## 5 Solution of the Surrounding Soil  
Performing a Laplace transform to Equation (2), it gives  
\[(\lambda_{j}^{\mathrm{s}}+2G_{j}^{\mathrm{s}})\frac{\partial^{2}}{\partial z ^{2}}U_{j}^{\mathrm{s}}(r,z,s)+C_{j}^{\mathrm{s}}(1/\frac{\partial}{\partial z }+\frac{\partial^{2}}{\partial z^{2}})U_{j}^{\mathrm{s}}(r,z,s)+c_{j}^{ \mathrm{s}}(\frac{\partial^{2}}{\partial z^{2}}+1/\frac{\partial}{\partial z }+\frac{\partial^{2}}{\partial z^{2}})U_{j}^{\mathrm{s}}(r,z,s)=\rho_{j}^{ \mathrm{s}}s^{2}U_{j}^{\mathrm{s}}(r,z,s) \tag{10}\]  
where \(U_{j}^{\mathrm{s}}(r,z,s)\) is the Laplace transform of \(u_{j}^{\mathrm{s}}(r,z,t)\). Setting \(U_{j}^{\mathrm{s}}(r,z,s)=R_{j}^{\mathrm{s}}(r)Z_{j}^{\mathrm{s}}(z)\) and substituting it into Equation (10) with rearrangement yields  
\[(\lambda_{j}^{\mathrm{s}}+2G_{j}^{\mathrm{s}}+c_{j}^{\mathrm{s}}s)\frac{1}{Z _{j}^{\mathrm{s}}}\frac{\partial^{2}Z_{j}^{\mathrm{s}}}{\partial z^{2}}-\rho_ {j}^{\mathrm{s}}s^{2}+(G_{j}^{\mathrm{s}}+c_{j}^{\mathrm{s}}s)\frac{1}{R_{j}^ {\mathrm{s}}}\Bigg{[}\frac{1}{r}\frac{\partial R_{j}^{\mathrm{s}}}{\partial r }+\frac{\partial^{2}R_{j}^{\mathrm{s}}}{\partial r^{2}}\Bigg{]}=0. \tag{11}\]  
Further splitting Equation (11), the following ordinary differential equations can be given:  
\[\frac{d^{2}Z_{j}^{\mathrm{s}}}{dz^{2}}+(h_{j}^{\mathrm{s}})^{2}Z_{j}^{\mathrm{ s}}=0 \tag{12a}\] \[\frac{d^{2}R_{j}^{\mathrm{s}}}{dr^{2}}+\frac{1}{r}\frac{dR_{j}^{\mathrm{s}}}{ dr}-(q_{j}^{\mathrm{s}})^{2}R_{j}^{\mathrm{s}}=0 \tag{12b}\]  
where \(h_{j}^{\mathrm{s}}\) and \(q_{j}^{\mathrm{s}}\) are undetermined coefficients that satisfy the following expression:  
\[-(\lambda_{j}^{\mathrm{s}}+2G_{j}^{\mathrm{s}}+c_{j}^{\mathrm{s}}s)(h_{j}^{ \mathrm{s}})^{2}+(G_{j}^{\mathrm{s}}+c_{j}^{\mathrm{s}}s)(q_{j}^{\mathrm{s}})^ {2}=\rho_{j}^{\mathrm{s}}s^{2}. \tag{13}\]Rearranging the terms of Equation (13) gives  
\[(q_{j}^{\rm s})^{2}=\frac{(\lambda_{j}^{\rm s}+2G_{j}^{\rm s}+c_{j}^{\rm s}s)(l_{ j}^{\rm s})^{2}+\rho_{j}^{\rm s}s^{2}}{(G_{j}^{\rm s}+c_{j}^{\rm s}s)}. \tag{14}\]  
Thus, the general solutions of Equations (12a) and (12b) are  
\[Z_{j}^{\rm s}(z)=C_{j}^{\rm s}\cos(l_{j}^{\rm s}z)+D_{j}^{\rm s}\sin(l_{j}^{ \rm s}z) \tag{15a}\] \[R_{j}^{\rm s}(r)=A_{j}^{\rm s}l_{0}(q_{j}^{\rm s}r)+B_{j}^{\rm s}K_{0}(q_{j}^{ \rm s}r) \tag{15b}\]  
where \(l_{0}(q_{j}^{\rm s}r)\) and \(K_{0}(q_{j}^{\rm s}r)\) are the first and second kind, respectively, modified Bessel functions of order zero; \(A_{j}^{\rm s}\), \(B_{j}^{\rm s}\), \(C_{j}^{\rm s}\) and \(D_{j}^{\rm s}\) are undetermined coefficients. Substituting \(U_{j}^{\rm s}(r,z,s)=R_{j}^{\rm s}(r)Z_{j}^{\rm s}(z)\) into Equations (5a) and (5b), respectively, gives  
\[D_{j}^{\rm s}=0 \tag{16a}\] \[\tan(l_{j}^{\rm s}H)=\frac{\overline{K}_{j}^{\rm s}}{l_{j}^{\rm s}H} \tag{16b}\]  
where \(\overline{K}_{j}^{\rm s}=K_{j}^{\rm s}H/E_{j}^{\rm s}\); \(K_{j}^{\rm s}=k_{j}^{\rm s}+s\delta_{j}^{\rm s}\) denotes the complex stiffness of viscoelastic supports beneath the pile toe. Solving the transcendental Equation (16b) yields the eigenvalues \(h_{jn}^{\rm s}(n{=}1,2,\dots\) ). Then, \(q_{jn}^{\rm s}\) can be further obtained with the substitution of \(h_{jn}^{\rm s}\) into Equation (14). Combining Equations (5a), (5b) and (6), the general solution \(U_{j}^{\rm s}\) can be obtained as  
\[U_{j}^{\rm s}=\left\{ \begin{array}{l}\sum\limits_{n=1}^{\infty}A_{jn}^{\rm s }K_{0}(q_{jn}^{\rm s}r)\cos(h_{jn}^{\rm s}z)\ \ \ \ (j=m+1)\\ \sum\limits_{n=1}^{\infty}\left[B_{jn}^{\rm s}I_{0}(q_{jn}^{\rm s}r)+C_{jn}^{ \rm s}K_{0}(q_{jn}^{\rm s}r)\right]\cos(l_{jn}^{\rm s}z)\ \ (j=m,\dots,2,1)\end{array} \right. \tag{17}\]  
where \(A_{jn}^{\rm s}\), \(B_{jn}^{\rm s}\) and \(C_{jn}^{\rm s}\) are undetermined coefficients. Hence, the shear stress at the inner interface of the \(j\)th sub-layer can be further expressed as  
\[\tau_{j}^{\rm s}=\left\{