SH-Waves in a Medium Containing a Disordered Periodic Array of Cracks

1995 ◽  
Vol 62 (2) ◽  
pp. 312-319 ◽  
Author(s):  
Y. Mikata
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Numerical Results ◽  
Crack Spacing ◽  
Periodic Array ◽  
Wave Number ◽  
Sh Wave ◽  
Reflection And Transmission ◽  
Dispersion And Attenuation

Reflection and transmission of an SH-wave by a disordered periodic array of coplanar cracks is investigated, and subsequently its application to the dispersion and attenuation of an SH-wave in a disorderedly cracked medium is also treated. This is a stochastic boundary value problem. The formulation largely follows Mikata and Achenbach (1988b). The problem is formulated for an averaged scattered field, and the governing singular integral equation is derived for a conditionally averaged crack-opening displacement using a quasi-crystalline-like approximation. Unlike our previous study (Mikata and Achenbach, 1988b) where a point scatterer approximation was used for the regular part of the integral kernel, however, no further approximation is introduced. The singular integral equation is solved by an eigenfunction expansion involving Chebyschev polynomials. Numerical results are presented for the averaged reflection and transmission coefficients of zeroth order as a function of the wave number for normal incidence, a completely disordered crack spacing, and various values of the ratio of crack length and average crack spacing. Numerical results are also presented for the dispersion and attenuation of an SH-wave in a disorderedly cracked medium.

scholarly journals Productivity Formulae of an Infinite-Conductivity Hydraulically Fractured Well Producing at Constant Wellbore Pressure Based on Numerical Solutions of a Weakly Singular Integral Equation of the First Kind

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Chaolang Hu ◽  
Jing Lu ◽  
Xiaoming He
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Constant Pressure ◽  
Numerical Results ◽  
Production Performance ◽  
Weakly Singular ◽  
Wellbore Pressure ◽  
Infinite Conductivity ◽  
Fractured Well

In order to increase productivity, it is important to study the performance of a hydraulically fractured well producing at constant wellbore pressure. This paper constructs a new productivity formula, which is obtained by solving a weakly singular integral equation of the first kind, for an infinite-conductivity hydraulically fractured well producing at constant pressure. And the two key components of this paper are a weakly singular integral equation of the first kind and a steady-state productivity formula. A new midrectangle algorithm and a Galerkin method are presented in order to solve the weakly singular integral equation. The numerical results of these two methods are in accordance with each other. And then the solutions of the weakly singular integral equation are utilized for the productivity formula of hydraulic fractured wells producing at constant pressure, which provide fast analytical tools to evaluate production performance of infinite-conductivity fractured wells. The paper also shows equipotential threads, which are generated from the numerical results, with different fluid potential values. These threads can be approximately taken as a family of ellipses whose focuses are the two endpoints of the fracture, which is in accordance with the regular assumption in Kuchuk and Brigham, 1979.

On the oscillations of harbours of arbitrary shape

1970 ◽  
Vol 42 (3) ◽  
pp. 447-464 ◽  
Author(s):  
Li-San Hwang ◽  
Ernest O. Tuck
Keyword(s):  
Integral Equation ◽  
Velocity Field ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Arbitrary Shape ◽  
Amplification Factor ◽  
Numerical Results ◽  
Constant Depth ◽  
Wave Amplification

A theory is developed for calculating oscillations of harbours of constant depth and arbitrary shape. This theory is based on the solution of a singular integral equation. Numerical results have been calculated for rectangular harbours so as to check the accuracy of the method. Examples for wave amplification factor and velocity field for both rectangular and actual complex-shaped harbours are given.

scholarly journals Unbounded solution of characteristic singular integral equation using differential transform method

2014 ◽  
Vol 9 (7) ◽  
pp. 2869-2881 ◽  
Author(s):  
Mohammad Abdulkawi Mahiub
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Numerical Results ◽  
Differential Transform Method ◽  
Cauchy Type ◽  
Unbounded Solution ◽  
Transform Method ◽  
Present Solution ◽  
Differential Transform

In this paper, The differential transform method is extended to solve the Cauchy type singular integral equation of the first kind. Unbounded solution of the Cauchy type singular  Integral equation is discussed. Numerical results are shown to illustrate the efficiency and accuracy of the present solution.

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

A semianalytical and finite-element solution to the unbonded contact between a frictionless layer and an FGM-coated half-plane

2017 ◽  
Vol 24 (2) ◽  
pp. 448-464 ◽  
Author(s):  
Jie Yan ◽  
Changwen Mi ◽  
Zhixin Liu
Keyword(s):  
Integral Equation ◽  
Finite Element ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Singular Integral ◽  
Material Properties ◽  
Elastic Layer ◽  
Coated Substrate ◽  
Receding Contact ◽  
Contact Size

In this work, we examine the receding contact between a homogeneous elastic layer and a half-plane substrate reinforced by a functionally graded coating. The material properties of the coating are allowed to vary exponentially along its thickness. A distributed traction load applied over a finite segment of the layer surface presses the layer and the coated substrate against each other. It is further assumed that the receding contact between the layer and the coated substrate is frictionless. In the absence of body forces, Fourier integral transforms are used to convert the governing equations and boundary conditions of the plane receding contact problem into a singular integral equation with the contact pressure and contact size as unknowns. Gauss–Chebyshev quadrature is subsequently employed to discretize both the singular integral equation and the force equilibrium condition at the contact interface. An iterative algorithm based on the method of steepest descent has been proposed to numerically solve the system of algebraic equations, which is linear for the contact pressure but nonlinear for the contact size. Extensive case studies are performed with respect to the coating inhomogeneity parameter, geometric parameters, material properties, and the extent of the indentation load. As a result of the indentation, the elastic layer remains in contact with the coated substrate over only a finite interval. Exterior to this region, the layer and the coated substrate lose contact. Nonetheless, the receding contact size is always larger than that of the indentation traction. To validate the theoretical solution, we have also developed a finite-element model to solve the same receding contact problem. Numerical results of finite-element modeling and theoretical development are compared in detail for a number of parametric studies and are found to agree very well with each other.

Sign in / Sign up

Export Citation Format

Share Document