The investigation of semi‐strip's stress state with a longitudinal crack

Abstract

The article is dedicated to the investigation of the stress state of a semi-strip weakened by a longitudinal crack. Two statements of the problem are considered. The integral Fourier transform is applied directly to the initial problem. The discontinuous boundary problem which is formulated in vector form is solved with the help of the matrix differential calculation and the Green's matrix-function's discontinuous properties. The solving of the problem is reduced to the solving of the system of three singular integral equations (SSIE). This system is solved by two methods regarding to the conditions at the semi-strip's short edge. The orthogonal polynomials method is used when the semi-strip is loaded at the center of its short edge, and the special generalized scheme, which allows consideration of fixed singularities, this is applied when the whole semi-strip's short edge is loaded. The stress intensity factors (SIF) are investigated for the two cases. K E Y W O R D Sfixed singularity, Green's function, longitudinal crack, semi-strip, singular integral equations INTRODUCTIONThe elasticity problem for a semi-strip with a longitudinal crack is a model problem. Its solving is important both for the development of theoretical methods and for the practical demand of engineering problems.The problems of elastic strips and semi-strips weakened by cracks have been solved by many methods: with the help of harmonic functions, simple and double layer potentials, integral transformations, asymptotic methods and others.The mixed elasticity problems were solved in [1]. Elastodynamic analysis of multiple crack problem in 3-D bi-materials was made in [2]. A new type of cracks adding to Griffith-Irwin cracks was investigated in [3]. In the case of the new type of cracks, fracture occurs due to an increase in the stress concentrations up to singular concentrations. A finite elastic wedge-shaped thick plate was considered in [4].The problem about determining the thermal stress in a thermoelastic strip with a collinear array of cracks which are parallel to the strip's sides was considered in [5]. The solving of the Duhamel-Neumann equation was constructed with the help of harmonic functions. The solving of the problem for the infinite strip with a semi-infinite crack was reduced to solving of singular integral equation by the use of simple layer and double layer potentials in [6]. The method for the elastic strip which is weakened by cracks and holes was proposed in [7]. The solution of this problem was reduced to singular integral equation with the help of complex potentials.The plane problem about the interfacial crack between the elastic strip and the elastic semi-plane from another material was solved in [8]. The regular asymptotic method was applied for the solving of the system of integral equations for the displacements' jumps. This method is efficient when the crack is sufficiently narrow. Two configurations of modeling of an interfacial crack with parallel free boundaries were investigated in [9]: a crack in the semi-plane...

Exact solution of the plane problems for poroelastic rectangle and semi‐strip

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Linear interface crack under harmonic shear: Effects of crack's faces closure and friction

“…The detailed study of thickness ratio, porosity, stress amplitude ratio and inhomogeneity parameters influence on stress intensity factors was conducted. The semi-strip with a longitudinal crack was considered in [35], using two different approaches and comparing the stress intensity factors computed for both models. Crack on the interface of dissimilar orthotropic strips was investigated in [36].…”