• Structural Engineering and Mechanics
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• v.85 no.5
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• pp.621-633
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• 2023

The major objective of this paper is to study the receding contact problem between two functional graded layers under a flat indenter. The gravity is assumed negligible, and the shear moduli of both layers are assumed to vary exponentially along the thickness direction. In the absence of body forces, the problem is reduced to a system of Fredholm singular integral equations with the contact pressure and contact size as unknowns via Fourier integral transform, which is transformed into an algebraic one by the Gauss-Chebyshev quadratures and polynomials of both the first and second kinds. Then, an iterative speediest descending algorithm is proposed to numerically solve the system of algebraic equations. Both semi-analytical and finite element method, FEM solutions for the presented problem validate each other. To improve the accuracy of the numerical result of FEM, a graded FEM solution is performed to simulate the FGM mechanical characteristics. The results reveal the potential links between the contact stress/size and the indenter size, the thickness, as well as some other material properties of FGM.

• The Pure and Applied Mathematics
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• v.27 no.4
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• pp.187-205
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• 2020

In this article, we prove coincidence point theorems for comparable 𝜓-contractive mappings in ordered non-Archimedean fuzzy metric spaces utilizing the recently established concept of 𝓣-comparability and relatively weaker order theoretic variants. With a view to show the usefulness and applicability of this work, we solve the system of ordered Fredholm integral equations as an application. In the process, this presentation generalize and improve some prominent recent results obtained in Mihet [Fuzzy Sets Syst., 159 (6), 739-744, (2008)], Altun and Mihet [ Fixed Point Theory Appl. 2010, 782680, (2010)], Alam and Imdad [Fixed Point Theory, 18(2), 415-432, (2017)] and several others in the settings of partially ordered non-Archimedean fuzzy metric spaces.

• Transactions of the Korean Society of Mechanical Engineers
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• v.7 no.2
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• pp.145-152
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• 1983

An adhesively bonded anisotropic structure containing a part-through crack subjected to in-plane mixed mode deformations is investigated. The problem is reduced to a pair of Fredholm integral equations of the second kind by mathematical analysis. By solving these equations numerically stress intensity factors k$_{1}$ and k$_{2}$ are presented. Two cases are considered with respect to fiber orientations. Case one is to fix the fiber orientations of sound plate bonded to cracked plate with various fiber orientations. The other is to vary fiber orientations for both plates. As boundary conditions, tension and shear loading respectively, are applied to bonded anisotropic plates to observe mixed mode deformations.

• Structural Engineering and Mechanics
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• v.18 no.6
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• pp.759-768
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• 2004

The torsion of a penny-shaped crack in a transversely isotropic strip is investigated in this paper. The shear moduli are functionally graded in such a way that the mathematics is tractable. Hankel transform is used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by taking the asymptotic behavior of Bessel function into account. The effects of material property parameters and geometry criterion on the stress intensity factor are investigated. Numerical results show that increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface can suppress crack initiation and growth, and that the stress intensity factor varies little with the increasing of the strip's height.

• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.487-500
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• 2013

In this paper we provide a brief introduction to the continuity of approximate point spectrum on the algebra B(X), using basic properties of Fredholm operators and the SVEP condition. Also, we give an example showing that in general it not holds that if the spectrum is continuous an operator T, then for each ${\lambda}{\in}{\sigma}_{s-F}(T){\setminus}\overline{{\rho}^{\pm}_{s-F}(T)}$ and ${\in}$ > 0, the ball $B({\lambda},{\in})$ contains a component of ${\sigma}_{s-F}(T)$, contrary to what has been announced in [J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity II, Integral Equations Operator Theory 4 (1981), 459-503] page 462.

• Journal of Mechanical Science and Technology
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• v.18 no.3
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• pp.419-431
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• 2004

The dynamic response of a cracked functionally graded piezoelectric material (FGPM) under transient anti-plane shear mechanical and in-plane electrical loads is investigated in the present paper. It is assumed that the electroelastic material properties of the FGPM vary smoothly in the form of an exponential function along the thickness of the strip. The analysis is conducted on the basis of the unified (or natural) crack boundary condition which is related to the ellipsoidal crack parameters. By using the Laplace and Fourier transforms, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, the electric field, FGPM gradation, crack length, and electromechanical coupling coefficient.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.9 no.1
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• pp.63-77
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• 2005

The Dirichlet-Neumann problem for the dissipative Helmholtz equation in a connected plane region bounded by closed curves and containing cuts is studied. The Neumann condition is given on the closed curves, while the Dirichlet condition is specified on the cuts. The existence of a classical solution is proved by potential theory. The integral representation of the unique classical solution is obtained. The problem is reduced to the Fredholm equation of the second kind and index zero, which is uniquely solvable. Our results hold for both interior and exterior domains.

• Tunnel and Underground Space
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• v.26 no.3
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• pp.201-211
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• 2016

A method for estimating rock joint size distribution using contained traces length distribution from 3D cylindrical window survey was suggested. To reduce the numerical error, an improved technique was applied. The accuracy was verified by referring to Monte-Carlo simulation and it was found that the error can be decreased with suitable gamma values.

• Bulletin of the Society of Naval Architects of Korea
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• v.13 no.3
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• pp.1-9
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• 1976

A numerical method for solving the boundary-value problem related to potential flows with a free surface and an experimental work are introduced in this paper. The forced heaving motion of cylinders with arbitrary shapes in water of finite depth are Considered here. The Fredholm integral equation of the first kind is employed in determining strengths of singularities distributed on the body surface. And the results obtained by the present method for the case of a heaving circular cylinder on water of finite depth agree well with existing results of earlier investigators.

• International Journal of Naval Architecture and Ocean Engineering
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• v.6 no.1
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• pp.175-185
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• 2014

This study presents the prediction of propagated wave profiles using the wave information at a fixed point. The fixed points can be fixed in either space or time. Wave information based on the linear wave theory can be expressed by Fredholm integral equation of the first kinds. The discretized matrix equation is usually an ill-conditioned system. Tikhonov regularization was applied to the ill-conditioned system to overcome instability of the system. The regularization parameter is calculated by using the L-curve method. The numerical results are compared with the experimental results. The analysis of the numerical computation shows that the Tikhonov regularization method is useful.