M. Kanoria

Study of Dynamic Response in a Functionally Graded Spherically Isotropic Hollow Sphere with Temperature Dependent Elastic Parameters

This paper is concerned with the investigation of thermoelastic interactions in a functionally graded spherically isotropic hollow sphere in which the thermophysical properties are temperature dependent in the context of the linear theory of generalized thermoelasticity (Green and Lindsay theory). Both the boundaries are stress free and are subjected to prescribed temperatures. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain which is then solved by eigenvalue approach. The numerical inversion of the transforms is carried out using a method of Bellman et al. The thermoelastic dynamic displacements, stresses and temperatures are computed numerically and presented graphically in a number of figures. The results, corresponding to the cases when the material properties are temperature independent and the outer radius of the sphere tends to infinity, agree with those of the existing literature.

Water-wave Scattering by Thick Vertical Barriers

This paper is concerned with two-dimensional scattering of a normally incident surface wave train on an obstacle in the form of a thick vertical barrier of rectangular cross section in water of uniform finite depth. Four different geometrical configurations of the barrier are considered. The barrier may be surface-piercing and partially immersed, or bottom-standing and submerged, or in the form of a submerged rectangular block not extending down to the bottom, or in the form of a thick vertical wall with a submerged gap. Appropriate multi-term Galerkin approximations involving ultraspherical Gegenbauer polynomials are used for solving the integral equations arising in the mathematical analysis. Very accurate numerical estimates for the reflection coefficient for each configuration of the barrier are then obtained. The reflection coefficient is depicted graphically against the wave number for each configuration. It is observed that the reflection coefficient depends significantly on the thickness for a wide range of values of the wave number, and as such, thickness plays a significant role in the modelling of efficient breakwaters.

Generalized Thermoelastic Problem of a Spherically Isotropic Infinite Elastic Medium Containing a Spherical Cavity

This paper is concerned with the determination of thermoelastic stresses and temperature in a spherically isotropic infinite elastic medium having a spherical cavity in the context of the linear theory of generalized thermoelasticity with two relaxation time parameters (Green and Lindsay theory). The surface of the cavity is stress free and is subjected to a thermal shock. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain, which is then solved by the eigenvalue approach. The numerical inversion of the transforms is carried out using the Bellman method. The stresses and temperature are computed and presented graphically. A comparison with isotropic body has also been studied.

Oblique wave scattering by submerged thin wall with gap in finite-depth water

Abstract The problem of oblique wave scattering by a submerged thin vertical wall with a gap in finite-depth water and its modification when another identical wall is introduced, are investigated in this paper. The techniques of both one-term and multiterm Galerkin approximations have been utilized in the mathematical analysis. The multi-term approximations in terms of appropriate Chebyshev polynomials provide extremely accurate numerical estimates for the reflection coefficient. The reflection coefficient is depicted graphically for a number of geometries. It is found that by the introduction of another identical wall, there occurs zero reflection for certain wave numbers. This may have some bearings on the modelling of a breakwater.

Generalized thermoelastic interaction in a functionally graded isotropic unbounded medium due to varying heat source with three-phase-lag effect

This paper deals with the problem of thermoelastic interactions in a functionally graded isotropic unbounded medium due to the presence of periodically varying heat sources in the context of the three-phase-lag thermoelastic models, GN ii (TEWOED) and GN iii (TEWED). The governing equations of three-phase-lag thermoelastic model (3P), generalized thermoelasticity without energy dissipation (GN ii) and with energy dissipation (GN iii) for a functionally graded material (i.e., a material with spatially varying material properties) are established. The governing equations are expressed in a Laplace–Fourier double transform domain and solved in that domain. Then the inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand are obtained numerically in a complex domain by using Laguerres method and the inversion of the Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the thermal displacement, temperature and thermal stress are obtained for a hypothetical material. The solution to the analogous problem for homogeneous isotropic material is obtained by taking a suitable non-homogeneity parameter. Finally, the results obtained are presented graphically to show the effect of non-homogeneity on thermal displacement, temperature and thermal stress. A comparison of the results for different theories (three-phase-lag model, GN ii and GN iii) is presented and the effect of non-homogeneity is also shown. In absence of non-homogeneity the results corresponding to the 3P model, GN ii and GN iii model agree with the results of the existing literature.

Water wave scattering by a submerged circular-arc-shaped plate

The problem of water wave scattering by a thin circular-arc-shaped plate submerged in infinitely deep water is investigated by linear theory. The circular-arc is not necessarily symmetric about the vertical through its center. The problem is formulated in terms of a hypersingular integral equation for a discontinuity of the potential function across the plate. The integral equation is solved approximately using a finite series involving Chebyshev polynomials of the second kind. The unknown constants in the finite series are determined numerically by using the collocation and the Galerkin methods. Both the methods ultimately produce very accurate numerical estimates for the reflection coefficient. The numerical results are depicted graphically against the wave number for a variety of configurations of the arc. Some results are compared with known results available in the literature and good agreement is achieved. The suitability of using a circular-arc-shaped plate as an element of a water wave lens has also been discussed on the basis of the present numerical results.

Study of Dynamical Response in a Two-Dimensional Transversely Isotropic Thick Plate Due to Heat Source

This paper deals with a two dimensional problem for a transversely isotropic thick plate having heat source. The upper surface of the plate is stress free with prescribed surface temperature while the lower surface of the plate rests on a rigid foundation and is thermally insulated. The study is carried out in the context of three-phase-lag thermoelastic model, GN model II (TEWOED) and GN model III (TEWED). The governing equations for displacement and temperature fields are obtained in Laplace–Fourier transform domain by applying Laplace and Fourier transform techniques. The inversion of double transform has been done numerically. The numerical inversion of Laplace transform is done by using a method based on Fourier Series expansion technique. Numerical computations have been done for magnesium (Mg) and a comparison of the results for different theories (three-phase-lag model, GN model II, GN model III) are presented graphically. The results for an isotropic material (Cu) have been deduced numerically and presented graphically to compare with those of transversely isotropic material (Mg).

Water wave scattering by thick rectangular slotted barriers

This paper is concerned with scattering of surface water waves by a thick vertical slotted barrier of rectangular cross-section with an arbitrary number of slots of unequal lengths along the vertical direction, and present in finite depth water. Four different geometrical configurations of the slotted barrier are considered. The barrier may be surface piercing and partially immersed, or bottom standing and submerged, or in the form of a submerged slotted thick block not extending down to the bottom, or in the form of a slotted thick wall extending from the free surface to the bottom. Galerkin approximations involving ultraspherical Gegenbauer polynomials are utilized in the mathematical analysis for solving first kind integral equations valid in the union of several disjoint intervals, to obtain very accurate numerical estimates for the reflection coefficient which is depicted graphically against the wave number in a number of figures for various configurations of the thick slotted barrier. Numerical codes prepared for this problem are valid for an arbitrary number of slots, the length of the slots as well as the wetted portions of the barrier for each configuration being unequal. However, to show the dependence of the reflection coefficient on the number of slots, results for a slotted wall with five slots are graphically displayed in one figure. Some results in the limiting cases have been compared with known results and good agreement is seen to have been achieved. New results are also presented showing total reflection at some moderate wave numbers for submerged slotted barriers.

One-dimensional problem of a fractional order two-temperature generalized thermo-piezoelasticity

This paper is concerned with the determination of the thermoelastic stress, strain and conductive temperature in a piezoelastic half-space body in which the boundary is stress free and subjected to thermal loading in the context of the fractional order two-temperature generalized thermoelasticity theory (2TT). The two-temperature three-phase-lag (2T3P) model, two-temperature Green–Naghdi model III (2TGNIII) and two-temperature Lord–Shulman (2TLS) model of thermoelasticity are combined into a unified formulation introducing unified parameters. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain that is then solved by the state-space approach. The numerical inversion of the transform is carried out by a method based on Fourier series expansion techniques. The numerical estimates of the quantities of physical interest are obtained and depicted graphically. The effect of the fractional order parameter, two-temperature and electric field on the solutions has been studied and comparisons among different thermoelastic models are made.

Study of Finite Thermal Waves in a Magnetothermoelastic Rotating Medium

This article deals with the problem of finite thermoelastic wave propagation in an unbounded rotating medium due to a periodically varying heat source under the influence of a magnetic field. The governing equations for generalized thermoelasticity with energy dissipation (GNIII) and without energy dissipation (GNII) have been solved by using the Laplace–Fourier double transform technique. The inversion of the Fourier transform has been done by using residual calculus, and the inversion of the Laplace transformation is carried out using Fourier series expansion technique. The physical quantities have been computed numerically and presented graphically to compare the results for different theories (GNII and GNIII) and to show the effects of rotation, magnetic field, and the damping coefficient on the physical quantities.