Complex Stress Functions Research Articles

Mutual Interference of Two Surface Cracks under Hertzian Contact Loading

Analysis model containing two inclined surface cracks on semi-infinite elastic body is established and analyzed on the basis of linear fracture mechanics to examine mutual interference of two surface cracks. Muskhelishvili's complex stress functions are introduced and a set of singular integral equations is obtained for a dislocation density function. The stress intensity factors at crack tip are obtained by using the Gerasoulis'method. When two surface cracks are parallel and have the same length, the values of and (variation of ) for crack 1 and crack 2 decrease by the mutual interference of two surface cracks as the distance between the two surface cracks shortens. The effect of mutual interference is remarkable in high friction coefficient. In case that two surface cracks are parallel, the values of and for crack 2 decrease as the length ratio ot crack 2 to crack 1 becomes small. As the crack inclination angle rises, the value of and the mutual interference of for crack 2 increase and the value of for crack 1 becomes smaller than that for crack 2.

Application of Mapping Theory to Integral Equation for Stress Analysis of Spur Gears. Remarks about Hirano's Method and Calculation at Internal Point.

The complex variable integral equation has been applied by Hirano to solve plane stress problems. This paper presents a new mathematical approach to the formulation of integro-differential equations and simple formulae for analyzing the stress and displacement distributions at internal points in gears using a conformal mapping technique. In addition to setting up a system of linear equations for calculating the complex stress functions, we show how the conformal mapping function that transforms the gear domain into a half-plane is analytically obtained from the viewpoint of complex variable analysis.

A Hybrid Method to Determine Individual Stresses in Orthotropic Composites Using Only Measured Isochromatic Data

Ability to determine the individual stresses photoelastically in orthotropic composites from isochromatic data, but without employing isoclinics or isopachics, is developed and demonstrated. This new capability is achieved by combining isochromatic information, complex stress functions and numerical nonlinear least-squares concepts into a new and effective hybrid method for stress analyzing composite materials. Unlike previous methods of photo-orthotropic-elasticity, the present technique simultaneously smoothes the recorded isochromatic data, provides reliable boundary information, and separates the isochromatics into individual stress components throughout the area of interest. The method is illustrated by evaluating the three individual stresses throughout a region adjacent to a hole in a uniaxially loaded orthotropic composite plate from the measured isochromatics.

Analysis of the orders of stress singularity at the corner point of a diamond-shape rigid inclusion or hole in an infinite plate under antiplane bending by conformal mapping

The orders of the stress singularity at the corner point of a diamond-shape rigid inclusion or hole in an infinite elastic plate are studied. Combining the complex stress functions in elasticity theory with the conformal mapping, the stress functions are expressed in power series with infinite terms. By virtue of the stress functions and the consideration of singular behavior of stress at the corner point, the orders of the stress singularity at the corner point are determined. It is found there are two kinds of order at the corner point, which correspond to Mode I and Mode II deformations, respectively. The stress intensities corresponding to the orders are defined, and the influence of Poisson's ratio of the plate on the values of the orders is discussed.

Bimaterial plane with elliptic hole under uniform tension normal to the interface

A bimaterial plane subject to uniform tensile traction normal to the interface is analyzed. The plane consists of two dissimilar elastic half-planes each with a semi-elliptic notch. The stress analysis is carried out by using a rational mapping function and complex stress ones. Stress Intensity of Debonding is defined as an index for the stress intensity at the tip of debonding and is investigated. The expression of energy release rate is derived by applying the residue theorem to a contour integral expressed by the complex stress function. Stress distributions around a circular hole and on the interface are shown. Stress concentration factors are studied for various oval shapes, material constants, and debonding lengths. Expressions of stress concentration factor are given.

箱形断面柱-H形断面はり接合部のシャパネルの複素応力関数による弾性解析

This paper presents the analytical studies to estimate the elastic stress distribution and displacement of shear panel on the box type steel column to H beam connection using complex stress function, when the connection is subject to the severe earthquake loading. A large axial force is introduced into the column for highrise building with bending stress. Therefore, the shear panel should be designed subject to not only shear force but also axial force and bending stress. Numerical example is demonstrated regarding the shear panel of actual highrise building. Finally, suitable design formulas are recommended.

The use of the cracked Brazilian disc geometry for rock fracture investigations

The cracked Brazilian disc geometry is receiving attention for rock fracture mechanics research due to its many advantages over other conventional geometrical configurations. A thorough study of the geometry is vitally important at this stage. A superimposition technique is presented to accurately solve the general cracked straight through Brazilian disc (CSTBD) fracture problem using the dislocation method and also the complex stress function method. The solution is valid for any crack inclination angle and any crack length. The solution for the cracked chevron notched Brazilian disc (CCNBD) is then discussed. The theoretical evaluation for the Mode I cracked Brazilian disc fracture problem, a special case of the general problem, has been numerically calibrated by both finite element and boundary element methods, and a simple and accurate method to calculate the Mode I critical (minimum) dimensionless stress intensity factor for the CCNBD specimen is given. The valid geometrical range and valid minimum specimen size for the CCNBD geometry is presented to allow a valid Mode I fracture toughness value to be determined. The theoretical and numerical solutions for the stress intensity factor and the valid geometrical range are also substantiated by extensive experimental validation by comparing the results with those obtained by the ISRM-suggested CB and SR methods.

Analysis of the Second Mixed Boundary Value Problem for a Thin Plate

The second mixed boundary value problem is solved by the classical theory of thin plate bending. The mixed boundary consists of a boundary (M) on which one respective component of external force and deflective angle are given, and on the remaining boundary the external forces are given. The boundary (M) is straight and the remaining boundary is arbitrary configuration. A closed solution is obtained. Complex stress functions and a rational mapping function are used. A half-plane with a crack is analyzed under a concentrated torsional moment. Stress distributions before and after the crack initiation, and stress intensity factors are obtained for from short to long cracks and for some Poisson’s ratio.

Thin plate bending problem of partially bonded bi-material strips

A general solution to the thin plate bending problem of partially bonded bi-material strips is obtained. The two strips are symmetrically bonded along a finite straight interface with debondings at both sides. Complex stress function approach together with the rational mapping function technique are utilized to obtain the general solution. A concentrated bending moment applied at eachtip of the strip is considered. Distributions of bending and torsional moments along the boundaries of the two strips as well as the Stress Intensity of Debonding (SID) at the debonding tips are demonstrated for different states of debonding lengths, material constants and rigidity ratios.

Plane elastic solution for the second mixed boundary value problem and its application

A solution is obtained for the second mixed boundary value problem with two boundaries loaded with external forces, and the other two boundaries simply supported. A rational mapping function formed as a sum of fractional expressions and complex stress functions is used for the analysis. A closed form solution can be obtained under uniform tensile and shearing loadings. These solutions are applicable to crack problems in linear elastic fracture mechanics. In order to illustrate the general solution, a double connected region is considered, in which interaction between a square hole with a crack extension and another crack is investigated. Stress distributions and stress intensity factors are obtained.

THE USE OF A SECONDARY STATE TO DETERMINE THE T‐STRESS

Abstract— This paper examines the determination of the T‐stress by making use of a secondary state and the complex‐variable forms of the work reciprocal and Rice's Jx integrals in terms of the Muskhelishvili complex stress functions. A simple relationship is found which relates the stress intensity factors and T‐stresses of the two states and is applied to the problem of a crack at the root of a notch.

Stress distribution at the rigid circular arc inclusion end

The elastic plane problem of a rigid co-circular arc inclusion under arbitrary loads is dealt with. Applying Schwarz's reflection principle integrated with the analysis of the singularity of complex stress functions, The general solution of the problem is found and several closed-form solutions to some problems of practical importance are given. Finally, the stress distribution at the arc inclusion end is examined and a comparison is made with that of the rigid line inclusion end to show the effect of curvature.

面内偶力と集中力を受ける異材接合部の破壊条件

A bi-material problem is analyzed for bonded dissimilar two half-planes using a rational mapping function and complex stress functions. The half-planes are bonded in the finite length with cracks and subjected to coupling force and external force applied at infinity. The cracks have the same length at an end of the interface and perpendicular to the interface in each material. This analytical model can be used for the investigation of the crack initiation from the interface debonding tip. The stress analysis is carried out and the stress distribution and stress intensity factors are obtained. By focusing on an interface debonding tip, it is possible to decide analytically the fracture phenomena. The conditions of occurrence of debonding extension and crack branching are examined under different coupling and external forces for various material constants.

On the mathematical problems of composite materials with a doubly periodic set of cracks

In this paper; the mathematical problem of the second fundamental problem of composite materials with a doubly periodic set of arbitrary shape cracks are investigated, and the interface are arbitrary smooth closed contours. At first, we establish mathematical models by using Muskhelisvili complex variable methods, change the primitive problems into searching complex stress functions which satisfy four boundary value problems and construct forms of the solution, them, under some general restrictions it is reduced to normal type singular integral equation, the unique solvability in proved mathematically.

Second Mixed‐Boundary‐Value Problem for Thin‐Plate Bending

The second mixed‐boundary‐value problem is solved by the classical theory of thin‐plate bending. The mixed boundary consists of a boundary of simple supported type and that of external forces. The boundary of simple support is straight and the remain is arbitrary configuration and a closed‐form solution is obtained. Complex stress functions and a rational mapping function are used for the analysis. As an example to derive a general solution, a half‐plane with a crack subject to uniform bending moment is considered. The crack initiates from an end of the simple support due to stress concentration. Stress distributions before and after the crack initiation are obtained. The influence of the simple support and a crack on the stress is investigated. Stress intensity factors are obtained from short to long crack and for some Poisson's ratio.

Boundary collocation method for analyzing perforated plate problems

The problem of a two-dimensional plate weakened by an array of holes of arbitrary location is analyzed using the Muskhelishvili complex variable formulation and least square boundary collocation method. Several sets of complex stress functions are proposed in this article in solving the perforated plate problem with and without the interaction effect between holes and crack. In addition, the method has been extended to the problem of cracks emanating from one of the holes in a perforated plate. The results in this article compare favorably with the existing solutions.

A mixed boundary value problem for debonding of a semielliptic rigid inclusion on the rim of a half plane

A general solution is obtained for a mixed boundary value problem of plane elasticity. In the present paper the components of the displacement are given on two segments of the boundary and those of the external force are given on the rest of the boundary. A rational mapping function and a complex variable method are used for the analysis. The solution which is obtained in closed form is the first derivative of the complex stress function and does not contain an integral form. A half plane with a semielliptic rigid inclusion is analyzed under uniform tension. The stress distribution, the stress intensity of debonding at the debonded tip of the semielliptic inclusion and the propagation of debonding are investigated.

Study of the stress intensity factor of preformed V shape fracture tip

This paper gives the complex stress function of preformed V shape fracture under the blasting load. With Westergaard's method, the stress field and displacement field of preformed V shape fracture tip are derived, and hence its stress intensity factor is obtained. The blasting test result shows that the formulas derived are correct and effective.

Bonded Bi-material Half-Planes With Semi-elliptical Notch Under Tension Along the Interface

A problem of two bonded, dissimilar half-planes containing an elliptical hole on the interface is solved. The external load is uniform tension parallel to the interface. A rational mapping function and complex stress functions are used and an analytical solution is obtained. Stress distributions are shown. Stress concentration factors are also obtained for arbitrary lengths of debonding and for several material constants. In addition, an approximate expression of the stress concentration factor is given for elliptical holes and the accuracy is investigated.

A Solution of the Mixed Boundary Value Problem for an Infinite Plate With a Hole Under Uniform Heat Flux

A general solution of the mixed boundary value problem with displacements and external forces given on the boundary is obtained for an infinite plate with a hole subjected to uniform heat flux. Complex stress functions, a rational mapping function, and the dislocation method are used for the analysis. The stress function is obtained in a closed form and the first derivative is given by such a form that does not contain the integral term. The mapping function is represented in the form of a sum of fractional expressions. A problem is solved for a crack initiating from a point of a circular hole on which the displacement is rigidly stiffened. Stress distributions and stress intensity factors are calculated.