Sum Of Fractional Expressions Research Articles

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.

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.

A Crack Initiating from Rhombic Rigid Inclusion

A crack initiating from a rhombic rigid inclusion in an infinite plane is analyzed as a mixed‐boundary‐value problem of plane elasticity. Complex‐variable method is used with the aid of the rational mapping function of a sum of fractional expressions. A rigorous closed‐form solution is obtained for the shape expressed by the mapping function. The uniform tension and uniform shear stress are considered as loading condition. Stress distributions and stress‐intensity factors are computed for various combinations of crack length and corner angles. The intensity of corner of rigid inclusion is also computed. Through these analyses, we investigate the possibility of cracking and debonding initiation at the corner top when the crack becomes longer.

Mixed boundary value problem of simple support type in plane elasticity

A general solution of a mixed boundary value problem in the plane elasticity is obtained for the boundary condition; one of two displacement components and one of two external force components are given on a part of the boundary. The part of the boundary is a straight line. A mapping function of a sum of fractional expressions and the complex stress functions are used. A closed solution can be obtained. As an example, a semi-infinite plate with a simple support on a part of the boundary is analyzed. The analysis is carried out for the states before and after the crack initiation at an end of the simple support. The loading condition is that the constant tensile load applies on the simple support. The stress distribution, the stress intensity factor and the resultant moment on the simple support are obtained.

A debonding and a crack on a circular rigid inclusion subjected to rotation

A model of debonding and crack occurring from a circular rigid inclusion in an infinite plate is analyzed under the loading condition of the inclusion rotation. A mapping function of a sum of fractional expressions and complex stress functions are used for the analysis of the mixed boundary value problem of the plane elasticity. The following values are obtained: values of stress singularity at the debonded tip where only a debonding exists i.e. without a crack; stress intensity factors just after the occurrence of a crack; and the energy release rates for the debonding propagation and the crack occurrence. Furthermore we discuss which phenomenon occurs, the debonding propagation or the crack occurrence at the debonded tip. We also discuss from which tip the debonding propagation occurs.

Cracks at joint of strip and semiinfinite plate

An elastic plate with cracks at a joint of a strip and a semi-infinite plate is analyzed as a plane elastic problem. Two cracks of equal length originate from the joint. The complex variable method and the rational mapping function are used for the stress analysis. The closed solution is obtained. The rational mapping function is formed as a sum of fractional expressions. By devising the determination of the coefficients of mapping function, the stress analysis for the shape with cracks from short to long can be carried out. Uniaxial tension, bending and uniform tension are considered as loading conditions. Stresses in the neighborhood of the joint before and after the occurrence of cracks are investigated. Stress intensity factors are obtained.

A Crack Originating from an Angular Corner of a Semi-infinite Plate With a Step

A semi-infinite plate with a step and a crack originating from its angular corner is analyzed as a plane elastic problem. Uniform tension is considered as the boundary condition. Stress distributions are analytically calculated for the state before and after the occurrence of a crack. Stress intensity factors which are important in linear fracture mechanics are analytically calculated and are investigated for some angles of the angular corner and various crack lengths. The influence of the step on the stress intensity factors is also investigated. A rational mapping function which maps the semiinfinite region with a step and a crack into a unit circle is formed as a sum of fractional expressions. The rational mapping function and Muskhelishvili's method are used for the analysis. A closed solution which is exact for the shape represented by the rational mapping function is obtained.

Uniform tension of a semi-infinite plate with a crack at an end of a stiffened edge

The semi-infinite plate which is rigidly stiffened at a part on the boundary and with a crack originating from an end of the stiffened edge is analyzed as a mixed boundary value problem in a plane elastic problem. A complex variable method and a rational mapping function are used for the analysis. A closed solution is obtained. The rational mapping function is formed as a sum of fractional expressions. Distributions of stress and displacement in the neighbourhood of the crack and the stiffened edge are investigated for the state before and after occuring of a crack. Stress intensity factors which are important in linear fracture mechanics are obtained for various crack lengths.

Bending of Strip with Semielliptic Notches or Cracks

Strips with semi-elliptical notches or edge cracks on both sides are analyzed as a thin plate bending problem. The rational mapping functions of a sum of fractional expressions are used for the analysis. The closed solutions are obtained. Stress distributions, stress intensity factors and stress concentration factors under transverse bending and torsion are investigated. The expressions for stress intensity factor and stress concentration factor are obtained with accuracy. It is also shown that stress intensity factor can be obtained from the expression of stress concentration factor.