# Strain-Softening Analyses of a Circular Bore under the Influence of Axial Stress
## 1 Introduction  
Many analytical solutions are presented while the excavation of circular and spherical cavities is considered in pre-stressed fields. Different yield criteria (including Mohr-Coulomb (M-C) and Hoek-Brown (H-B) criteria) are employed, and the corresponding EP, EBP and strain-softening constitutive relations are taken into account.  
For elasto plastic and elastic-brittle-plastic constitutive relations, [PERSON] et al. [1], [PERSON] [2], [PERSON] and [PERSON] [3], [PERSON] [4], [PERSON] [5], [PERSON] and [PERSON] [6], [PERSON] [7], [PERSON] and [PERSON] [8] and [PERSON] [9] presented analytical and semi-analytical solutions for predicting the stresses and displacements around the cavities. [PERSON] and [PERSON] [8] commented on this kind of problem and derived closed-form solutions on the basis of M-C and H-B yield criteria.The displacements obtained by different assumptions were compared. Recently, [PERSON] [10] presented an analytical model dealing with an expanded cylindrical cavity in an elasto plastic medium with a variable plastic potential of flow. [PERSON] et al. [11] analysed the earth pressure around the cylindrical shafts in soft ground by means of experimental and numerical method and discussed the shaft-soil interaction.  
With consideration of the strain-softening behavior of geomaterial, [PERSON] et al. [12], [PERSON] et al. [13], [PERSON] et al. [14], [PERSON] and [PERSON] [15] and [PERSON] et al. [16] presented different forms of solutions for the circular bore in M-C and H-B rock and ground reaction or response curve (GRC). Their results were in good agreement with each other. [PERSON] and [PERSON] [17] studied the shear bands ahead of the advancing cylindrical and spherical penetrometers. [PERSON] et al. [18] predicted the evolution of the plastic region around a wellbore in Drucker-Prager rocks.  
Normally, the problem is treated as an axisymmetric plane strain to derive analytical or semi-analytical solutions. The geostress is presumed to be hydrostatic and the geometrical isotropic. The out-of-plane stress is assumed to be the second principle stress with \(\sigma_{z}=\
u(\sigma_{x}+\sigma_{y})\).  
[PERSON] [19] studied the abovementioned axisymmetric plane strain problem in which the initial stress was of equal magnitude in three axes, and he found that the axial stress did not remain the intermediate one The effect of the out-of-plane stress on the cavity wall displacement was significant for geomaterial with a large drop in strength at yield.  
[PERSON] et al. [20] analyzed the circular bore with different magnitudes of the out-of-plane stress. The numerical analyses show that the results are much influenced by the out-of-plane stress.  
[PERSON] et al. [21] presented the theoretical solution for predicting stresses and displacements around a circular bore in M-C rock mass, subjected to a hydrostatic stress field in the cross section and out-of-plane stress along the axis of the bore. The EBP relationship with a non-associated flow rule was used in the analysis. The comparison of the results illustrated that the effect of the out-of-plane stress on the displacements and plastic zones cannot be ignored.  
As an extension of the brittle-plastic solution of the circular bore [21], the strain-softening analysis is performed and the corresponding solutions were obtained. Numerical schemes were presented and verified, and the results obtained by different constitutive relationships were compared.