Coordinate-free derivation and weak formulation of a general imperfect interface model for thermal conduction in composites

In a great number of situations of practical interest, the interfaces between the constituent phases of a composite turn out to be imperfect. In the context of thermal conduction, an interface is said to be imperfect if the requirement that both the temperature and the normal heat flux be continuous across the interface is not satisfied. A powerful method based on mathematical asymptotic analysis has been proposed and developed in the literature by several authors for the derivation of linear imperfect interface models of thermal conduction. This method consists in replacing an interphase of small uniform thickness between two-phases by an imperfect interface of null thickness characterized by the temperature and normal heat flux jump relations deduced by carrying out an appropriate asymptotic analysis. The objective of the present work is threefold. Firstly, it aims to explicitly show and emphasize the key role played by Hadamard’s relation in the method. Secondly, it has the purpose of using a coordinate-free differential geometry theory and Hadamard’s relation to render the method coordinate-free. Thirdly and most importantly, the present work gives a weak formulation for the problem concerning the steady thermal conduction in a composite with the interfaces described by the general temperature and normal heat flux jump relations derived. This weak formulation is a key step toward solving the problem by the extended finite element method (XFEM) presented in a companion paper.     

Feeble interfaces in bimaterials

Summary In the present paper, the thermal and thermo-elastic response of a bi-material to temperature changes is analyzed, when its interface exhibits a simultaneous weakness in traction transferring and heat flow conducting (feeble interface). Such a pathological behavior of an interface is described by two sets of constitutive relationships relating the heat flow passing through the interface to the temperature jump and the interfacial components of the traction to those of the displacement jump. The bimaterial model considered is that of a circular inhomogeneity in an elastic matrix with linear forms of the constitutive relationships. When the solutions of both heat conduction and thermoelastic problems with a perfect interface are known, the corresponding problems with a feeble interface are reduced to the solution of two dislocation problems: a heat conduction problem with an appropriate temperature dislocation applied across the interface, and an elasticity problem with an appropriate displacement dislocation of Somigliana type acting across the interface. For both dislocation problems, general representations of their solutions in terms of two-phase potential functions of complex variables are provided. Detailed analytical results are given for a circular inhomogeneity with a feeble interface disturbing a linear distribution of the temperature change in the matrix. In this case, the stress field within the inhomogeneity has a linear distribution and it vanishes for the limiting case of a sliding interface. For a specific value of the interface parameter H, which characterizes the thermal imperfection, there are no shear stresses within the inhomogeneity. Finally, since the constitutive laws describing the thermal and mechanical interface behavior correlate tensors of different order, the resulting fields in the system are drastically affected by the inhomogeneity size.     

Thermal Conductivity and Interfacial Thermal Resistance in Bilayered Nanofilms by Nonequilibrium Molecular Dynamics Simulations

A nonequilibrium molecular dynamics study of the cross-plane thermal conductivity and interfacial thermal resistance of nanoscale bilayered films is presented. The films under study are composed of argon and another material that is identical to argon except for its atomic mass. The results show that a large temperature jump occurs at the interface and that the interfacial thermal resistance plays an important role in heat conduction for the whole films. The cross-plane thermal conductivity is dependent on the average temperature. The interfacial thermal resistance is found to be dependent apparently on the atomic mass ratio of the two materials and the temperature, but to be independent of the film thickness. A linear relationship is observed between the reciprocal of the cross-plane thermal conductivity and that of the film thickness with the film thickness between 5.4 nm and 64.9 nm, which is in good agreement with results in the literature for a single film.     

Theoretical analysis of heat conduction problems of nonhomogeneous functionally graded materials for a layer sandwiched between two half-planes

Functionally graded materials are the materials whose material properties are smoothly varying along one axis, and they are used as buffer layers to connect two dissimilar materials. By choosing proper functionally graded parameters, the material properties at the interface can be identical to prevent the interfacial fracture problem. This study analyzes the heat conduction problem of nonhomogeneous functionally graded materials for a layer sandwiched between two half-planes. From the Fourier transform method, the full-field solutions of temperature and heat flux are obtained in explicit forms. Numerical calculations based on the analytical solutions are performed and are discussed in detail. The continuous characteristics of the temperature and heat flux along the interface are emphasized, and some interesting phenomena are presented in this study. It is noted that the temperature and heat flux fields along the interface for nonhomogeneous functionally graded materials are continuous if the conductivities are identical at the interface. Furthermore, the temperature and heat flux q _y have the identical contour slopes across the interface.     

Transient heat conduction in a medium with multiple circular cavities and inhomogeneities

A two‐dimensional transient heat conduction problem of multiple interacting circular inhomogeneities, cavities and point sources is considered. In general, a non‐perfect contact at the matrix/inhomogeneity interfaces is assumed, with the heat flux through the interface proportional to the temperature jump. The approach is based on the use of the general solutions to the problems of a single cavity and an inhomogeneity and superposition. Application of the Laplace transform and the so‐called addition theorem results in an analytical transformed solution. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. Several numerical examples are given to demonstrate the accuracy and the efficiency of the method. The approximation error decreases exponentially with the number of the degrees of freedom in the problem. A comparison of the companion two‐ and three‐dimensional problems demonstrates the effect of the dimensionality. Copyright © 2009 John Wiley & Sons, Ltd.     

Non-steady state heat conduction across an imperfect interface: A dual-reciprocity boundary element approach

The two-dimensional problem of determining the time-dependent temperature in a bimaterial with a homogeneously imperfect interface is considered. A temperature jump which is proportional to the thermal heat flux is assumed across the imperfect interface. Through the use of the corresponding steady-state Green's function for the imperfect interface, a dual-reciprocity boundary element method is derived for the numerical solution of the problem under consideration. To assess the validity and accuracy of the proposed method of solution, some specific problems are solved.     

Coupled conduction-convection problem for a cylinder in an enclosure

The coupled heat conduction/convection problem for a solid cylinder in either a rectangular or a circular enclosure filled with air is solved by an operator-splitting pseudo-time-stepping finite element method, which automatically satisfies the continuity of the interfacial temperature and heat flux. The temperature distribution in the cylinder and in the fluid is obtained showing that the usual practice of prescribing a uniform heat flux boundary condition at the interface may not lead to an accurate solution. From the profile of the local Nusselt number, which is strongly dependent on the Rayleigh number but weakly dependent on the thermal conductivity ratio, it is concluded that most of the heat transfer takes place in the lower half of the cylinder through a convective mode.     

层间弱粘贴复合材料层板的热弹性脱层

本文建立了一个层间弱粘贴复合材料层板热弹性脱层模型。该模型建立在两个描述层间弱粘贴的基本假设基础上。层间位移不连续由层间粘贴的物理关系来描述，表现为层间位移跳跃值与层间残余横向应力的关系；层间温度不连续由层间传热薄层来描述，并据此给出一个计算层间温度跳跃值的计算公式，表现为温度跳跃值与层间横向张开量之间的关系。在此假设基础上，根据平衡方程和静态传热方程导出了正交柱面弯曲层板热弹性脱层解。算例显示了层间弱粘贴对层板热弹性响应的影响。     

Thermal Green’s functions in plane anisotropic bimaterials with spring-type and Kapitza-type imperfect interface

By means of the extended version of the Stroh formalism for uncoupled thermo-anisotropic elasticity, two-dimensional Green’s function solutions in terms of exponential integrals are derived for the thermoelastic problem of a line heat source and a temperature dislocation near an imperfect interface between two different anisotropic half-planes with different thermo-mechanical properties. The imperfect interface investigated here is modeled as a generalized spring layer with vanishing thickness: (1) the normal heat flux is continuous at the interface, whereas the temperature field undergoes a discontinuity which is proportional to the normal heat flux; (2) the tractions are continuous across the interface, whereas the displacements undergo jumps which are proportional to the interface tractions. This kind of imperfect interface can be termed a thermally weakly conducting and mechanically compliant interface. In the Appendix we also present the isothermal Green’s functions in anisotropic bimaterials with an elastically stiff interface to demonstrate the basic ingredients in the analyses of a stiff interface.     

A homogenization technique for heat transfer in periodic granular materials

A homogenization technique is proposed to simulate the thermal conduction of periodic granular materials in vacuum. The effective thermal conductivity (ETC) and effective volumetric heat capacity (EVHC) can be obtained from the granular represent volume element (RVE) via average techniques: average heat flux and average temperature gradient can be formulated by the positions and heat flows of particles on the boundaries of the RVE as well as of the contact pairs within the RVE. With the thermal boundary condition imposed on the border region around the granular RVE, the ETC of the granular RVE can be computed from the average heat flux and average temperature gradient obtained from thermal discrete element method (DEM) simulations. The simulation results indicate that the ETC of the granular assembly consisting of simple-cubic arranged spheres coincides with the theoretical prediction. The homogenization technique is performed to obtain the ETC of the RVE consisting of random packed particles and the results exhibit the anisotropy of the thermal conduction properties of the RVE. Both the ETC and EVHC obtained are then employed to simulate the thermal conduction procedure in periodic granular materials with finite element analyses, which give the similar results of temperature profile and conduction properties as the DEM simulations.     

Estimation of heat flux at metal-mold interface during solidification of cylindrical casting

For modeling solidification process of casting accurately, the correct information about the heat flux boundary condition is required. In this study, an inverse heat conduction model is established to determine the interfacial heat flux at metal-mold in the process of casting with a cylindrical geometry. The numerically calculated temperature is compared with the exact solution and simulation solution obtained by commercial software ProCAST to investigate the accuracy of forward heat conduction model. The analysis of calculated heat flux indicates that the inverse model may be taken as a feasible and effective tool for the estimation of the metal-mold interfacial heat flux during solidification of cylindrical casting.     

Reformulation of the Nosé–Hoover thermostat for heat conduction simulation at nanoscale

This paper investigates the heat conduction phenomena at nanoscale by reformulating the Nosé–Hoover thermostat for nonequilibrium molecular dynamics simulation. Inspired by the Nosé–Hoover mechanics, the feedback temperature force caused by the temperature control is reformulated to a more general level, aiming at (1) controlling the temperature locally at several distinct spots and (2) eliminating the rigid-body translation and rotation which are unexpectedly introduced into the system due to the feedback temperature force, so that accurate trajectories of atoms can be generated and the heat conduction simulation at nanoscale can be performed successfully. Correspondingly, the definition of temperature is modified; the expression of the Hamiltonian is generalized. To demonstrate the capability and feasibility of this newly formulated algorithm, we studied heat conduction phenomenon in a beam-like finite size specimen via our in-house developed computer code. The results, temperature distributions across the specimen, are shown to reach steady state after a period of time which cannot be achieved by the original Nosé–Hoover thermostat. Also, it is concluded that the heat conduction at nanoscale exhibits the same feature of Fourier’s law at macroscopic scale, namely that the heat flux is linearly proportional to the temperature gradient, if the temperature is averaged over a sufficiently large time interval and large spatial region. The thermal conductivity can thereafter be calculated based on the linear relation between the heat flux and the temperature gradient. It is found that the obtained numerical value of thermal conductivity matches the experimental result very well.     

An inverse heat conduction problem with heat flux measurements

Using heat flux measurements as additional information to solve inverse heat conduction problems was and is still rarely employed. Lot of disadvantages linked to heat flux measurement specificities (local disturbance, intrusive measurement, lack of knowledge and proficiency, etc.) make people prefer temperature measurements which are well documented and very widespread. Solving inverse heat conduction problems with heat flux measurements is quite different than the one which uses temperatures and need to be investigated deeply. In this work, this problem is approached through the solution of a bioengineering problem consisting in the development of a non‐invasive blood perfusion probe. The effort here is focused on the development of a methodology for the estimation of time‐dependent blood perfusion from heat flux measurements. The physical probe incorporates a thin heat flux sensor, which is placed in contact with the tissue region where the perfusion is to be measured. The sensor records the heat flux due to an imposed thermal event, which is achieved by air flow. A one‐dimensional mathematical model is used to simulate the thermal event occurring at the contact region holding between the probe and the tissue. A combined parameter and function estimation procedure is developed to estimate simultaneously time‐dependent blood perfusion and thermal contact conductance between the probe and the tissue. The robustness of the method was demonstrated through several test cases using simulated data. The presented examples include various functional changes in the time evolution of blood perfusion. Results from this study have shown the feasibility of solving inverse problems with heat flux measurements and the two unknowns are estimated with no a priori information about their functional forms. Copyright © 2006 John Wiley & Sons, Ltd.     

Thermal convection in a fluid layer sandwiched between two porous layers of different permeabilities

Summary A linear stability analysis is carried out for the convective instability problem in a horizontal fluid layer sandwiched between two porous layers of different permeabilities. The velocity boundary condition is a general one and is via-media between the free and rigid boundary conditions. The thermal condition at the porous-fluid interface is assumed to be neither constant heat flux nor constant temperature, but a condition leading to a third-type of boundary condition. The principle of exchange of stability is valid for the problem and the critical eigenvalue is obtained for the general boundary condition using the single-term Rayleigh-Ritz technique. The results of several works are recovered as limiting cases of the present study.     

Flux Jumps and H-T Diagram of Instability for MgB2

We present a study of magneto-thermal instabilities in polycrystalline MgB₂ superconductor, by magnetic hysteresis loop measurements and by investigations of magnetic flux dynamics with a miniature Hall probe. Temperature and magnetic field ranges where the flux jumps may be observed have been determined. On the basis of measurements of the magnetic flux dynamics, an average magnetic diffusivity describing the process of the flux jump is estimated. This parameter is compared with the thermal and magnetic diffusivities calculated on the basis of available data for thermal conductivity, heat capacity and resistivity. It is shown that the estimated value of the field of the first flux jump is influenced significantly by the field dependence of specific heat. In order to explain the observed phenomenon, the temperature reached by the sample during the flux jump at different magnetic fields is calculated.     

Analytical study for the estimation of thermal properties of processed meat based on hyperbolic heat conduction model

This study proposes an analytical method in conjunction with existing experimental temperature to estimate the unknown relaxation time and thermal diffusivity of processed meat based on the hyperbolic heat conduction model. This analytical method is a combination of the Laplace transform and least squares methods. The thermal contact resistance at the interface between adjacent samples at different temperatures is assumed to be negligible. The relaxation time is estimated from the temperature jump at a specific measurement location. The thermal diffusivity is determined from the definition of the dimensionless spatial coordinate and the resulting relaxation time. The results show that the relaxation time and thermal diffusivity obtained are in good agreement with the existing results. The obtained dimensionless temperature history at a specific measurement location is close to the experimental temperature data. This means that the Cattaneo–Vernottee (CV) model can be suitable for this study. The proposed analytical inverse method can be applied to determine a more accurate estimate of such problems. A comparison of the estimate obtained from CV and dual phase lag models is made.     

BEM simulation of compressible fluid flow in an enclosure induced by thermoacoustic waves

The problem of unsteady compressible fluid flow in an enclosure induced by thermoacoustic waves is studied numerically. Full compressible set of Navier–Stokes equations are considered and numerically solved by boundary-domain integral equations approach coupled with wavelet compression and domain decomposition to achieve numerical efficiency. The thermal energy equation is written in its most general form including the Rayleigh and reversible expansion rate terms. Both, the classical Fourier heat flux model and wave heat conduction model are investigated.The velocity–vorticity formulation of the governing Navier–Stokes equations is employed, while the pressure field is evaluated from the corresponding pressure Poisson equation. Material properties are taken to be for the perfect gas, and assumed to be pressure and temperature dependent.     

A coupled conduction convection and radiation problem for three insulated cables suspended in air

The coupled heat conduction, convection and radiation problem for three heated insulated cables suspended in air is solved by an operator-splitting pseudo-time-stepping finite element method, which automatically satisfies the continuity of the interfacial temperature and heat flux. The main feature of the solution procedure is that the multi-phases are treated as a single computational domain with unknown interfacial boundary conditions. The temperature distribution in the heated metal cylinder, in the insulating layer and in the open air together with the convective flow pattern are obtained simultaneously.     

A hypersingular boundary integral analysis of axisymmetric steady-state heat conduction across a non-ideal interface between two dissimilar materials

Steady-state axisymmetric heat conduction across a non-ideal interface between two dissimilar materials is considered. The non-ideal interface may be either low or high conducting. The relevant interfacial conditions are formulated in terms of hypersingular boundary integral equations. A simple boundary element procedure based on the hypersingular boundary integral formulations is proposed for solving numerically the axisymmetric heat conduction problem under consideration. Numerical results for some specific problems are obtained.