## 2 Numerical Model  
SPH is a particle-based method, so the continuum is represented by the particles. Using the SPH interpolation, the continuity equation in the Lagrangian approach for the generic particle \(i\) can be represented as [36; 37]. \[\frac{d\rho_{a}}{dt}=-\rho_{a}\sum_{b=1}^{N}\
u_{b}V_{ab}.\
abla W_{ab}+0.1 hc_{0 }\sum_{b=1}^{N}\forall_{b}\
i_{ab}.\
abla W_{ab} \tag{1}\]where \(\frac{d\alpha_{a}}{dt}\) is the material derivative of the density of \(a\)th particle, \(m_{b}\) is the mass of \(b\)th particle, \(\mathbf{V}_{ab}\) is the velocity difference between particles \(a\) and \(b\) and \(\
abla W_{ab}\) represents the gradient of the kernel function. Kernel function is originally shown by \(W(\mathbf{x}_{a}-\mathbf{x}_{b},h)\) where \(\mathbf{x}_{a}-\mathbf{x}_{b}\) is the position difference between two neighboring particles and \(h\) is the smoothing length. In the present study, Wendland C2 kernel with a support domain of \(2h\) is used. \(N\) is the number of particles and \(c_{0}\) is the speed of the sound. The second term at the right-hand side of the continuity equation is added to remove the numerical oscillations in the pressure field as proposed in [38]. \(\mathfrak{D}_{ab}\) is the diffusive term and the equation is given as:  
\[\mathfrak{D}_{ab}\!=\!2(\rho_{b}-\rho_{a})\frac{\mathbf{r}_{ba}}{|\mathbf{r}_{ab}|^{2}} -\left[\left\langle\
abla\rho\right\rangle_{a}{}^{L}+\left\langle\
abla\rho \right\rangle_{b}{}^{L}\right] \tag{2}\]  
where \(\mathbf{r}_{ab}\) is the difference of the position vectors between particles \(a\) and \(b\) and \(\left\langle\
abla\rho\right\rangle^{L}\) is the renormalized density gradient and calculated by  
\[\left\langle\
abla\rho\right\rangle_{a}{}^{L}=\sum_{b}^{N}\mathbf{\
u}_{b}(\rho_ {b}-\rho_{a})\mathbf{L}_{a}\
abla_{a}W_{ab} \tag{3}\]  
\[\mathbf{L}_{a}=[\sum_{b}^{N}\mathbf{\
u}_{b}\mathbf{r}_{ab}\bigotimes\
abla_{a}W_{ab}]^{-1} \tag{4}\]  
SPH approximation of momentum equation can be represented for each direction as:  
\[\frac{dV_{a}}{dt}=\sum_{b=1}^{N}m_{b}\Bigg{(}\Bigg{(}\frac{\sigma^{ij}_{a}}{ \rho_{a}{}^{2}}+\frac{\sigma^{ij}_{b}}{\rho_{b}{}^{2}}\Bigg{)}+\pi_{ab}\delta^ {ij}-\frac{\omega}{A}C_{ab}\Bigg{)}\
abla W_{ab}+g_{a} \tag{5}\]  
where \(\delta^{ij}\) is the Kronecker tensor, \(g\) is the gravitational acceleration and \(\sigma^{ij}\) is the stress tensor and is represented as  
\[\sigma^{ij}=-p\delta^{ij}+S^{ij} \tag{6}\]  
where the first term and the second term of the right side are the isotropic and deviatoric parts of the stress tensor, respectively. Pressure, \(p\), can be calculated from equation of state in which pressure of a particle is found from the variation of density [39]. \[p=\frac{K}{\rho_{0}}(\rho-\rho_{0}) \tag{7}\]  
where \(K\) is the bulk modulus of the material. Deviatoric stress can be calculated as:  
\[\frac{DS^{ij}}{Dt}=E\bigg{(}D^{ij}-\frac{1}{3}\delta^{ij}D^{ij}\bigg{)} \tag{8}\]  
where \(E\) is the Young's modulus and \(D^{ij}\) is the rate of transformation vector and calculated as:  
\[D^{ij}=\frac{1}{2}\Bigg{(}\frac{\partial V^{i}}{\partial x^{j}}+\frac{\partial V ^{j}}{\partial x^{i}}\Bigg{)} \tag{9}\]  
\(D^{ij}\) is defined for a global coordinate system. In local coordinates of each particle, \(D^{ij}\) has only unidirectional strain for a cable model. In the momentum equation, \(\pi_{ab}\) is the artificial viscosity [40] which is used to reduce the unphysical oscillations in the numerical model. \[\pi_{ab}=\frac{\varphi{\mu_{ab}}^{2}-\phi{\mu_{ab}}c_{ab}}{\rho_{ab}} \tag{10}\]  
\[\mathfrak{D}_{ab}\!=\!2(\rho_{b}-\rho_{a})\frac{\mathbf{r}_{ba}}{|\mathbf{r}_{ab}|^{2}} -\left[\left\langle\
abla\rho\right\rangle_{a}{}^{L}+\left\langle\
abla\rho \right\rangle_{b}{}^{L}\right] \tag{2}\]0  
where \(\varphi\) and \(\phi\) are empirical coefficients taken as 0.2 and 1, respectively. A damping term denoted as \(\frac{\omega}{A}C_{ab}\) is added into the momentum equation. In this term, \(A\) is the unstretched cross-sectional area of the cable and \(\omega\) is a constant representing the rate of the damping.