In this section, the finite difference method is used to obtain the numerical solution of nodal values of the displacement function satisfying the governing equation. It is obvious that the governing equation in terms of the displacement function is a fourthorder elliptical partial differential equation with variable coefficients. At the same time, the stress expression expressed in terms of the displacement function is a thirdorder partial differential equation, and the displacement expression expressed in terms of the displacement function is a secondorder partial differential equation.
All of these partial differential equations are transformed into their corresponding algebraic equations by using the finite difference method. The numerical calculation process is divided into three steps: Firstly, the values of the displacement function at each point of the domain are solved by the algebraic equations of the governing equations and the boundary conditions. Secondly, the partial derivative values of the displacement functions at each point are obtained by their difference equations. Finally, the displacement components and the stress components at each point are solved by the partial derivative values of the displacement function and the values of the displacement function.
5.1 Difference scheme of governing equation
The governing equation in terms of displacement function is suitable for solving the internal mesh points of the domain. According to Eq. (8), the governing equation is composed of total eight different partial derivatives of the displacement function of order ranging from one to four together with the displacement function itself. All the individual derivatives of the governing equation are replaced by their corresponding central difference expressions having local truncation errors of \(o(h^{2})\) and \(o(k^{2} )\). The mesh length in rdirection (idirection) is recorded as h, and the mesh length in θdirection (jdirection) is recorded as k.
$$\begin{aligned}& \biggl( \frac{\partial ^{4}\psi }{\partial r^{4}} \biggr)_{i,j} = \frac{1}{h ^{4}} [ \psi _{i + 2,j}  4\psi _{i + 1,j} + 6\psi _{i,j}  4 \psi _{i  1,j} + \psi _{i  2,j} ], \end{aligned}$$
(16a)
$$\begin{aligned}& \biggl( \frac{\partial ^{4}\psi }{\partial \theta ^{4}} \biggr)_{i,j} = \frac{1}{k^{4}} [ \psi _{i,j + 2}  4\psi _{i,j + 1} + 6\psi _{i,j}  4\psi _{i,j  1} + \psi _{i,j  2} ] , \end{aligned}$$
(16b)
$$\begin{aligned}& \biggl( \frac{\partial ^{4}\psi }{\partial r^{2}\partial \theta ^{2}} \biggr) _{i,j} = \frac{1}{h^{2}k^{2}}\left [ \textstyle\begin{array}{l} \psi _{i + 1,j + 1}  2\psi _{i + 1,j} + \psi _{i + 1,j  1}  2 \psi _{i,j + 1} + 4\psi _{i,j} \\ {} 2\psi _{i,j  1} + \psi _{i  1,j + 1}  2\psi _{i  1,j} + \psi _{i  1,j  1} \end{array}\displaystyle \right ] , \end{aligned}$$
(16c)
$$\begin{aligned}& \biggl( \frac{\partial ^{3}\psi }{\partial r^{3}} \biggr)_{i,j} = \frac{1}{2h ^{3}} [ \psi _{i + 2,j}  2\psi _{i + 1,j} + 2\psi _{i  1,j}  \psi _{i  2,j} ] , \end{aligned}$$
(16d)
$$\begin{aligned}& \biggl( \frac{\partial ^{3}\psi }{\partial r\partial \theta ^{2}} \biggr) _{i,j} = \frac{1}{2hk^{2}} [ \psi _{i + 1,j + 1}  2\psi _{i + 1,j} + \psi _{i + 1,j  1}  \psi _{i  1,j + 1} + 2\psi _{i  1,j}  \psi _{i  1,j  1} ], \end{aligned}$$
(16e)
$$\begin{aligned}& \biggl( \frac{\partial ^{2}\psi }{\partial r^{2}} \biggr)_{i,j} = \frac{1}{h ^{2}} [ \psi _{i + 1,j}  2\psi _{i,j} + \psi _{i  1,j} ], \end{aligned}$$
(16f)
$$\begin{aligned}& \biggl( \frac{\partial ^{2}\psi }{\partial \theta ^{2}} \biggr)_{i,j} = \frac{1}{k^{2}} [ \psi _{i,j + 1}  2\psi _{i,j} + \psi _{i,j  1} ], \end{aligned}$$
(16g)
$$\begin{aligned}& \biggl( \frac{\partial \psi }{\partial r} \biggr)_{i,j} = \frac{1}{2h} [ \psi _{i + 1,j}  \psi _{i  1,j} ] . \end{aligned}$$
(16h)
Substituting Eqs. (16a)–(16h) into Eq. (8), the governing equation for solving the internal mesh points of the domain is written in terms of nodal unknowns of the displacement function ψas follows:
$$ \begin{aligned}[b] &\xi _{1}\psi ( i + 2,j ) + \xi _{2}\psi ( i + 1,j + 1 ) + \xi _{3}\psi ( i + 1,j ) + \xi _{2}\psi ( i + 1,j  1 ) + \xi _{4}\psi ( i,j + 2 ) \\ &\quad {}+ \xi _{5}\psi ( i,j + 1 ) + \xi _{6}\psi ( i,j ) + \xi _{5}\psi ( i,j  1 ) + \xi _{4}\psi ( i,j  2 ) + \xi _{7}\psi ( i  1,j + 1 ) \\ &\quad {}+ \xi _{8}\psi ( i  1,j ) + \xi _{7}\psi ( i  1,j  1 ) + \xi _{9}\psi ( i  2,j ) = 0 ,\end{aligned} $$
(17)
where
$$\begin{aligned}& \xi _{1} = r_{i}^{3}k^{4} ( r_{i}  h ), \\& \xi _{2} = r_{i}h^{2}k^{2}(2r_{i}  3h), \\& \xi _{3} = r_{i}k^{2} \bigl[  4r_{i}^{3}k^{2}  4r_{i}h^{2} + 2r_{i} ^{2}hk^{2} + 6h^{3} + 5r_{i}h^{2}k^{2}  4.5h^{3}k^{2} \bigr], \\& \xi _{4} = h^{4}, \\& \xi _{5} = 2h^{2} \bigl(  2h^{2}  2r_{i}^{2}k^{2} + 5h^{2}k^{2} \bigr), \\& \xi _{6} = 6r_{i}^{4}k^{4} + 6h^{4} + 8r_{i}^{2}h^{2}k^{2}  10r_{i} ^{2}h^{2}k^{2}  20h^{4}k^{2} + 9h^{4}k^{4}, \\& \xi _{7} = r_{i}h^{2}k^{2} ( 2r_{i} + 3h ), \\& \xi _{8} = r_{i}k^{2} \bigl(  4r_{i}^{3}k^{2}  4r_{i}h^{2}  2r_{i} ^{2}hk^{2}  6h^{3} + 5r_{i}h^{2}k^{2} + 4.5h^{3}k^{2} \bigr), \\& \xi _{9} = r_{i}^{3}k^{4} ( r_{i} + h ). \end{aligned}$$
The finite difference scheme of the governing equation at one node is symmetric about both r and θaxes, and the computational domain at one node involves thirteen neighboring nodes. Obviously, when the node (\(i,j\)) is close to the real boundary, the computational domain does not only involve the real boundary, but also involves a layer of imaginary nodes. The boundary formed by a layer of imaginary nodes is called an imaginary layer which is outside the real boundary.
5.2 Difference scheme of displacement components
It can be seen that the radial displacement component and the hoop displacement component are the secondorder partial derivatives of the displacement function. Unlike the case of governing equations, the central difference method has been avoided for the displacement components because most of time they are found to include nodes exterior to the imaginary layer. Therefore, on the basis of keeping the order of local truncation error also to be \(o(h ^{2})\) or \(o(k ^{2})\), different finite differencing schemes (for example, forward difference, backward difference, and center difference) are adopted for different derivatives present in the displacement components. It should be noted that the expression of the displacement component has two Forms (FormI and FormII), and in the following section, only the difference formula of the displacement components in FormI is given. The difference formula of displacement components in FormII is similar to that in FormI.
For radial displacement, four different versions of finite difference formulas have been developed for points on different regions of the boundary. These versions of finite difference formulas are obtained by adapting different combinations of forward and backward differencing schemes in both r and θ directions. Here, the differential formulas of four radial displacements are given. It is observed that the radial displacement component contains nine nodes in the computational domain, but no nodes beyond the imaginary layer.

(a)
rforward difference, θforward difference:
$$ \begin{aligned}[b] u_{r}(i,j) ={}& a_{1}\psi ( i + 2,j + 2 )  4a_{1}\psi ( i + 2,j + 1 ) + 3a_{1}\psi ( i + 2,j ) \\ &{} 4a_{1}\psi ( i + 1,j + 2 ) + 16a_{1}\psi ( i + 1,j + 1 )  12a_{1}\psi ( i + 1,j ) \\ &{}+ ( 3a_{1}  b_{1} )\psi ( i,j + 2 )  ( 12a_{1}  4b_{1} )\psi ( i,j + 1 )\\ &{} + ( 9a_{1}  3b_{1} )\psi ( i,j ), \end{aligned} $$
(18)
where \(a_{1} =  \frac{1}{8r_{i}hk(1  \mu )}\), \(b_{1} = \frac{5  4 \mu }{4r_{i}^{2}k(1  \mu )}\);

(b)
rforward difference, θbackward difference:
$$ \begin{aligned}[b] u_{r}(i,j) ={}& {} 3a_{1}\psi ( i + 2,j ) + 4a_{1}\psi ( i + 2,j  1 )  a_{1}\psi ( i + 2,j  2 ) \\ &{}+ 12a_{1}\psi ( i + 1,j )  16a_{1}\psi ( i + 1,j  1 ) + 4a_{1}\psi ( i + 1,j  2 ) \\ &{} ( 9a_{1}  3b_{1} )\psi ( i,j ) + ( 12a_{1}  4b_{1} )\psi ( i,j  1 ) \\ &{} ( 3a _{1}  b_{1} )\psi ( i,j  2 ), \end{aligned} $$
(19)
where \(a_{1} =  \frac{1}{8r_{i}hk(1  \mu )}\), \(b_{1} = \frac{5  4 \mu }{4r_{i}^{2}k(1  \mu )}\);

(c)
rbackward difference, θforward difference:
$$ \begin{aligned}[b] u_{r}(i,j) ={}& {} ( 3a_{1} + b_{1} )\psi ( i,j + 2 ) + ( 12a_{1} + 4b_{1} )\psi ( i,j + 1 )  ( 9a_{1} + 3b_{1} )\psi ( i,j ) \\ &{}+ 4a_{1}\psi ( i  1,j + 2 )  16a_{1}\psi ( i  1,j + 1 ) + 12a_{1}\psi ( i  1,j ) \\ &{} a_{1}\psi ( i  2,j + 2 ) + 4a_{1}\psi ( i  2,j + 1 )  3a_{1}\psi ( i  2,j ), \end{aligned} $$
(20)
where \(a_{1} =  \frac{1}{8r_{i}hk(1  \mu )}\), \(b_{1} = \frac{5  4 \mu }{4r_{i}^{2}k(1  \mu )}\);

(d)
rbackward difference, θbackward difference:
$$ \begin{aligned}[b] u_{r}(i,j) ={}& ( 9a_{1} + 3b_{1} )\psi ( i,j )  ( 12a_{1} + 4b_{1} )\psi ( i,j  1 ) + ( 3a_{1} + b_{1} )\psi ( i,j  2 ) \\ &{} 12a_{1}\psi ( i  1,j ) + 16a_{1}\psi ( i  1,j  1 )  4a_{1}\psi ( i  1,j  2 ) \\ &{}+ 3a_{1}\psi ( i  2,j )  4a_{1}\psi ( i  2,j  1 ) + a_{1}\psi ( i  2,j  2 ), \end{aligned} $$
(21)
where \(a_{1} =  \frac{1}{8r_{i}hk(1  \mu )}\), \(b_{1} = \frac{5  4 \mu }{4r_{i}^{2}k(1  \mu )}\).
For the hoop displacement component in the computational domain, only five nodes are involved, and the five node positions are symmetric about both r and θdirections. Therefore, only one difference formula is given for the hoop displacement component in the computational domain. It can be applied for points on any region of the boundary without the inclusion of nodes exterior to the imaginary layer.
$$ \begin{aligned}[b] u_{\theta } (i,j) ={}& ( a_{2} + c_{2} )\psi ( i + 1,j ) + b_{2}\psi ( i,j + 1 ) + (  2a_{2}  2b _{2} + d_{2} )\psi ( i,j ) \\ &{}+ b_{2}\psi ( i,j  1 ) + ( a_{2}  c_{2} ) \psi ( i  1,j ), \end{aligned} $$
(22)
where \(a_{2} = \frac{1}{h^{2}}\), \(b_{2} = \frac{1  2\mu }{2r_{i}^{2}k ^{2}(1  \mu )}\), \(c_{2} =  \frac{3}{2r_{i}h}\), \(d_{2} = \frac{3}{r _{i}^{2}}\).
5.3 Difference scheme of stress components
For the stress components, only the difference formula of the stress components in Form I is given. Here, two different finite difference formulas have been developed using the various combinations of central difference, forward difference, and back difference schemes for the individual derivatives. It should be mentioned that the difference schemes for stress components are divided into four situations: r center difference–θ forward difference, r center difference–θ backward difference, r forward difference–θ center difference, and r backward difference–θ center difference. In order to ensure that the nodes involved in the computational domain do not exceed the imaginary layer, the combination of different difference schemes is also adopted for some partial derivatives.
For example, in the difference scheme of r center difference–θ forward difference, although the forward difference in θdirection is specified, the combination of center difference for a secondorder derivative of the displacement function \(\psi (r, \theta )\) and the forward difference for a firstorder derivative of the displacement function \(\psi (r, \theta )\) is used for the difference scheme for a thirdorder derivative of the displacement function \(\psi (r, \theta )\). This can ensure that the number of difference algebraic equations is equal to the number of nodes in the computational domain.

(1)
Difference equations of radial stress component \(\sigma _{r}\) and circumferential stress component \(\sigma _{\theta }\).

(a)
rcenter difference, θforward difference:
$$\begin{aligned} \sigma _{r} ( i,j ) =& (  A_{1}  C_{1} ) \psi ( i + 1,j + 2 ) + ( 4A_{1} + 4C_{1} ) \psi ( i + 1,j + 1 ) \\ &{} + (  3A_{1}  3C_{1} ) \psi ( i + 1,j )  B_{1}\psi ( i,j + 3 ) + ( 2A_{1} + 6B_{1} ) \psi ( i,j + 2 ) \\ &{}+ (  8A_{1}  12B_{1} + D_{1} ) \psi ( i,j + 1 ) + ( 6A_{1} + 10B_{1} )\psi ( i,j ) \\ &{}+ (  3B_{1}  D_{1} )\psi ( i,j  1 ) + (  A _{1} + C_{1} )\psi ( i  1,j + 2 ) \\ &{}+ ( 4A_{1}  4C_{1} )\psi ( i  1,j + 1 ) + (  3A_{1} + 3C_{1} )\psi ( i  1,j ), \end{aligned}$$
(23)
where \(A_{1} =  \frac{E}{4r_{i}h^{2}k(1 + \mu )}\), \(B_{1} = \frac{\mu E}{4r _{i}^{3}k^{3}(1  \mu ^{2})}\), \(C_{1} = \frac{E(6  5\mu )}{8r_{i} ^{2}hk(1  \mu ^{2})}\), \(D_{1} =  \frac{E(10  9\mu )}{4r_{i}^{3}k(1  \mu ^{2})}\),
$$\begin{aligned} \sigma _{\theta } ( i,j ) =& (  A_{2}  C_{2} ) \psi ( i + 1,j + 2 ) + ( 4A_{2} + 4C_{2} ) \psi ( i + 1,j + 1 ) \\ &{}+ (  3A_{2}  3C_{2} ) \psi ( i + 1,j ) B_{2}\psi ( i,j + 3 ) + ( 2A_{2} + 6B_{2} ) \psi ( i,j + 2 ) \\ &{}+ (  8A_{2}  12B_{2} + D_{2} ) \psi ( i,j + 1 )+ ( 6A_{2} + 10B_{2} )\psi ( i,j ) \\ &{} + (  3B_{2}  D_{2} )\psi ( i,j  1 ) + (  A _{2} + C_{2} )\psi ( i  1,j + 2 ) \\ &{}+ ( 4A_{2}  4C_{2} )\psi ( i  1,j + 1 ) + (  3A_{2} + 3C_{2} )\psi ( i  1,j ), \end{aligned}$$
(24)
where \(A_{2} = \frac{E(2  \mu )}{4r_{i}h^{2}k(1  \mu ^{2})}\), \(B_{2} = \frac{E}{4r _{i}^{3}k^{3}(1 + \mu )}\), \(C_{2} =  \frac{E(7  5\mu )}{8r_{i}^{2}hk(1  \mu ^{2})}\), \(D_{2} = \frac{E(11  9\mu )}{4r_{i}^{3}k(1  \mu ^{2})}\);

(b)
rcenter difference, θbackward difference:
$$\begin{aligned} \sigma _{r} ( i,j ) =& ( 3A_{1} + 3C_{1} ) \psi ( i + 1,j ) + (  4A_{1}  4C_{1} ) \psi ( i + 1,j  1 ) \\ &{}+ ( A_{1} + C_{1} ) \psi ( i + 1,j  2 )+ ( 3B_{1} + D_{1} )\psi ( i,j + 1 ) \\ &{} + (  6A_{1}  10B_{1} )\psi ( i,j )+ ( 8A_{1} + 12B_{1}  D_{1} )\psi ( i,j  1 ) \\ &{} + (  2A_{1}  6B_{1} )\psi ( i,j  2 ) + B _{1}\psi ( i,j  3 ) + ( 3A_{1}  3C_{1} ) \psi ( i  1,j ) \\ &{} + (  4A_{1} + 4C_{1} )\psi ( i  1,j  1 ) + ( A_{1}  C_{1} )\psi ( i  1,j  2 ), \end{aligned}$$
(25)
where \(A_{1} =  \frac{E}{4r_{i}h^{2}k(1 + \mu )}\), \(B_{1} = \frac{\mu E}{4r _{i}^{3}k^{3}(1  \mu ^{2})}\), \(C_{1} = \frac{E(6  5\mu )}{8r_{i} ^{2}hk(1  \mu ^{2})}\), \(D_{1} =  \frac{E(10  9\mu )}{4r_{i}^{3}k(1  \mu ^{2})}\),
$$\begin{aligned} \sigma _{\theta } ( i,j ) =& ( 3A_{2} + 3C_{2} ) \psi ( i + 1,j ) + (  4A_{2}  4C_{2} ) \psi ( i + 1,j  1 ) \\ &{} + ( A_{2} + C_{2} ) \psi ( i + 1,j  2 )+ ( 3B_{2} + D_{2} )\psi ( i,j + 1 ) \\ &{}+ (  6A_{2}  10B_{2} )\psi ( i,j )+ ( 8A_{2} + 12B_{2}  D_{2} )\psi ( i,j  1 ) \\ &{}+ (  2A_{2}  6B_{2} )\psi ( i,j  2 ) + B _{2}\psi ( i,j  3 ) + ( 3A_{2}  3C_{2} ) \psi ( i  1,j ) \\ &{}+ (  4A_{2} + 4C_{2} )\psi ( i  1,j  1 ) + ( A_{2}  C_{2} )\psi ( i  1,j  2 ), \end{aligned}$$
(26)
where \(A_{2} = \frac{E(2  \mu )}{4r_{i}h^{2}k(1  \mu ^{2})}\), \(B_{2} = \frac{E}{4r_{i}^{3}k^{3}(1 + \mu )}\), \(C_{2} =  \frac{E(7  5\mu )}{8r_{i}^{2}hk(1  \mu ^{2})}\), \(D_{2} = \frac{E(11  9\mu )}{4r_{i}^{3}k(1 \mu ^{2})}\);

(2)
Difference equation of shear stress \(\tau _{r \theta }\)

(a)
rforward difference, θcenter difference:
$$\begin{aligned} \tau _{r\theta } (i,j) =&  A_{3}\psi ( i + 3,j )  B_{3} \psi ( i + 2,j + 1 ) + ( 6A_{3} + 2B_{3} ) \psi ( i + 2,j ) \\ &{}  B_{3}\psi ( i + 2,j  1 )+ 4B_{3}\psi ( i + 1,j + 1 ) \\ &{} + (  12A_{3}  8B _{3} + C_{3} + E_{3} )\psi ( i + 1,j )+ 4B_{3} \psi ( i + 1,j  1 ) \\ &{}+ (  3B_{3} + D_{3} )\psi ( i,j + 1 ) + ( 10A_{3} + 6B_{3}  2C_{3}  2D_{3} + F_{3} )\psi ( i,j ) \\ &{}+ (  3B_{3} + D_{3} )\psi ( i,j  1 ) + (  3A_{3} + C_{3}  E_{3} )\psi ( i  1,j ), \end{aligned}$$
(27)
where \(A_{3} = \frac{E}{4h^{3}(1 + \mu )}\), \(B_{3} =  \frac{\mu E}{4r _{i}^{2}hk^{2}(1  \mu ^{2})}\), \(C_{3} =  \frac{2E}{r_{i}h^{2}(1 + \mu )}\), \(D_{3} = \frac{E}{2r_{i}^{3}k^{2}(1  \mu )}\), \(E_{3} = \frac{9E}{4r_{i}^{2}h(1 + \mu )}\), \(F_{3} =  \frac{9E}{2r _{i}^{3}(1 + \mu )}\);

(b)
rbackward difference, θcenter difference:
$$\begin{aligned} \tau _{r\theta } (i,j) =& ( 3A_{3} + C_{3} + E_{3} )\psi ( i + 1,j ) + ( 3B_{3} + D_{3} )\psi ( i,j + 1 ) \\ &{}+ (  10A_{3}  6B_{3}  2C_{3}  2D_{3} + F_{3} )\psi ( i,j ) + ( 3B_{3} + D_{3} )\psi ( i,j  1 ) \\ &{} 4B_{3}\psi ( i  1,j + 1 ) + ( 12A_{3} + 8B_{3} + C_{3}  E_{3} )\psi ( i  1,j ) \\ &{} 4B_{3}\psi ( i  1,j  1 ) + B_{3}\psi ( i  2,j + 1 )  ( 6A_{3} + 2B_{3} )\psi ( i  2,j ) \\ &{}+ B_{3}\psi ( i  2,j  1 ) + A_{3}\psi ( i  3,j ), \end{aligned}$$
(28)
where \(A_{3} = \frac{E}{4h^{3}(1 + \mu )}\), \(B_{3} =  \frac{\mu E}{4r _{i}^{2}hk^{2}(1  \mu ^{2})}\), \(C_{3} =  \frac{2E}{r_{i}h^{2}(1 + \mu )}\), \(D_{3} = \frac{E}{2r_{i}^{3}k^{2}(1  \mu )}\), \(E_{3} = \frac{9E}{4r_{i}^{2}h(1 + \mu )}\), \(F_{3} =  \frac{9E}{2r _{i}^{3}(1 + \mu )}\).