The effect of a nonlinear Mohr–Coulomb criterion on borehole stresses and damage-zone estimate

1994 ◽  
Vol 31 (1) ◽  
pp. 104-109 ◽  
Author(s):  
Yarlong Wang
Keyword(s):  
Oil Sands ◽  
Damage Zone ◽  
Closed Form Solution ◽  
Numerical Approach ◽  
Solution Procedure ◽  
Form Solution ◽  
Far Field ◽  
Circular Opening ◽  
Oil Sand ◽  
Coulomb Criterion

To investigate the possible error introduced by the assumption of a linear Mohr–Coulomb criterion on the stress distribution near a circular opening, a numerical approach is used to calculate both the damage-zone radius and the stresses near a circular opening. A general solution procedure for the stress calculation under a uniform far-field loading is presented and a closed-form solution for the near-opening stresses is given for a cohesionless medium. Based on the numerical results calculated, it is concluded that the linear assumption is acceptable for a strong rock with an unconfined compressive strength that is of similar magnitude to the far-field stress. However, the tangential stress calculated with this linear assumption may be overestimated, but the damage zone may be underestimated in weaker rock or soil such as an oil sand formation. Key words : Mohr–Coulomb criterion, circular opening, nonlinearity, oil sands, Gauss–Legendre method.

An Investigation on Forced Convection in Clearance of Corotating Disks With Diverging Laminar Flow

2020 ◽  
Vol 143 (3) ◽  
Author(s):  
Achhaibar Singh ◽  
D. K. Singh
Keyword(s):  
Reynolds Number ◽  
Temperature Field ◽  
Closed Form ◽  
Forced Convection ◽  
Energy Equation ◽  
Closed Form Solution ◽  
Solution Procedure ◽  
Form Solution ◽  
Speed Ratio ◽  
Angular Velocities

Abstract In this work, forced convection in the laminar outflow between two disks is predicted. The disks can be corotating or contrarotating with equal or with different angular velocities. A published closed form solution is used to model the forced convection. The simplified energy equation is discretized using a finite difference method. The discretized equation is solved using the tridiagnal matrix algorithm (TDMA). Results are obtained for a heat flux as well as a specified temperature boundary condition and are presented for various flow parameters such as throughflow Reynolds number, rotational Reynolds number, gap ratio and speed ratio. The parameters except the speed ratio affect the temperature field strongly. The computed Nusselt number is in good agreement with the published experimental data. The coupling of closed form solution and energy equation predicts temperature field accurately. Additionally, the solution procedure is simplified considerably and the solution is obtained with lesser computational resources.

A Symbolic-Numerical Approach for the Synthesis of Planar Rigid-Body Guidance

1994 ◽  
Author(s):  
Yong Chen ◽  
Jinyuan Fu
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Fourth Order ◽  
Closed Form Solution ◽  
Numerical Approach ◽  
Form Solution ◽  
System Of Nonlinear Equations ◽  
Single Variable ◽  
All Solutions ◽  
Order Algebraic

Abstract Based on the pole method and the computer symbolic manipulating technique, a fourth order algebraic equation with single variable has been derived for the closed form solution of planar rigid-body guidance. The newly developed method eliminates the requirement of sloving system of nonlinear equations. All solutions of the problem can be obtained directly and easily without the need of initial values. A numerical example is given demonstrating the efficiency and advantages of the method.

A Quantitative Evaluation of Two Classical Approximations Used to Predict the Extent of Vertical Hydraulic Fractures

1983 ◽  
Vol 105 (4) ◽  
pp. 512-527 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Hydraulic Fracture ◽  
Closed Form Solution ◽  
Solution Procedure ◽  
Form Solution ◽  
Critical Parameters ◽  
Hydraulic Fractures ◽  
Fracture Width ◽  
Special Case

An integral equation was developed to predict the critical parameters (fracture width and length) associated with the propagation of a vertical hydraulic fracture and a numerical solution procedure was developed. The effects of the classical approximations of pressure and fracture width were investigated both separately and together. It was found that the effects associated with the pressure approximation were relatively insignificant, whereas those associated with the fracture width approximation were significant, particularly when the formation was only moderately permeable. Finally, an exact closed-form solution of the integral equation was developed for a special case. It was shown that when the formation is only moderately permeable, this solution provides a better approximation of the exact solution than the classical solution of Carter [2].

Efficient Numerical Modeling of Hertzian Line Contact for Material With Inhomogeneities

2012 ◽  
Author(s):  
Xiaoqing Jin ◽  
Zhanjiang Wang ◽  
Qinghua Zhou ◽  
Leon M. Keer ◽  
Qian Wang
Keyword(s):  
Half Plane ◽  
Closed Form Solution ◽  
Numerical Approach ◽  
General Purpose ◽  
Form Solution ◽  
Inclusion Problem ◽  
Computational Domain ◽  
Line Contact ◽  
Equivalent Inclusion Method ◽  
Parametric Studies

The present work proposes an efficient and general-purpose numerical approach for handling two-dimensional inhomogeneities in an elastic half plane. The inhomogeneities can be of any shape, at any location, with arbitrary material properties (which can also be non-homogeneous). To perform the numerical analysis, we first derive an explicit closed-form solution for a rectangular inclusion with uniform eigenstrain components, where the inclusion is aligned with the surface of the half plane. In view of the equivalent inclusion method, an inhomogeneity problem can be converted to a corresponding inclusion problem. In order to determine the distribution of the equivalent eigenstrain, the computational domain is meshed into rectangular elements whose resultant contributions can be efficiently computed using an efficient algorithm based on fast Fourier transform (FFT). In principle, there is no specific limitation on the type of the external load, although our major concern is the contact analysis. Parametric studies are performed and typical results highlighting the deviation of the current solution from the classical Hertzian line contact theory are presented.

scholarly journals Linearized dynamics of spherical bubble clouds

1989 ◽  
Vol 199 ◽  
pp. 155-176 ◽  
Author(s):  
Luca D'Agostino ◽  
Christopher E. Brennen
Keyword(s):  
Energy Dissipation ◽  
Bubble Dynamics ◽  
Bubble Radius ◽  
Closed Form Solution ◽  
Form Solution ◽  
Dynamical Analysis ◽  
Far Field ◽  
Fourier Synthesis ◽  
Spherical Bubble ◽  
Pressure Excitation

The present work investigates the dynamics of the one-dimensional, unsteady flow of a spherical bubble cloud subject to harmonic far-field pressure excitation. Bubble dynamics effects and energy dissipation due to viscosity, heat transfer, liquid compressibility and relative motion of the two phases are included. The equations of motion for the average flow and the bubble radius are linearized and a closed-form solution is obtained. The results are then generalized by means of Fourier synthesis to the case of arbitrary far-field pressure excitation. The flow displays various regimes (sub-resonant, trans-resonant and super-resonant) with different properties depending on the value of the relevant flow parameters. Examples are discussed in order to show the effects of the inclusion of the various energy dissipation mechanisms. Finally the results for the case of Gaussian-shaped far-field pressure change are presented and the most important limitations of the theory are briefly discussed. The simple linearized dynamical analysis developed so far clearly demonstrates the importance of the complex phenomena connected to the interaction of the dynamics of the bubbles with the flow and provides an introduction to the more realistic study of the same flows with nonlinear bubble dynamics.

Sign in / Sign up

Export Citation Format

Share Document