Anti-Plane Deformation Uniformly Piecewise Homogeneous Space with a Periodic System of Semi-Infinite Internal Cracks

2020 ◽  
pp. 13-20
Author(s):  
V.N. Hakobyan ◽  
A.A. Grigoryan
Keyword(s):  
Integral Equation ◽  
Numerical Calculation ◽  
Homogeneous Space ◽  
Plane Deformation ◽  
Contact Stresses ◽  
Antiplane Deformation ◽  
Intensity Factors ◽  
Shear Moduli ◽  
Mechanical Quadrature ◽  
Equal Thickness

In this paper, we have constructed a solution for the problem of antiplane deformation of a uniformly piecewise homogeneous space of two alternately repeating heterogeneous layers of equal thickness from different materials, which are relaxed on their median planes by two semi-infinite, periodic parallel tunneling cracks. A system of defining equations of the problem is derived in the form of a system of two singular equations of the first kind, with respect to contact stresses acting in the contact zones on the median planes of heterogeneous layers, the solution of which, in the general case, is constructed by the method of mechanical quadrature. In the particular case when the cracks in the heterogeneous layers are the same, the solution of the problem is reduced to the solution of two independent equations and their closed solutions are constructed. The defining singular integral equation of the problem is also obtained in the case when there are no cracks in one of the heterogeneous layers. In the general case, a numerical calculation was carried out and patterns of changes in contact stresses and intensity factors of destructive stresses at the end points of cracks were determined depending on the physical and mechanical and geometric parameters of the problem, which are the ratios of the shear moduli of the layers and the ratio of the layer thickness and crack lengths.

Analysis of Crack Propagation in Roller Bearings Using the Boundary Integral Equation Method—A Mixed-Mode Loading Problem

1988 ◽  
Vol 110 (3) ◽  
pp. 408-413 ◽  
Author(s):  
L. J. Ghosn
Keyword(s):  
Integral Equation ◽  
Stress Intensity ◽  
Crack Propagation ◽  
Boundary Integral Equation ◽  
Stress Intensity Factors ◽  
High Speed ◽  
Roller Bearing ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Intensity Factors

Crack propagation in a rotating inner raceway of a high-speed roller bearing is analyzed using the boundary integral method. The model consists of an edge plate under plane strain condition upon which varying Hertzian stress fields are superimposed. A multidomain boundary integral equation using quadratic elements was written to determine the stress intensity factors KI and KII at the crack tip for various roller positions. The multidomain formulation allows the two faces of the crack to be modeled in two different subregions making it possible to analyze crack closure when the roller is positioned on or close to the crack line. KI and KII stress intensity factors along any direction were computed. These calculations permit determination of crack growth direction along which the average KI times the alternating KI is maximum.

Circumferential Edge Crack in a Cylindrical Cavity

1977 ◽  
Vol 44 (2) ◽  
pp. 250-254 ◽  
Author(s):  
L. M. Keer ◽  
V. K. Luk ◽  
J. M. Freedman
Keyword(s):  
Integral Equation ◽  
Stress Intensity ◽  
Cylindrical Cavity ◽  
Edge Crack ◽  
Numerical Scheme ◽  
Crack Opening ◽  
Integral Transforms ◽  
Intensity Factors ◽  
Physical Quantities ◽  
Elastostatic Problem

The elastostatic problem of a circumferential edge crack in a cylindrical cavity is investigated. The problem is formulated by means of integral transforms and reduced to a singular integral equation. The numerical scheme of Erdogan, Gupta, and Cook is used to obtain the relevant physical quantities and the stress-intensity factors, and crack opening displacements are computed for several values of crack length.

Interface Stress Analysis of the Bending Cylinder Containing Inclusion

2011 ◽  
Vol 201-203 ◽  
pp. 951-955
Author(s):  
Xin Yan Tang
Keyword(s):  
Integral Equation ◽  
Stress Intensity ◽  
Singular Integral Equation ◽  
Stress Analysis ◽  
Singular Integral ◽  
Stress Intensity Factors ◽  
Integral Equation Method ◽  
Equation Method ◽  
Intensity Factors ◽  
Engineering Structures

Using the elasticity and the singular integral equation method, an analysis of a bending cylinder containing inclusions is carried out. The disturbing interface stresses on the inclusion sides and the stress intensity factors at the inclusion tips are obtained. The results given in this paper are useful for the strength design of the engineering structures or mechanical components containing inclusions.

Behavior of a Horizontal Fluid Filled Crack Under Moving Hertzian Load

2005 ◽  
Author(s):  
X. Jin ◽  
L. M. Keer ◽  
E. L. Chez
Keyword(s):  
Half Plane ◽  
Continuous Distribution ◽  
Contact Stresses ◽  
Hertzian Contact ◽  
Subsurface Crack ◽  
Intensity Factors ◽  
Crack Volume ◽  
Rolling Body ◽  
Soft Inclusion ◽  
Edge Dislocations

Numerical analysis is presented for a fluid filled subsurface crack in an elastic half plane loaded by Hertzian contact stresses. The opening volume of the horizontal Griffith crack is fully occupied by an incompressible fluid. In the presence of friction, a moving Hertzian line contact load is applied at the surface of the half plane. The stress intensity factors at the tips of the fluid filled crack are analyzed on condition that the change of the opening crack volume vanishes due to the fluid incompressibility. The method used is that of replacing the crack by a continuous distribution of edge dislocations. As a cycle of rolling can be viewed as shifting the Hertzian contact stresses across the surface of the half plane, the results of this analysis may prove useful in the prediction of rolling fatigue of an elastic rolling body containing a soft inclusion.

scholarly journals Subsurface Cracks Under Conditions of Slip, Stick, and Separation Caused by a Moving Compressive Load

1987 ◽  
Vol 54 (2) ◽  
pp. 393-398 ◽  
Author(s):  
S. D. Sheppard ◽  
J. R. Barber ◽  
M. Comninou
Keyword(s):  
Integral Equation ◽  
Stress Intensity ◽  
Coulomb Friction ◽  
Compressive Load ◽  
Mode I ◽  
Subsurface Crack ◽  
Intensity Factors ◽  
Subsurface Cracks ◽  
Load Location ◽  
Crack Faces

The Mode I and II stress intensity factors (KI, KII) at the two tips of a subsurface crack subjected to a moving compressive load are studied. Coulomb friction along the crack faces results in a number of history dependent slip-stick configurations and nonsymmetric variation in KI and KII. The formulation used to study this variation involves a singular integral equation in two variables which must be solved numerically, and because of the history dependence, requires an incremental solution. Crack lengths and coefficients of friction that result in as many as three zones for any load location are considered in this paper, while a previous paper (Sheppard et al., in press) was limited to configurations involving two zones only.

A New Boundary Integral Equation Method for Cracked Piezoelectric Bodies

2006 ◽  
Vol 306-308 ◽  
pp. 465-470 ◽  
Author(s):  
Kuang-Chong Wu
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Present Method ◽  
Boundary Integral Equations ◽  
Integral Formula ◽  
Electric Displacement ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Equation Method ◽  
Intensity Factors

A novel integral equation method is developed in this paper for the analysis of two-dimensional general piezoelectric cracked bodies. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh’s formalism for anisotropic elasticity in conjunction with Cauchy’s integral formula. The proposed boundary integral equations contain generalized boundary displacement (displacements and electric potential) gradients and generalized tractions (tractions and electric displacement) on the non-crack boundary, and the generalized dislocations on the crack lines. The boundary integral equations can be solved using Gaussian-type integration formulas without dividing the boundary into discrete elements. The crack-tip singularity is explicitly incorporated and the generalized intensity factors can be computed directly. Numerical examples of generalized stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.

scholarly journals ON THE KOLMOGOROV-WIENER FILTER FOR RANDOM PROCESSES WITH A POWER-LAW STRUCTURE FUNCTION BASED ON THE WALSH FUNCTIONS

2021 ◽  
pp. 39-47
Author(s):  
V. N. Gorev ◽  
A. Yu. Gusev ◽  
V. I. Korniienko ◽  
A. A. Safarov
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Numerical Calculation ◽  
Weight Function ◽  
Structure Function ◽  
Random Process ◽  
Power Law ◽  
Wiener Filter ◽  
Walsh Function ◽  
Better Than

Context. We investigate the Kolmogorov-Wiener filter weight function for the prediction of a continuous stationary random process with a power-law structure function. Objective. The aim of the work is to develop an algorithm of obtaining an approximate solution for the weight function without recourse to numerical calculation of integrals. Method. The weight function under consideration obeys the Wiener-Hopf integral equation. A search for an exact analytical solution for the corresponding integral equation meets difficulties, so an approximate solution for the weight function is sought in the framework of the Galerkin method on the basis of a truncated Walsh function series expansion. Results. An algorithm of the weight function obtaining is developed. All the integrals are calculated analytically rather than numerically. Moreover, it is shown that the accuracy of the Walsh function approximations is significantly better than the accuracy of polynomial approximations obtained in the authors’ previous papers. The Walsh function solutions are applicable in wider range of parameters than the polynomial ones. Conclusions. An algorithm of obtaining the Kolmogorov-Wiener filter weight function for the prediction of a stationary continuous random process with a power-law structure function is developed. A truncated Walsh function expansion is the basis of the developed algorithm. In opposite to the polynomial solutions investigated in the previous papers, the developed algorithm has the following advantages. First of all, all the integrals are calculated analytically, and any numerical calculation of the integrals is not needed. Secondly, the problem of the product of very small and very large numbers is absent in the framework of the developed algorithm. In our opinion, this is the reason why the accuracy of the Walsh function solutions is better than that of the polynomial solutions for many approximations and why the Walsh function solutions are applicable in a wider range of parameters than the polynomial ones. The results of the paper may be applied, for example, to practical traffic prediction in telecommunication systems with data packet transfer.

scholarly journals A Mechanical Quadrature Method for Solving Delay Volterra Integral Equation with Weakly Singular Kernels

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Li Zhang ◽  
Jin Huang ◽  
Yubin Pan ◽  
Xiaoxia Wen
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Richardson Extrapolation ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Proportional Delays ◽  
Weakly Singular ◽  
Uniqueness Of The Solution ◽  
Singular Kernels ◽  
Mechanical Quadrature Method

In this work, a mechanical quadrature method based on modified trapezoid formula is used for solving weakly singular Volterra integral equation with proportional delays. An improved Gronwall inequality is testified and adopted to prove the existence and uniqueness of the solution of the original equation. Then, we study the convergence and the error estimation of the mechanical quadrature method. Moreover, Richardson extrapolation based on the asymptotic expansion of error not only possesses a high accuracy but also has the posterior error estimate which can be used to design self-adaptive algorithm. Numerical experiments demonstrate the efficiency and applicability of the proposed method.

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