## Appendix A  
Some gradient, divergence and curl operators on the potentials are as follows:  
\[\mathbf{\
abla}\mathbf{\Phi}=\frac{\partial\Phi}{\partial t}\mathbf{\dot{r}}+\frac{1}{r} \frac{\partial\Phi}{\partial\theta}\mathbf{\dot{\theta}}+\frac{\partial\Phi}{ \partial z}\mathbf{z}, \tag{1}\]  
Figure 5: Angular dependences of the maximum displacements at the (0.5 m, 45\({}^{\circ}\), 1 m) position: radial (red line), axial (green line), and tangential (blue line) displacements generated by (**a,c**) \(P_{j}\) and (**b,d**) \(P_{z}\) of 1.0 N, located at (**a,b**) (0 m, 0 m, 1 m) and (**c,d**) (0 m, 0.25 m, 1 m) positions. \[\mathbf{\
abla}\times(X\hat{\mathbf{e}}_{z})=\left[\frac{1}{r}\frac{\partial(X\hat{\mathbf{e}}_ {z})}{\partial\theta}-\frac{\partial(X\hat{\mathbf{e}}_{z})}{\partial z}\hat{\mathbf{r} }+\left[\frac{\partial(X\hat{\mathbf{e}}_{z})}{\partial z}-\frac{\partial(X\hat\mathbf{{e}}_{z})}{\partial r}\right]\hat{\mathbf{\theta}}\right.\] \[\left.+\frac{1}{r}\left\{\frac{\partial(r(X\hat{\mathbf{e}}_{z})_{ \theta})}{\partial r}-\frac{\partial(X\hat{\mathbf{e}}_{z})}{\partial\theta}\right\} \hat{\mathbf{z}}=\frac{1}{r}\frac{\partial X}{\partial\theta}\hat{\mathbf{r}}-\frac{ \partial X}{\partial r}\hat{\mathbf{\theta}},\right. \tag{100}\] \[\mathbf{\
abla}\times\mathbf{\
abla}\times(\
abla\hat{\mathbf{e}}_{z})=\mathbf{ \
abla}\times\left(\frac{1}{r}\frac{\partial\mathbf{\
abla}}{\partial\theta}\hat{ \mathbf{r}}-\frac{\partial\mathbf{\
abla}}{\partial\theta}\hat{\mathbf{\theta}}\right)\] \[=\left[-\frac{\partial\left(-\frac{\partial\mathbf{\
abla}}{\partial z }\right)}{\partial z}\right]\hat{\mathbf{r}}+\left[\frac{\partial\left(1\frac{ \partial\mathbf{\
abla}}{\partial z}\right)}{\partial z}\right]\hat{\mathbf{\theta}}+ \frac{1}{r}\left[\frac{\partial\left(r\left(-\frac{\partial\mathbf{\
abla}}{ \partial r}\right)\right)}{\partial r}-\frac{\partial\left(\frac{1}{r}\frac{ \partial\mathbf{\
abla}}{\partial\theta}\right)}{\partial\theta}\right]\hat{\mathbf{z}}\] \[=\left(\frac{\partial^{2}\Psi}{\partial r\partial z}\right)\hat{ \mathbf{r}}+\left(\frac{1}{r}\frac{\partial^{2}\Psi}{\partial\theta\theta\theta }\right)\hat{\mathbf{\theta}}-\frac{1}{r}\left(\frac{\partial\mathbf{\
abla}}{ \partial r}+\frac{r^{2}\Psi}{\partial r^{2}}+\frac{1}{r}\frac{\partial^{2} \Psi}{\partial\theta^{2}}\right)\hat{\mathbf{z}}\]  
Substituting the above equations into Equation (44) gives  
\[\mathbf{u}=\left(\frac{\partial\Phi}{\partial r}+\frac{1}{r}\frac{ \partial X}{\partial\theta}+a\frac{\partial^{2}\Psi}{\partial r\partial z} \right)\hat{\mathbf{r}}+\left(\frac{1}{r}\frac{\partial\Phi}{\partial\theta}+ \frac{\partial X}{\partial r}+a\frac{1}{r}\frac{\partial^{2}\Psi}{\partial \theta\theta\theta}\right)\hat{\mathbf{\theta}}\] \[+\left[\frac{\partial\Phi}{\partial z}-a\left(\frac{\partial^{2} \Psi}{\partial r^{2}}+\frac{1}{r}\frac{\partial\Psi}{\partial r}+\frac{1}{r} \frac{\partial^{2}\Psi}{\partial\theta^{2}}\right)\right]\hat{\mathbf{z}}. \tag{101}\]  
From Equation (100), we obtain the displacement components as  
\[u_{r}=\frac{\partial\Phi}{\partial r}+\frac{1}{r}\frac{\partial X}{\partial \theta}+a\frac{\partial^{2}\Psi}{\partial r\partial z}, \tag{102}\]  
Figure 5:0  
Figure 5:1  
in which \(u_{r}\), \(u_{\theta}\) and \(u_{z}\) denote the radial, tangential, and axial displacements, respectively. The radial component can be obtained by substituting Equations (71), (77), and (79) into Equation (103) in the \((\xi,\theta,\eta)\) coordinates. Figure 5:2 Figure 5:3 Figure 5:4  
where \(\frac{\partial\Xi}{\partial\xi}\cong 0\) and \(\frac{\partial\Sigma}{\partial\xi}\cong 0\). By rearranging the above equation in terms of the \(A_{m}\), \(B_{m}\) and \(C_{m}\) coupling constants, we obtain  
Figure 5:5  
where  
Figure 5:6  
Figure 5:7  
Figure 5:8  
Figure 5:9For TIC, the stress-strain [18] and the strain-displacement [31] relations are given as  
\[\left[ \begin{array}{c}\sigma_{rr}\\ \sigma_{\theta\theta}\\ \sigma_{zz}\\ \sigma_{\theta z}\\ \sigma_{\tau z}\\ \sigma_{\tau\theta}\end{array} \right]=\left[