Heat Conductivity Tensor Research Articles

Bilayer thermal metadevices that mold transient heat flows

Thermal manipulation has been widely researched due to its potential in novel functions, such as cloaking, illusion and sensing. To avoid the spatially inhomogeneous and anisotropic heat conductivity tensors (introduced by transformation thermodynamics), several thermal cloaks based on scattering cancellation (SC) approach are designed and experimentally demonstrated. However, the current SC cloak is only effective in the steady-state heat transfer condition as the heat capacity terms are neglected by the requirement of constant external fields. In order to expand the SC cloak to the transient regime, a bilayer scheme is introduced to achieve the accurate solutions of the camouflage equations considering time factors. Indeed, there are two equations to solve with two unknown quantities: first the thermal conductivity and second the product of specific heat capacity by the density. We construct thermal camouflage devices and verify their ability to mold dynamic heat flows both by simulations and experiments. Our proposed method paves an efficient avenue to extend SC approach to dynamic heat transfer regime.

On multiphysics concurrent multiscale topology optimization for designing porous heat-activated compliant mechanism under convection for additive manufacture

In this research, we investigated the use of concurrent multiscale topology optimization to design additively manufacturable lightweight porous compliant mechanisms that enable harnessing heat energy as the energy source to perform mechanical actions. By establishing two independent dimensionless representations of the design problem, i.e., macro and microscale domains, a concurrent multiscale topology optimization framework is implemented, and the effective properties of the microscale (i.e., elastic and heat conductivity tensors, as well as the thermal expansion coefficient) are calculated and used as the thermo-elastic modeling effective properties of the macroscale. For heat transfer physics, heat transport in solids, as well as heat convection, are simultaneously considered in this study. Sensitivity analysis on the suggested concurrent optimization scheme was derived to address the macro and microstructure coupling as well as the thermo-elastic physics coupling. Several numerical cases are studied with single and multiple microstructure systems. To investigate the macrostructure dependency on the microstructure systems, a study was performed for multiple microstructure subsystems. Incorporating several microstructures into a single macro design domain increased design freedom and improved the performance-to-weight ratio. Furthermore, and for achieving good additive manufacturability we investigated the connectivity of the multi-microstructure optimization and implement boundary competitivity technique to attain fully connected design for attaining good additive manufacturability.

Graph-based metamaterials: Deep learning of structure-property relations

The effective properties of metamaterials are tailored through the design of their internal structures. According to their main building block, the family of porous three-dimensional metamaterials is divided into truss-, plate- and shell-lattices. The exploration of their full design-space is hampered in practice by a lack of a systematic method to represent their topologies. Here, we demonstrate for the first time that graph models provide an effective representation of shell-lattices. This new graph representation is then leveraged to obtain deep learning-based structure–property models. Using finite element simulations, the stiffness and heat conductivity tensors are established for more than 40,000 microstructural configurations. We find that a modified crystal graph convolutional neural network model provides an accurate description of the structure–property relations. We anticipate the proposed graph-based modeling framework to be applicable to any man-made periodic microstructure, thereby enabling the design and discovery of new materials exhibiting exceptional mechanical, thermal, electrical or magnetic properties.

Investigation of concurrent multiscale topology optimization for designing lightweight macrostructure with high thermal conductivity

Lightweight and high heat conductive solid structures are playing important role in various fields of engineering. To maximize the design performance for such structures, we investigated multiscale topology optimization for excessive lightweight heat-conductive porous structures and introduced a mathematical optimization model formulation for concurrently optimizing the structures (macrostructure) and the constitutive pores (microstructure). The microscale is considered a representative volume element and designed using the asymptotic homogenization method. For each iteration, the effective heat conductivity tensor of the microstructure is evaluated during the optimization process and used as the heat conductivity of the macrostructure. Sensitivity analysis on this concurrent optimization scheme was derived to address the macro and microstructure coupling. To broaden the scope of the research applicability, three topology optimization methods, i.e., SIMP, level set and ESO are investigated, and the results are compared and discussed. The suggested formulations showed a successful application of the concurrent multiscale optimization formulations and good coupling on the macro and microscale. Also, the formulations demonstrated a strong influence between the macro and the microscale of the design problem for the topology optimization methods. Increasing the design freedom by introducing various microstructures for the macro design domain showed superior performance associated with attaining high weight reduction. In addition, the concurrent optimization scheme has enabled the microstructures to attain a good spatial layout of materials while taking into account the weight reduction constraint. The spatial arrangements of the designed microstructures have achieved conducting heat in a shorter path toward the heatsink zone of macrostructure design. This allows attaining good performance with high weight reduction. Furthermore, numerical examples of different mesh numbers were used to study mesh dependency of multiscale topology optimization. The scope of the study was broadened by the inclusion of 3D case studies. Implementing Isosurface technique to achieve high detail model was also used to attain high detailed concurrently optimized design with minimal mesh number to minimize the computational cost. The 3D optimized case was investigated experimentally.

Multifield constitutive identification of non-conventional thermo-viscoelastic periodic Cauchy materials

The present work proposes a variational-asymptotic homogenization technique for non-conventional thermo-viscoelastic periodic microstructured materials. According to second sound theories, the heat flux vector linearly depends upon the history of temperature gradient and the heat conduction tensor represents the kernel of the relative hereditary integral, thus overcoming the paradox inferred from the usual Fourier’s law of thermal waves propagating at infinite speed. Down-scaling relations have been provided in the frequency domain, relating the transformed micro displacement and relative temperature fields to the corresponding macro variables and their gradients through frequency-dependent perturbation functions which convey the influence of the underlying microstructural heterogeneity. Average field equations of infinite order have been derived. Transformed field equations of the equivalent first-order medium have been obtained as Euler–Lagrange equations of a suitable functional whose first variation is properly truncated. They are characterized by frequency-dependent overall constitutive tensors, whose closed form has been provided. A benchmark test assesses the capabilities of the proposed homogenization method, where the solutions relative to a periodic, bi-phase, thermo-viscoelastic material and to the equivalent homogenized medium match under the hypothesis of periodic source terms.

A Multiple-Relaxation-Time Lattice Boltzmann Model for Natural Convection in a Hydrodynamically and Thermally Anisotropic Porous Medium under Local Thermal Non-Equilibrium Conditions

To investigate the natural convective process in a hydrodynamically and thermally anisotropic porous medium at the representative elementary volume (REV) scale, the present work presented a multiple-relaxation-time lattice Boltzmann method (MRT-LBM) based on the assumption of local thermal non-equilibrium conditions (LTNE). Three sets of distribution function were used to solve the coupled momentum and heat transfer equations. One set was used to compute the flow field based on the generalized non-Darcy model; the other two sets were used to solve the temperature fields of fluid and solid under the LTNE. To describe the anisotropy of flow field of the porous media, a permeability tensor and a Forchheimer coefficient tensor were introduced into the model. Additionally, a heat conductivity tensor and a special relaxation matrix with some off-diagonal elements were selected for the thermal anisotropy. Furthermore, by selecting an appropriate equilibrium moments and discrete source terms accounting for the local thermal non-equilibrium effect, as well as choosing an off-diagonal relaxation matrix with some specific elements, the presented model can recover the exact governing equations for natural convection under LTNE with anisotropic permeability and thermal conductivity with no deviation terms through the Chapman-Enskog procedure. Finally, the proposed model was adopted to simulate several benchmark problems. Good agreements with results in the available literatures can be achieved, which indicate the wide practicability and the good accuracy of the present model.

Calculation of thermal conductivity coefficients of electrons in magnetized dense matter

The solution of Boltzmann equation for plasma in magnetic field, with arbitrarily degenerate electrons and nondegenerate nuclei, is obtained by Chapman–Enskog method. Fully ionized plasma is considered. The tensor of the heat conductivity coefficients in nonquantized magnetic field is calculated. For strongly degenerate plasma the asymptotic analytic formulas are obtained. The Lorentz approximation, with neglecting of electron–electron encounters, is asymptotically exact for strongly degenerate plasma. For strongly degenerate electrons we obtain an asymptotically exact analytical solution for the heat conductivity tensor in presence of a magnetic field. This solution has considerably more complicated dependence on the magnetic field than those in previous publications, and gives several times smaller relative value of a thermal conductivity across the magnetic field at [Formula: see text].

Calculation of Thermal Conductivity Coefficients of Electrons in Magnetized Dense Matter

The solution of Boltzmann equation for plasma in magnetic field, with arbitrarily degenerate electrons and non-degenerate nuclei, is obtained by Chapman-Enskog method. Functions, generalizing Sonin polynomials are used for obtaining an approximate solution. Fully ionized plasma is considered. The tensor of the heat conductivity coefficients in non-quantized magnetic field is calculated. For non-degenerate and strongly degenerate plasma the asymptotic analytic formulas are obtained, which are compared with results of previous authors. The Lorentz approximation, with neglecting of electron-electron encounters, is asymptotically exact for strongly degenerate plasma. We obtain, for the first time, in three polynomial approximation, with account of electron-electron collisions, analytical expressions for the heat conductivity tensor for non-degenerate electrons, in presence of a magnetic field. Account of the third polynomial improved substantially the precision of results. In two polynomial approximation our solution coincides with the published results. For strongly degenerate electrons we obtain, for the first time, an asymptotically exact analytical solution for the heat conductivity tensor in presence of a magnetic field. This solution has considerably more complicated dependence on the magnetic field than those in previous publications, and gives several times smaller relative value of a thermal conductivity across the magnetic field at $\omega\tau\gtrsim 0.8$.

A difference scheme for a conjugate-operator model of a heat conduction problem in polar coordinates

In polar coordinates, a discrete analog of the conjugate-operator model of a heat conduction problem is formulated to hold the structure of the original model. The difference scheme converges with second-order accuracy in the case of discontinuous parameters of the medium in the Fourier law and irregular grids. An efficient algorithm for solving the discrete conjugate-operator model when heat conduction tensor is a unit operator is proposed.

Uniqueness results for a boundary value problem in dipolar thermoelasticity to model composite materials

Abstract Composites with microstructure display nonlocal effects and can be effectively modeled through dipolar elasticity. A mixed initial boundary problem is addressed for dipolar thermoelasticity. The basic equations and conditions of the problem are set-up and uniqueness results are proven, which are obtained without introducing definiteness restrictions on the elastic coefficients and, also, without the usual condition imposed to the heat conductivity tensor of being positive definite. An additional result is finally obtained, namely, a reciprocal identity of the Betti's type which underlies another uniqueness result obtained again under weak restrictions.

Electron heat conductivity of epitaxial graphene on silicon carbide

The diagonal component of the electron heat conductivity tensor of epitaxial graphene formed in a semiconductor has been investigated within a simple analytical model. It is shown that the heat conductivity sharply changes at a chemical potential close to the substrate band gap edge. Low-temperature expressions for the heat conductivity are derived.

Analytical Investigation of Heat Transfer in an Anisotropic Band with Heat Fluxes Assigned at the Boundaries

An analytical solution of a nonstationary problem of the theory of heat conduction in an anisotropic band under heat transfer conditions at the boundaries has been obtained for the first time by applying the Fourier transformation with respect to the longitudinal variable and the Laplace transformation for time. The problem is formulated and solved in regions with anisotropy of general form when the principal axes of the heat conduction tensor are set at an angle that orients these axes relative to the Cartesian coordinate system.

Magnetothermoelectric response from holography

In this note we study the effects of a magnetic field on transport using holographic models with broken translational invariance. We show that, after carefully subtracting off non-trivial magnetisation currents, it is possible to express the DC transport currents of the boundary theory in terms of properties of a black hole horizon. This allows us to obtain simple analytic expressions for the electrical, thermoelectric and heat conductivity tensors. Our results apply to both isotropic and anisotropic models, including holographic Q-lattices and to certain theories where translational invariance is broken by linear sources for axions.

Causal dissipation and shock profiles in the relativistic fluid dynamics of pure radiation.

CURRENT THEORIES OF DISSIPATION IN THE RELATIVISTIC REGIME SUFFER FROM ONE OF TWO DEFICITS: either their dissipation is not causal or no profiles for strong shock waves exist. This paper proposes a relativistic Navier-Stokes-Fourier-type viscosity and heat conduction tensor such that the resulting second-order system of partial differential equations for the fluid dynamics of pure radiation is symmetric hyperbolic. This system has causal dissipation as well as the property that all shock waves of arbitrary strength have smooth profiles. Entropy production is positive both on gradients near those of solutions to the dissipation-free equations and on gradients of shock profiles. This shows that the new dissipation stress tensor complies to leading order with the principles of thermodynamics. Whether higher order modifications of the ansatz are required to obtain full compatibility with the second law far from the zero-dissipation equilibrium is left to further investigations. The system has exactly three a priori free parameters χ,η,ζ, corresponding physically to heat conductivity, shear viscosity and bulk viscosity. If the bulk viscosity is zero (as is stated in the literature) and the total stress-energy tensor is trace free, the entire viscosity and heat conduction tensor is determined to within a constant factor.

Experiments on Transformation Thermodynamics: Molding the Flow of Heat

It was recently shown theoretically that the time-dependent heat conduction equation is form invariant under curvilinear coordinate transformations. Thus, in analogy to transformation optics, fictitious transformed space can be mapped onto (meta)materials with spatially inhomogeneous and anisotropic heat-conductivity tensors in the laboratory space. On this basis, we design, fabricate, and characterize a microstructured thermal cloak that molds the flow of heat around an object in a metal plate. This allows for transient protection of the object from heating while maintaining the same downstream heat flow as without object and cloak.

Solution of the inverse problems of heat admittance in order to derive characteristics of anisotropic materials

A solution of the inverse problem in order to determine the principal coefficients and principal axes angles of the heat conduction tensor of an anisotropic material is proposed in this paper. It is based upon a previously found solution of the heat conduction problem in an anisotropic half-space heated with a thermal flow. The technique is based upon expansion of the discrepancy composite function into Taylor series and determination of differential vectors of the target coefficients, which are then used for application of iterative gradient descent algorithms. The results have proved fine convergence even if the initial approximation of the coefficients vector is different from the target by more than one hundred percent. The proposed method can be applied to determination of thermophysical characteristics of composite materials used for thermal protection of hypersonic flying vehicles.

Several Results on Uniqueness and Continuous Data Dependence in Thermoelasticity of Type III

In the present paper we consider the linear theory of thermoelasticity of type III for anisotropic and inhomogeneous materials as developed by Green and Naghdi (1992, 1995). We consider the general initial-boundary value problems associated with the theory in concern and then we identify some mild constitutive hypotheses that lead to appropriate energetic measures of solutions. On this basis we are able to prove some new uniqueness and continuous data dependence theorems. When the boundary surface is maintained at zero thermal displacement, we are able to obtain uniqueness and continuous data dependence results by using only the positiveness of the density mass, the positive definiteness of the elasticity tensor and the positive definiteness of the heat conductivity tensor, without any sign defined assumption upon the specific heat. These results are established by means of new techniques which are essentially different from those previously used for the classical theory of thermoelasticity.

On a Theory of Thermoviscoelastic Mixtures

We derive a continuum theory for a thermoviscoelastic composite with the help of an entropy production inequality proposed by Green and Laws. The composite is modelled as a mixture of an elastic solid and a Kelvin-Voigt material. The theory is derived in lagrangian description. It is shown that the linear heat conduction tensor is symmetric and that the theory allows for “second sound” effects. In contrast with other theories of solid-fluid mixtures, in the present theory the diffusive force depends on both relative displacement and relative velocity. A uniqueness result in the linearized theory is presented. Finally, the effect of a concentrated heat source is investigated.

Sensitivity of the macroscopic thermal conductivity tensor to topological microstructural changes

This paper proposes a closed form expression for the sensitivity of the macroscopic heat conductivity tensor for two-dimensional problems to topological microstructural changes of the underlying material. The sensitivity formula is remarkably simple. It is derived by applying the concept of topological derivative within a variational multi-scale framework for steady-state heat conduction where the macroscopic temperature gradient and heat flux are defined as volume averages of their microscopic counterparts over a representative volume element (RVE) of material. The classical Fourier law is assumed to hold at the scale referred to as microscopic (the RVE). The derived sensitivity – a symmetric second order tensor field over the RVE domain – measures how the estimated macroscopic conductivity tensor changes when a small circular inclusion is introduced at the micro-scale. The proposed formula finds potential application in the design and optimisation of heat conducting materials.

Friction tensor concept for textured surfaces

Directionality of grinding marks influences the coefficient of friction during sliding. Depending on the sliding direction the coefficient of friction varies between maximum and minimum for textured surfaces. For random surfaces without any texture the friction coefficient becomes independent of the sliding direction. This paper proposes the concept of a friction tensor analogous to the heat conduction tensor in anisotropic media. This implies that there exists two principal friction coefficients µ1,2 analogous to the principal conductivities k 1,2. For symmetrically textured surfaces the principal directions are orthogonal with atleast one plane of symmetry. However, in the case of polished single crystalline solids in relative sliding motion, crystallographic texture controls the friction tensor.