2.1 Establishment of constitutive relationships
In the erosion process, the fluid exerts hydraulic shear stress (τ ) on the soil pipe wall at the solid-fluid interface, causing continuous soil loss at the rate of \(\dot{\varepsilon}\) (erosion rate). Previous studies (Arulanandan and Perry1983; Briaud et al. 2001; Wan and Fell 2004a; Bonelli and Brivois 2008) have confirmed the erosion constitutive law as given by
\begin{equation} \dot{\varepsilon}=K\left(\tau-\tau_{c}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [1]\nonumber \\ \end{equation}
where \(K=\) erosion coefficient, and \(\tau_{c}=\) critical shear stress. The erosion coefficient, also known as the coefficient of erodibility, is usually taken as a constant parameter describing the soil properties (Wan and Fell 2004a; Haghighi et al. 2013). Wan and Fell (2004a) proposed laboratory testing method to determine the erosion coefficient in the pre-formed soil pipe. The erosion law is generally employed to explain stress-strain relationship on the soil pipe periphery (Nieber et al. 2019). The erosion coefficient reflects the binding force of the inner soil layer to resist wall shear stress. A higher magnitude of erosion coefficient indicates a weak cohesion in the soil, that is easy to be detached. Critical shear stress is considered as the surficial shear strength influenced by roughness features. When hydraulic shear stress (i.e., external stress) exceeds the critical shear stress applied on the soil-water contact surface, the soil body is sheared with a linear deformable response in the form of particle detachment (Parron Vera et al. 2014). As shown in Fig. 1 (a), the path of the fluid is extended due to the presence of hydraulic shear stress. The viscosity of the fluid and the roughness of soil surface result in the velocity gradient, which generates shear action (Benaissa et al. 2012).
The movement of soil particles (i.e., erosion) is regarded as the corresponding shear strain. The erosion rate (\(\dot{\varepsilon}\), kg/m2/s) is defined as the mass loss (\(m\), kg) of soil per area (A , m2) and time (\(t\), s).
\begin{equation} \dot{\varepsilon}=\frac{m}{A\bullet t}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [2]\nonumber \\ \end{equation}
The fluid erodes the soil layer with a certain rate, defined as eroded depth (\(d_{\varepsilon}\), m/s). Based on the eroded soil volume, the eroded depth can be determined by
\begin{equation} d_{\varepsilon}=\frac{m}{\rho_{s}\bullet A\bullet t}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [3]\nonumber \\ \end{equation}
The eroded depth and erosion rate can both be utilized to estimate the rate of soil loss. Shear strain (\(\gamma\), dimensionless) induced by hydraulic shear stress in soil body, therefore, can be expressed as
\begin{equation} \gamma=\frac{V}{d_{\varepsilon}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [4]\nonumber \\ \end{equation}
where V is the velocity of the fluid.