Elastic crack-tip stress field in a semi-strip

Reut, Victor; Vaysfeld, Natalya; Zhuravlova, Zinaida

doi:10.3221/igf-esis.44.07

Cited by 3 publications

(3 citation statements)

References 22 publications

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“…The generalized method of SIE solving [41] was applied for the solving of SSIE (24). Accordingly to it the function̂( ) is searched in the form̂(…”

Section: Casementioning

confidence: 99%

“…The generalized method of SIE solving was applied for the solving of SSIE . Accordingly to it the function

\hat{χ} false(ξ false)

is searched in the form

\hat{χ} (ξ) = \sum_{k = 0}^{N - 1} ([], s_{k} ρ_{k}^{-} ((), ξ) + s_{k + N} ρ_{k}^{+} ((), ξ)), ξ \in ([], - 1; 1)

where

\begin{matrix} ρ_{2 k}^{\mp} false(ξ false) = {(1 \pm ξ)}^{Re λ_{k}} \cdot \cos false(Im λ_{k} \ln false(1 \pm ξ false) false), \\ ρ_{2 k + 1}^{\mp} false(ξ false) = {(1 \pm ξ)}^{Re λ_{k}} \cdot \sin false(Im λ_{k} \ln false(1 \pm ξ false) false), \end{matrix} k = \bar{0, N - 1}

.…”

Section: The Solving Of the Ssiementioning

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 generalized method of SIE solving [41] was applied for the solving of SSIE (24). Accordingly to it the function̂( ) is searched in the form̂(…”

Section: Casementioning

confidence: 99%

“…The generalized method of SIE solving was applied for the solving of SSIE . Accordingly to it the function

\hat{χ} false(ξ false)

is searched in the form

\hat{χ} (ξ) = \sum_{k = 0}^{N - 1} ([], s_{k} ρ_{k}^{-} ((), ξ) + s_{k + N} ρ_{k}^{+} ((), ξ)), ξ \in ([], - 1; 1)

where

\begin{matrix} ρ_{2 k}^{\mp} false(ξ false) = {(1 \pm ξ)}^{Re λ_{k}} \cdot \cos false(Im λ_{k} \ln false(1 \pm ξ false) false), \\ ρ_{2 k + 1}^{\mp} false(ξ false) = {(1 \pm ξ)}^{Re λ_{k}} \cdot \sin false(Im λ_{k} \ln false(1 \pm ξ false) false), \end{matrix} k = \bar{0, N - 1}

.…”

Section: The Solving Of the Ssiementioning

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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“…The roots of the transcendental equation are found numerically. The generalized method[11] is used for the solving of SSIE(8).Accordingly to it the unknown functions are searched in the form ( here ( ) are Chebyshev polynomials of the second kind. The segment [−1; 1] is divided on 2N segments by the points : 2 −1 0 ,−0.5 ( ) = 0, = 0,2 − 1.…”

mentioning

confidence: 99%

On the Stress State of the Semi-Strip with a Longitudinal Crack

Vaysfeld¹,

Zhuravlova²

2018

J. Model. Optim.

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The stress state of the elastic semi-strip is investigated in the paper. The lateral sides of the semi-strip are fixed and the semi-strip’s short edge is under the mechanical load. The longitudinal crack is located inside the semi-strip. The problem is reduced to the one-dimensional problem with the help of Fourier sin-, cos- transformation, which was applied directly to the Lame’s equilibrium equations and the boundary conditions. The one-dimensional problem is formulated is a vector form. Its solution is constructed with the help of the matrix differential calculation and the Green matrix-function, which was constructed in the bilinear form. The solution of the problem is reduced to the solving of three singular integral equations. The first equation in this system contains two fixed singularities in its kernel. To consider them the corresponding transcendental equation is constructed, and its roots are found. The special generalized method is applied to solve the system of singular integral equations. The stress intensity factors are calculated.

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