On one new approach to the solving of an elasticity mixed plane problem for the semi-strip

Vaysfeld, Natalya; Zhuravlova, Zinaida

doi:10.1007/s00707-015-1452-x

Cited by 17 publications

(17 citation statements)

References 14 publications

(10 reference statements)

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“…The general solution was constructed with the help of matrix differential calculation [37] in the following form ⃗ 0 ( ) = 1 ( )…”

Section: Reducing the Problem To A One-dimensional Boundary Problemmentioning

confidence: 99%

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

Vaysfeld

Zhuravlova

2020

Z Angew Math Mech

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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...

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“…The general solution was constructed with the help of matrix differential calculation [37] in the following form ⃗ 0 ( ) = 1 ( )…”

Section: Reducing the Problem To A One-dimensional Boundary Problemmentioning

confidence: 99%

“…Green's matrix-function ( , ) is constructed for the problem (13) in the bilinear form. [37] So, the partial solution of the problem (10) has the following form…”

Section: The Construction Of the Discontinuous Solutionmentioning

confidence: 99%

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

Vaysfeld

Zhuravlova

2020

Z Angew Math Mech

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“…It leaded the initial problem to one-dimensional boundary problem in the transformation's domain. The last one was formulated as the vector boundary valued problem and solved exactly with the apparatuses of the matrix differential calculations and Green's matrix function [18]. The problem was reduced to the singular integral equation's solving.…”

Section: Introductionmentioning

confidence: 99%

On the stress investigation at the edges of the fixed elastic semi-strip

Vaysfeld¹,

Kryvyi²,

Zhuravlova³

2016

Frattura Ed Integrità Strutturale

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The stress state of the elastic fixed semi-strip with the regarding of the singularities at its edge is investigated in the article. The initial boundary problem is reduced to a vector boundary problem in the transformation's domain by the use of integral Fourier transformation. The one-dimensional vector boundary problem is solved exactly with the help of matrix differential calculations and Green's matrix apparatus. The problem's solving was focused at the solving of the singular integral equation (SIE) with the two fixed singularities at the ends of the integration's interval. The symbol of SIE was constructed and the generalized method of the SIE solving was applied. The stress' distributions of the semi-strip are investigated.

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“…The solution of the vector boundary problem was obtained in the form [27,28] A Green's matrix function [29]. The expression (10) can be rewritten in scalar form …”

Section: General Solving Scheme For the Semi-strip Stress State Estimmentioning

confidence: 99%

“…ccording to the approach [29], the Fourier's transformation was applied to the system of Lame's equilibrium Eqs.…”

Section: General Solving Scheme For the Semi-strip Stress State Estimmentioning

confidence: 99%

Elastic crack-tip stress field in a semi-strip

Reut

Vaysfeld

Zhuravlova

2018

Frattura Ed Integrità Strutturale

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ABSTRACT. In this article the plain elasticity problem for a semi-strip with a transverse crack is investigated in different cases of boundary conditions at the semi-strip's end. Unlike many works dedicated to this subject, the fixed singularities in the singular integral equation's kernel are considered. The integral transformations' method is applied by a generalized scheme to reduce the initial problem to a one-dimensional problem. The one-dimensional problem is formulated as a vector boundary value problem which is solved with the help of matrix differential calculations and Green's matrix apparatus. The problem is reduced to solve the system of three singular integral equations. Depending on the conditions given on the short edge of the semistrip, the obtained singular integral equation can have one or two fixed singularities. A special method is applied to solve this equation in regard to the singularities existence. Hence, the system of the singular integral equations (SSIE) is solved with the help of the generalized method. The stress intensity factors (SIF) are investigated for different lengths of crack. The novelty of this work is the application of a new approach allowing the consideration of fixed singularities in the problem of a transverse crack in the elastic semi-strip. The comparison of the accuracy of numerical results during the use of different approaches to solve the SSIE is calculated.

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On one new approach to the solving of an elasticity mixed plane problem for the semi-strip

Cited by 17 publications

References 14 publications

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

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

On the stress investigation at the edges of the fixed elastic semi-strip

Elastic crack-tip stress field in a semi-strip

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