2.2 Deformation and friction-slip behavior of the rough contact surface
Sliding of geomaterials is a process of slow accumulation of internal contact and friction, which is consistent with creep characteristics. Therefore, we still describe the deformation of single contacting asperities based on the velocity creep theory proposed by E. Aharonov and C H. Scholz (2018), as follows
,
,
,
where, and are normal and tangential stresses on the contact asperities interface. All parameters included in the equation are shown in Table 1. Equations (2)-(4) illustrate that the deformation of contact asperities is a creep process that is related to temperature, creep activation energy, and creep velocity. Further, the frictional force between individual contacting asperities can be expressed as , where is the real contact area between individual contacting asperities.
The pressure on single contacting asperities is certain, which satisfies , where is the nominal contact area of a single asperity and is the normal stress acting on this nominal contact area. In addition, the sum of the nominal contact areas () of all contacting asperities is equal to the nominal contact area () of the entire contact surface at the time of full contact, i.e., . Then, can be further expressed as
.
The frictional force at the rough contact surface can be considered to be equal to the sum of the shear forces of each asperity (), as follows
.
The friction coefficient of the rough contact surface can be defined as the friction force divided by the positive pressureP i.e. , where positive pressure equals to . Therefore, the can be expressed as
.
Equation (7) includes porosity , which is an inherent structural property of the geomaterials. Their pores are closely related to the seepage coefficient and fluid viscosity, which are important factors affecting the friction-slip behavior. Based on the hydraulic diffusivity () (Wibberley, 2002) and the specific storage capacity m () (Renner and Steeb, 2014), we can obtain the expression for the porosity as follows
,
where, k is the permeability, η is the fluid viscosity,m is the specific storage capacity, cf is the compressibility of the pore fluid, and cpp is the compressibility of the pore space. Substituting equation (8) into equation (7), the friction coefficient can be expressed as
.
Further, based on the relationship between permeability coefficient and saturation( K : hydraulic conductivity; WS : degree of saturation; L , U : fitting parameters ) (Li , 2021), the friction coefficientµs can be expressed as
Equation (10) describes the friction coefficient of the macroscopic rough contact surface, which is based on the creep accumulation of microscopic asperities and includes random contact processes. Previous models considered the normal stress (or shear stress) to be the same across the entire contact surface, which was an average treatment. However, equation (10) only considers that the deformation mode of each micro-contact asperity is the same, but the number of contact asperities is random (in accordance with the exponential relationship), which is closer to the real situation.