Theory and Numerical Simulation of Deep Rock Mass Based on a Non-Euclidean Model

Yingji Bao,Binsong Jiang

doi:10.1155/2022/4492406

Abstract

Rock masses existing in nature may experience closed internal stress under the action of geological structures. This closed internal stress makes the deformation of the rock mass incompatible, so classic continuum theory is not suitable for analyzing the stress and deformation of rock masses. In this study, a non-Euclidean model for rock masses was established based on differential geometry. By choosing the non-Euclidean parameter as the internal variable, a thermodynamic model was constructed. Then, numerical computation based on the non-Euclidean model was applied to a circular tunnel in a deep rock mass. The distribution of internal stress and the effects of rock parameters on the distribution of internal stress were analyzed. Through our research, we conclude that the stress field of the deep rock mass consists of classical stress and internal stress and that the internal stress shows a distinct wavy behavior. The radius of the fractured zone decreases with an increasing Young’s modulus E and Poisson’s ratio v , but increases with increasing non-Euclidean parameter ξ.

Highlights

In classical continuum theories of rock mechanics, a rock is a continuous, connected body that undergoes deformation, and the deformation satisfies the equations of compatibility. e metric tensor for measuring the distances between material particles in the reference configuration is Euclidean
Rock masses existing in nature may contain closed internal stress. e internal stress makes the deformation of the rock incompatible. erefore, the classical continuum theory is not suitable for analyzing the incompatible deformation of rock masses [1,2,3,4,5]
E relationship between the continuum theory and mathematical theory of differential geometry has been studied by many researchers

Summary

Introduction

In classical continuum theories of rock mechanics, a rock is a continuous, connected body that undergoes deformation, and the deformation satisfies the equations of compatibility. e metric tensor for measuring the distances between material particles in the reference configuration is Euclidean. Rock masses existing in nature may contain closed internal stress. It is necessary to introduce non-Euclidean space to describe the incompatible deformation, and differential geometry will be used to describe the closed internal stress. E plate is a two-dimensional Riemannian space immersed in three-dimensional Euclidean space R3 According to this scenario, the undeformed rock mass with closed internal stress should be regarded as a manifold. It is assumed that the undeformed rock mass with closed internal stress was a manifold. Π: E⟶N is a smooth surjective map called a bundle projection, and V Rs is an s-dimensional vector space. Due to the existence of closed internal stress, the deformation of rock masses is not compatible. Where R is the tensor contraction of Rij, and R 2R1212

The Thermodynamics Model and Discussion

Analysis Based on Numerical Computations

Findings

Conclusions