Two-Dimensional Mixed Problem of Thermoelasticity for a Semistrip

Вайсфельд, Н. Д.; Zhuravlova, Zinaida

doi:10.1007/s10958-017-3609-8

Cited by 10 publications

(5 citation statements)

References 1 publication

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“…Accordingly to the approach, [36] the Fourier's transformation was applied to the system of Lame's equations (2) and to the boundary conditions (3)-(5) by the scheme…”

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

confidence: 99%

“…Accordingly to the approach, the Fourier's transformation was applied to the system of Lame's equations and to the boundary conditions by the scheme

[\begin{matrix} u_{β} (x), φ_{1 β} \\ v_{β} (x), φ_{2 β} \end{matrix}] = \int_{0}^{\infty} ([], \begin{matrix} u (x, y), φ_{1} (y) \\ v (x, y), φ_{2} (y) \end{matrix}) ([], \begin{matrix} \cos β y \\ \sin β y \end{matrix}) dy

with the inverse formulae

[\begin{matrix} u ((), x, y), φ_{1} ((), y) \\ v ((), x, y), φ_{2} ((), y) \end{matrix}] = \frac{2}{π} \int_{0}^{\infty} ([], \begin{matrix} u_{β} false(x false), φ_{1 β} \\ v_{β} false(x false), φ_{2 β} \end{matrix}) ([], \begin{matrix} \cos β y \\ \sin β y \end{matrix}) d β

After it the initial problem was reduced to a discontinuous vector boundary problem

({, \begin{matrix} L_{2} {\overset{y}{true}}_{β} (x) = \vec{f} (x), \\ {\overset{y}{true}}_{β} (0) = 0, {\overset{y}{true}}_{β} () \end{matrix})

…”

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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“…Accordingly to the approach, [36] the Fourier's transformation was applied to the system of Lame's equations (2) and to the boundary conditions (3)-(5) by the scheme…”

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

confidence: 99%

“…Accordingly to the approach, the Fourier's transformation was applied to the system of Lame's equations and to the boundary conditions by the scheme

[\begin{matrix} u_{β} (x), φ_{1 β} \\ v_{β} (x), φ_{2 β} \end{matrix}] = \int_{0}^{\infty} ([], \begin{matrix} u (x, y), φ_{1} (y) \\ v (x, y), φ_{2} (y) \end{matrix}) ([], \begin{matrix} \cos β y \\ \sin β y \end{matrix}) dy

with the inverse formulae

[\begin{matrix} u ((), x, y), φ_{1} ((), y) \\ v ((), x, y), φ_{2} ((), y) \end{matrix}] = \frac{2}{π} \int_{0}^{\infty} ([], \begin{matrix} u_{β} false(x false), φ_{1 β} \\ v_{β} false(x false), φ_{2 β} \end{matrix}) ([], \begin{matrix} \cos β y \\ \sin β y \end{matrix}) d β

After it the initial problem was reduced to a discontinuous vector boundary problem

({, \begin{matrix} L_{2} {\overset{y}{true}}_{β} (x) = \vec{f} (x), \\ {\overset{y}{true}}_{β} (0) = 0, {\overset{y}{true}}_{β} () \end{matrix})

…”

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

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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“…One needs to solve the boundary value problems (1)-(4), (1)-(2), (5)-(6) and (1)-(2), (7) to estimate the stress state of the semi-strip in three cases. The initial problem is reduced to the vector boundary problem [47] by the use of the approach [48]. This approach, as it was shown earlier in the monograph [49], allows to present the general solution of the inhomogeneous vector equation as the superposition of the general and partitial solutions of the vector equation.…”

Section: The Statement Of a Problemmentioning

confidence: 99%

The plane mixed problem for an elastic semi-strip under different load types at its short edge

Menshykov¹,

Reut

et al. 2018

International Journal of Mechanical Sciences

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The mixed problem for the fixed semi-strip is investigated in this article for the three cases of the applied mechanical load. The solution of the boundary problem is reduced to the solution of the singular integral equation (SIE) with regard to the unknown displacements derivative. Three cases of SIE are investigated: when the mechanical load is applied on the center of the semi-strips edge, when the mechanical load is distributed near the left lateral side and when the mechanical load is distributed on the whole semi-strip's edge. In the first case SIE is solved by the using of the orthogonal polynomials method. In the second and third cases the characteristical equations to SIE are constructed, and the SIE are solved with the help of the generalized method. The stress state of the semi-strip is investigated for the three cases. Keywords semi-strip • singular integral equation • fixed singularity • orthogonal polynomials method • generalized method

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“…The initial problem (1)-( 5) is reduced to the one-dimensional problem with the help of Fourier sin-, costransformation applied by the variable y [7].…”

Section: The General Solving Schemementioning

confidence: 99%

“….. is applied to the problem (7) by the generalized scheme [9]. The problem (7) in the transformation domain can be written as…”

Section: The Construction Of the Partial And Discontinuous Solutionsmentioning

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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Two-Dimensional Mixed Problem of Thermoelasticity for a Semistrip

Cited by 10 publications

References 1 publication

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

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

The plane mixed problem for an elastic semi-strip under different load types at its short edge

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

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