## 1 Introduction
The cohesion \(c\) at the interface between the elastic and plastic zones in the surrounding rock of a tunnel could be expressed by the unconfined compressive strength \(\sigma_{\text{c}}\) and internal-friction angle \(\varphi\). Therefore, the cohesion \(c\) at the interface between the elastic and plastic zones was appropriately considered in the Kastner formula. By comparing the difference between the Fenner and Kastner formulas, a modified Fenner formula of elastic-plastic stress and the plastic radius in a circular tunnel, which considered the formula at the interface between the elastic and plastic zones, was derived on the basis of M-C strength criterion [13]. As proven, the modified Fenner formula based on the M-C strength criterion was equivalent to the Kastner formula. Fenner, Kastner and modified Fenner formulas are widely used in tunnel engineering and are mainly used to determine the plastic zone of a tunnel and to analyse the radial displacement of a tunnel. The stress and displacement of surrounding rock of the tunnel were solved by Park, and on the basis of M-C criterion and the large strain similarity solution, the effect of small and large strain solutions on displacement was studied [14]. [PERSON] et al. studied the effect of the hydraulic pressure on the direction of the principal stress in plastic zone of surrounding rock of the circular rock tunnel; then, they analysed the relationship between the hydraulic pressure of tunnel wall and the plastic zone of surrounding rock of the tunnel based on Fenner formula [15]. On the basis of the M-C strength criterion, [PERSON] et al. deduced the analytical solutions of stress displacement and the range of the plastic zone in the elastic, elastoplastic and rheological stages [16]. [PERSON] et al. presented a semi-analytical solution for the tunnel excavation problem based on an extended D-P model [17]. Some elastic-plastic studies were conducted on the basis of the H-B strength criterion to better analyse the elastic-plastic mechanical properties of a rock tunnel. For the rock tunnel, [PERSON] et al. considered the structural plane of the surrounding rock, performed an elastic-plastic analysis of the surrounding rock of a circular tunnel and obtained the Fenner formula based on the H-B strength criterion [18]. [PERSON] et al. studied the theoretical solutions of stress, displacement and the plastic radius of a circular tunnel based on the H-B strength criterion and the non-associated flow law of elastic-brittle-plastic rock [19]. [PERSON] et al. believed that the intermediate principal stress was an important factor to determine the plastic zone; thus, the H-B strength criterion was modified by considering the intermediate principal stress, and the theoretical formula of the plastic zone radius of a rock tunnel was proposed [20]. [PERSON] et al. developed a new analytical-numerical procedure based on the H-B strength criterion to provide the stresses and strains around a circular tunnel in rock masses exhibiting different stress-strain behaviours; a simple relationship was also presented to determine the closure of a circular tunnel in terms of the rock mass strength and tunnel depth [21]. [PERSON] compared the applicability of the M-C and H-B strength criteria in a rock tunnel, analysed the elastic-brittle-plastic behaviour of tunnel and evaluated the stress difference of the surrounding rock in a tunnel between the expansive and non-expansive rocks using a numerical method [22]. On the basis of the difference in the analytical solutions of displacement of the surrounding rock between the M-C and H-B strength criteria, [PERSON] et al. used the non-associated flow rule to compare the displacement difference of the surrounding rock in a tunnel between soft and hard rocks with the effect of the angle of dilatancy [23]. [PERSON] et al. deduced a mathematical expression to determine the radius of the plastic zone based on M-C and H-B strength criteria; then, they compared the convergence of circular tunnels in elastoplastic rock masses [24].