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Transform Methods
Published in Matthew N. O. Sadiku, Sudarshan R. Nelatury, Analytical Techniques in Electromagnetics, 2015
Matthew N. O. Sadiku, Sudarshan R. Nelatury
With a restriction on the domain of definition of Fourier transform we can conceive of Fourier Sine and Cosine transforms. These are particularly suitable for semi-infinite regions. We shall give the definitions and properties of these transforms, and illustrate their applications using examples.
Thermal wave propagation in a two-dimensional problem under gravitational field due to time-dependent thermal loading and memory effect
Published in Waves in Random and Complex Media, 2023
Pallabi Purkait, Abhik Sur, M. Kanoria
Using the inversion formulae of Fourier Sine and Cosine transforms, we have Using the results we have Substituting (52)–(54) into (36)–(39), the stress components are given by We consider the problem of a half-space Ω defined by which is subjected to the following boundary conditions.
Investigation of interactions among collinear Griffith cracks situated in a functionally graded medium under thermo-mechanical loading
Published in Journal of Thermal Stresses, 2020
In this article, three collinear Griffith cracks are assumed in an infinite functionally graded medium along the x-axis. One of the cracks is placed at the center and the other two cracks are located symmetrically on either side of the central one (see Figure 1). These cracks are partially insulated and steady-state heat flux is applied in y-direction away from the cracks’ region. Mechanical stresses in the forms of crack surface tractions are applied on collinear cracks. It is assumed that material properties are varying in the direction perpendicular to the plane of cracks and are approximated by exponential functions. The medium considered here may be treated as a non-homogeneous elastic medium under the condition of the plane strain or plane stress. Here, the isotropic stress-strain law is followed while ignoring the coupling effects between mechanical and thermal loadings. Using the principle of superposition, equations of elasticity and heat conduction are solved using Fourier sine and cosine transforms. Those equations are analytically reduced to a system of integral equations of the first kind with Cauchy type singularity. The numerical solutions of these systems of integral equations yield temperature and displacement fields in the medium subject to thermal and mechanical loadings. The thermal crack surface stresses are also obtained. Mode I SIFs and stress magnification factors (SMFs) under particular cases of thermal, mechanical, and both thermal and mechanical loadings are calculated. The numerical computation and pictorial representations of the SMFs to find the possibilities of shielding and amplification of the cracks are the key features of the present article. Graphical presentations of temperature distribution and thermal crack surface tractions are shown in sections 5.1 and 5.2 respectively.