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.