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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.

A problem is solved for a bimaterial plane that consists of partially bonded dissimilar half‐planes subject to uniform traction parallel to the interface at infinity. Fracture conditions at a debonded tip are investigated, i.e., a crack initiation into a material and/or the debonding propagation along the interface. A crack initiation is specified by the following three conditions: (1) Stress distribution before a crack initiation; (2) stress‐intensity factor for opening mode immediately after the crack initiation; and (3) energy release rate for the crack. The debonding propagation is specified by the following two conditions: (1) Stress distribution before the crack initiation; and (2) energy release rate for the debonding. There are chances at the same time that both the crack and the debonding may occur and/or that two cracks may initiate into both two materials. However, one phenomenon of the failure must occur in general at the debonded tip. In these cases, the fracture phenomenon is evaluated and s...

A solution is derived for the elastic problem of two bonded dissimilar planes. The bonded planes are symmetric with respect to the interface, which is straight. A rational mapping function and complex stress functions are used for the analysis. The problem is reduced to a Riemann-Hilbert problem. Elastic constants of the solution is expressed by Dundur’s parameters. The first derivative of the loading term is obtained as a closed form, which does not include an integral term. Therefore, there is no need of a numerical integration to obtain components of stress. As an example, the analysis is carried out for bonded half-planes subjected to concentrated couples at infinities and with cracks at an end of the finite interface. Stress distributions before and after initiation of the crack are investigated. Stress-intensity factors are investigated for some elastic constants. In particularly stress-intensity factors for short cracks are obtained, and it is also investigated that a crack initiates at either material.

A solution is found for a cracked half‐plane loaded by a wedge‐shaped rigid punch with friction. It is assumed that Coulomb's friction law is applied between the punch and the matrix and that the contact region is straight. Since the punch ends are sharp corners, stresses become singular and a crack begins from the punch end. An analytical solution is obtained by means of a rational mapping function and complex stress functions. Stress distributions, stress‐intensity factors, and resultant moments are shown for various angles of the wedge‐shaped punch. The conditions for crack initiation are also investigated for the various combinations of coefficient of friction, loading, and angle. When a flat‐ended punch is subjected to rotation, the stress‐intensity factor of the opening mode vanishes for a certain frictional coefficient μB. If a coefficient of friction is larger than μB and the stress‐intensity factor is larger than the fracture‐toughness value, then a crack begins. When it is smaller than μB, the i...

A bimaterial problem is considered for shapes that are symmetrical with respect to an interface that is finite and straight. A rational mapping function and complex stress functions are used to obtain a general solution for bonded half‐planes with cracks at an end of the interface. A stress analysis is carried out for a state of uniform tension at infinity, which is either symmetrical or antisymmetrical with respect to the interface. Stress distributions and stress‐intensity factors are obtained for various crack length and elastic constants. When one material is in tension and the other is free from traction, stress‐intensity factors are also investigated. The bond effects on the stress‐intensity factors are studied. The elastic constant's effects on the stress‐intensity factor for a short crack and the stress function for special material constants are also investigated.

A general solution to the frictional punch problem is obtained for any shape, but the part of the punch is a straight boundary. As an example, a crack initiating from an end of the punch in an elastic half‐plane is analyzed. A rational mapping function and complex stress functions are used for the analysis. The stress distributions, the stress intensity factors and the resultant moment to keep the punch horizontal are evaluated for some Poisson's ratio and frictional coefficient. For a short crack, the effect of the frictional coefficient to the stress intensity factor is investigated.

A thermo‐elastic problem is analyzed for an infinite plate with a kinked crack, which occurs from an end of a linear initial crack. The plate is subjected to a uniform heat flow in an arbitrary direction. The rational mapping function of a sum of fractional expressions, thermal dislocation and the complex variable method are used in the analysis. A closed solution is obtained for the shape represented by the rational mapping function. Distributions of heat flow, temperature and stress, the stress intensity factor and singularity of heat flow are investigated, both before and after the occurrence of a kinked crack.

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.

The relation between the intensity of a corner and the stress concentration factor is investigated for the bending problem of a thin plate with a fixed edge on the boundary. Its relation is establi...

A propatgation direction of a kinked crack in an infinite plate is investigated for heat flux in an arbitrary direction. Criteria based on a stress state before the occurrence of a kinked crack, i.e., the maximum circumferential stress theory, the maximum principal stress theory, and the minimum strain energy density theory Smin are used. On the other hand, criteria based on the stress state immediately after the occurrence of a kinked crack, i.e., KI max theory, Kll=O theory, and the maximum energy release rate theory gmax are also used. Results of thermoelasticity with regard to a kinked crack are used in these criteria. Criteria based on the stress state after the occurrence of a kinked crack almost give the same direction of propagation: the angle between 75° and 77°. The path of propagation is investigated by criteria of Smin and gmax for heat flux perpendicular to an initial crack.

The stress concentration of the clamped edge is investigated in the longitudinal shear problem for isotropy and orthogonal anisotropy. The value of the stress concentration at the end of a rounded corner can be expressed by an infinite series of terms of the radius of curvature and the order of stress singularity at the corner. The accuracy of the expression that takes the first few terms in the series is investigated, and it is proved that the expression is effective. The relation of the stress concentration between isotropy and orthogonal anisotropy can be obtained.

In the thin plate bending problem, a formula is derived for the stress concentration at the end of a rounded notch in a clamped edge. Two different stress states in the vicinity of the notch are considered, i.e., the stress states symmetric and antisymmetric to the bisector of the notch angle. Three kinds of stress concentration at the end of the notch are investigated. It is shown that the stress concentration can be given by an infinite series expressed in terms of the radius of curvature and Williams order of stress singularity. A rational mapping function and complex stress functions are used for the analysis of the displacement boundary value problem. Examples of the stress distribution are also presented.