Vector-matrix Differential Equation Research Articles

Analytical study of micropolar thermoelastic rectangular plate under three theories

A three-dimensional micropolar thermoelastic problem is considered for an isotropic rectangular plate that is subjected to simply support and isothermal on its four adjacent edges. Normal mode analysis is applied to the governing equations to form a vector-matrix differential equation and the analytical solution of the vector-matrix differential equation is obtained by using eigenvalue approach methodology. Numerical results are presented graphically along with theoretical analysis for sinusoidal variation of the stresses applied on the top surface of the rectangular plate.

A study of the 3-phase lag model to a two-dimensional isotropic micro-polar thermoelastic medium with memory-dependent properties

This article enlightens the two-dimensional (2D) isotropic micro-polar thermoelastic problem of the three-phase-lag (3PHL) model and heat conduction equation is formulated in the context of memory-dependent derivative (MDD). The equations are converted into the domain of the Laplace transform vector matrix differential equation form and solved by using the eigenvalue technique. To obtain an analytical solution of displacement, temperature, and stress components Silicon material properties are used. Inversion of the Laplace transform with Fourier series expansion technique is used to obtain the numerical solution. For obtaining graphical results Mathematica software is used. For the purpose of exhibiting the beauty of MDD in the present model comparisons are made between the time delay parameters and kernel functions (constant, linear, and non-linear kernels), respectively, over the micro-polar panel. The results obtained have a valuable impact on structural analysis, especially in the design of rotating machinery structures using accurate material properties.

Photothermal effects of semiconducting medium with non-local theory

A generalized magnetothermoelastic half-space problem is considered for a homogeneous, isotropic and semiconducting medium under the non-local heat equation with dual-phase-lag model. The boundary surface of the medium is subjected to a prescribed time-dependent and exponential order compression along with a prescribed temperature and carrier intensity gradient. Two integral transformations, Laplace transform for time variable and Fourier transform for space variable, are employed to the equations of motion and the heat conduction equation for formulation of a vector-matrix differential equation which is then solved by using eigenvalue approach. Inversion processes for the two integral transforms are carried out numerically. Finally, the effects of physical field variables and stress components are analyzed and illustrated graphically under the variation of different physical parameters.

Wave propagation in a non-local magneto-thermoelastic medium permeated by heat source

In this paper, we consider the new concept of non-local heat conduction equation to generalized magneto-thermoelastic problem of two dimensional isotropic and homogeneous half-space in presence of heat-flux at the boundary surface. By using the harmonic plane waves, the governing equations are transformed to the vector matrix differential equation which is then solved by eigenvalue method. The analytical closed form solutions for displacement component, temperature distribution and stress components have been made and comparisons are also illustrated graphically with the theory of non-local dual-phase-lag (NLDPL) and non-local Lord-Shulman (NLLS) theory for different values of physical parameters. The significant effects of non-local variables as well as phase lagging parameters on displacements, temperature distribution and stress components are studied by means of graphically and concluding remarks are drawn.

Modeling of a homogeneous isotropic half space in the context of multi-phase lag coupled thermoelasticity

A two-dimensional multi-phase lag model in the context of generalized thermoelasticity is established for an isotropic half-space medium. A vector-matrix differential equation is obtained from the governing equations using normal mode analysis. The eigenvalue approach is applied to obtain the solutions. The temperature-dependent displacements, stresses, strains are calculated numerically and represented graphically to show the accuracy of the solution under mechanical and thermal loads.

Eigen value approach with memory dependant derivative on homogeneous isotropic infinitely extended rotating plate of a finite thickness in absence of heat source

Our present manuscript is an attempt to derive a model of generalized thermoelasticity with dual phase lag heat conduction by using the methodology of memory dependent derivative for a isotropic rotating plate subject to the prescribed boundary conditions with constant magnetic and electric intensities. Two integral transform such as Laplace transform for time variable and Fourier transform for space variable are employed to the governing equations to formulate vector-matrix differential equation which is then solved by eigenvalue approach methodology. The inversion of two integral transformations is carried out using suitable numerical techniques. Numerical computations for displacement, thermal strain and stress component, temperature distribution are evaluated and presented graphically under influences of different physical parameters.

Memory response in dual-phase-lag thermoelastic medium due to instantaneous heat source

The present work deals with thermoelastic interactions in an isotropic infinite medium with memory-dependent derivative under a dual-phase-lag model. The governing equations are expressed in vector–matrix differential equation (VMDE) form in the Laplace–Fourier transform domain and solved using the eigenvalue approach. Numerical estimations of different thermophysical quantities such as displacement, temperature, and stresses are presented graphically for a coper material by employing the Gaussian quadrature formula and Honig–Hirdes method. Several remarkable points have been mentioned in the graphical representation of the field variables for various kernel functions, time-delay parameters, and magnitude of instantaneous heat source.

Moore-Gibson-Thompson theory of a non-local excited semiconductor medium with stability studies

The present study introduces a theoretical framework according to the Moore-Gibson-Thompson (MGT) Model in the context of generalized thermoelasticity theory. The excited semiconductor material is utilized to formulate the governing equations in one dimensional (1D) with a non-local parameter. According to the photo- thermoelasticity theory, the overlapping between the electronic (optical) deformation and elastic deformation is taken into account. During the photo-excitation processes, the non-local equation of motion and the heat equation is introduced according to the model of Moore-Gibson-Thompson (MGT). The novel model describes the interaction between plasma and thermomechanical waves in a non-dimensional form. Laplace transform for time variable is applied to convert the governing equations into system of ordinary differential equations. The vector-matrix differential equation with the eigenvalues approach method is employed to obtain the solutions of the physical quantities analytically. Laplace transform invers numerically is utilized when some boundary conditions are applied at the semiconductor free surface to obtain the complete general of the main physical quantities. The numerical computations for the input parameters of Silicon as semiconductor material are used to illustrate the obtained results graphically and discussed.

Eigenvalue approach to study Rayleigh waves in nonlocal orthotropic layer lying over nonlocal orthotropic half-space with dual-phase-lag model

In this present article, we deal with the propagation of Rayleigh surface waves in the nonlocal orthotropic thermoelastic layer which is lying over a nonlocal orthotropic thermoelastic half-space in the context of the dual-phase-lag model. This problem is considered in the context of Eringen’s nonlocal elasticity theory. A vector-matrix differential equation is obtained which is solved by an eigenvalue approach. The frequency equation of Rayleigh surface waves is derived and various particular cases are discussed. The different characteristics namely attenuation coefficient, specific loss, penetration depth, and phase velocity are presented graphically with respect to frequency.

State space approach to characterize Rayleigh waves in a layer lying over a half-space with nonlocal thermoelasticity

The paper deals with a new model based on Eringen's nonlocal thermoelasticity. The propagation of Rayleigh waves in a nonlocal thermoelastic layer which is lying over a nonlocal thermoelastic half-space is considered in the context of energy dissipation theory. The normal mode analysis is employed to the considered equations to obtain vector matrix differential equation which is then solved by state space approach.The frequency equation of Rayleigh surface wave is derived. The effect of the nonlocal parameter on phase velocity, attenuation coefficient, specific loss and penetration depth of Rayleigh surface waves is presented graphically.

Rayleigh waves in porous orthotropic medium with phase lags

The present article deals with the propagation of Rayleigh surface waves in homogeneous orthotropic medium. This thermoelastic problem is studied under the purview of three-phase-lag model of hyperbolic thermoelasticity in the presence of voids. The normal mode analysis is employed to obtain a vector matrix differential equation which is then solved by eigenvalue approach. The frequency equations for different cases are derived. In order to illustrate the analytical developments, the numerical solution is carried out and the computer simulated results in respect of phase velocity and attenuation coefficient are presented graphically. Phase velocity and attenuation coefficient decreases in the presence of voids. The present problem is the most general one as other problems can be obtained as special cases from it.

Study of an isotropic slab in context of memory-dependent derivative due to heat source

A model of generalized thermoelasticity is proposed here for an isotropic slab. The governing equations of motion and heat conduction equation are written in context of memory-dependent derivative with two-phase lag variables in the presence of heat source. A vector matrix differential equation is formulated as a compact form of the governing equations in transformed domain. The corresponding solution is obtained with the help of eigenvalue approach and integral transformation. Numerical computations for the important physical variables like displacement, thermal strain, stress components, and temperature distribution with respect to space variables are evaluated and represented graphically.

Surface waves in porous thermoelastic medium with two relaxation times

In this paper, the propagation of Rayleigh waves in homogeneous isotropic thermoelastic medium in the presence of voids with Green-Lindsay model is considered. 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. The phase velocity, attenuation coefficient, specific loss and penetration depth are computed numerically and presented graphically and the effect of the voids on the considered parameters is shown in the figures.

Eigenvalue approach to hyperbolic thermoelastic problem in porous orthotropic medium with Green-Lindsay model

The present article deals with the thermal shock response in homogeneous orthotropic medium under the purview of Green-Lindsay model in the presence of voids. The normal mode analysis is used to obtain vector matrix differential equation which is then solved by eigenvalue approach. In order to illustrate the analytical developments, the numerical solution is carried out and the effect of voids on stresses, displacements and temperature are presented graphically.

Memory-dependent derivative under generalized three-phase-lag thermoelasticity model with a heat source

PurposeThis study aims to attempt to construct a new mathematical model of the generalized thermoelasiticity theory based on the memory-dependent derivative (MDD) considering three-phase-lag effects. The governing equations of the problem associated with kernel function and time delay are illustrated in the form of vector matrix differential equations. Implementing Laplace and Fourier transform tools, the problem is sorted out analytically by an eigenvalue approach method. The inversion of Laplace and Fourier transforms are executed, incorporating series expansion procedures. Displacement component, temperature and stress distributions are obtained numerically and illustrated graphically and compared with the existing literature.Design/methodology/approachThis study is to analyze the influence of MDD of three-phase-lag heat conduction interaction in an isotropic semi-infinite medium. The current model has been connected to generalize two-dimensional (2D) thermoelasticity problem. The governing equations are shown in vector matrix form of differential equation concerning Laplace-transformed domain and solved by using the eigenvalue technique. The combined Laplace Fourier transform is applied to find the analytical interpretations of temperature, stresses, displacement for silicon material in a non-dimensional form. Inverse Laplace transform has been found by applying Fourier series expansion techniques introduced by Honig and Hirdes (1984) after performing the inverse Fourier transform.FindingsThe main conclusion of this current study is to demonstrate an innovative generalized concept for heat conducting Fourier’s law associated with moderation of time parameter, time delay variable and kernel function by applying the MDDs. However, an important role is played by the time delay parameter to characterize the behavioral patterns of the physical field variables. Further, a new categorization for materials may be created rendering to this new idea along MDD for the time delay variables to develop a new measure of its potential to regulate heat in the medium.Originality/valueGeneralized thermoelasticity is hastily undergoing modification day-by-day from basic thermoelasticity. It has been progressed to get over from the limitations of fundamental thermoelasticity, for instance, infinite velocity components of thermoelasticity interference, in the adequate thermoelastic response of a solid to short laser pulses and deprived illustrations of thermoelastic performance at low temperature. In the past few decades, the fractional calculus is used to change numerous existing models of physical procedure, and its applications are used in various fields of physics, continuum mechanics, fluid mechanics, biology, viscoelasticity, biophysics, signal and image processing, control theory, engineering fields, etc.

Finite-Time Stabilization and Destabilization Analysis of Quaternion-Valued Neural Networks with Discrete Delays

In this paper, the finite-time stabilization and destabilization of a class of quaternion-valued neural networks (QVNNs) with discrete delays are investigated. In order to surmount the difficulty of noncommutativity of quaternion, a new vector matrix differential equation (VMDE) is proposed by employing decomposition method. And then, a nonlinear controller is designed to stabilize the VMDE in a finite-time interval. Furthermore, under that controller, the finite-time stability and instability of the QVNNs are analyzed via Lyapunov function approach, and two criteria are derived, respectively; furthermore, the settling time is also estimated. At last, by two illustrative examples we verify the correctness of the conclusions.

Three Dimensional Thermal Shock Problem in Magneto-Thermoelastic Orthotropic Medium

The paper is concerned with the study of magneto-thermoelastic interactions in three dimensional thermoelastic medium under the purview of three-phase-lag model of generalized thermoelasticity. The medium under consideration is assumed to be homogeneous orthotropic medium. The fundamental equations of the three-dimensional problem of generalized thermoelasticity are obtained as a vector-matrix differential equation form by employing normal mode analysis which is then solved by eigenvalue approach. Stresses and displacements are presented graphically for different thermoelastic models.

Three-dimensional nonlocal thermoelasticity in orthotropic medium based on Eringen’s nonlocal elasticity

ABSTRACT The present article deals with thermoelastic boundary value problem in three-dimensional orthotropic medium based on Eringen’s nonlocal elasticity. A time-dependent thermal shock is applied on the traction-free surface of homogeneous orthotropic solid and the problem is treated in the context of Green-Naghdi model type III. The governing equations of three-dimensional problem of generalized thermoelasticity are obtained as a vector matrix differential equation form by employing normal mode analysis to the considered equations which are then solved by eigenvalue approach. The effect of nonlocal parameter on the displacements and temperature are presented graphically.

Effect of hydrostatic pressure and memory effect on magneto-thermoelastic materials with two-temperatures

The present article is aimed at studying the effect of an induced magnetic field in an initially stressed homogeneous, isotropic, thermoelastic half-space. The corresponding mathematical modeling has been formulated in the context the memory-dependent heat transport equation in the context of three-phase lag model of generalized thermoelasticity involving two-temperature theory. The bounding plane of the medium is subjected to different types of prescribed temperature and variable mechanical loading. Employing the Laplace transform, the analytical results for the distributions of the thermophysical quantities have been derived and the corresponding vector-matrix differential equation is solved with the help of state space approach. The numerical inversion of the Laplace transform is carried out with the help of an efficient and pragmatic algorithm based on the Riemann-sum approximation technique. Numerical computations for a copper material is performed and have been demonstrated graphically. The result provides a motivation to investigate the problem and draw conducting remarks due to the influence of the magnetic field, memory effect and time-delay parameter also.

Calculation and Modeling of Transition Processes in DC Resonant Converters of LLC Type

For a linear equivalent circuit of a dc–dc converter of LLC type, a generalized vector-matrix equation is obtained for individual intervals of circuit linearity. When solving the third-order vector-matrix differential equation for an LC circuit based on the method of dividing processes into fast (in the LC circuit) and slow (in the converter output circuit) ones, the output voltage of the converter is assumed to be constant over the circuit half-cycle operation. This voltage is calculated by averaging over the average value of the converter output current obtained in the previous half-cycle. The processes in the dc–dc converter are calculated using the fitting method, in which the value of the LC circuit state vector at the end of the circuit linearity interval is used as its initial value in the next linearity interval. This method includes a standard algorithm for calculating the process over a half-cycle. A half-cycle can consist of several intervals of linearity. The start and end moments of these intervals are determined, as well as the voltage values on the output diagonal of the inverter bridge and on the secondary winding of the output transformer. The integral of the converter output current is calculated. The comparison of the calculated results with the results of simulation in the MATLAB/Simulink environment at the start confirms the correctness of the proposed method for calculating transients.