Vector-matrix Differential Equation Research Articles

Eigenvalue Approach to Fractional-Order Dual-Phase-Lag Thermoviscoelastic Problem of a Thick Plate

The present paper deals with the problem of thermoviscoelastic interactions in a homogeneous isotropic thick plate whose upper surface is stress-free and is subjected to a known temperature distribution, while the lower surface rests on a rigid foundation and is thermally insulated. The problem is treated on the basis of fractional-ordered dual-phase-lag model of thermoelasticity. To study the viscoelastic nature of the material, Kelvin–Voigt model of linear viscoelasticity is employed. The governing equations are transformed into a vector-matrix differential equation with the use of joint Laplace and Fourier transforms, which is then solved by the eigenvalue approach. Numerical estimates of displacements, stresses and temperature are computed for copper material by using a numerical inversion technique. Finally, all the physical fields are represented graphically to estimate and highlight the effects of the fractional parameter, viscosity and time.

Generalized Magnetothermoelastic Interaction for a Rotating Half Space

A generalized magnetothermoelasticity, in the context of Lord-Shulman theory, is employed to investigate the interaction of a homogeneous and isotropic perfect conducting half space with rotation. The Laplace transform for time variable is used to formulate a vector-matrix differential equation which is then solved by eigenvalue method. The continuous solution of displacement component while the discontinuous solutions of stress components, temperature distribution, induced magnetic and electric field have been analyzed in an approximate manner using assymptotic expansion for small time. The graphical representations also prove this continuity and discontinuity of the solutions.

Eigenfunction expansion method to characterize Rayleigh wave propagation in orthotropic medium with phase lags

ABSTRACTThe present paper deals with the propagation of Rayleigh surface waves in homogeneous, orthotropic thermoelastic half space in the context of three-phase lag model of thermoelasticity. A vector matrix differential equation is formed by employing normal mode analysis to the considered equations which is then solved by eigenfunction expansion method. The frequency equations for different cases are derived and the path of surface particles during Rayleigh wave propagation is found elliptical. Effect of phase lags on phase velocity, attenuation coefficient, and specific loss are presented graphically with respect to frequency as well as wave number.

A Study on the Generalized Thermoelastic Problem for an Anisotropic Medium

A vector–matrix differential equation is formulated using normal mode analysis from the governing equations of a three-dimensional anisotropic half space in presence of heat source and gravity. The corresponding solution is obtained with the help of eigenvalue approach. Numerical computations for displacement, thermal strain and stress component, temperature distribution are evaluated and presented graphically.

Generalized thermoelastic interaction in an isotropic solid cylinder without energy dissipation

In this paper, we constructed the generalized thermoelastic equations of an isotropic solid cylinder. The formulation is applied in the context of Green and Naghdi theory of types II (without energy dissipation). The material of the cylinder is supposed to be homogeneous isotropic both mechanically and thermally. The governing equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain, which is then solved by an eigenvalue approach. Numerical results for the temperature distribution, displacement and radial stress are represented graphically.

The influence of thermal and conductive temperatures in a nanoscale resonator

Abstract In this work, the thermoelastic interaction in a nano-scale resonator based on two-temperature Green-Naghdi model is established. The nanoscale resonator ends were simply supported. In the Laplace’s domain, the analytical solution of conductivity temperature and thermodynamic temperature, the displacement and the stress components are obtained. The eigenvalue approach resorted to for solutions. In the vector-matrix differential equations form, the essential equations were written. The numerical results for all variables are presented and are illustrated graphically.

ПРИЗНАКИ РЕГУЛЯРНОСТИ И УСТОЙЧИВОСТИ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ ВЫСШЕГО ПОРЯДКА

On the basis of previous works of authors new signs of regularity and stability of vector-matrix differential equations with a variable main part are specified.

Eigenfunction expansion method to analyze thermal shock behavior in magneto-thermoelastic orthotropic medium under three theories

ABSTRACTThis article is concerned with a study of thermal shock response in infinite thermoelastic medium under the purview of Lord–Shulman model, Green–Naghdi theory III, and three-phase-lag model of generalized thermoelasticity. The medium under consideration is assumed to be homogeneous, orthotropic, and thermally conducting. The fundamental equations of the two-dimensional problem of generalized thermoelasticity with three-phase-lag model in an infinite elastic medium under the influence of magnetic field are obtained as a vector–matrix differential equation form using normal mode analysis which is then solved by the Eigenfunction expansion method. Numerical results for the temperature, displacements and thermal stress distribution are presented graphically.

Eigen Value Approach to Generalized Thermoelastic Interactions in an Unbounded Body with Circular Cylindrical Cavity without Energy Dissipation

The theory of generalized thermoelasticity in the context of the Green-Naghdi model -II (thermoelasticity without energy dissipation) is studied for an infinite circular cylindrical cavity subjected to two different cases of thermoelastic interactions when the radial stress is zero for (a) maintaining constant temperature and (b) temperature is varying exponentially with time. The Laplace transform from time variable is used to the governing equations to formulate a vector matrix differential equation which is then solved by the eigen value approach. Numerical computations for the displacement component, temperature distribution and components of thermal stress have been made and presented graphically.

Two-dimensional generalized thermo-elastic problem for anisotropic half-space

This paper concerns with the study of wave propagation in fibre reinforced anisotropic half space under the influence of temperature and hydrostatic initial stress. Lord-Shulman theory is applied to the heat conduction equation. The resulting equations are written in the form of vector matrix differential equation by using Normal Mode technique, finally which is solved by Eigen value approach.

A Note on the Generalized Thermoelasticity Theory With Memory-Dependent Derivatives

In this note, two aspects in the theory of heat conduction model with memory-dependent derivatives (MDDs) are studied. First, the discontinuity solutions of the memory-dependent generalized thermoelasticity model are analyzed. The fundamental equations of the problem are expressed in the form of a vector matrix differential equation. Applying modal decomposition technique, the vector matrix differential equation is solved by eigenvalue approach in Laplace transform domain. In order to obtain the solution in the physical domain, an approximate method by using asymptotic expansion is applied for short-time domain and analyzed the nature of the waves and discontinuity of the solutions. Second, a suitable Lyapunov function, which will be an important tool to study several qualitative properties, is proposed.

A discontinuity analysis of generalized thermoelasticity theory with memory-dependent derivatives

In this article, the discontinuities of the theory of heat conduction model with memory-dependent derivatives are emphasized. To analyse the discontinuities, the memory-dependent model is applied to a transient thermo-mechanical process. The fundamental equations of the problem are expressed in the form of a vector matrix differential equation. Applying modal decomposition technique the vector matrix differential equation is solved by an eigenvalue approach in the Laplace transform domain. In order to obtain the solution in the physical domain an approximate method by using asymptotic expansion is applied for short time domain and to analyse the nature of the waves and discontinuity of the solutions. Finally, a suitable Lyapunov function, which will be an important tool to study several qualitative properties, is proposed.

Analytical Solutions of a Two-Dimensional Generalized Thermoelastic Diffusions Problem Due to Laser Pulse

In this paper, we apply the generalized thermoelastic theory with mass diffusion to a two-dimensional problem for a half-space. The surface of the half-space is taken to be traction-free and heated by laser pulse. The analytical solution is adopted for the temperature, the displacement components, concentration, the stress components and chemical potential. The nonhomogeneous basic equations have been written in the form of a vector–matrix differential equation, which is then solved using the eigenvalue approach. A comparison is made in the case of the absence and presence of a mass diffusion between the coupled and Lord–Shulman theories. The results obtained are presented graphically for the effects of the laser pulse and the mass diffusion to display the phenomena of physical significance.

Eigenvalue Approach in a Generalized Thermal Shock Problem for a Transversely Isotropic Half-Space

In the present work, the investigating of the disturbances in a homogeneous, transversely isotropic elastic medium with generalized thermoelastic theory has been concerned. The formulation is applied to generalized thermoelasticity based on three different theories. Laplace and Fourier transforms are used to solve the problem analytically. The essential equations have been written as a vector-matrix differential equation in the Laplace transform domain, then solved by an eigenvalue approach. The inverses of Fourier transforms are obtained analytically. The result is used to solve a specific two-dimensional problem. The technique is illustrated by means of several numerical experiments performed. The results were verified numerically and are plotted.

Effect of Rotation in an Orthotropic Elastic Slab

The fundamental equations of the two dimensional generalized thermoelasticity (L-S model) with one relaxation time parameter in orthotropic elastic slab has been considered under effect of rotation. The normal mode analysis is used to the basic equations of motion and heat conduction equation. Finally, the resulting equations are written in the form of a vector-matrix differential equation which is then solved by the eigenvalue approach. The field variables in the space time domain are obtained numerically. The results corresponding to the cases of conventional thermoelasticity CTE), extended thermoelasticity (ETE) and temperature rate dependent thermoelasticity (TRDTE) are compared by means of graphs.

Vector-Matrix Models of a Salient-Pole Synchronous Machine with Permanent Magnets in Terms of Different State Vector Components

Different forms of mathematical description of the electromagnetic processes occurring in a salient-pole synchronous machine with permanent magnets are considered. In this case, the machine parameters are set in the rotating coordinate system with the longitudinal d axis orientation by the rotor angular position. The presented vector-matrix differential equations of the stator electrical equilibrium and the formulas for moment calculation according to the problem class allows for transition from one two-dimensional basis to another one without the direct recalculation of inductances from the d, q coordinates, which in its turn simplifies the further analysis and design procedures. In this case, all transformations of the state variables are performed in compliance with the scalar values invariance condition; as a result, the energy characteristics of the real three-phase electrical machine and its mathematical models have the same numerical values and do not require the application of special matching coefficients.

Plane wave propagation in a 3D anisotropic half-space under Green-Naghdi theory II

In this paper, the theory of coupled thermoelasticity in three dimension is employed for triclinic half-space, subjected to time dependent heat source on the boundary of the space which is traction free and is considered in the context of Green-Naghdi model of type II (thermoelasticity without energy dissipation) of generalized thermoelasticity. Normal mode analysis is used to the non-dimensional coupled equations. Finally, the resulting equations are written in the form of a vector-matrix differential equation which is then solved by eigenvalue approach. Numerical results for the temperature, thermal stresses, and displacements are presented graphically and analyzed. Mathematical results shown in thermoelastic curves were supplemented by tectonic movements of elastic lithospheric plates.

Analytical solution of thermoelastic interaction in a half-space by pulsed laser heating

In this article, we consider the problem of a two-dimensional thermoelastic half-space in the context of generalized thermoelastic theory with one relaxation time. The surface of the half-space is taken to be traction free and thermally insulated. The solution of the considered physical quantity can be broken down in terms of normal modes. The nonhomogeneous basic equations have been written in the form of a vector-matrix differential equation, which is then solved by an eigenvalue approach. The exact analytical solution is adopted for the temperature, the components of displacement and stresses. The results obtained are presented graphically for the effect of laser pulse to display the phenomena physical meaning. The graphical results indicate that the thermal relaxation time has a great effect on the temperature, the components of displacement and the components of stress.

Analytical solution of a two-dimensional thermoelastic problem subjected to laser pulse

In this article, the problem of a two-dimensional thermoelastic half-space are studied using mathematical methods under the purview of the generalized thermoelastic theory with one relaxation time is studied. The surface of the half-space is taken to be thermally insulated and traction free. Accordingly, the variations of physical quantities due to by laser pulse given by the heat input. The nonhomogeneous governing equations have been written in the form of a vector-matrix differential equation, which is then solved by the eigenvalue approach. The analytical solutions are obtained for the temperature, the components of displacement and stresses. The resulting quantities are depicted graphically for different values of thermal relaxation time. The result provides a motivation to investigate the effect of the thermal relaxation time on the physical quantities.

Generalized thermoelastic problem of an infinite body with a spherical cavity under dual-phase-lags

The aim of the present contribution is the determination of the thermoelastic temperatures, stress, displacement, and strain in an infinite isotropic elastic body with a spherical cavity in the context of the mechanism of the two-temperature generalized thermoelasticity theory (2TT). The two-temperature Lord–Shulman (2TLS) model and two-temperature dual-phase-lag (2TDP) model of thermoelasticity are combined into a unified formulation with unified parameters. The medium is assumed to be initially quiescent. The basic equations are written in the form of a vector matrix differential equation in the Laplace transform domain, which is then solved by the state-space approach. The expressions for the conductive temperature and elongation are obtained at small times. The numerical inversion of the transformed solutions is carried out by using the Fourier-series expansion technique. A comparative study is performed for the thermoelastic stresses, conductive temperature, thermodynamic temperature, displacement, and elongation computed by using the Lord–Shulman and dual-phase-lag models.