Propagation Of Thermoelastic Waves Research Articles

The Characteristic of Thermoelastic Waves in Transversely Isotropic Finite Cylinders

A theoretical as well as a numerical investigation of the propagation of thermoelastic waves and vibration of transversely isotropic cylinders of finite length is discussed. Lord-Shulman theory is adopted here to model the thermoelastic deformation of cylinders. A semi analytical finite element (SAFE) method is employed to study dispersion of thermoelastic waves and natural frequencies of vibration of finite cylinders with traction free curved surfaces having both ends insulated and constrained by frictionless rigid walls. Numerical results obtained by the SAFE method for the frequencies of vibration of a sapphire rod are found to be in excellent agreement with published results. Natural frequencies of vibration for first three axisymmetric and asymmetric modes are presented graphically for a silicon nitride thermoelastic cylinder. Also, numerical results showing dispersion of both propagating and evanescent circumferential waves in infinite and finite cylinders are presented also.

The Analytical Form of the Dispersion Equation of Elastic Waves in Periodically Inhomogeneous Medium of Different Classes of Crystals

The investigation of thermoelastic wave propagation in elastic media is bound to have much influence in the fields of material science, geophysics, seismology, and so on. The heat conduction equations and bound equations of motions differ by the difficulty level and presence of many physical and mechanical parameters in them. Therefore thermoelasticity is being extensively studied and developed. In this paper by using analytical matrizant method set of equation of motions in elastic media are reduced to equivalent set of first-order differential equations. Moreover, for given set of equations, the structure of fundamental solutions for the general case has been derived and also dispersion relations are obtained.

Thermoelastic wave propagation in a transversely isotropic thick plate under Green–Naghdi theory due to gravitational field

ABSTRACTThe aim of this article is to study the gravitational response in an infinite, homogeneous, transversely isotropic thick plate. The upper surface of the plate is free from traction, and the surface temperature distribution is imposed on it. The lower surface of the plate is kept on a rigid base and is thermally insulated. The heat conduction equations in the presence of heat source are of hyperbolic types among them (i) without energy dissipation (GN-II model) and (ii) with energy dissipation (GN-III model). The Laplace–Fourier double-transform technique has been developed to solve the nondimensional coupled field equations. The physical quantities are obtained in the space–time domain using a numerical inversion method. Numerical calculations are performed for a thick plate made of magnesium (Mg) metal and have been depicted graphically to estimate the effect of gravity. The results for the isotropic material (Cu) and that of the transversely isotropic material (Mg) have been compared for both the models.

Analytical solution of generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads

ABSTRACTIn this article, a fully analytical solution of the generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads, based on Lord–Shulman model, is presented. The general forms of axisymmetric thermal and mechanical boundary conditions as arbitrary time-dependent heat transfer and traction, respectively, are considered at the inner and outer radii of the disk. The governing equations are solved analytically using the principle of superposition and the Fourier–Bessel transform. The general closed form solutions are presented for temperature and displacement fields. To validate the solutions, the results of this study are compared with the numerical results available in the literature, which show good agreement. For the temperature, displacement and stresses, radial distributions, and time histories are plotted and discussed. The propagation of thermoelastic waves and their reflection from the boundary of the disk are clearly shown. Moreover, effects of relaxation time and angular velocity on temperature, displacement, and stress fields are investigated.

Nonlocal thermo-elastic wave propagation in temperature-dependent embedded small-scaled nonhomogeneous beams

In this paper, the thermo-elastic wave propagation analysis of a temperature-dependent functionally graded (FG) nanobeam supported by Winkler-Pasternak elastic foundation is studied using nonlocal elasticity theory. The nanobeam is modeled via a higher-order shear deformable refined beam theory which has a trigonometric shear stress function. The temperature field has a nonlinear distribution called heat conduction across the nanobeam thickness. Temperature-dependent material properties change gradually in the spatial coordinate according to the Mori-Tanaka model. The governing equations of the wave propagation of the refined FG nanobeam are derived by using Hamilton’s principle. The analytic dispersion relation of the embedded nonlocal functionally graded nanobeam is obtained by solving an eigenvalue problem. Numerical examples show that the wave characteristics of the functionally graded nanobeam are related to the temperature distribution, elastic foundation parameters, nonlocality and material composition.

Circumferential Waves in Transradially Isotropic Thermoelastic Spherical Curved Plates

This paper presents the propagation of thermoelastic waves along circumferential direction in homogeneous, transradially isotropic spherical curved plates. Mathematical modeling of the problem to obtain dispersion curves for transversely isotropic thermally conducting spherically curved elastic plates leads to coupled differential equations. The coupled differential equations of motion and heat conductions equation in conjunction with stress free, rigidly fixed thermal boundary conditions on the inner and outer surfaces of a spherical curved plate are solved with matrix Frobenius method. In order to illustrate theoretical development, numerical solutions are obtained and presented graphically for a zinc, cobalt and silicon nitride plate. The graphical results are compared with the available literature. Thermal variations for the non-axially symmetric case of plane strain vibrations, these modes remain coupled and are affected by temperature change. Moreover, these vibration modes are found to be dispersive in nature.

Propagation of nonlinear thermoelastic waves in porous media within the theory of heat conduction with memory: physical derivation and exact solutions

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid‐saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time‐fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time‐fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.

Three-Dimensional Analysis of a Thermo-Viscoelastic Half-Space due to Thermal Shock in Temperature-Rate-Dependent Thermoelasticity

AbstractThe present paper is aimed at studying the effects of viscosity and time on the propagation of thermoelastic waves in a homogeneous and isotropic three-dimensional medium whose surface is acted upon by a thermal load under the purview of temperature-rate-dependent thermoelasticity. The normal mode analysis technique has been employed to solve the resulting non-dimensional coupled field equations and hence the exact expressions for displacement component, stress, temperature field and strain are obtained. The problem is further illustrated by computing the numerical values of the field variables for a copper- like material and depicting them graphically. Numerical results predict finite speed of propagation for thermoelastic waves.

Numerical and analytical study of the propagation of thermoelastic waves in a medium with heat-flux relaxation

The thermoelastic problem of laser exposure of metals and dielectrics is studied taking into account the finite speed of propagation of thermal waves and using a numerical finite-difference algorithm. The resulting numerical solution is compared with the analytical one. The problem is solved in coupled and uncoupled formulations. The solutions of the hyperbolic thermoelastic problem are compared with the solutions of the classical problem. Analytical expressions are obtained for the propagation speeds of the thermoelastic wave components. Times are determined at which the difference between the solutions of the hyperbolic and classical thermoelastic problems can be detected experimentally.

Reflection of thermoelastic wave on the interface of isotropic half-space and tetragonal syngony anisotropic medium of classes 4, 4/m with thermomechanical effect

The thermoelastic wave propagation in a tetragonal syngony anisotropic medium of classes 4, 4/m having heterogeneity along z axis has been investigated by employing matrizant method. This medium has an axis of second-order symmetry parallel to z axis. In the case of the fourth-order matrix coefficients, the problems of wave refraction and reflection on the interface of homogeneous anisotropic thermoelastic mediums are solved analytically.

Numerical and Analytical Studies of the Propagation of Thermoelastic Waves in a Medium Taking into Account Heat Flow Relaxation

Numerical and Analytical Studies of the Propagation of Thermoelastic Waves in a Medium Taking into Account Heat Flow Relaxation

Generalized thermoelastic wave band gaps in phononic crystals without energy dissipation

We present a theoretical investigation of the thermoelastic wave propagation in the phononic crystals in the context of Green–Nagdhi theory by taking thermoelastic coupling into account. The thermal field is assumed to be steady. Thermoelastic wave band structures of 3D and 2D are derived by using the plane wave expansion method. For the 2D problem, the anti-plane shear mode is not affected by the temperature difference. Thermoelastic wave bands of the in-plane x–y mode are calculated for lead/silicone rubber, aluminium/silicone rubber, and aurum/silicone rubber phononic crystals. The new findings in the numerical results indicate that the thermoelastic wave bands are composed of the pure elastic wave bands and the thermal wave bands, and that the thermal wave bands can serve as the low boundary of the first band gap when the filling ratio is low. In addition, for the lead/silicone rubber phononic crystals the effects of lattice type (square, rectangle, regular triangle, and hexagon) and inclusion shape (circle, oval, and square) on the normalized thermoelastic bandwidth and the upper/lower gap boundaries are analysed and discussed. It is concluded that their effects on the thermoelastic wave band structure are remarkable.

Estimates of azimuthal numbers associated with elementary elliptic cylinder wave functions

The paper deals with issues related to the construction of solutions, 2 π-periodic in the angular variable, of the Mathieu differential equation for the circular elliptic cylinder harmonics, the associated characteristic values, and the azimuthal numbers needed to form the elementary elliptic cylinder wave functions. A superposition of the latter is one possible form for representing the analytic solution of the thermoelastic wave propagation problem in long waveguides with elliptic cross-section contour. The classical Sturm-Liouville problem for the Mathieu equation is reduced to a spectral problem for a linear self-adjoint operator in the Hilbert space of infinite square summable two-sided sequences. An approach is proposed that permits one to derive rather simple algorithms for computing the characteristic values of the angular Mathieu equation with real parameters and the corresponding eigenfunctions. Priority is given to the application of the most symmetric forms and equations that have not yet been used in the theory of the Mathieu equation. These algorithms amount to constructing a matrix diagonalizing an infinite symmetric pentadiagonal matrix. The problem of generalizing the notion of azimuthal number of a wave propagating in a cylindrical waveguide to the case of elliptic geometry is considered. Two-sided mutually refining estimates are constructed for the spectral values of the Mathieu differential operator with periodic and half-periodic (antiperiodic) boundary conditions.

Application of a hybrid mesh-free method for shock-induced thermoelastic wave propagation analysis in a layered functionally graded thick hollow cylinder with nonlinear grading patterns

Abstract This article exploits a hybrid mesh-free method for coupled thermoelasticity analysis (without energy dissipation) and thermoelastic wave propagation analysis in layered FGMs subjected to shock loading. The presented hybrid mesh-free method is based on generalized finite difference (GFD) and Newmark finite difference (NFD) methods. The Green–Naghdi (GN) theory of coupled thermoelasticity is assumed to derive the governing equations for FG thick hollow cylinder. The layered FG cylinder is assumed to be under thermal shock loading. The mechanical properties of layered FG cylinder are considered to vary along the radial direction as nonlinear functions in terms of volume fraction. Thermoelastic wave propagations are studied in details at various time instants for various grading patterns of mechanical properties. The effects of nonlinear grading patterns on thermoelastic wave propagations are obtained and discussed using the presented effective mesh-free method.

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.

On the existence of motion-induced heat flux due to thermoelastic waves in a one-dimensional solid rod

In this Letter, we show the existence of motion-induced heat transfer in homogeneous isotropic solids due to the propagation of thermoelastic waves. In particular, using the linearized governing equations for thermoelastic waves, we show that heat transfer in a one-dimensional rod is not only due to conduction but also to the local particle displacement a phenomenon which, in principle, is similar to advective heat transfer in fluids. It is found that the time-averaged heat transfer is dependent on both the material properties and the external excitation parameters. This mechanism can potentially be useful in ultrasonic welding and in the development of solid state refrigerators or heat pumps.

Stochastic analysis of elastic wave and second sound propagation in media with Gaussian uncertainty in mechanical properties using a stochastic hybrid mesh-free method

The main objective of this article is the exploitation of a stochastic hybrid mesh-free method based on stochastic generalized finite difference (SGFD), Newmark finite difference (NFD) methods and Monte Carlo simulation for thermoelastic wave propagation and coupled thermoelasticity analysis based on GN theory (without energy dissipation). A thick hollow cylinder with Gaussian uncertainty in mechanical properties is considered as an analyzed domain for the problem. The effects of uncertainty in mechanical properties with various coefficients of variations on thermo-elastic wave propagation are studied in details. Also, the time histories and distribution on thickness of cylinder of maximum, mean and variance values of temperature and radial displacement are studied for various coefficients of variations (COVs).

Influence of microstructure on thermoelastic wave propagation

Numerical simulations of the thermoelastic response of a microstructured material on a thermal loading are performed in the one-dimensional setting to examine the influence of temperature gradient effects at the microstructure level predicted by the thermoelastic description of microstructured solids (Berezovski et al. in J. Therm. Stress. 34:413–430, 2011). The system of equations consisting of a hyperbolic equation of motion, a parabolic macroscopic heat conduction equation, and a hyperbolic evolution equation for the microtemperature is solved by a finite-volume numerical scheme. Effects of microtemperature gradients exhibit themselves on the macrolevel due to the coupling of equations of the macromotion and evolution equations for macro- and microtemperatures.

Application of the nonlocal Darcy law to the propagation of nonlinear thermoelastic waves in fluid saturated porous media

In the propagation of nonlinear waves in fluid saturated porous media, a key role is played by the Darcy law. In many cases one should take into account a realistic spatial or temporal variability of rock parameters, as permeability, porosity, diffusivity…To this purpose we here introduce a nonlocal Modified Darcy Law (MDL) by considering a fractional derivative generalization of the classical case. As an application we here study the effect of a generalized Darcy law on the propagation of nonlinear thermoelastic waves in porous media. In more details, we discuss an application of this MDL to an early Natale and Salusti model about quick and strong pressure and temperature waves in fluid saturated rocks. The original model leads to the Burgers equation, while with the MDL here we obtain a nonlocal formulation of the Burgers equation. We moreover find the analytic solution in the case that diffusion plays a secondary role. With this nonlocal model we obtain a richer analysis of realistic characteristic of such transient phenomena, in particular the spatial delays in wave propagation.

Shock-induced thermoelastic wave propagation analysisin a thick hollow cylinder without energy dissipation using mesh-free generalized finite difference (GFD) method

The main objective of this article is the exploitation of the generalized finite difference method to study the thermoelastic wave propagation, that is, the dynamic behaviors of displacement and temperature field in a thick hollow cylinder. The thermoelasticity governing equations are derived based on Green–Naghdi coupled thermoelasticity theory (without energy dissipation). The generalized finite difference (GFD) method is used to approximate the space variables, and Newmark finite difference (NFD) is employed to obtain the behaviors of parameters in time domain. The time histories of displacement and temperature fields across the thickness of the cylinder are obtained and the propagations of thermal and elastic waves are illustrated at various times. Using the GFD method, the wave front in temperature and elastic domains can be tracked, and the comparison between results based on GFD and other numerical methods shows very good agreement. The application of GFD method in coupled thermoelasticity problems has a high capability because it does not require a mesh generation. A comparison between the presented mesh-free GFD method and meshless local Petrov–Galerkin (MLPG) method shows a good agreement of the results.