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1
Content available Fourier, Laguerre, Laplace Transforms with applications
Aghili Arman
Journal of Mathematics and Applications
|
2021
|
Vol. 44
5--17
EN
In this article, the author considered certain time fractional equations using joint integral transforms. Transform method is a powerful tool for solving singular integral equations, integral equation with retarded argument, evaluation of certain integrals and solution of partial fractional differential equations. The obtained results reveal that the transform method is very convenient and effective. Illustrative examples are also provided.
2
Content available Uniform stress field inside a non-parabolic open inhomogeneity interacting with a mode III crack
Wang X., Schiavone P.
Archives of Mechanics
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2021
|
Vol. 73, nr 1
67--81
EN
Using conformal mapping techniques, analytic continuation and the theory of Cauchy singular integral equations, we prove that a non-parabolic open inhomogeneity embedded in an elastic matrix subjected to a uniform remote anti-plane stress nevertheless admits an internal uniform stress field despite the presence of a finite mode III crack in its vicinity. Our analysis indicates that: (i) the internal uniform stress field is independent of the specific shape of the inhomogeneity and the presence of the finite crack; (ii) the existence of the finite crack plays a key role in the non-parabolic open shape of the inhomogeneity and in the non-uniform stresses in the surrounding matrix; (iii) the two-term asymptotic expansion at infinity of the stress field in the matrix is independent of the presence of the finite crack. Detailed numerical results are presented to demonstrate the proposed theory.
3
Content available Study of the anti-plane problem of a Dugdale-Barenblatt crack in a welded strip using the integral equation method
Chaouche Amine Brick, Ferdjani Hicheme, Tala-Ighil Nacer
Journal of Theoretical and Applied Mechanics
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2019
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Vol. 57 nr 2
475--487
EN
The elasto-static anti-plane problem of a Dugdale-Barenblatt crack in a welded infinite strip is formulated in terms of a singular integral equation (SIE). The weld joint is modeled as a tri-material structure: the weld metal (WM), the base metal (BM) and the heat-affected zone (HAZ). The HAZ is modeled with an exponentially variable shear modulus. The SIE is solved using Tchebychev polynomials. The influence of the elastic mismatching (the ratio between the shear modulus of WM and BM) and width of the HAZ on the fracture load and on the crack propagation is investigated.
4
Content available remote Solution of the plane contact problem between a finite-thickness laterally graded solid and a rigid stamp of an arbitrary tip-profile
Arslan O.
Archives of Mechanics
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2019
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Vol. 71, nr 6
531--565
EN
A singular integral equation (SIE) approach and a finite element method are developed for the solution of the frictional sliding contact problem between a finite-thickness laterally graded solid and a rigid stamp of an arbitrary tip-shape considering the plane strain assumption. An exponential shear modulus variation is introduced through the lateral direction. The field variables are obtained applying the Fourier transformation techniques on the governing partial differential equations. A surface displacement gradient is then utilized to derive a SIE of the second kind. A numerical solution of the SIE is performed by using a collation method and the Gauss quadrature integration techniques for the flat, triangular and circular stamp profiles. Finite element analyses (FEA) of the same contact problems are also performed upon selection of the augmented Lagrange contact-solver in ANSYS. For the incomplete (triangular and circular) stamp problems, an iterative algorithm is developed in order to obtain practically computational solutions for any desired contact lengths. Successful convergence of the SIE results and excellent consistency between the SIE and FEA results are attained, that indicate the reliability of both methods. The change in the thickness is shown to alter the contact behavior of the laterally graded solid significantly.
5
Content available Mathematical modelling of stationary thermoelastic state for a plate with periodic system of inclusions and cracks
Zelenyak Volodymyr
Acta Mechanica et Automatica
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2019
|
Vol. 13, no. 1
11--15
EN
Two-dimensional stationary problem of heat conduction and thermoelasticity for infinite elastic body containing periodic system of inclusions and cracks is considered. Solution of the problem is constructed using the method of singular integral equations (SIEs). The numerical solution of the system integral equations are obtained by the method of mechanical quadrature for a plate heated by a heat flow, containing periodic system elliptic inclusions and thermally insulated cracks. There are obtained graphic dependences of stress intensity factors (SIFs), which characterise the distribution of intensity of stresses at the tops of a crack, depending on the length of crack, elastic and thermoelastic characteristics inclusion, relative position of crack and inclusion.
6
Content available remote Plane receding contact problem for a functionally graded layer supported by two quarter-planes
Çömez İ., Kahya V., Erdöl R.
Archives of Mechanics
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2018
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Vol. 70, nr 6
485--504
EN
In this study, the plane receding contact problem for a functionally graded (FG) layer resting on two quarter-planes is considered by using the theory of linear elasticity. The layer is indented by a rigid cylindrical punch that applies a concentrated force in the normal direction. While the Poisson’s ratio is kept constant, the shear modulus is assumed to vary exponentially through-the-thickness of the layer. It is assumed that the contact at the layer-punch interface and the layer-substrate interface is frictionless, and only the normal tractions can be transmitted along the contact regions. Applying the Fourier integral transform, the plane elasticity equations are converted to a system of two singular integral equations, in which the contact stresses and the contact widths are unknowns. The singular integral equations are solved numerically by Gauss–Jacobi integration formula. Effects of the material inhomogeneity, the distance between quarter-planes and the punch radius on the contact stresses, the contact widths, and the stress intensity factors at the sharp edges are shown. Although the theoretical analysis is formulated with respect to elastic quarter planes, the numerical studies are carried out only for rigid ones.
7
Content available On 3D anticrack problem of thermoelectroelasticity
Kaczyński A.
Acta Mechanica et Automatica
|
2018
|
Vol. 12, no. 2
109--114
EN
A solution is presented for the static problem of thermoelectroelasticity involving a transversely isotropic space with a heat-insulated rigid sheet-like inclusion (anticrack) located in the isotropy plane. It is assumed that far from this defect the body is in a uniform heat flow perpendicular to the inclusion plane. Besides, considered is the case where the electric potential on the anticrack faces is equal to zero. Accurate results are obtained by constructing suitable potential solutions and reducing the thermoelectromechanical problem to its thermomechanical counterpart. The governing boundary integral equation for a planar anticrack of arbitrary shape is obtained in terms of a normal stress discontinuity. As an illustration, a closed-form solution is given and discussed for a circular rigid inclusion.
8
Content available Antiplane deformation of a bimaterial with thin interfacial nonlinear elastic inclusion
Sulym H., Piskozub Y., Polanski J.
Acta Mechanica et Automatica
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2018
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Vol. 12, no. 3
190--195
EN
The problem of longitudinal shear of bimaterial with thin nonlinear elastic inclusion at the interface of matrix materials is considered. Solution of the problem is constructed using the boundary value problem of combining analytical functions and jump functions method. The model of the thin inclusion with nonlinear resilient parameters is built. Solution of the problem is reduced to a system of singular integral equations with variable coefficients. The convergent iterative method for solving such a system is offered for various nonlinear strain models, including Ramberg-Osgood law. Numerical calculations are carried out for different values of non-linearity characteristic parameters for the inclusion material. Their parameters are analysed for the tensely-deformed matrix under loading a uniformly distributed shear stresses and for a balanced system of the concentrated forces.
9
Content available remote Supplementing the numerical solution of singular/hypersingular integral equations/inequalities with parametric inequality constraints with applications to crack problems
Ioakimidis N. I.
Computer Assisted Methods in Engineering and Science
|
2017
|
Vol. 24, no. 1
41--65
EN
Singular and hypersingular integral equations appear frequently in engineering problems. The approximate solution of these equations by using various numerical methods is well known. Here we consider the case where these equations are supplemented by inequality constraints-mainly parametric in equality constraints, but also the case of singular/hypersingular integral inequalities. The approach used here is simply to employ the computational method of quantifier elimination efficiently implemented in the computer algebra system Mathematica and derive the related set of necessary and sufficient conditions for the validity of the singular/hypersingular integral equation/inequality together with the related in equality constraints. The present approach is applied to singular integral equations/inequalities in the problem of periodic arrays of straight cracks under loading- and fracture-related inequality constraints by using the Lobatto-Chebyshev method. It is also applied to the hypersingular integral equation/inequality of the problem of a single straight crack under a parametric loading by using the collocation and Galerkin methods and parametric inequality constraints.
10
Content available remote Continuous contact problem of a functionally graded layer resting on an elastic half‐plane
Öner E., Adiyaman G., Birinci A.
Archives of Mechanics
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2017
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Vol. 69, nr 1
53--73
EN
In this study, the continuous contact problem of a functionally graded layer resting on an elastic half-plane and loaded by a rigid rectangular stamp is examined. The problem is solved assuming that the functionally graded (FG) layer is isotropic and the shear modulus and mass density vary exponentially throughout the layer’s thickness. However, the body force of the elastic half-plane is neglected. In addition, it is assumed that all surfaces are frictionless and only compressive stress is transferred along the contact surfaces. The mathematical problem is reduced to a singular integral equation in which the contact stress under the rigid stamp is unknown using the Fourier integral transform and boundary conditions related to the problem. This singular integral equation is solved numerically using the Gauss–Chebyshev integration formula. The dimensionless contact stress under the rigid stamp, the initial separation loads and the initial separation distances between the FG layer and the elastic half-plane are obtained for various dimensionless quantities.
11
A boundary-value problem for the equation of mixed type with generalized operators of fractional differentiation in the boundary conditions
Repin O. A., Kumykova S. K.
Journal of Applied Analysis
|
2016
|
Vol. 22, nr 1
27--36
EN
The paper is devoted to the study of a boundary-value problem for an equation of mixed type with generalized operators of fractional differentiation in boundary conditions. We prove uniqueness of solutions under some restrictions on the known functions and on the different orders of the operators of generalized fractional differentiation appearing in the boundary conditions. Existence of solutions is proved by reduction to a Fredholm equation of the second kind, for which solvability follows from the uniqueness of the solution of our original problem.
12
Content available An Analysis Of Stress Intensity Factor Due To Normal Stress For A Cracked Plate Reinforced With A Sheet By Seam Welding
Kim O.-H., Kim Y. C.
Archives of Metallurgy and Materials
|
2015
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Vol. 60, iss. 2B
1441--1444
EN
It is essential in damage tolerance design to determine the stress intensity factor theoretically. The stress intensity factor for a cracked plate that is reinforced with a sheet by seam welding is determined theoretically and plotted as function of the seam welding location and stiffness ratio. The singular integral equation is derived based on the compatibility condition between the cracked plate and the reinforcement plate, and it is solved by means of Erdogan and Gupta‘s method. The theory is verified by comparing the results of the present analysis with those of a numerical analysis. The results from the present analysis show that the reinforcement effect improves as the welding line is situated closer to the crack and as the stiffness ratio of the cracked plate and the reinforcement plate increases.
13
Content available Doubly periodic cracks in the anisotropic medium with the account of contact of their faces
Maksymovych O., Pasternak I., Sulym H., Kutsyk S.
Acta Mechanica et Automatica
|
2014
|
Vol. 8, no. 3
160--164
EN
The paper presents complex variable integral formulae and singular boundary integral equations for doubly periodic cracks in anisotropic elastic medium. It utilizes the numerical solution procedure, which accounts for the contact of crack faces and produce accurate results for SIF evaluation. It is shown that the account of contact effects significantly influence the SIF of doubly periodic curvilinear cracks both for isotropic and anisotropic materials.
14
Content available remote Singular integral equations with multiplicative Cauchy-type kernels
Kaczmarek P., Pylak D., Wójcik P.
Fasciculi Mathematici
|
2013
|
Nr 50
77--90
EN
In this paper we consider singular integral equations of the first kind with multiplicative Cauchy-type kernels defined on n-dimensional domains. We give their general solutions in the class of Holder continuous functions and propose the statements of uniqueness problem.
15
Content available Singular integral equation with a multiplicative Cauchy kernel in the half-plane
Karczmarek P.
Opuscula Mathematica
|
2008
|
Vol. 28, no. 1
63-72
EN
In this paper the explicit solutions of singular integral equation with a multiplicative Cauchy kernel in the half-plane are presented.
16
Content available Generalized characteristic singular integral equation with Hilbert kernel
Karczmarek P.
Opuscula Mathematica
|
2008
|
Vol. 28, no. 3
287-294
EN
In this paper an explicit solution of a generalized singular integral equation with a Hilbert kernel depending on indices of characteristic operators is presented.
17
Content available Approximate solutions of a singular integral equation with Cauchy kernels in the quarter plane
Pylak D.
Opuscula Mathematica
|
2008
|
Vol. 28, no. 2
179-194
EN
In the paper, we present explicit formulae for the solution of the singular integral equation with Cauchy kernels in the quarter plane. Next, Jacobi and Chebyshev polynomials are used to derive approximate solutions of this equation.
18
Content available Application of Chebyshew and trigonometric polynomials to the approximation of a solution of a singular integral equation with a multiplicative Cauchy kernel in the half-plane
Karczmarek P.
Opuscula Mathematica
|
2008
|
Vol. 28, no. 2
129-136
EN
In this article Chebyshev and trigonometric polynomials are used to construct an approximate solution of a singular integral equation with a multiplicative Cauchy kernel in the half-plane.
19
Two collinear griffith cracks in an orthotropic strip with punches
Singh P. C., Patra B., Maulik T. N.
Applied Mechanics and Engineering
|
1998
|
Vol. 3, no 4
621-653
EN
Integral transform technique is employed to solve the elastodynamic problem of steady-state propagation of two symmetrically situated identical collinear Griffith cracks along the mid plane of orthotropic strip of finite thickness 2h with centrally situated moving punches along the boundaries of the layer. The problem is reduced to the solution to a pair of simultaneous singular integral equations with Cauchy type singularities which have finally been solved through finite Hilbert transform technique. For large h, analytical expressions for the local stress field near the crack tip and the stress intensity factors are obtained. Graphical plots of the numerical results are also presented.
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