Investigation of the stress state of the elastic semi-strip with a transverse crack

Reut, V. V.; Vaysfeld, Natalya; Zhuravlova, Zinaida

doi:10.1016/j.tafmec.2019.01.005

Cited by 4 publications

(3 citation statements)

References 23 publications

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“…and not from the corresponding Equation (10). We then obtain formula (16), while the Lagrange series in functions 𝐹 𝑆 𝑘 (𝑦), whose coefficients are determined using the functions 𝑓 𝑆 𝑘 (𝑦) (16), will uniformly converge (at least at |𝑦| < 1) with the trigonometric series in functions sin 𝑘𝜋𝑦 (𝑘 = 1, 2, …).…”

Section: Biorthogonal Functionsmentioning

confidence: 99%

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A strip with constant stresses on the cut: Exact solutions

et al. 2022

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We construct exact solutions of three boundary value problems in the theory of elasticity for an infinite strip with a central transverse cut on which constant normal stresses are specified (even-symmetric deformation). We consider three variants of homogeneous boundary conditions on the strip sides: (1) free sides, (2) firmly clamped sides, and (3) there are identical stiffeners on the strip sides. The solutions of all problems are represented as series in Papkovich-Fadle eigenfunctions whose coefficients are determined from simple closed formulas.

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Section: Biorthogonal Functionsmentioning

confidence: 99%

“…Note one feature related to the determination of the functions 𝑓 𝑆 𝑘 (𝑦). Using the corresponding formula (10) and the biorthogonality relation following from it, we find the Lagrange expansion of the entire function 𝐹 𝑆 (𝜆, 𝑦) by assuming the parameter 𝜆 to be real:…”

Section: Biorthogonal Functionsmentioning

confidence: 99%

A strip with constant stresses on the cut: Exact solutions

et al. 2022

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“…The stress intensity factors for two cases can be found as,

\begin{matrix} K_{I -} = \sum_{k = 0}^{2 N - 1} s_{k}^{1} \frac{\sqrt{π (b_{1} - b_{0})} (n + 1) {((), - 1)}^{k}}{\sqrt{2}}, K_{I +} = \sum_{k = 0}^{2 N - 1} s_{k}^{1} \frac{\sqrt{π (b_{1} - b_{0})} (n + 1)}{\sqrt{2}} \\ K_{I I -} = \sum_{k = 0}^{2 N - 1} s_{k}^{2} \frac{\sqrt{π (b_{1} - b_{0})} (n + 1) {((), - 1)}^{k}}{\sqrt{2}}, K_{I I +} = \sum_{k = 0}^{2 N - 1} s_{k}^{2} \frac{\sqrt{π (b_{1} - b_{0})} (n + 1)}{\sqrt{2}} \end{matrix}

…”

Section: The Stress Intensity Factorsmentioning

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 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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An exact solution for a partially clamped rectangle with a crack

Коваленко

Menshova

Kerzhaev

2021

J. Phys.: Conf. Ser.

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The article deals with a boundary value problem for a rectangle whose horizontal sides are rigidly clamped, and the ends are free. In the centre of the rectangle, a vertical cut is made on which a discontinuity of the longitudinal displacements is given. An exact solution to the problem is constructed in the form of series in Papkovich–Fadle eigenfunctions. First, the corresponding boundary value problem for an infinite clamped strip is solved, then the solution for a rectangle is superimposed on this solution, with the help of which the boundary conditions at its ends are satisfied. Examples are given in which discontinuities of three types are considered which differ in the smoothness of the discontinuity contour near its ends.

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Investigation of the stress state of the elastic semi-strip with a transverse crack

Cited by 4 publications

References 23 publications

A strip with constant stresses on the cut: Exact solutions

A strip with constant stresses on the cut: Exact solutions

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

An exact solution for a partially clamped rectangle with a crack

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