Analytical Solution For Stress Distribution Research Articles

Study on stress distribution and extrusion load threshold of compressed filled rock joints

Abstract The distribution of stress and the normal extrusion load threshold in weak interlayer are crucial for direct shear test of filled rock joints, but there is a lack of theoretical research in this area. First, an analytical solution for stress distribution was derived using a semi-inverse method. Then, it is compared by the numerical simulation method. Finally, the influence of the width and thickness of weak interlayer on the extreme values of stress components was analyzed, and the distribution pattern of the normal extrusion load was discussed. The results show that under the same conditions, the analytical solution and the numerical simulation results are in good agreement. The maximum horizontal stress in the weak interlayer decreases with increasing width and increases with increasing thickness, while the change of the minimum is opposite. The normal extrusion load increases first and then decreases along the width direction of the weak interlayer. By comparing the normal extrusion load with the empirical value, the mechanism of extrusion failure in the weak interlayer is revealed.

Prediction of hydraulic fracture initiation pressure in a borehole based on a neural network model considering plastic critical distance

Prediction of the hydraulic fracture initiation pressure is essential for in-situ hydraulic fracturing design and stress measurement. The prediction models based on the theory of critical distance have been proofed to fit the abnormal initiation pressure caused by the bore size. However, the critical distance used by the previous models is not physics-based, which may result in errors for prediction. In addition, the laboratory determination of the critical distance neglected the boundary loading effect, causing an underestimating of the critical distance. This work aims to provide an efficient and accurate model for predicting the in-situ hydraulic fracture initiation pressure with consideration of a physic-based plastic critical distance. We first establish a universal analytical solution for stress distribution and plastic zone around a borehole in a cylindrical sample through ideal elastoplastic theory. Using the analytical solution, the plastic critical distance of shale rock is analyzed based on a hydraulic fracturing experiment, and compared with the one estimated by the previous methods. Then, a set of numerical simulation are carried out to build a data set of hydraulic fracture initiation pressure under different stresses, tensile strength, and ratio of plastic critical distance to borehole radius. Finally, the nonlinear relationship between these parameters is mapped by a deep neural network model. According to this study, the following conclusions are obtained: 1) The plastic critical distance increases as the loading stress increases. 2) The established neural network trained by numerical simulation results is plausible, meanwhile, with high precision and high efficiency. 3) Increase in stresses and tensile strength can strengthen the influence of critical plastic distance. 4) Kirsch’s approximate stress solution can bring an error for determination of the critical distance in lab-scale experiments.

Water inrush and failure characteristics of coal seam floor over a confined aquifer

Failure behaviors of the floor rocks under coal seam mining in the conditions of hard magma rock roof and confined aquifer are studied. Based on the theory of rock stresses and elasticity mechanics, the combined effects of the abutment pressure induced by the hard roof and by the water pressure under the thin aquicludes of the floor rocks were considered, and a mechanical model was constructed along the strike of the working face. An analytical solution of stress distribution was derived in the floor rocks, the distributions of vertical, horizontal and shear stresses were calculated. In combination with the in-situ measurement, the results show that: 1) when the periodic pressure caused by the roof collapse occurs on the working face, and the maximum stress concentration in the floor appears at the elastic–plastic junction in the direction of the strike of the working face. With the increase of the depth of the floor, the horizontal stress coefficient tends to decrease, and the corresponding shear stress coefficient isoline shows a “symmetric spiral” distribution and propagates downward to the floor at a certain angle with the vertical direction. This causes the floor rocks to generate compression and shear or tension and shear failure. 2) when the immediate roof of coal seam is the magma rock, the abutment pressure shows a trend of a slow increase initially and then a rapid increase later. The peak value of abutment pressure appears at the location of 4–6 meters from the coal wall of the working face, and the concentration coefficient of the abutment pressure is between 1.4 and 1.8. 3) according to the measurement and calculation of the failure depths of the floor at different positions under the same coal seam, it is found that the maximum failure depth appears near the coal wall of the working face. The failure depth reduces by 11.6% after the floor goes through “the roof caving and re-compaction”, which causes the fractures in the floor to close and the thickness of the effective aquiclude increases. In the un-mined area of the working face, the failure depth is 55% of the maximum failure depth. 4) both the theoretical calculation and the numerical simulation show that the failure depth of the floor increases obviously under the combined action of high vertical stress and the water pressure. Under the condition that the thickness of the aquiclude is relatively thin, the water pressure of the floor and pressure intensity of the roof are the sensitive factors to affect the maximum failure depth of the floor. As the mining depth increases, the coal seam is getting closer to the Ordovician Limestone. This study provides an improved understand and theoretical support for the safe mining of coal seams under similar conditions.

Analytical solution for stress distribution around deep lined pressure tunnels under the water table

Existing models on the stress distribution around deep buried tunnels pay little attention to the impact of rock-liner interaction and seepage variation. The conventional solution for deep lined tunnels has an inherent mathematical restriction since it is limited to thin liners. In this paper, we solve the poro-mechanical problem of stress and pore pressure distribution using the complex variable method and the conformal mapping technique. The proposed solution relaxes the assumptions of thin liner and negligible stresses that limit the usefulness of existing approaches. To derive the solution, the stress function is expanded into Laurent series in the complex plane. Analytic expressions for both with and without gravity conditions are obtained. The proposed solution is compared with the Finite Element Method; a good agreement has been obtained between the numerical simulations and proposed analytical solution. It is found that with the liner installation, the effective maximum principal stress (MPS) decreases by about 50% for the given case condition. In addition, the pore pressure distribution around the tunnel increases sharply compared to the no liner condition. The parametric study shows that the effective MPS value is highly affected by the liner relative thickness, stiffness and permeability. Its location on the tunnel boundary moves from the horizontal direction to the invert with the increase of the liner thickness and stiffness. Furthermore, the results show that low liner permeability results in high pore pressure around the tunnel, which reduces the effective stress within the rock mass.

Analytical Solution for Stress Distribution around Arbitrary Stopes Using Evolutionary Complex Variable Methods

AbstractIn this study, we proposed an evolutionary complex variable method (ECVM) to find the analytical solution for the stress distribution in an infinite homogeneous, isotropic, and elastic rock...

An analytical solution of stress distribution around underground gas storage cavern in bedded salt rock

The main purpose of this study is to investigate the stress distribution around an underground gas storage cavern in a bedded salt rock formation. First, considering the characteristics of the salt rock formations in China, the mechanical model was simplified into a hollow cylinder which contains a non-salt interlayer. Subsequently, based on the theory of elasticity, the Love displacement function was established, and the general solution of stress components was obtained using the boundary and continuity conditions for this problem. To verify the validity of the analytical solution, a numerical simulation was performed. Comparison between the analytical solution results and the numerical simulation results shows a relatively good agreement. Thus, the analytical solution could be used to describe the characteristics of stress distribution around the gas storage cavern containing the non-salt layer, which is constructed in the bedded salt rock.

An Analytical Solution for Microannulus Cracks Developed Around a Wellbore

In situations like blowout or hydraulic fracturing, excessive fluid pressure may cause leaking in the casing in shallower parts of the formation. The resulting high pressure may form a cylindrical crack around the wellbore. An analytical solution for stress distribution and displacement along a cylindrical crack formed between the casing and the formation is provided in this paper. The crack is assumed to be opened by uniform fluid pressure exerted on both sides of the crack. This solution has a wide range of applications from failure analysis of fibers in manufacturing composite materials to wellbore integrity in petroleum and subsurface engineering problems.

Mechanistic Analysis of Rock Damage Anisotropy and Rotation Around Circular Cavities

We used the differential stress-induced damage (DSID) model to predict anisotropic crack propagation under tensile and shear stress. The damage variable is similar to a crack density tensor. The damage function and the damage potential are expressed as functions of the energy release rate, defined as the thermodynamic force that is work-conjugate to damage. Contrary to the previous damage models, flow rules are obtained by deriving dissipation functions by the energy release rate, and thermodynamic consistency is ensured. The damage criterion is adapted from the Drucker–Prager yield function. Simulations of biaxial stress tests showed that: (1) three-dimensional states of damage can be obtained for three-dimensional states of stress; (2) no damage propagates under isotropic compression; (3) crack planes propagate in the direction parallel to major compression stress; (4) damage propagation hardens the material; (5) stiffness and deformation anisotropy result from the anisotropy of damage. There is no one-to-one relationship between stress and damage. We demonstrated the effect of the loading sequence in a two-step simulation (a shear loading phase and a compression loading phase): the current state of stress and damage can be used to track the effect of stress history on damage rotation. We finally conducted a sensitivity analysis with the finite element method, to explore the stress conditions in which damage is expected to rotate around a circular cavity subject to pressurization or depressurization. Simulation results showed that: (1) before damage initiation, the DSID model matches the analytical solution of stress distribution obtained with the theory of elasticity; (2) the DSID model can predict the extent of the tensile damage zone at the crown, and that of the compressive damage zone at the sidewalls; (3) damage generated during a vertical far-field compression followed by a depressurization of the cavity is more intense than that generated during a depressurization of the cavity followed by a vertical far-field compression.

Bearing strength of carbon/epoxy composite laminates

Fracture behavior of carbon/epoxy laminates under pin loading is studied experimentally and analytically. Effects of ratios of specimen width to hole diameter and edge distance to hole diameter on bearing strength are investigated. Characteristic length of the laminates obtained using the HK model is in good agreement with the experimental data. The larger the hole size induced, the lower the bearing strength measured under pin loading. The bearing strength and failure mode could be predicted using the HK model and Zhang's analytical solution of stress distribution around a pin load hole. Chamis' method for prediction of bearing strength is also considered to predict failure mode and bearing strength. A modification of Chamis' method is made using a factor of rupture. The bearing strength predicted by the modified method is reasonably close to the experimental data.