Heat Flux Vector Research Articles

A thermodynamically consistent formulation of the Johnson–Cook model

The model of Johnson and Cook, which includes a viscoplastic flow rule and a damage criterion, is widely used to describe the mechanical behaviour of metallic materials subjected to severe loading conditions, such as those encountered during fabrication operations or impact. This model has been built on empirical, rather than physical, grounds. The present paper therefore aims at revisiting the model of Johnson and Cook from the view point of thermodynamics with internal variables. The interest of this approach is twofold. First, it provides a guide for the construction of a complete thermomechanical constitutive model, with some constitutive relations not only for the stress tensor but also specific internal energy, specific entropy and heat flux vector. Second, it allows highlighting some possible limitations of the original model of Johnson and Cook. Such limitations can be circumvented with an alternative model, which is described in the present work. For illustration purpose, some applications of both the original and alternative models are presented in the final section.

Finite element analysis of effect of grain orientation on the thermal conduction of h-BN ceramics

Study of thermal conduction process of ceramics by experimental and theoretical model analysis methods is complicated and non-intuitive. In this work, to analyze the influence mechanism of h-BN grain orientation on the thermal conduction of h-BN ceramics, new mesoscale finite element models considering the anisotropy in the thermal conductivity of h-BN grain were established based on random sequential adsorption method and periodic boundary condition. And the accuracy of the finite element models was verified by theoretical model and experimental results. The heat flux vector and temperature distributions intuitively show that the orientation of h-BN grains will significantly affect the thermal conduction process of the ceramics by affecting the heat flow path. The heat flow will propagate mainly along the “thermal conducting pathways” formed by the h-BN grains when the h-BN layer direction shows small angle with the external temperature difference direction, otherwise, h-BN grains will act as barriers to the heat conduction.

Studies of weak external force on thermal conductivity of complex liquids using homogenous perturbed simulations

A modified Evan-Gillan homogenous perturbed scheme was employed to measure the thermal response and corresponding normalized plasma thermal conductivity (λ0) of strongly coupled dusty plasmas (SCDPs). Homogenous perturbed molecular dynamics (HPMD) approach was employed to analyze the efficiency and compared the obtained outcomes of perturbed heat flux vector to the outcomes computed through equilibrium molecular dynamics (EMD) approach. Our new outcomes show that the thermal response of heat energy current through HPMD and EMD approaches is an excellent agreement with each other for SCDPs at much low value of normalized external perturbation (F*). The obtained HPMD simulations demonstrate that the presented approach provides good outcomes with fast convergence and small system size for low-tointermediate plasma couplings (Γ) at very weak F*. It was found that the measured plasma λ0 outcomes at nearly equilibrium weak F* (= 0.00005) are in reasonable agreement with earlier results obtained from EMD, homogenous and inhomogeneous nonequilibrium molecular dynamics simulations and theoretical predictions and generally overpredicted the plasma λ0 by ~1%–~24% depending on the arrangement of plasma parameters (Г, к). It was shown that the HPMD is an excellent approach to calculate the plasma λ0 and to recognize the elementary patterns in 3D SCDPs.

A novel generalized thermoelasticity with higher-order time-derivatives and three-phase lags

PurposeIn this work, a modified thermoelastic model of heat conduction, including higher order of time derivative, is constructed by extending the Roychoudhuri model (TPL) (Choudhuri, 2007). In this new model, Fourier’s law of heat conduction is replaced by using Taylor series expansions, including three different phase lags for the heat flux, the thermal displacement and the temperature gradient. The generalized thermoelasticity models of Lord–Shulman (Lord and Shulman, 1967), Green and Naghdi (1991), dual-phase lag (Tzou, 1996) and three-phase lag (TPL) (Choudhuri, 2007) are obtained as special cases. The paper aims to discuss these issues.Design/methodology/approachThe aim of this work is to establish a new generalized mathematical model of thermoelasticity that includes TPL in the vector of heat flux, and in the thermal displacement and temperature gradients extending TPL model (Li et al., 2019e). In this model, Fourier law of heat conduction is replaced by using Taylor series expansions to a modification of the Fourier law with introducing three different phase lags for the heat flux vector, the temperature gradient, and the thermal displacement gradient and keeping terms up with suitable higher orders.FindingsThe established high-order three-phase-lag heat conduction model reduces to the previous models of thermoelasticity as special cases.Originality/valueIn this paper, a TPL thermoelastic model is developed by extending the Roychoudhuri (Sherief and Raslan, 2017) model (TPL) considering the Taylor series approximation of the equation of heat conduction. This model is an alternative construction to the TPL model. The new model includes high order of TPL in the vector of heat flux, and in the thermal displacement and temperature gradients.

An introductory guide to fluid models with anisotropic temperatures. Part 1. CGL description and collisionless fluid hierarchy

We present a detailed guide to advanced collisionless fluid models that incorporate kinetic effects into the fluid framework, and that are much closer to the collisionless kinetic description than traditional magnetohydrodynamics. Such fluid models are directly applicable to modelling the turbulent evolution of a vast array of astrophysical plasmas, such as the solar corona and the solar wind, the interstellar medium, as well as accretion disks and galaxy clusters. The text can be viewed as a detailed guide to Landau fluid models and it is divided into two parts. Part 1 is dedicated to fluid models that are obtained by closing the fluid hierarchy with simple (non-Landau fluid) closures. Part 2 is dedicated to Landau fluid closures. Here in Part 1, we discuss the fluid model of Chew–Goldberger–Low (CGL) in great detail, together with fluid models that contain dispersive effects introduced by the Hall term and by the finite Larmor radius corrections to the pressure tensor. We consider dispersive effects introduced by the non-gyrotropic heat flux vectors. We investigate the parallel and oblique firehose instability, and show that the non-gyrotropic heat flux strongly influences the maximum growth rate of these instabilities. Furthermore, we discuss fluid models that contain evolution equations for the gyrotropic heat flux fluctuations and that are closed at the fourth-moment level by prescribing a specific form for the distribution function. For the bi-Maxwellian distribution, such a closure is known as the ‘normal’ closure. We also discuss a fluid closure for the bi-kappa distribution. Finally, by considering one-dimensional Maxwellian fluid closures at higher-order moments, we show that such fluid models are always unstable. The last possible non Landau fluid closure is therefore the ‘normal’ closure, and beyond the fourth-order moment, Landau fluid closures are required.

Three-phase-lag thermoelastic heat conduction model with higher-order time-fractional derivatives

In the last few years, the theory of fractional calculus has been successfully used in thermoelasticity theories and many models of thermoelasticity with fractional order are established by several authors. In the present article, a new model of three-phase-lag thermoelastic heat conduction of higher-order time-fractional derivatives has been derived based on fractional calculus. Using the approach of the Taylor series expansion of time-fractional order developed by Jumarie (Comput Math Appl 59:1142, 2010), an alternative construction model is established extending Ezzat and others (Arch Appl Mech 82:557, 2012) and Roychoudhuri (J Therm Stress 30:231, 2007) models. This new model includes high-order time-fractional derivative approximations of three-phase-lags in the heat flux vector, the temperature gradient and in the thermal displacement gradient. We applied the resulting formulation to an infinite non-homogeneous orthotropic thermoelastic functionally graded medium having a spherical cavity with a power-law distribution of material properties along the radial direction. The effects of high-order time-fractional derivative parameters and non-homogeneity index on various distributions are discussed in detail and represented graphically and tabular forms. Finally, to illustrate the validity and accuracy of the proposed model, a comparison was made with various previous models, which are considered as special cases of our model.

Enhancing thermal conductivity of Diamond/Cu composites by regulating distribution of bimodal diamond particles

Copper matrix composites reinforced with bimodal diamond particles were prepared via pressure infiltration. The effects of diamond particles mixing mode on the microstructure and thermal conductivity of the composites were analyzed. The heat flux vector distributions of the composites with or without the secondary diamond were simulated using finite element models based on the experimental microstructure images. The results show that the thermal conductivity of Diamond/Cu composites under different diamond particle mixing schemes all exceeded 700 W/mK. The obtained maximum thermal conductivity of Diamond/Cu composite is 853 W/mK when 25 vol% of secondary diamond particles with a particle size ratio of 10 were introduced. Such a result is explained by the optimization effect of secondary diamond particles on heat conduction channel and interfacial thermal resistance. Furthermore, the simulation results also prove that the introduction of secondary diamond can effectively improve the heat flux conduction along the diamond reinforced phase.

Theoretical investigation on the thermo-mechanical responses of the human skin during thermal therapy

Laser ablation is an efficient technique of thermal treatment for tumor, during which process, temperature of the tumor tissue will be elevated by the laser beam. Irreversible burn and thermal deformation appear in the tissue, which can induce the protein denaturation and cell membrane looseness to kill the cancer cells. In order to investigate the thermo-mechanical responses of the skin during therapy process, a mechanical model consisting of an elastic plate resting on viscoelastic foundation was proposed to describe the irradiated skin tissue which is subjected to a moving laser beam. The DPL bio-heat conduction model, Kirchhoff hypothesis and the Kelvin-Voigt model were employed in the present study. The governing equations were analytically solved with the usage of Green's function method and the closed-form solutions for the temperature, deformation and stress distribution in the tissue were obtained. A numerical example was given to illustrate the analytical solutions and representative results were shown in figures to make it more intuitive. The influences of laser moving speed and the phase lags of heat flux vector and temperature gradient were discussed in detail.

Two temperature fractional order thermoelasticity theory in a spherical domain

This article is an application of fractional thermoelasticity in association with two-temperature theory. The fractional heat conduction model has been proposed to investigate the thermal variations within the bounded spherical region. The corresponding heat conduction equation has been derived in the context of the generalized two-temperature theory of fractional thermoelasticity. The analytical solutions of thermal variations have been obtained in the Laplace domain, which are inverted using the Gaver–Stehfest algorithm in the time domain. Kuznetsov convergence criterion has been discussed for the bounded variations and stability of the problem. The delay time translations used in the heat flux vector and the temperature gradient result in the finite speed of thermal wave propagation. As a special case of time fractional derivative, the classical and generalized thermoelasticity theories have been recovered.

Thermal analysis and phase transformation behaviour during additive manufacturing of Ti–6Al–4V alloy

ABSTRACTDespite the fact that the additive manufacturing (AM) technique has been established for almost two decades, its optimisation is still performed by trial and error experimentation. In the present work, a finite element modelling approach was used to study both the temperature distribution and heat flux vector characteristics during multi-layer deposition of a Ti–6Al–4V alloy that take place in the AM process. The results revealed the difference between different powder deposition time intervals on thermal cycles, heat flux vectors and the resulting microstructures. Good agreement between the numerical and experimental results was found. The results obtained can be used for process optimisation.

The generic model of general relativity

We develop a generic spacetime model in general relativity which can be used to build any gravitational model within general relativity. The generic model uses two types of assumptions: (a) geometric assumptions in addition to the inherent geometric identities of the Riemannian geometry of spacetime and (b) assumptions defining a class of observers by means of their four-velocity u^{a} which is a unit timelike vector field. The geometric assumptions as a rule concern symmetry assumptions (the so called collineations). The latter introduces the 1+3 decomposition of tensor fields in spacetime. The 1+3 decomposition results in two major results. The 1+3 decomposition of u_{a;b} defines the kinematic variables of the model (expansion, rotation, shear and four-acceleration) and defines the kinematics of the gravitational model. The 1+3 decomposition of the energy momentum tensor representing all gravitating matter introduces the dynamic variables of the model (energy density, the isotropic pressure, the momentum transfer or heat flux vector and the traceless tensor of the anisotropic pressure) as measured by the defined observers and defines the dynamics of the model. The symmetries assumed by the model act as constraints on both the kinematical and the dynamical variables of the model. As a second further development of the generic model we assume that in addition to the four-velocity of the observers u_{a} there exists a second universal vector field n_{a} in spacetime so that one has a so-called double congruence (u_{a},n_{a}) which can be used to define the 1+1+2 decomposition of tensor fields. The 1+1+2 decomposition leads to an extended kinematics concerning both fields building the double congruence and to a finer dynamics involving more physical variables. After presenting and discussing the results in their full generality we show how they are applied in practice by considering in a step by step approach the case of a string fluid in Bianchi I spacetime for the comoving observers.

On the Increase in Signal Depth due to High-Order Effects in Micro- and Nanosized Deformable Conductors

With regard to the transmission of a thermomechanical signal on extremely short temporal and spatial scales, which represents an issue particularly important dealing with micro- and nanosized electromechanical systems, it is well known that the so-called non-Fourier effects become not negligible. In addition, it has to be considered that the interaction among multiple energy carriers has, as a direct consequence, the involvement of high-order terms in the time-differential formulation of the dual-phase lag heat conduction constitutive equation linking the heat flux vector with the temperature variation gradient. Accepting that the deformations caused by the temperature variations are small enough to be modeled under the assumptions typical of the linear thermoelasticity, in the present article we take into account the highest Taylor expansion orders able to guarantee (under appropriate assumptions) stability conditions, thermodynamic consistency, and at the same time the existence of an influence domain of the external data linked to the energy transmission as thermal waves. To this aim, a cylindrical domain filled by an anisotropic and inhomogeneous thermoelastic material is investigated, although the results obtained will be independent from the considered geometry: for such a reason, we will be able to consider as illustrative examples some simulations referred to single-layer graphene and to show how the expansion orders selected strongly influence the domain of influence depth.

On the time decay in phase–lag thermoelasticity with two temperatures

<abstract><p>The aim of this paper is to study the time decay of the solutions for two models of the one-dimensional phase-lag thermoelasticity with two temperatures. The first one is obtained when the heat flux vector and the inductive temperature are approximated by a second-order and first-order Taylor polynomial, respectively. In this case, the solutions decay in a slow way. The second model that we consider is obtained taking first-order Taylor approximations for the inductive thermal displacement, the inductive temperature and the heat flux. The decay is, therefore, of exponential type.</p></abstract>

Calculation of Gas Flow Between Two Coaxial Cylinders in a Free-Molecular Regime Under Specular-Diffuse Boundary Conditions

Using the method of characteristics, within the framework of the kinetic approach, an analytical solution of the problem on gas heat and mass transfer in a long channel formed by two coaxial cylinders in a free-molecular regime has been constructed. As the basic equation describing the kinetics of the process, use is made of the Boltzmann kinetic equation for a collisionless gas and as a boundary condition of the channel walls — a model of specular-diffuse reflection. The profile of the heat flux vector in the channel has been constructed, and the gas heat and mass flows through the channel cross section have been calculated as functions of the accommodation coefficients of the gas molecule pulses tangent to the surfaces of the cylinders.

Thermal Transport of Graphene Sheets with Fractal Defects.

Graphene combined with fractal structures would probably be a promising candidate design of an antenna for a wireless communication system. However, the thermal transport properties of fractal graphene, which would influence the properties of wireless communication systems, are unclear. In this paper, the thermal transport properties of graphene with a Sierpinski fractal structure were investigated via the reverse non-equilibrium molecular dynamics simulation method. Simulation results indicated that the thermal conductivity of graphene with fractal defects decreased from 157.62 to 19.60 (W m−1 K−1) as the fractal level increased. Furthermore, visual display and statistical results of fractal graphene atomic heat flux revealed that with fractal levels increasing, the real heat flux paths twisted, and the angle distributions of atomic heat flux vectors enlarged from about (−30°, 30°) to about (−45°, 45°). In fact, the fractal structures decreased the real heat flow areas and extended the real heat flux paths, and enhanced the phonon scattering in the defect edges of the fractal graphene. Analyses of fractal graphene thermal transport characters in our work indicated that the heat transfer properties of fractal graphene dropped greatly as fractal levels increased, which would provide effective guidance to the design of antennae based on fractal graphene.

Characteristics of Rayleigh wave propagation in orthotropic magneto-thermoelastic half-space: An eigen function expansion method

• This problem is more general as other problem with different thermoelastic models. • Effect of phase lag due to heat flux vector is more dominating in comparison with other phase lags. • The secular equation magnitude increases with an increasing of wave number. • Rayleigh wave velocity increases with the increase of wave number and magnetic field. • The attenuation coefficient decreases with the increase of wave number and magnetic field. In this article, we theoretically demonstrate the characteristics of Rayleigh surface wave propagation in a homogeneous and orthotropic thermoelastic half-space in the context of three-phase-lag model of generalized thermoelasticity. The influence of magnetic field on Rayleigh wave is analyzed in the framework of two-temperature model. A vector matrix differential equation is formed by employing normal mode analysis, which is then solved by the eigen function expansion method. The frequency equations in closed form are derived and the path of surface particles during Rayleigh wave propagation is found to be elliptical. The results show appreciable differences in phase velocity, attenuation coefficient and specific loss due to the presence of heat-flux phase-lag and is more dominating in comparison with other phase lags.

Assessment of mean and fluctuating velocity and temperature of CuO/water nanofluid in a horizontal channel: Large eddy simulation

A comprehensive investigation applying the large eddy simulation approach to turbulent forced convection of CuO/water nanofluid flowing through a horizontal channel is carried out. Dealing with the sub-grid scale stress tensor and heat flux vector, the wall-adopting local eddy-viscosity model is employed. The periodic boundary condition is imposed to the streamwise and spanwise directions, while the no-slip and constant heat flux are applied to the walls. The results indicate that adding nanoparticles into the base fluid increases the dimensionless mean velocity and fluctuations of velocity and temperature. This increment is more evident for turbulent Reynolds stress and turbulent heat flux in the streamwise direction than the other directions. Therefore, higher energy is transferred between nanofluid layers which results in a higher amount of heat transfer than the pure water. It is also observed that the nanoparticles enhance the turbulence energy at all frequencies, and the decay in the fluctuations occurs at the higher wavenumbers.

Investigation on Thermal Conductivity of Steel Fiber Reinforced Concrete Using Mesoscale Modeling

A mesoscale model was developed to investigate the effect of steel fiber on the thermal conductivity of steel fiber-reinforced concrete (SFRC). Delaunay triangulation was employed to generate the unstructured mesh for SFRC materials. The model was validated using the existing experimental data. Then, it was used to study how model thickness affected simulation outcomes of thermal conductivity of models with different fiber lengths, by which an appropriate thickness was determined for the later analyses. The validated and optimized model was applied to the study of relationships between thermal conductivity and factors such as fiber content, fiber aspect ratio and different parts of an SFRC block by conducting steady-state heat analyses with the finite element analysis software ANSYS. The simulation results reveal that adding steel fiber increases thermal conductivity considerably, while fiber aspect ratio only has an insignificant effect. Besides, the presence of steel fibers has an obvious impact on the distribution of temperature and heat flux vector of the SFRC blocks.

Dual-phase-lag thermoelastic problem in finite cylindrical domain with relaxation time

Purpose The purpose of this paper is to frame a dual-phase-lag model using the fractional theory of thermoelasticity with relaxation time. The generalized Fourier law of heat conduction based upon Tzou model that includes temperature gradient, the thermal displacement and two different translations of heat flux vector and temperature gradient has been used to formulate the heat conduction model. The microstructural interactions and corresponding thermal changes have been studied due to the involvement of relaxation time and delay time translations. This results in achieving the finite speed of thermal wave. Classical coupled and generalized thermoelasticity theories are recovered by considering the various special cases for different order of fractional derivatives and two different translations under consideration. Design/methodology/approach The work presented in this manuscript proposes a dual-phase-lag mathematical model of a thick circular plate in a finite cylindrical domain subjected to axis-symmetric heat flux. The model has been designed in the context of fractional thermoelasticity by considering two successive terms in Taylor’s series expansion of fractional Fourier law of heat conduction in the two different translations of heat flux vector and temperature gradient. The analytical results have been obtained in Laplace transform domain by transforming the original problem into eigenvalue problem using Hankel and Laplace transforms. The numerical inversions of Laplace transforms have been achieved using the Gaver−Stehfast algorithm, and convergence criterion has been discussed. For illustrative purpose, the dual-phase-lag model proposed in this manuscript has been applied to a periodically varying heat source. The numerical results have been depicted graphically and compared with classical, fractional and generalized thermoelasticity for various fractional orders under consideration. Findings The microstructural interactions and corresponding thermal changes have been studied due to the involvement of relaxation time and delay time translations. This results in achieving the finite speed of thermal wave. Classical coupled and generalized thermoelasticity theories are recovered by considering the various special cases for different order of fractional derivatives and two different translations under consideration. This model has been applied to study the thermal effects in a thick circular plate subjected to a periodically varying heat source. Practical implications A dual-phase-lag model can effectively be incorporated to study the transient heat conduction problems for an exponentially decaying pulse boundary heat flux and/or for a short-pulse boundary heat flux in long solid tubes and cylinders. This model is also applicable to study the various effects of the thermal lag ratio and the shift time. These dual-phase-lag models are also practically applicable in the problems of modeling of nanoscale heat transport problems of semiconductor devices and accordingly semiconductors can be classified as per their ability of heat conduction. Originality/value To the authors’ knowledge, no one has discussed fractional thermoelastic dual-phase-lag problem associated with relaxation time in a finite cylindrical domain for a thick circular plate subjected to an axis-symmetric heat source. This is the latest and novel contribution to the field of thermal mechanics.

Analysis of an augmented fully-mixed formulation for the coupling of the Stokes and heat equations

We introduce and analyse an augmented mixed variational formulation for the coupling of the Stokes and heat equations. More precisely, the underlying model consists of the Stokes equation suggested by the Oldroyd model for viscoelastic flow, coupled with the heat equation through a temperature-dependent viscosity of the fluid and a convective term. The original unknowns are the polymeric part of the extra-stress tensor, the velocity, the pressure, and the temperature of the fluid. In turn, for convenience of the analysis, the strain tensor, the vorticity, and an auxiliary symmetric tensor are introduced as further unknowns. This allows to join the polymeric and solvent viscosities in an adimensional viscosity, and to eliminate the polymeric part of the extra-stress tensor and the pressure from the system, which, together with the solvent part of the extra-stress tensor, are easily recovered later on through suitable postprocessing formulae. In this way, a fully mixed approach is applied, in which the heat flux vector is incorporated as an additional unknown as well. Furthermore, since the convective term in the heat equation forces both the velocity and the temperature to live in a smaller space than usual, we augment the variational formulation by using the constitutive and equilibrium equations, the relation defining the strain and vorticity tensors, and the Dirichlet boundary condition on the temperature. The resulting augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Schauder and Banach theorems, combined with the Lax-Milgram theorem and certain regularity assumptions, are applied to prove the unique solvability of the continuous system. As for the associated Galerkin scheme, whose solvability is established similarly to the continuous case by using the Brouwer fixed-point and Lax–Milgram theorems, we employ Raviart–Thomas approximations of orderkfor the stress tensor and the heat flux vector, continuous piecewise polynomials of order ≤k+ 1 for velocity and temperature, and piecewise polynomials of order ≤ k for the strain tensor and the vorticity. Finally, we derive optimala priorierror estimates and provide several numerical results illustrating the good performance of the scheme and confirming the theoretical rates of convergence.