overline{3},\overline{3})+9F_9^{(\alpha )}(1,3,\overline{1},\overline{1},\overline{2},\overline{2},\overline{2},\overline{3},\overline{3})\\ & +9F_9^{(\alpha )}(2,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{2},\overline{3},\overline{3})+3F_9^{(\alpha )}(1,1,\overline{1},\overline{2},\overline{2},\overline{2},\overline{3},\overline{3},\overline{3})+3F_9^{(\alpha )}(2,2,\overline{1},\overline{1},\overline{1},\overline{2},\overline{3},\overline{3},\overline{3})\\ & +3F_9^{(\alpha )}(3,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{2},\overline{2},\overline{3})+27F_{10}^{(\alpha )}(1,2,3,\overline{1},\overline{1},\overline{2},\overline{2},\overline{3},\overline{3})+9F_{10}^{(\alpha )}(1,1,2,\overline{1},\overline{2},\overline{2},\overline{3},\overline{3},\overline{3})\\ & +9F_{10}^{(\alpha )}(1,2,2,\overline{1},\overline{1},\overline{2},\overline{3},\overline{3},\overline{3})+9F_{10}^{(\alpha )}(1,1,3,\overline{1},\overline{2},\overline{2},\overline{2},\overline{3},\overline{3})+9F_{10}^{(\alpha )}(2,2,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{3},\overline{3})\\ & +9F_{10}^{(\alpha )}(2,3,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{2},\overline{3})+9F_{10}^{(\alpha )}(1,3,3,\overline{1},\overline{1},\overline{2},\overline{2},\overline{3},\overline{3})+F_{10}^{(\alpha )}(1,1,1,\overline{2},\overline{2},\overline{2},\overline{3},\overline{3},\overline{3})\\ & +F_{10}^{(\alpha )}(2,2,2,\overline{1},\overline{1},\overline{1},\overline{3},\overline{3},\overline{3})+F_{10}^{(\alpha )}(3,3,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{2},\overline{2})+27F_{11}^{(\alpha )}(1,1,2,3,\overline{1},\overline{2},\overline{2},\overline{3},\overline{3})\\ & +27F_{11}^{(\alpha )}(1,2,2,3,\overline{1},\overline{1},\overline{2},\overline{3},\overline{3})+27F_{11}^{(\alpha )}(1,2,3,3,\overline{1},\overline{1},\overline{2},\overline{2},\overline{3})+9F_{11}^{(\alpha )}(1,1,2,2,\overline{1},\overline{2},\overline{3},\overline{3},\overline{3})\\ & +9F_{11}^{(\alpha )}(1,1,3,3,\overline{1},\overline{2},\overline{2},\overline{2},\overline{3})+9F_{11}^{(\alpha )}(2,2,3,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{3})+3F_{11}^{(\alpha )}(1,1,1,2,\overline{2},\overline{2},\overline{3},\overline{3},\overline{3})\\ & +3F_{11}^{(\alpha )}(1,1,1,3,\overline{2},\overline{2},\overline{2},\overline{3},\overline{3})+3F_{11}^{(\alpha )}(1,2,2,2,\overline{1},\overline{1},\overline{3},\overline{3},\overline{3})+3F_{11}^{(\alpha )}(1,3,3,3,\overline{1},\overline{1},\overline{2},\overline{2},\overline{2})\\ & +3F_{11}^{(\alpha )}(2,2,2,3,\overline{1},\overline{1},\overline{1},\overline{3},\overline{3})\phantom{\frac{-\pi }{32{\beta }_1^3{\beta }_2^2{\beta }_3^2}}+3F_{11}^{(\alpha )}(2,3,3,3,\overline{1},\overline{1},\overline{1},\overline{2},\overline{2})].\end{array}$$ 5.11c

Here again, to remove the terms p_i−p_j in the denominators, the same procedure is used as for rearranging $X_m^{(\alpha )}\hspace{0.17em}(m=1,2,3;\alpha =0,1)$ arising in the displacement fundamental solutions. The explicit expressions for the functions $F_m^{(\alpha )}(\cdots )$ (m=4,5,…,11;α=0,1) are quite lengthy and thus they are given in appendix B. By substituting equation (5.10) into equation (5.7), and then equations (3.15), (4.8) and (5.7) into equation (5.3), we obtain explicit expressions for the second derivatives of the displacement fundamental solutions, from which the derivatives of the stress fundamental solutions can be subsequently obtained by using equation (2.9).

Similar to the displacement and stress fundamental solutions, the final explicit expressions of the derivatives of the stress fundamental solutions or the higher order derivatives of the displacement fundamental solutions have no factors like p_i−p_j in the denominators, implying they remain valid even in degenerate and nearly degenerate cases. Like the stress fundamental solutions or the first derivatives of the displacement fundamental solutions presented in the last section, the present unified explicit expressions for the derivatives of the stress fundamental solutions or the second derivatives of the displacement fundamental solutions have the following distinct features compared with some previous works (e.g. [20,26,28]):

• — they are completely explicit and contain only elementary functions;

• — they are unified and consistently valid for non-degenerate, partially degenerate, fully degenerate and nearly degenerate cases;

• — they are given in Cartesian coordinates instead of spherical coordinates; and

• — they involve no higher order tensors.

We are reasonably convinced that some of these special features could be advantageous and efficient for the numerical evaluation and implementation of the stress fundamental solutions and their derivatives in the BEM.

6. Verification examples

In order to verify the correctness and the accuracy of the novel displacement fundamental solutions and their derivatives for general anisotropic materials, some available analytical fundamental functions are used. For the non-degenerate case, a general transversely isotropic material is considered. If the symmetry axis of the considered transversely isotropic material coincides with the x₃-axis, then the non-zero components of the elasticity tensor can be written in contracted notation as

$$\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ & C_{22}& C_{23}& 0& 0& 0\\ & & C_{33}& 0& 0& 0\\ & & & C_{44}& 0& 0\\ & \text{sym.}& & & C_{55}& 0\\ & & & & & C_{66}\end{array}\right],$$ 6.1

where

$$C_{11}=C_{22},C_{13}=C_{23},C_{44}=C_{55},C_{66}=\frac{(C_{11}-C_{12})}{2}.$$ 6.2

In particular, the following elastic constants used by Tonon et al. [37] and Lee [26] are taken in the present example

$$\begin{array}{rl}{C}_{11}& =88,C_{12}=72,C_{13}=40,\\ C_{33}& =24,C_{44}=16(\text{unit:}\hspace{0.17em}{10}^7\hspace{0.17em}\mathrm{N}\hspace{0.17em}{\mathrm{m}}^{-2}).\end{array}\}$$ 6.3

It should be mentioned here that nearly degenerate and fully degenerate cases are also included in the present example, since it is fully degenerate in the direction along the material symmetry axis (see appendix A). For the partially degenerate case, the elastic constant C₃₃ from the non-degenerate case is modified according to

$$C_{33}=\frac{{(C_{13}+2C_{44})}^2}{C_{11}}\to p_1\ne p_2=p_3,$$ 6.4

or

$$C_{33}=\frac{4C_{11}{C}_{44}^2+2C_{13}(C_{11}-C_{12})(C_{13}+2C_{44})}{C_{11}^2-C_{12}^2}\to p_1=p_2\ne p_3.$$ 6.5

For the fully degenerate case, the isotropic material alumina (Al) is considered, and the independent elastic constants are given by

$$C_{11}=468\hspace{0.17em}\mathrm{GPa}\mathrm{and}{C}_{12}=156\hspace{0.17em}\mathrm{GPa}.$$ 6.6

In all cases, the displacement fundamental solutions and their derivatives evaluated by using the novel unified and explicit expressions are compared with available analytical fundamental solutions. For transversely isotropic materials, the analytical expressions of Pan & Chou [41] for the displacement fundamental solutions are used. For isotropic materials, the analytical fundamental solutions can be found in many textbooks. The derivatives of the analytical displacement fundamental solutions are evaluated by the computer software Mathematica using the algebraic algorithm.

The evaluated points are chosen as

$$\mathit{\text{x}}=(r\cos \varphi \sin \theta ,\hspace{0.17em}r\sin \varphi \sin \theta ,\hspace{0.17em}r\cos \theta ),r=1,\varphi =\frac{\pi }{4},$$ 6.7

with θ varying from 0° to 90°; see figure 1 for the illustration of the evaluation point.

Figure 1.

Definition of the evaluation point.

Figures 2–4 show the results for the non-degenerate transversely isotropic material considered. To keep the clarity of the figures, only some components of the displacement fundamental solutions and their derivatives are presented. Figures 5–7 present the results for a partially degenerate transversely isotropic material. The results for an isotropic material (fully degenerate case) are illustrated in figures 8–10. These figures demonstrate clearly that the present novel displacement fundamental solutions and their derivatives for generally anisotropic materials agree perfectly with other available analytical solutions for transversely isotropic and isotropic materials. This verifies that the present novel displacement fundamental solutions and their derivatives are correct and universally valid for non-degenerate, partially degenerate and fully degenerate cases.

Figure 3.

$u_{ij,k}^G$ for a non-degenerate transversely isotropic material (solid lines, solutions of this paper; circles, analytical solutions). (Online version in colour.)

Figure 6.

$u_{ij,k}^G$ for a partially degenerate transversely isotropic material (solid lines, solutions of this paper; circles, analytical solutions). (Online version in colour.)

Figure 9.

$u_{ij,k}^G$ for a fully degenerate isotropic material (solid lines, solutions of this paper; circles, analytical solutions). (Online version in colour.)

Figure 2.

$u_{ij}^G$ for a non-degenerate transversely isotropic material (solid lines, solutions of this paper; circles, analytical solutions). (Online version in colour.)

Figure 4.

$u_{ij,kl}^G$ for a non-degenerate transversely isotropic material (solid lines, solutions of this paper; circles, analytical solutions). (Online version in colour.)

Figure 5.

$u_{ij}^G$ for a partially degenerate transversely isotropic material (solid lines, solutions of this paper; circles, analytical solutions). (Online version in colour.)

Figure 7.

$u_{ij,kl}^G$ for a partially degenerate transversely isotropic material (solid lines, solutions of this paper; circles, analytical solutions). (Online version in colour.)

Figure 8.

$u_{ij}^G$ for a fully degenerate isotropic material (solid lines, solutions of this paper; circles, analytical solutions). (Online version in colour.)

Figure 10.

$u_{ij,kl}^G$ for a fully degenerate isotropic material (solid lines, solutions of this paper; circles, analytical solutions). (Online version in colour.)

7. Conclusions

Novel explicit analytical displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions are presented for three-dimensional generally anisotropic and linear elastic materials in terms of the Stroh eigenvalues p_i (i=1,2,3). The fundamental solutions are first expressed by appropriate infinite integrals. Then, the Cauchy residue theorem is applied to analytically evaluate the corresponding infinite integrals in terms of the elementary functions of the Stroh eigenvalues without terms like p_i−p_j in the denominators of the resulting analytical expressions. Therefore, the novel explicit fundamental solutions are universally valid for non-degenerate, partially degenerate, fully degenerate and nearly degenerate cases. The novel unified analytical fundamental solutions without terms like p_i−p_j in their denominators may have significant advantages in the numerical implementation of the three-dimensional anisotropic fundamental solutions into the BEM to avoid possible numerical errors in degenerate and nearly degenerate cases, where considerable errors in the numerical evaluation may occur. Nearly degenerate cases exist, for instance, near the direction along the symmetry axis of the transversely isotropic materials, where all three Stroh eigenvalues are nearly identical. The correctness and the accuracy of the present unified and analytical fundamental solutions are verified by using some other available analytical fundamental solutions for non-degenerate, partially degenerate and fully degenerate cases.

Even though the present fundamental solutions are explicit and generally valid, their convenience and efficiency compared with other methods are of particular interest. After the submission of this article, we started our investigation into the efficiency of the present unified explicit expressions in comparison with another two methods, one of which is based on the direct numerical integration of the infinite integrals and the other on the Cauchy residue theorem applied to the infinite integrals with the assumption of distinct Stroh eigenvalues. Our results reveal that the present unified explicit expressions are in fact more efficient and accurate than the other two methods under consideration. The corresponding results will be reported in a forthcoming paper [42]. Indeed, an efficiency assessment of the present unified explicit expressions compared with many other different formulations as presented in [20–29,38–40], just to mention a few of them known in the literature, is also highly desirable, but it demands further comprehensive investigations. This topic will be pursued by the authors in future works.

Appendix A. Stroh eigenvalues

The Stroh eigenvalues are the roots of the following characteristic equation:

$$\mid \mathit{\Gamma }(p)\mid =0$$ A 1

with positive imaginary part. Because Γ_ij(p)=C_ikjlξ′_kξ′_l and ξ′=n+pm, the Stroh eigenvalues are dependent on the elasticity tensor, the direction of the observation point and the chosen unit base vectors in the oblique plane. For generally anisotropic materials, closed-form analytical solutions can unfortunately not be found for the Stroh eigenvalues because the characteristic equation from equation (A 1) is sextic.

For transversely isotropic materials, closed-form analytical solutions can be found for the Stroh eigenvalues. If the symmetry axis of the material coincides with the x₃-axis, then the non-zero components of the elasticity tensor are given by equation (6.1). By choosing (figure 1)

$$\overline{\mathit{\text{x}}}=\left(\begin{array}{c}\sin \theta \cos \varphi \\ \sin \theta \sin \varphi \\ \cos \theta \end{array}\right),\mathit{\text{n}}=\left(\begin{array}{c}\cos \theta \cos \varphi \\ \cos \theta \sin \varphi \\ -\sin \theta \end{array}\right)\mathrm{and}\mathit{\text{m}}=\left(\begin{array}{c}-\sin \varphi \\ \cos \varphi \\ 0\end{array}\right),$$ A 2

one of the Stroh eigenvalues can be obtained as

$$p_1=i\sqrt{{\cos }^2\theta +\frac{{\mathrm{C}}_{44}{\sin }^2\theta }{{\mathrm{C}}_{66}}}.$$ A 3

The other two Stroh eigenvalues, p₂ and p₃, satisfy

$$\begin{array}{rl}& {\mathrm{C}}_{11}{\mathrm{C}}_{44}{({\cos }^2\theta +p^2)}^2+({\mathrm{C}}_{11}{\mathrm{C}}_{33}-{\mathrm{C}}_{13}^2-2{\mathrm{C}}_{13}{\mathrm{C}}_{44})({\cos }^2\theta +p^2)\\ & \times {\sin }^2\theta +{\mathrm{C}}_{33}{\mathrm{C}}_{44}{\sin }^4\theta =0.\end{array}$$ A 4

Based on the theory of polynomials, the following three cases may arise:

• (i) Non-degenerate case when

  $$\theta \ne 0\to p_1\ne p_2\ne p_3.$$ A 5

  When θ=0 and π, we obtain three identical eigenvalues p₁=p₂=p₃=i. This means that the general transversely isotropic materials have three distinct Stroh eigenvalues everywhere except in the direction along the material symmetry axis where all three Stroh eigenvalues are i. In spite of this fact, general transversely isotropic materials are referred to as non-degenerate materials for simplicity.
• (ii) Partially degenerate case when

  $$C_{33}=\frac{{(C_{13}+2C_{44})}^2}{C_{11}}\to p_1\ne p_2=p_3$$ A 6
  or

  $$C_{33}=\frac{4C_{11}{C}_{44}^2+2C_{13}(C_{11}-C_{12})(C_{13}+2C_{44})}{C_{11}^2-C_{12}^2}\to p_1=p_2\ne p_3.$$ A 7

  In this case, there are two distinct eigenvalues changing with the direction except the direction along the symmetry axis where there is only one eigenvalue.
• (iii) Fully degenerate case when

  $$C_{33}=\frac{C_{11}{C}_{13}^2}{C_{12}^2},C_{44}=\frac{C_{66}{C}_{13}}{C_{12}}\to p_1=p_2=p_3.$$ A 8

  In this case, both (A 6) and (A 7) are satisfied simultaneously. In the special case of isotropic materials, we have p₁=p₂=p₃=i, which means that the isotropic materials are fully degenerate everywhere and the three Stroh eigenvalues are identical and constant.

Appendix B. Explicit expressions for $F_m^{(\alpha )}$

The functions $F_m^{(\alpha )}(\cdots )$ for m=4,5,…,11 and α=0,1 arising in equations (5.11a–c) are given in the following:

$$\begin{array}{rl}{F}_4^{(0)}(z_1,\dots ,z_7)& ={[\prod _{i=3}^7(z_1-z_i)(z_2-z_i)]}^{-1}\\ & \times [v_1^4+v_2^2-v_1^3{v}_3-v_2{v}_4+v_1^2(-3v_2+v_4)+v_1(2v_2{v}_3-v_5)+v_6],\\ F_5^{(0)}(z_1,\dots ,z_7)& ={[\prod _{i=4}^7(z_1-z_i)(z_2-z_i)(z_3-z_i)]}^{-1}\\ & \times [v_2^3+v_3^2+v_2^2(-v_1{v}_4+v_4^2-2v_5)+v_3{v}_4(v_1^2-v_1{v}_4+v_5)-2v_3{v}_6\\ & +v_6(-v_1(v_1^2-v_1{v}_4+v_5)+v_6)-(v_1^2-v_1{v}_4+v_5)v_7\\ & +v_2(v_3{v}_4+v_1^2{v}_5+v_5^2-v_1(2v_3+v_4{v}_5-3v_6)-2v_4{v}_6+v_7)]\\ \text{and}{F}_m^{(1)}(z_1,\dots ,z_7)& =F_{m-2}^{(0)}(z_1,\dots ,z_6)+z_7{F}_m^{(0)}(z_1,\dots ,z_7),m=4,5.\end{array}\}$$ B 1

In equation (B 1), v_i in $F_4^{(0)}(z_1,\dots ,z_7)$ are defined by

$$v_i=\{\begin{array}{ll}{e}_i(z_1,z_2),& i=1,2,\\ e_{i-2}(z_3,\dots ,z_7),& i=3,4,\dots ,7,\end{array}$$ B 2

and v_i in $F_5^{(0)}(z_1,\dots ,z_7)$ are defined by

$$v_i=\{\begin{array}{ll}{e}_i(z_1,z_2,z_3),& i=1,2,3,\\ e_{i-3}(z_4,\dots ,z_7),& i=4,5,\dots ,7.\end{array}$$ B 3

Here and in what follows, e_i(⋯ ) is the elementary symmetric polynomial defined by equation (4.13),

$$\begin{array}{rl}{F}_6^{(0)}(z_1,\dots ,z_8)& ={[\prod _{i=3}^8(z_1-z_i)(z_2-z_i)]}^{-1}[v_1^5-v_1^4{v}_3+v_1^3(-4v_2+v_4)+v_1^2(3v_2{v}_3-v_5)\\ & +v_2(-v_2{v}_3+v_5)+v_1(3v_2^2-2v_2{v}_4+v_6)-v_7],\\ F_7^{(0)}(z_1,\dots ,z_8)& ={[\prod _{i=4}^8(z_1-z_i)(z_2-z_i)(z_3-z_i)]}^{-1}[v_2^4+v_3^2{v}_4^2+v_2^3(-v_1{v}_4+v_4^2-2v_5)\\ & -v_3^2{v}_5+v_3{v}_5{v}_6+v_1^4{v}_7-2v_3{v}_4{v}_7+v_2^2(v_3{v}_4+v_1^2{v}_5+v_5^2\\ & -v_1(3v_3+v_4{v}_5-3v_6)-2v_4{v}_6+2v_7)+v_3{v}_8+v_1^3(v_8-v_3{v}_5-v_4{v}_7)\\ & +v_1^2(v_3^2+v_3{v}_4{v}_5+v_5{v}_7-v_4{v}_8)-v_6{v}_8+v_7^2-v_1(2v_3^2{v}_4+v_3(v_5^2-3v_7)\\ & +v_6{v}_7-v_5{v}_8)+v_2(2v_3^2-2v_5{v}_7+v_3(2v_1(v_1-v_4)v_4+(2v_1+v_4)v_5-3v_6)\\ & +v_6^2+v_1^2(v_4{v}_6-4v_7)-v_1(v_5{v}_6-3v_4{v}_7+2v_8)-v_1^3{v}_6+v_4{v}_8)],\\ F_8^{(0)}(z_1,\dots ,z_8)& ={[\prod _{i=5}^8(z_1-z_i)(z_2-z_i)(z_3-z_i)(z_4-z_i)]}^{-1}\\ & \times [v_3^3+v_2^2{v}_4{v}_5-v_4^2{v}_5+v_2{v}_4{v}_5^3-2v_2{v}_4{v}_5{v}_6+v_4{v}_5{v}_6^2-v_2^3{v}_7+2v_2{v}_4{v}_7\\ & -v_2^2{v}_5^2{v}_7-v_4{v}_5^2{v}_7+2v_2^2{v}_6{v}_7-2v_4{v}_6{v}_7+2(v_4{v}_5+(-v_2+v_6)v_7)v_8\\ & -v_3^2(v_2{v}_5-v_1{v}_5^2+v_5^3+2v_1{v}_6-3v_5{v}_6+3v_7)-v_2{v}_6^2{v}_7+2v_2{v}_5{v}_7^2\\ & +v_1^2(v_4(v_5{v}_6-2v_7)-v_7(v_2{v}_6+v_5{v}_7)+(v_5(v_2+v_6)+v_7)v_8)-v_7^3\\ & +v_1(v_4^2-v_4{v}_5(v_5(v_2+v_6)-3v_7)+v_7(v_2^2{v}_5+v_2{v}_5{v}_6-3v_2{v}_7+v_6{v}_7)\\ & -2v_4{v}_8-(v_2^2+v_2(v_5^2-2v_6)+v_6^2+v_5{v}_7)v_8+v_8^2)+v_3(-v_4{v}_5^2+v_2^2{v}_6\\ & +2v_4{v}_6+v_6^3-3v_5{v}_6{v}_7+3v_7^2-v_2(2v_4+v_1{v}_5{v}_6-v_5^2{v}_6+2v_6^2-3v_1{v}_7\\ & +v_5{v}_7-2v_8)+2v_5^2{v}_8-2v_6{v}_8+v_1^2(v_6^2-2v_5{v}_7+v_8)\\ & +v_1(v_4{v}_5+2v_5^2{v}_7+v_6{v}_7-v_5(v_6^2+3v_8)))-v_5{v}_8^2+v_1^3(v_7^2-v_6{v}_8)],\\ F_m^{(1)}(z_1,\dots ,z_8)& =F_{m-2}^{(0)}(z_1,\dots ,z_7)+z_8{F}_m^{(0)}(z_1,\dots ,z_8),m=6,7\\ \text{and}{F}_8^{(1)}(z_1,\dots ,z_8)& =F_5^{(0)}(z_5,z_6,z_7,z_1,\dots ,z_4)+z_8{F}_8^{(0)}(z_1,\dots ,z_8).\end{array}\}$$ B 4

In equation (B 4), v_i in $F_6^{(0)}(z_1,\dots ,z_8)$ are defined by

$$v_i=\{\begin{array}{ll}{e}_i(z_1,z_2),& i=1,2,\\ e_{i-2}(z_3,\dots ,z_8),& i=3,4,\dots ,8,\end{array}$$ B 5

v_i in $F_7^{(0)}(z_1,\dots ,z_8)$ are defined by

$$v_i=\{\begin{array}{ll}{e}_i(z_1,z_2,z_3),& i=1,2,3,\\ e_{i-3}(z_4,\dots ,z_8),& i=4,5,\dots ,8,\end{array}$$ B 6

and v_i in $F_8^{(0)}(z_1,\dots ,z_8)$ are defined by

$$v_i=\{\begin{array}{ll}{e}_i(z_1,\dots ,z_4),& i=1,2,3,4,\\ e_{i-4}(z_5,\dots ,z_8),& i=5,6,7,8.\end{array}$$ B 7

Furthermore, we have

$$\begin{array}{rl}{F}_9^{(0)}(z_1,\dots ,z_9)& ={[\prod _{i=3}^9(z_1-z_i)(z_2-z_i)]}^{-1}[v_1^6-v_1^5{v}_3+v_1^4(-5v_2+v_4)+v_1^3(4v_2{v}_3-v_5)+v_1^2(6v_2^2-3v_2{v}_4+v_6)\\ & -v_2(v_2^2-v_2{v}_4+v_6)-v_1(3v_2^2{v}_3-2v_2{v}_5+v_7)+v_8],\\ F_{10}^{(0)}(z_1,\dots ,z_9)& ={[\prod _{i=4}^9(z_1-z_i)(z_2-z_i)(z_3-z_i)]}^{-1}[v_2^5+v_3^3{v}_4+v_2^4(v_4^2-v_1{v}_4-2v_5)+v_3^2{v}_5^2-v_3^2{v}_4{v}_6-v_3^2{v}_7\\ & +v_3{v}_6{v}_7+v_8^2+v_2^3(v_3{v}_4+v_1^2{v}_5+v_5^2-v_1(4v_3+v_4{v}_5-3v_6)-2v_4{v}_6+2v_7)-v_1^5{v}_8\\ & -2v_3{v}_5{v}_8-v_7{v}_9+v_1^2(v_3^2(v_4^2+v_5)+v_3(v_5{v}_6-4v_8)+v_6{v}_8-v_5{v}_9)\\ & +v_2^2(3v_3^2+v_3(3v_1(v_1-v_4)v_4+(4v_1+v_4)v_5-3v_6)-v_1^3{v}_6+v_6^2\\ & +v_1^2(v_4{v}_6-4v_7)-2v_5{v}_7+2v_4{v}_8-v_1(v_5{v}_6-3v_4{v}_7+5v_8)-v_9)\\ & -v_1^3(v_3{v}_4(v_3+v_6)+v_5{v}_8-v_4{v}_9)+v_1^4(v_3{v}_6+v_4{v}_8-v_9)+v_3{v}_4{v}_9\\ & -v_1(2v_3^3+v_3^2(2v_4{v}_5-3v_6)+v_7{v}_8-v_6{v}_9+v_3(v_6^2-3v_4{v}_8+2v_9))\\ & +v_2(v_3^2(2v_4^2-3v_5)+v_1^4{v}_7+v_7^2-v_1^3(2v_3{v}_5+v_4{v}_7-5v_8)-2v_6{v}_8\\ & +v_5{v}_9+v_1^2(3v_3^2+2v_3{v}_4{v}_5-3v_3{v}_6+v_5{v}_7-4v_4{v}_8+3v_9)\\ & -v_1(4v_3^2{v}_4+2v_3(v_5^2-v_4{v}_6-2v_7)+v_6{v}_7-3v_5{v}_8+2v_4{v}_9)+v_3(v_5{v}_6-3v_4{v}_7+4v_8))],\\ F_{11}^{(0)}(z_1,\dots ,z_9)& ={[\prod _{i=5}^9(z_1-z_i)(z_2-z_i)(z_3-z_i)(z_4-z_i)]}^{-1}\\ & \times [v_3^4-v_4^3+v_1^2{v}_4^2{v}_5^2-v_1{v}_4^2{v}_5^3-v_1^2{v}_4^2{v}_6+v_1{v}_4^2{v}_5{v}_6+v_4^2{v}_5^2{v}_6-v_4^2{v}_6^2\\ & -2v_4^2{v}_5{v}_7+v_1^3{v}_4{v}_6{v}_7-v_1^2{v}_4{v}_5{v}_6{v}_7+v_1{v}_4{v}_6^2{v}_7-v_4{v}_6{v}_7^2+3v_4^2{v}_8\\ & -v_3^3(v_2{v}_5-v_1{v}_5^2+v_5^3+2v_1{v}_6-3v_5{v}_6+3v_7)+v_2^4{v}_8+2v_1^2{v}_4{v}_5^2{v}_8\\ & +v_1^2{v}_4{v}_6{v}_8-3v_1{v}_4{v}_5{v}_6{v}_8+v_4{v}_6^2{v}_8+2v_4{v}_5{v}_7{v}_8+v_1^4{v}_8^2-3v_4{v}_8^2\\ & -v_1{v}_7{v}_8^2+v_8^3+(-v_1^4{v}_7+v_1^3(v_4+v_5{v}_7+v_8)-v_1^2(3v_4{v}_5+v_6{v}_7\\ & +v_5{v}_8)+v_1(v_4(2v_5^2+v_6)+v_7^2+v_6{v}_8)-2(v_4{v}_5{v}_6-v_4{v}_7+v_7{v}_8))v_9\\ & +(v_1^2-v_1{v}_5+v_6)v_9^2+v_2^3(v_5(-v_1+v_5)v_8-v_6(v_4+2v_8)+v_1{v}_9)\\ & +v_2^2(v_4^2+v_4{v}_6((v_1-v_5)v_5+2v_6)-3v_4{v}_8+v_8(v_6(v_1^2-v_1{v}_5+v_6)\\ & +(3v_1-2v_5)v_7+2v_8)+v_1(-v_1{v}_5+v_5^2-2v_6)v_9)+v_3^2(-v_4{v}_5^2+v_2^2{v}_6\\ & +2v_4{v}_6+v_6^3-3v_5{v}_6{v}_7+3v_7^2-v_2(3v_4+v_1{v}_5{v}_6-v_5^2{v}_6+2v_6^2-3v_1{v}_7\\ & +v_5{v}_7-4v_8)+3v_5^2{v}_8-3v_6{v}_8+v_1^2(v_6^2-2v_5{v}_7+2v_8)-2v_5{v}_9\\ & +v_1(v_4{v}_5+2v_5^2{v}_7+v_6{v}_7-v_5(v_6^2+5v_8)+3v_9))+v_2(v_8(v_7^2-2v_6{v}_8)\\ & +v_1^2(3v_4{v}_8-v_4{v}_6^2+v_5{v}_7{v}_8-4v_8^2-v_5{v}_6{v}_9+3v_7{v}_9)+v_1^3(v_6{v}_9-v_7{v}_8)\\ & -v_4(v_6^3+2v_5^2{v}_8-2v_6(v_5{v}_7+v_8))+v_4^2{v}_5^2+v_1(v_4{v}_5{v}_6^2-v_6{v}_7{v}_8\\ & -2v_4^2{v}_5-3v_4{v}_6{v}_7+v_4{v}_5{v}_8+3v_5{v}_8^2+(v_4+v_6^2-2v_5{v}_7-3v_8)v_9)\\ & -v_9^2+2v_5{v}_8{v}_9)+v_3(v_4{v}_5{v}_6^2-v_4^2{v}_5-v_2^3{v}_7-2v_4{v}_5^2{v}_7-v_4{v}_6{v}_7-v_7^3\\ & +4v_4{v}_5{v}_8+3v_6{v}_7{v}_8-3v_5{v}_8^2+3v_5{v}_6{v}_9-2(v_4+v_6^2-v_5{v}_7-v_8)v_9\\ & +v_2^2(2v_4{v}_5-v_5^2{v}_7+2v_6{v}_7+v_5{v}_8-2v_9)+v_1^3(v_7^2-2v_6{v}_8+v_5{v}_9)\\ & +v_2(2v_5{v}_7^2-v_6^2{v}_7+v_4(2v_5^3-5v_5{v}_6+3v_7)+v_5{v}_6{v}_8-5v_7{v}_8-2v_5^2{v}_9\\ & +4v_6{v}_9)+v_1^2(v_4{v}_5{v}_6-3v_4{v}_7-v_2{v}_6{v}_7-v_5{v}_7^2+3v_2{v}_5{v}_8+2v_5{v}_6{v}_8\\ & +v_7{v}_8-(2v_2+v_5^2+2v_6)v_9)+v_1(2v_4^2+v_6{v}_7^2+v_2^2(v_5{v}_7-4v_8)\\ & -2v_6^2{v}_8-v_5{v}_7{v}_8+4v_8^2-v_4(2v_2(v_5^2-v_6)+v_5^2{v}_6-5v_5{v}_7+6v_8)\\ & -4v_7{v}_9+v_2(v_5{v}_6{v}_7-3v_7^2-3v_5^2{v}_8+4v_6{v}_8+3v_5{v}_9)))\\ & +v_1{v}_4^2{v}_7-2v_1^3{v}_4{v}_5{v}_8-v_1^3{v}_5{v}_8^2+v_1^2{v}_6{v}_8^2]\\ \text{and}{F}_m^{(1)}(z_1,\dots ,z_9)& =F_{m-3}^{(0)}(z_1,\dots ,z_8)+z_9{F}_m^{(0)}(z_1,\dots ,z_9),m=9,10,11.\end{array}\}$$ B 8

In equation (B 8), v_i in $F_9^{(0)}(z_1,\dots ,z_9)$ are defined as

$$v_i=\{\begin{array}{ll}{e}_i(z_1,z_2),& i=1,2,\\ e_{i-2}(z_3,\dots ,z_9),& i=3,4,\dots ,9,\end{array}$$ B 9

v_i in $F_{10}^{(0)}(z_1,\dots ,z_9)$ are defined as

$$v_i=\{\begin{array}{ll}{e}_i(z_1,z_2,z_3),& i=1,2,3,\\ e_{i-3}(z_4,\dots ,z_9),& i=4,5,\dots ,9\end{array}$$ B 10

and v_i in $F_{11}^{(0)}(z_1,\dots ,z_9)$ are defined as

$$v_i=\{\begin{array}{ll}{e}_i(z_1,\dots ,z_4),& i=1,2,3,4,\\ e_{i-4}(z_5,\dots ,z_9),& i=5,6,\dots ,9.\end{array}$$ B 11

It should be mentioned here that v_i in different $F_m^{(\alpha )}(\cdots )$ have different definitions.

Authors' contributions

L.X. carried out the study, derived the formulae, wrote the computer program, prepared the figures and drafted the manuscript; C.Z. conceived and designed the study, supervised the study and revised the draft of the manuscript; J.S. and V.S. participated in the study, participated in the discussion of the results and helped improve the draft of the manuscript. All authors gave final approval for publication.

Competing interests

We have no competing interests.

Funding

This work is supported by the China Scholarship Council (CSC) (project no. 2011626148) and the German Research Fundation (DFG, project nos. ZH 15/14-1 and ZH 15/20-1), which are gratefully acknowledged.