Green Naghdi Model Research Articles

Memory Response of Refined GN Models on a Rotating Fiber-Reinforced Magneto-Thermoelastic Medium Due to Pulsed Laser and Inclined Load

With the aid of the Caputo fractional derivative, this study constructs a novel mathematical model of magneto-thermoelasticity to investigate the transient phenomena for a fiber-reinforced rotating half-space under the influence of pulsed laser as heat source and inclined load in the context of refined three-phase-lag (TPL) Green–Naghdi (GN) models of generalized thermoelasticity, which is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. Together with the temperature gradient (Seebeck effect) and charge density influence, the generalized magneto-thermoelasticity model’s equations now include the modified Ohm’s law. An intense non-Gaussian laser beam heats the bounding plane’s surface. By using the normal mode analysis, the analytical expressions of the thermophysical quantities are produced in the physical domain. All physical quantities have been graphically depicted for different GN models (refined GN-II and GN-III models) to indicate the effect of memory-dependent derivative, Seebeck parameter, rotation and reinforcement of the medium under the influence of inclined load and pulsed laser.

Analysis of an initially stressed functionally graded thermoelastic medium (type III) without energy dissipation

The current investigation deals with the deformation of a non-homogeneous thermoelastic half space under hydrostatic initial stress for the Green–Naghdi model III. The medium is supposed to be rotating with a constant angular velocity. The non-homogeneous properties of the material are along the x-direction. At the first instance, the problem has been solved analytically to obtain stress and displacement components. Further, the numerical values of these expressions are evaluated using a computer program for a particular medium. The numerical values obtained are then presented graphically to show the effect of initial stress parameter and non-homogeneity parameter on the quantities.

Thermoelastic diffusion based on a nonlocal three-phase-lag diffusion model with double porosity structure

This work aims to develop a new diffusion-elasticity-based model using Eringen’s nonlocal elasticity theory under the purview of the three-phase-lag (TPL) model of hyperbolic thermoelasticity. This problem is treated in the context of a double porosity structure in a homogeneous, isotropic thermoelastic medium. By considering the phase laggings of the diffusion flux vector and using nonlocal continuum mechanics, new constitutive and field equations are derived. The problem is solved by employing the normal mode analysis technique which gives the exact results of all the physical variables. To illustrate the theoretical results, different physical quantities are calculated numerically and presented graphically with respect to distance and time. The influences of different thermoelastic models like TPL (three-phase-lag model), GN-III (Green-naghdi model), DPL (dual-phase-lag model), LS (Lord-Shulman model), and CT (coupled thermoelasticity) on the physical quantities are shown graphically. Additionally, the effects of nonlocal parameters, double voids, and diffusion on the physical quantities are represented graphically. Some limiting and particular cases are also derived from the governing equations and those results have been compared with the existing results available in the literature. Some 3D surface curves are also presented to study the impact of diffusion, nonlocality, and double voids on various physical quantities.

High-level Green–Naghdi model for large-amplitude internal waves in deep water

In this paper, a high-level Green–Naghdi (HLGN) model for large-amplitude internal waves in a two-layer fluid system, where the upper-fluid layer is of finite depth and the lower-fluid layer is of infinite depth, is developed under the rigid-lid free-surface approximation. The equations of the present HLGN model follow Euler's equations under the sole assumption that the horizontal and vertical velocity distributions along the vertical column are presented by known shape functions for each layer. The linear dispersion relations of the HLGN model for different levels are presented and compared with those obtained by other strongly nonlinear models for deep water, including the fully nonlinear models that include the dispersion effects $O(\mu )$ (where $\mu$ is the ratio of the upper-fluid layer depth to a typical wavelength) derived by Choi & Camassa (Phys. Rev. Lett., vol. 77, 1996, pp. 1759–1762) and $O(\mu ^2)$ derived by Debsarma et al. (J. Fluid Mech., vol. 654, 2010, pp. 281–303). It is shown that the HLGN model has a wider application range than other models. Solutions of travelling large-amplitude internal solitary waves in the absence and presence of background shear-current are then investigated by using the HLGN model. For the no-current cases, results obtained by the HLGN model show better agreement with Euler's solution on wave profile, velocity profile at the maximum interface displacement and wave speed compared with those obtained by other models. For the background shear-current cases, results obtained by the HLGN model also show good agreement with those obtained by solving the Dubreil-Jacotin–Long equation.

On the two-layer high-level Green-Naghdi model in a general form

On the two-layer high-level Green-Naghdi model in a general form

Thermoelastic Interaction in a Functionally Graded Medium Due to Refined Three-Phase-Lag Green–Naghdi Model under Gravitational Field

This study investigates the wave propagation of a two-dimensional functionally graded material (FGM) thermoelastic half-space under the influence of inclined load, gravitational force and thermal loading from a laser pulse in the context of refined three-phase-lag Green–Naghdi (GN) model. Normal mode analysis has been used to obtain the analytical expression of the physical quantities. All the outcomes of the physical quantities have been illustrated graphically to analyze the effect of non-homogeneity, angle of inclination of the inclined load and the gravitational force for different GN models (simple and refined GN-II and GN-III models).

Scattering of Long Waves by Freely Oscillating Submerged Plates

Abstract We consider a horizontal, submerged plate in shallow water that is allowed to oscillate in the vertical direction due to the wave loads. The plate is attached to a linear spring and damper to control the oscillations. The focus of the study is on the transformation of the wave field by the submerged oscillating plate. To estimate energy scattering, wave reflection and transmission coefficients are determined from four wave gauges: two placed upwave and two placed downwave of the oscillating plate. The flow is governed by the nonlinear level I Green–Naghdi (GN) equations, coupled with the equations of the vertical oscillations of the plate. Time series of water surface elevation recorded at gauges upwave and downwave of the plate obtained by the GN model are compared with the available laboratory experiments and other data, and very good agreement is observed. Wave reflection and transmission coefficients are then determined for a range of involved parameters, including wave conditions (wavelength and wave height), initial submergence depth of the plate, plate length, and the spring-damper system attached to the plate. It is found that a submerged oscillating plate can have a remarkable effect on the wave field and that nonlinearity plays an important role in this wave–structure interaction problem. Discussion is provided on how the wave reflection and transmission coefficients vary with the wave conditions, plate characteristics, initial submergence depth, and spring-damper system properties.

Influence of the fractional-order strain on an infinite material with a spherical cavity under Green-Naghdi hyperbolic two-temperature thermoelasticity theory

In this work, a novel mathematical model of thermoelastic, homogenous, isotropic, and infinite medium with a spherical cavity has been constructed. Under the hyperbolic two-temperature Green-Naghdi theory of thermoelasticity type-I and type-III with fractional-order strain, the governing equations have been established. The bounding surface of the cavity has been thermally loaded by a ramp-type heat and is connected to a rigid foundation which prevents volumetric strain. Different values of the fractional-order and two-temperature parameters have shown numerical results for the dynamical and conductive temperature increment, strain, displacement, and average of principal stresses, which are graphically applicable to all the functions studied. The fractional-order parameter has significant effects on stress and strain distributions, while it has a limited effect on the dynamical and conductive temperatures increment. The hyperbolic two-temperature parameter has significant effects on all studied functions based on Green-Naghdi models of type-1 and type-II. Moreover, the ramp-time heat parameter has a significant impact on all the studied functions under all the studied models of thermoelasticity.

State space approach to study thermal shock problem in nonlocal thermoelastic medium with double porosity

In this article, a two-dimensional problem is studied in a homogeneous isotropic thermoelastic medium with double porosity in the context of nonlocal thermoelasticity and the problem is treated in the presence of the Green–Naghdi model of type-III. Constitutive relations and field equations are derived and the state space approach is employed to solve this problem. Thermal shock is applied on the boundary of the surface. From these relations and expressions, we have calculated stress components, displacement components, and temperature for different values of distance and time numerically and plotted graphically to demonstrate and compare theoretical results. The influences of void parameters, locality, and nonlocality on the various physical quantities are also analyzed. Some particular and special cases are also deduced from this study and compared with the existing results.

Dynamic crack propagation under thermal impact

This paper addresses the dynamic crack growth in an isotropic media subjected to a non-classical thermal shock by implementing the Green-Naghdi (GN) generalized thermoelasticity theory in the eXtended Finite Element Method (XFEM). To satisfy the crack tip singularity for modeling dynamic crack growth via XFEM, proper enrichment functions for field variables are needed, which are extracted from the asymptotic fields for a dynamically growing crack in a thermoelastic media. These near tip fields of a dynamically moving crack are also used in the formulations of the maximum circumferential stress criterion and the equivalent stress intensity factor. Finally, numerical examples of the dynamic crack propagation under thermal shock are given out, and the effect of the dissipation coefficient in the GN model and the thermal shock amplitude on the crack growth path is studied in detail. In addition, the crack propagation under GN thermal shock is compared with that under classic thermal shock. The main difference between them is discussed in a numerical example.

Numerical scheme for the generalised Serre–Green–Naghdi model

A numerical scheme for solving the recently derived generalised Serre–Green–Naghdi equations which produce a family of equations modelling waves in shallow water with varying dispersion relationships, is described. The numerical scheme extends schemes applied to the classical Serre–Green–Naghdi equations written in conservation law form and is the first validated scheme for all admitted dispersion relationships. The consistency of the described numerical scheme for all members of the family of equations is in contrast to other numerical schemes which require specialised modifications for individual members of the family of equations. A typical second-order implementation of this numerical scheme is then demonstrated and validated using known analytic solutions; the travelling solitary wave and the dam-break problem. The numerical method is further validated for general members of the family of equations using forced solutions. Select members of this family are then used to simulate the evolution of rectangular waves of depression for which there are experimental results. The numerical method is shown to be conservative, robust and second-order accurate for the entire family of equations. The validated numerical solutions support a classification of the family of equations based on their linear dispersive properties which includes advancing dispersive wave trains in contrast to classical trailing dispersive wave trains.

Parameter Sensitivity for Wave-Breaking Closures in Boussinesq-Type Models

We consider the issue of wave-breaking closure for the well-known Green–Naghdi model and attempt at providing some more understanding of the sensitivity of some closure approaches to the numerical set-up. More precisely and based on Kazolea and Ricchiuto (Ocean Model 123:16–39, 2018), we used two closure strategies for modeling wave-breaking of a solitary wave over a slope. The first one is the hybrid method consisting of suppressing the dispersive terms in a breaking region and the second one is an eddy viscosity approach based on the solution of a turbulent kinetic energy model. The two closures use the same conditions for the triggering of the breaking mechanisms. Both the triggering conditions and the breaking models themselves use case depended/ad/hoc parameters which are affecting the numerical solution while changing. The scope of this work is to make use of sensitivity indices computed by means of analysis of variance to provide the sensitivity of wave-breaking simulation to the variation of parameters such as the mesh size and the breaking parameters specific to each breaking model. The sensitivity analysis is performed using the UQlab framework for uncertainty quantification (Marelli et al., UQLab user manual—sensitivity analysis, Technical report, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Switzerland, 2019).

Bore Impact on Decks of Coastal Structures

Bore impact on horizontal fixed decks of coastal structures is studied by use of the Level I Green–Naghdi (GN) equations and the Navier–Stokes (NS) equations. The bore is generated by the breaking of a water reservoir, and may represent the propagation of a tsunami on land or broken storm waves. The bore-induced horizontal and vertical forces are determined and their variation with the bore and deck conditions is studied in this work. Various conditions of deck location with respect to the water level are considered, including cases with the deck under or above the still-water level. Two types of bore are considered, namely (i) a bore generated by a dam break, where the reservoir water depth is substantially larger than the downstream depth, and (ii) a bore generated by an initial mound of water, where the reservoir water depth is comparable to the downstream depth. It is shown that these mechanisms result in the formation of significantly different bore shapes. It is also shown that the relative height of the reservoir and the downstream water depth play a significant role in the bore generation and its impact on coastal structures. It is also found that the bore-induced forces vary almost linearly with the change in amplitude of the reservoir, while a change in the length of the reservoir has little effect on the loads. The horizontal force on submerged decks is shown to be independent of the submergence depth of the deck; this is due to the uniform velocity distribution over the water column of the bore. Results of the GN and NS models are compared with each other for submerged cases and the limitations, accuracy, and efficiency of these models in studying this problem are discussed. Results of the GN equations are in close agreement with the NS equations, making them a computationally efficient alternative for the study of this problem.

Eigenvalues approach on thermo-elastic diffusions problem for an infinite material containing spherical holes

In this article, the thermoelastic diffusion interaction in an unbounded medium containing spherical holes in the context Green-Naghdi model of type III are studied. The inner surface of hole is taken to be tractions free and due to an exponent of the decayed heat flux. The Laplace transforms techniques are used. The eigenvalues approaches are adopted for the solutions of applications for infinite media with a spherical hole. The numerical results for the concentration, the displacement, the chemical potential, the temperature, and the stress are presented analytical and graphically.

The patterns of traveling wave on shallow water modeled by Green-Naghdi equation

The Green-Naghdi equations are a shallow water waves model which play important roles in nonlinear wave fields. By using the trial equation method and the Complete discrimination system for the polynomial we obtained the classification of travelling wave patterns. Among those patterns, new singular patterns and double periodic patterns are obtained in the first time. And we draw the graphs which help us to understand the dynamics behaviors of the Green-Naghdi model intuitionally.

Thermo-Optical Mechanical Waves in a Rotating Solid Semiconductor Sphere Using the Improved Green–Naghdi III Model

The current study investigates thermophotovoltaic interactions using a new mathematical model of thermoelasticity established on a modification of the Green–Naghdi model of type III (GN-III). The basic equations, in which the heat transfer is in the form of the Moore–Gibson–Thompson (MGT) equation, are derived by adding a single delay factor to the GN-III model. The impact of temperature and electrical elastic displacement of semiconductors throughout the excited thermoelectric mechanism can be studied theoretically using this model. The proposed model was used to investigate the interactions between the processes of thermoelastic plasma in a rotating semiconductor solid sphere that was subjected to a thermal shock and crossed to an externally applied magnetic field. The influence of rotation parameters on various photothermal characteristics of silicon solid was presented and explored using the Laplace technique.

The vibration of thermoelastic silicon nitride Nanobeam based on green-naghdi theorem type-II subjected to mechanical damage and ramp-type heat

In this work, an analysis for thermoelastic homogeneous isotropic nanobeams under damage mechanics consideration was built. Under easily supported boundary conditions with fixed side ratios, the Green-Naghdi model type-II, an extended thermoelasticity theory model, has been utilized. For the governing differential equations, the Laplace transforms technique was used on the time variable. The answers were found in the domain of the Laplace transform. Tzou’s approximation approach based on an iteration formula was used to calculate the Laplace transform inversions numerically. The numerical findings for a rectangular silicon nitride thermoelastic nanobeam have been obtained and validated. As a case study, we assumed that the beam is thermally loaded with ramp-type heat and that its two edges are simply supported. Figures representing different scenarios have been used to display the numerical results. Mechanical damage value, ramp-time heat parameter and beam thickness are all reported to have a substantial influence on all of the examined functions.

Numerical Simulations of a Dispersive Model Approximating Free-Surface Euler Equations

In some configurations, dispersion effects must be taken into account to improve the simulation of complex fluid flows. A family of free-surface dispersive models has been derived in Fernández-Nieto et al. (Commun Math Sci 16(05):1169–1202, 2018). The hierarchy of models is based on a Galerkin approach and parameterised by the number of discrete layers along the vertical axis. In this paper we propose some numerical schemes designed for these models in a 1D open channel. The cornerstone of this family of models is the Serre – Green-Naghdi model which has been extensively studied in the literature from both theoretical and numerical points of view. More precisely, the goal is to propose a numerical method for the LDNH_2 model that is based on a projection method extended from the one-layer case to any number of layers. To do so, the one-layer case is addressed by means of a projection-correction method applied to a non-standard differential operator. A special attention is paid to boundary conditions. This case is extended to several layers thanks to an original relabelling of the unknowns. In the numerical tests we show the convergence of the method and its accuracy compared to the LDNH_0 model.

A study on thermoelastic interactions in fiber-reinforced mediums containing spherical cavities

This research pertains to the evaluation of thermoelastic interactions in fiber-reinforced material with spherical cavities. In the context of Green–Naghdi model (GNIII), the numerical solutions are obtained with the use of the finite element scheme. The discretization of the structure is done by the finite element method where the element is quadratic. The internal surface of the spherical hole is loaded by a uniform step in temperature. The results are graphically presented to show the effects in the absence and presence of reinforcement.

Thermo-viscoelastic orthotropic constraint cylindrical cavity with variable thermal properties heated by laser pulse via the MGT thermoelasticity model

Abstract In the past few decades, many models have been proposed to address the shortcomings found in the classical theories of thermoelasticity and to allow limited speeds of heat waves. In this context, in the current paper a new generalized model of thermoelasticity based on the Moore–Gibson–Thompson (MGT) equation has been introduced. This new model can be derived by introducing the relaxation time factor into the third type of Green–Naghdi model (GN-III). In contrast to the previous works, it was taken into account that the physical properties of the material are dependent on temperature and on the viscous type. The viscoelastic medium has been assumed to obey the Kelvin–Voigt model. On the basis of the present model, thermo-viscoelastic interactions have been investigated in an unbounded orthotropic body with a cylindrical cavity. The surface of the cavity is restricted and exposed to a pulse-formed heat flow that dissolves exponentially. The characteristic thermal modulus of the material is assumed to be a linear function of temperature. The Laplace transform can be used to eliminate time dependency from control equations. Using a suitable approximate method, the transformed equations have been finally inverted by numerical inversion of the Laplace transform. Certain comparisons have been introduced to estimate the effects of the viscosity, pulsed heat, and thermal temperature-independent properties on all studied fields. A comparison with previous models of thermoelasticity is also performed in tables to verify the accuracy of the proposed model. We found from the results that the physical fields strongly depend on the viscoelastic parameter, the change of the thermal conductivity, and pulsed heat, so it is not possible to neglect their effect on the manufacturing process of machines and devices.