Dynamic Borehole Pressure of Geological Material Finite in Radial Extent

— In recent years, the construction of tunnels and subways has increased significantly. Current researches related to the underground engineering always focus on quasi-static force analysis of semi-infinite media, thus it is of great significance to analysis the borehole pressure during tunnel construction. In this paper, a finite cavity expansion model in finite diameter geological media is proposed considering the influence of finite radial extent of geological media and drilling velocity, and theoretical and computational analysis is carried out.


Introduction
During the construction and operation of underground engineering, it is affected by the stress of the surrounding rock, so the dignified stress state analysis is of great significance. At present, the cavity expansion theory has been widely used in the stress analysis of important issues such as pile foundation, ground anchor, tunnel excavation and deep underground engineering construction [1] . Marshall [2] and others considered the impact of construction on surrounding pile foundation through spherical cavity expansion model. However, the above literature mostly considers the quasi-static stress problem in semi infinite geological media, and cannot consider the influence of the free surface of finite diameter media and the influence of dynamic loading on the stress state of surrounding rock during construction. At present, research on the dynamic forces acting on finite diameter media focuses on the penetration resistance of metal target plates. Jiang [3] , Song [4] and others established a finite cylindrical cavity expansion theory based on the Von Mises yield criterion, and applied the research results to projectile penetration into finite diameter metal targets. On this basis, Wang [5] used the unified strength theory to analyze the mechanism of rigid projectiles penetrating finite diameter metal targets. The model used was compared with the experiment and achieved good results., However, the Von Mises yield criterion is only applicable to isotropic metals, and the literature does not consider the effect of a unified strength coefficient on the strengthening section. Based on the unified strength theory, this article proposes a finite diameter geological medium cylindrical cavity expansion model suitable for borehole pressure analysis during the construction process. In the second section, we propose dynamic cavity expansion models for finite diameter geological materials in the elastic-plastic and plastic stages by assuming that the material in the plastic region satisfies the unified strength theory, and provide control equations. The third section analyzes the relationship between borehole pressure and cavity radius ratio, cavity expansion speed, and material parameter selection based on this model.

Model Description
The model in this paper is shown in Fig. 1, where r c is the cavity radius, r p is the plastic region radius, and r t is the finite target plate radius. Assuming that the material is incompressible, the elastic wave velocity is infinite and Poisson's ratio is 0.5. Therefore, the target plate can be divided into the plastic region (r c <r<r p ) and the elastic region (r p <r<r t ). The drilling process can be approximated as the process of increasing cavity radius from zero, so the borehole pressure can be approximated as the cavity wall radial pressure. The expansion process can be divided into two stages: the elastic-plastic stage (r p <r t ) and the plastic stage (r p ≡r t ).
Cylindrical coordinate system is employed in this paper, which stipulates that the radial coordinate r, radial displacement s and radial velocity v are positive in the outward direction, and the radial stress σ r , radial strain ε r , circumferential stress σ θ and circumferential strain ε θ are taken positive in the compressive. Since this problem is axisymmetric, equations of stress and strain components can be obtain:

Constitutive Model
A linear hardening model is used in the analysis [4] : The Hook's law provides the following relation in elastic region: The yield criterion and equivalent stress is provided by unified strength theory in the plastic region: where σ eq is the equivalent stress and b is a constant in unified strength theory. By combining equations (7) and (8) and considering the stress continuity condition and t the influence of b on the strengthening stage of the material the following relation can be obtained:

Governing Equation
The law of conservation of momentum at Cylindrical coordinate system is: The law of conservation of mass for incompressible materials gives the following velocity field: In addition, the conservation of momentum and mass should be satisfied on the wavefront: The subscript represents the two regions on both sides of the wavefront. Equations (13) and (14) can be simplified by the incompressible material assumptions to: Equations (7)-(16) form the basic equations of the model in this paper.

Plastic-Elastic Response(rp<rt)
1)Solution for the elastic region(rp<r<rt) Substituting equations (9) and (12) into equation (11) yields the equation of motion for the elastic region: Neglecting the acceleration term, integrating the above equation and bringing in the boundary conditions (σ r =0，r =r t ) gives: Substituting r =r p into equation (18), the radial stress of the elastic-plastic interface is found to be: 2)Solution for the plastic region(rp<r<rt) Substituting equations (10) and (12) where C is the constant of integration. Combining (19) 、 (21) and r =r p yields: r r r r

Discussion of Solution
Substituting r p =r t into equation (24) yields: Plastic-elastic response occurs at 0<r c <r c1 , while plastic response occurs at r c >r c1 . In addition, when the strain on the free surface of reaches the ultimate strain value ε f the solutions in this paper are no longer applicable, due to the incompressibility assumption. Combining equations (3) and (8) the maximum cavity radius can be obtained:

Numerical Experiments and Discussions
The material parameters of hornblende(a kind of the plastic elastic rocks) are presented in Tab. 1 [6] which is the basic parameters in numerical computations. And the calculation is limited at the range 0<r c /r t <0.2.  Fig. 2 shows that the borehole pressure increases with speed at smaller r c /r t and vice versa at larger. Fig. 3 shows that the borehole pressure is approximately linearly related to compressive strength. Fig. 4 shows that the modulus of elasticity has almost no effect on the borehole pressure. The borehole pressure is approximately linearly related to the tangential modulus as can be seen in Figure 5. Fig. 6 shows that the value of b is proportional to the borehole pressure which indicates that the unified strength model used in this paper can represent a variety of yield criteria and can be effectively extended to a wide range of geological materials. In summary, tangential modulus ， compressive strength and b are the three parameters that have the greatest effect on borehole pressure.

Conclusions
To address the difficulties encountered in dynamic borehole pressure in underground engineering construction, this paper combines unified strength theory and dynamic columnar cavity expansion theory, based on the assumption of incompressible material, derives the analytical solution of borehole pressure during drilling and gives the formula of radial stress distribution. Finally, the influence of various factors on the borehole pressure is discussed in detail based on the numerical experiments carried out on the plastic elastic rocks. It is found that the tangential modulus and compressive strength have a large influence on the borehole pressure and attention should be paid to the selection of these two material parameters in engineering design; the unified strength theory adopted in this paper can be extended to a variety of geological material models through the selection of different parameters.