Green Naghdi Model Research Articles

Finite Crack in a Thermoelastic Transversely Isotropic Medium Under Green-Naghdi Theory

In this paper, we have studied a model of finite linear Mode-I crack in a thermoelastic transversely isotropic medium under Green Naghdi theory. The crack is subjected to a prescribed temperature and a known tensile stress. The plane boundary surface is considered as isothermal and all the field variables are sufficiently smooth. The heat conduction equation is written under two temperature theory (2TT) for Green Naghdi model which contains absolute temperature as well as conductive temperature. The analytical expressions of displacement components, stress components and temperature variables are obtained by normal mode analysis and matrix inversion method. Comparisons have been made within Green Naghdi (G-N) theory of type I, type II and type III for displacement, stress and absolute temperature variables against the crack width for a transversely isotropic material (Cobalt) by virtues of graphs. Also, Comparison have been made among displacement, thermal stress and absolute temperature for different depths.

Thermoelastic diffusion of a solid cylinder in the context of modified Green–Naghdi models

The generalized thermoelastic diffusion model is introduced and applied to an isotropic cylinder. The cylinder is under the effect of thermal flux and chemical potential. Some modifications are made to the Green and Naghdi model. Creative multi-phase-lag models with and without energy dissipation are proposed to discuss the thermoelastic diffusion of the circular solid cylinders. The analytical solutions of thermoelastic diffusion governing equations of the solid cylinders have been deduced. Comparing examples are presented between the simple and refined Green–Naghdi of types II and III models. The validity of results is worthy and benchmark outcomes are accounted for to help different specialists in their future examinations.

Modified Moore–Gibson–Thompson photo-thermoelastic model for a rotating semiconductor half-space subjected to a magnetic field

This paper focuses on studying thermal, elastic and coupled plasma waves, in the sense of a photo-thermal process transport within an infinite semiconductor medium. In order to study photo-thermal interactions in two-dimensional semiconducting materials, a new mathematical model based on the Moore–Gibson–Thompson equation (MGTE) is implemented. The MGTE model involving the Green–Naghdi model of type III as well as the heat transport equation proposed by Lord and Shulman. We consider the semi-conductor half-space is rotated at a uniform angular speed and magnetized. The analysis of the distribution of thermophysical fields has been extracted by a normal mode method, represented graphically and discussed. The results predicted by the new and improved model have been compared with the generalized and classic ones. In addition, all field quantities have been examined for effects of rotation, a lifetime of the photo-generated, and the applied magnetic field.

The propagation of plane waves in nonlocal visco-thermoelastic porous medium based on nonlocal strain gradient theory

The present paper deals with propagation of plane harmonic waves in nonlocal visco-thermoelastic porous medium based on the nonlocal strain gradient theory. For the first time, the assumptions of Green-Naghdi model type III of hyperbolic thermoelasticity with voids in conjunction with nonlocal strain gradient theory are taken into account in formulation of the problem. The constitutive relations and field equations for nonlocal thermoelastic materials with voids are presented. In addition to transverse waves, three longitudinal waves (elastic, thermal, volume fraction) can be observed. The fundamental solution to equations for steady oscillations is derived. The effects of nonlocal length scale parameter, voids, viscosity on phase velocities, attenuation coefficients and penetration depths of waves with diverse frequencies are comprehensively studied and graphically presented.

Thermomechanical Waves in an Axisymmetric Rotating Disk Using Refined Green–Naghdi Models

In this paper, four thermoelasticity models are applied to investigate the thermo-mechanical waves in an axisymmetric rotating disk. For this purpose, some modifications of Green and Naghdi’s generalized thermoelasticity theory have been made. These novel dual- and triple-phase-lag models are presented. The closed-form solution governing equations of thermomechanical waves in the rotating disk has been presented. Some validation results are tabulated to compare the simple and refined dual- and triple-phase lag models based upon the Green–Naghdi theory. The effects of dimensionless time and angular speed on all variables are investigated. The validity of results is acceptable and benchmark results are tabulated to aid other researchers in their upcoming comparisons.

On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

This paper is on arbitrary high order fully discrete one-step ADER discontinuous Galerkin schemes with subcell finite volume limiters applied to a new class of first order hyperbolic reformulations of nonlinear dispersive systems based on an extended Lagrangian approach introduced by Dhaouadi et al. (Stud Appl Math 207:1–20, 2018), Favrie and Gavrilyuk (Nonlinearity 30:2718–2736, 2017). We consider the hyperbolic reformulations of two different nonlinear dispersive systems, namely the Serre–Green–Naghdi model of dispersive water waves and the defocusing nonlinear Schrödinger equation. The first order hyperbolic reformulation of the Schrödinger equation is endowed with a curl involution constraint that needs to be properly accounted for in multiple space dimensions. We show that the original model proposed in Dhaouadi et al. (2018) is only weakly hyperbolic in the multi-dimensional case and that strong hyperbolicity can be restored at the aid of a novel thermodynamically compatible GLM curl cleaning approach that accounts for the curl involution constraint in the PDE system. We show one and two-dimensional numerical results applied to both systems and compare them with available exact, numerical and experimental reference solutions whenever possible.

Thermal diffusion of an unbounded solid with a spherical cavity via refined three-phase-lag Green–Naghdi models

This article deals with thermal diffusion vibration analysis in an unbounded medium including a spherical cavity. Multi single/dual-phase-lag models based upon the Green–Naghdi theory are obtained to discuss the thermoelastic diffusion of the spherical cavity. The surface of the cavity is traction and chemical potential free, not isolated, and subjected to a flux varying harmonically with time. A comparison of results is made, and the validity of results is acceptable. Benchmark outcomes are tabulated to aid other researchers in their future investigations.

Thermo-mechanical disturbances in a nonlocal rotating elastic material with temperature dependent properties

Purpose The purpose of this paper is to study two-dimensional deformations in a nonlocal, homogeneous, isotropic, rotating thermoelastic medium with temperature-dependent properties under the purview of the Green-Naghdi model II of generalized thermoelasticity. The formulation is subjected to a mechanical load. Design/methodology/approach The normal mode analysis technique is adopted to procure the exact solution of the problem. Findings For isothermal and insulated boundaries, discussions have been made to highlight the influences of rotational speed, nonlocality, temperature-dependent properties and time on the physical quantities. Originality/value The exact expressions for the displacement components, stresses and temperature field are obtained in the physical domain. These are also calculated numerically for a magnesium crystal-like material and depicted through graphs to observe the variations of the considered physical quantities. The present study is useful and valuable for the analysis of problems involving mechanical shock, rotational speed, nonlocal parameter, temperature-dependent properties and elastic deformation.

On the Green-Naghdi equations with surface tension in the Camassa-Holm scale

The aim of this paper is to analyze the water waves problem for uneven bottom under the influence of surface tension. For that, we consider an asymptotic model of the 1D Green-Naghdi equations in Camassa-Holm scale and derive in a formal way by using the Whitham technique the Camassa-Holm equation under the influence of surface tension. After that, the well-posdeness of the obtained Camassa-Holm equation is proved by using the Picard iterative scheme which proves that there is no loss of regularity of the solution relative to the initial condition. Also, the $$H^s$$ -consistency, the stability and the convergency of the solution with the 1D Green-Naghdi model in Camassa-Holm scale are showed. Finally, the aspect of breaking wave for the Camassa-Holm equation is discussed in the presence of surface tension.

Transversely isotropic Euler Bernoulli thermoelastic nanobeam with laser pulse and with modified three phase lag Green Nagdhi heat transfer

This investigation aims to examine the vibration phenomenon in 2D transversely isotropic homogeneous Euler-Bernoulli nonlocal nanobeam with laser pulse with new modified three-phase lag Green Naghdi (TPL GN) model. The model contains a material length scale parameter that can capture the size effect, using the nonlocal theory of thermoelasticity. Temperature is assumed to vary sinusoidally. Laplace Transforms are used to derive the non-dimensional expressions for lateral deflection, axial displacement, temperature distribution, axial stress, and thermal moment in the transformed domain, and numerical inversion techniques are used to find the expressions in the physical domain. The ends of the nanobeam are considered to be simply supported and have a constant temperature. Effect of new modified TPL GN heat transfer and nonlocal parameter is represented graphically for lateral deflection, axial displacement, temperature distribution, axial stress, and thermal moment using the MATLAB software. Few specific cases are also derived.

Influence of several fields on Rayleigh waves propagation in a fiber-reinforced orthotropic half-space material under four thermoelastic models

The objective of this paper is to investigate the fiber-reinforced with effect of rotation, initial stress, heat flux, and gravity field on Rayleigh waves in electro-magnetoelastic homogeneous orthotropic material half-space under four models of generalized thermoelasticity: (i) classical and dynamical coupled model (CD), (ii) the Lord–Shulman model (LS), (iii) the Green–Lindsay model (GL), and (iv) the Green–Naghdi model (GN of type II) is devoted to study the problem. Lame’s potentials are used to obtain the frequency equation, which determines the velocity of Rayleigh waves (real part) and the attenuation coefficient (imaginary part) of the determinant under rotation, magnetic field, electric field, heat flux, initial stress, and gravity field. The numerical results have been obtained and illustrated graphically for each case. A comparison of the results was obtained with the previous theories of thermoelasticity in the absence of fiber-reinforced. The results obtained indicated the significant effect of the fiber-reinforced on the propagation of surface waves that has good contributions to industries, engineering, structures, and other topics related to the phenomenon.

Surface waves in piezothermoelastic transversely isotropic layer lying over piezothermoelastic transversely isotropic half-space

In this article, the propagation of Rayleigh surface waves in a piezothermoelastic transversely isotropic layer lying over a piezothermoelastic transversely isotropic half-space is investigated in the context of the Green–Naghdi model type III of hyperbolic thermoelasticity. The secular equation of Rayleigh surface waves is derived, and different cases are discussed. Phase velocity, attenuation coefficient and specific loss of surface waves are computed and presented graphically with respect to frequency, and a comparison of different wave characteristics for classical and generalized thermoelastic models is presented in the figures.

A unified formulation for thermoviscoelasticity of hollow sphere based on the second sound theories

By proposing the unifier factors, a unified formulation for the generalized coupled thermo-visco-elasticity response of hollow spheres based on the Lord–Shulman, Green–Lindsay, and Green–Naghdi models is developed. Kelvin–Voigt model is used in the current research to consider the viscoelastic response of the sphere. This formulation is used to capture the thermoviscoelastic response of finite isotropic hollow sphere under thermal shock. After transforming the one-dimensional radial equations into the non-dimensional form, finite element method (FEM) based on the Galerkin weak formulation is implemented. The governing equations are discretized through the radial domain utilizing FEM. The determined system of ordinary differential equations is traced using the average acceleration method as a type of Newmark time marching scheme. After validation, novel numerical results are demonstrated to analyze the influences of viscoelastic damping on the response of hollow sphere under rapid surface heating based on various types of generalized theories.

Electro–magneto–thermoelastic interactions in initially stressed orthotropic medium with Green–Naghdi model type-III

This article is concerned with a study of electro–magneto–thermoelastic interactions under Green–Naghdi theory-III of generalized thermoelasticity in the presence of initial stress. The fundamental equations of the two-dimensional problem in orthotropic medium under the influence of electric and magnetic field are obtained in the form of a vector–matrix of differential equation by employing normal mode analysis which is then solved by eigenfunction expansion method. The solution for temperature distribution, displacements, and stress components is obtained. The effect of magnetic field, electric field and phase velocity on the considered parameters is observed in the figures.

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.

Types of flow in Green-Naghdi Thermoelasticity Theory

In this paper we will show a review of the Green-Naghdi thermoelasticity theory.Such model have beautiful foundations which contains since the law of heat conduction proposed byFourier, fundamentals variables such as Helmholtz free energy, entropy, flux of heat and etc. and the constitutive hypothesis. The nature of partial differential equation of parabolic (PDE) type derived from classical thermoelasticity lead us to a mathematical inconsistency well-know as thermal signal speed paradox. We will show how Green-Naghdi model solved those mathematical inconsistency introducing different types of flow and giving us a hyperbolic PDE.

Refined dual-phase-lag Green–Naghdi models for thermoelastic diffusion in an infinite medium

This article has been devoted to investigating the thermoelastic diffusion problem of an infinite medium. The simple and refined single/dual-phase-lag Green–Naghdi models with and without energy dissipation are proposed. The exact solution of thermodiffusion coupling equations has been obtained considering the regularity conditions. The distributions of the non-dimensional temperature, dilatation, displacement, stresses, concentration and chemical potential are obtained and illustrated graphically. The validity of results is acceptable by comparing all outcomes. Benchmark results are reported to help other investigators in their future comparisons.

On solitary wave in nonuniform shear currents

In this paper, steady solutions of solitary waves in the presence of nonuniform shear currents are obtained by use of the high-level Green-Naghdi (HLGN) model. We focus on large-amplitude solitary waves in strong opposing shear currents. The linear-type currents, quadratic-type currents and cubic-type currents are considered. In particular, the wave speed, wave profile, velocity field, particle trajectories and vorticity distribution are studied. It is demonstrated that presence of the nonuniform shear current modifies the velocity field and vorticity field of the solitary wave.

Rayleigh waves in a nonlocal thermoelastic layer lying over a nonlocal thermoelastic half-space

The paper deals with Rayleigh wave propagation in a nonlocal thermoelastic layer, and the layer is lying over a nonlocal thermoelastic half-space. The problem is treated in the context of Eringen’s nonlocal thermoelasticity and Green–Naghdi model type III of hyperbolic thermoelasticity. The frequency equation of Rayleigh waves is derived, and different cases are also discussed. The effect of the nonlocal parameter on phase velocity, attenuation coefficient, specific loss, and penetration depth is presented graphically.

Hyperbolic model of internal solitary waves in a three-layer stratified fluid

We derive a new hyperbolic model describing the propagation of internal waves in a stratified shallow water with a non-hydrostatic pressure distribution. The construction of the hyperbolic model is based on the use of additional instantaneous variables. This allows one to reduce the dispersive multi-layer Green–Naghdi model to a first-order system of evolution equations. The main attention is paid to the study of three-layer flows over an uneven bottom in the Boussinesq approximation with the additional assumption of hydrostatic pressure in the intermediate layer. The hyperbolicity conditions of the obtained equations of three-layer flows are formulated, and solutions in the class of travelling waves are studied. Based on the proposed hyperbolic and dispersive models, numerical calculations of the generation and propagation of internal solitary waves are carried out and their comparison with experimental data is given. Within the framework of the proposed three-layer hyperbolic model, a numerical study of the propagation and interaction of symmetric and non-symmetric soliton-like waves is performed.