Time-dependent heat transfer in a plate with anisotropy of general form under the action of pulsed heat sources

An analytical solution to the problem of the theory of heat conduction in an anisotropic band under the pulsed (point) action of heat sources has been obtained for the first time by constructing the boundary influence function from using the Fourier and Laplace integral transforms. An arbitrary orientation of the principal axes of the thermal conductivity tensor and arbitrary (including negative) values of the off-diagonal components of the thermal conductivity tensor are taken into account. The found solution is extended to piecewise continuous densities of heat fluxes at the free boundaries of an anisotropic plate. The influences of the principal components and the orientation of the principal axes of the thermal conductivity tensor on time-dependent temperature fields in the anisotropic plate are determined. It is established that there are saddle points and separatrices dividing the temperature field into regions of influence of the boundary heat fluxes. The results are used to solve problems involving the thermal state of thermal protection made of anisotropic materials.     

An Analytical Study into Conjugate Heat Transfer on the Boundaries of Anisotropic Bodies

A simultaneous analytical solution of the second initial boundary-value problem of thermal conductivity in an anisotropic plate and of the simplest problem of the boundary layer on this plate is used to investigate conjugate heat transfer. The reciprocal influence of heat transfer during a variation of the orientation of the principal axes and coefficients of the thermal conductivity tensor is analyzed. A significant decrease in the heat fluxes from the boundary layer to the plate in the case of the principal axis with a high coefficient of thermal conductivity oriented along the body is demonstrated.     

Thermal Transport Properties of Layered Materials: Identification by a New Numerical Algorithm for Transient Measurements

Transient methods are widely used to determine thermal transport properties. In some situations they can be used for homogeneous media to measure several properties either simultaneously or separately. In this context an analytic model is available and a well-posed inverse problem of parameter identification has to be solved. The examination of composite media is more complicated. The algorithm proposed here allows simultaneous determination of the thermal conductivity and thermal diffusivity of layered dielectrics by transient measurements. It is based on a plane source that acts both as a resistive heater and temperature sensor. For the technique to be successful two essential aspects have to be considered: firstly, the mathematical modeling of the measured data (the forward problem) and secondly, the problem of ill-posedness of the inverse problem. For the proposed measurement configuration, a new fast data analysis algorithm based on an analytic solution for the forward problem is presented. In principle, a numerical solution such as an FEM solution of the heat conduction equation can be used instead of the analytical one, but the computational effort is much greater. The inverse problem is formulated as an output-least-squares problem, which leads to a transcendent algebraic system of equations. The method was successfully tested for different situations.Paper presented at the Fifteenth Symposium on Thermophysical Properties, June 22--27, 2003, Boulder, Colorado, U.S.A.     

Inverse problem of simultaneously estimating the thermal conductivity and boundary shape

An inverse analysis is used to simultaneously estimate the thermal conductivity and the boundary shape in steady-state heat conduction problems. The numerical scheme consists of a body-fitted grid generation technique to mesh the heat conducting body and solve the heat conduction equation – a novel, efficient, and easy to implement sensitivity analysis scheme to compute the sensitivity coefficients, and the conjugate gradient method as an optimization method to minimize the mismatch between the computed temperature distribution on some part of the body boundary and the measured temperatures. Using the proposed scheme, all sensitivity coefficients can be obtained in one solution of the direct heat conduction problem, irrespective of the large number of unknown parameters for the boundary shape. The obtained results reveal the accuracy, efficiency, and robustness of the proposed algorithm.     

Numerical solution of fuzzy boundary value problems using Galerkin method

This paper proposes a new technique based on Galerkin method for solving nth order fuzzy boundary value problem. The proposed method has been illustrated by considering three different cases depending upon the sign of coefficients with benchmark example problems. To show the applicability of the proposed method, an application problem related to heat conduction has also been studied. The results obtained by the proposed methods are compared with the exact solution and other existing methods for demonstrating the validity and efficiency of the present method.     

On inverse boundary heat-conduction problems for recovery of heat fluxes to anisotropic bodies with nonlinear heat-transfer characteristics

A closed method is proposed for recovering heat fluxes to anisotropic bodies under conditions of aero-gasdynamic heating from experimental temperature data at spatial-temporal nodes. The thermal protection of a body is made of anisotropic materials with components of thermal-conductivity tensor, which are dependent of temperature, i.e., are nonlinear. The method is based on approximating a spatial dependence of a heat flux by a linear combination of basis functions with sought coefficients (parameters), which are found by minimization of a quadratic functional of the residual (the discrepancy between experimental and theoretical temperature values) using the implicit method of gradient descent, as well as on constructing and numerically solving problems for the determination of sensitivity coefficients. To increase the degree of correctness of an inverse problem, along with a main functional, the regularizing functionals have been constructed and utilized on the basis of smoothness requirements for spatial functions of heat fluxes to have continuous first and second derivatives, which allowed heat fluxes with the coupled heat transfer to be recovered in the form of arbitrary functions: monotonic, nonmonotonic, having extrema, inflection points, etc. Numerous results of recovering heat fluxes to anisotropic bodies are obtained and discussed, with the regularization parameter being selected for every case.     

Microscale heat conduction in homogeneous anisotropic media: a dual-reciprocity boundary element method and polynomial time interpolation approach

In this paper, a dual-reciprocity boundary element method based on some polynomial interpolations to the time-dependent variables is presented for the numerical solution of a two-dimensional heat conduction problem governed by a third order partial differential equation (PDE) over a homogeneous anisotropic medium. The PDE is derived using a non-Fourier heat flux model which may account for thermal waves and/or microscopic effects. In the analysis, discontinuous linear elements are used to model the boundary and the variables along the boundary. The systems of algebraic equations are set up to solve all the unknowns. For the purpose of evaluating the proposed method, some numerical examples with known exact solutions are solved. The numerical results obtained agree well with the exact solutions.     

A method of determining nonstationary heat flux and heat conduction using parametric identification

The formulation and solution of the combined (boundary and coefficient) inverse heat-conduction problem of the simultaneous determination of the thermal characteristics of a material and the heat flux density entering the heat receiver are considered. The solution is based on a parameterization of this problem and the use of an extended digital Kalman filter. The results of a simulation of the simultaneous reestablishment of the heat flux and the heat conduction in a single experiment to measure the temperature of a surface are presented.     

Suggestions regarding thermal diffusivity measurements on pyrolytic graphite and pyrolytic boron nitride by the laser pulse method

The laser pulse method can be successfully applied to the measurement of thermal diffusivity of isotropic materials subject to some assumptions. For anisotropic materials, this method is applicable to the measurement of principal thermal diffusivity only on the condition that there is no difference in direction between the principal axis and that of the temperature gradient. After analyzing the heat conduction process in an anisotropic solid, it has been shown that large errors in the measurement of thermal diffusivity would exist if the direction of the principal axis deviates inconspicuously from that of the temperature gradient. The experimental results of thermal diffusivity of highly oriented pyrolytic graphite (HOPG) samples with various deviation angles have been compared with the analytical results. The laser pulse method is not applicable to measurements on semitransparent pyrolytic boron nitride (PBN). We adopted a two-layer composite sample to measure the thermal diffusivity of PBN in the c direction and a particular graphite-PBN composite sample has been prepared which has a very low thermal resistance at the interface. The thermal diffusivity and thermal conductivity of PG (below 2300°C) and PBN (below 1000°C) are given.Invited paper presented at the Ninth Symposium on Thermophysical Properties, June 24–27, 1985, Boulder, Colorado, U.S.A.     

Analysis of multi-layered filament-wound composite pipes under combined internal pressure and thermomechanical loading with thermal variations

The composite pipes manufactured by filament winding technology have anisotropic behavior owing to different reinforced ply angles. Composite pipes can be exposed to the thermomechanical loading due to hot fluid that flows into them. In this paper, based on the three-dimensional anisotropic elasticity, an exact elastic solution for thermal stresses and deformations of the pipes under internal pressure and a temperature gradient has been studied. Giving heat convection conditions the variation of temperature field within the pipe is obtained by solving the conduction equation at the wall. The influence of temperature field in the governing equations of thermoelasticity has been considered via a constitutive law. The shear extension coupling is also considered because of lay-up angles. Stress, strain and deformation distributions for different angle-ply pipe designs are investigated using the present theory.     

Heat transfer under conditions of phase transitions in bodies with anisotropic properties

An investigation is made of heat transfer in anisotropic bodies under conditions of phase transformations using the numerical solution of multidimensional unsteady-state problems of the Stefan type. A procedure of solving such problems is suggested. It is demonstrated that the characteristics of thermal conductivity tensors of both phases affect significantly both the temperature fields and the shape and velocity of motion of the boundary of phase transformations. Numerous results are analyzed of the effect which is made on the thermal state of anisotropic body by the orientation of principal axes and principal coefficients of thermal conductivity tensors of both phases.     

Residual stresses formation during the manufacturing process of epoxy matrix composites: resin yield stress and anisotropic chemical shrinkage

A study of residual stresses during the curing process of thermosetting resin composites is presented. A methodology is proposed for predicting the formation and the development of the manufacturing residual stresses, this approach is based on the determination of the resin yield stress. A self-consistent model is used to determine the cure-dependent effective mechanical properties, thermal expansion and chemical shrinkage coefficients of the composite material. This model allows considering for the composite material behaviour an anisotropic chemical shrinkage, which is not represented by a classical linear model. Finally, a one dimensional cure simulation and a modelling of residual stresses formation in composite plate are made by using a finite elements code. The effects of the cure-dependent material properties and of the resin yield stress on the residual stress formation are studied.     

An analytical solution for hydraulic conductivity of concrete considering properties of the Interfacial Transition Zone (ITZ)

As a composite material, hydraulic conductivity of concrete depends on conductivity of its components that are the mortar, aggregates and the Interfacial Transition Zone (ITZ). Since hydraulic conduction is analogous to heat and electrical conduction, analytical models from these analogous areas relating effective conductivity of composite to conductivity of its components can be used to find the effective hydraulic conductivity of concrete as a function of properties of its components, i.e., aggregate, mortar and the ITZ. However, effect of the conduction in the ITZ has not been considered in these models. This paper presents an analytical solution for the hydraulic conductivity of concrete as a three-phase composite material. The solution is an extension to the model originally proposed for conduction of composite media with randomly suspended spheres. Results of the proposed model compare well against the experimental results and those obtained from rigorous numerical analysis using the Finite Element (FE) method. The principal significance of this study lies in the development of a versatile analytical model that can be employed as a quick tool for assessment of hydraulic conductivity of concrete without the need for sophisticated FE models at the meso-scale level. It offers more insight into effect of different components of concrete on its overall conductivity.     

Parameter estimation in heat conduction using a two-dimensional inverse analysis

This article is concerned with a two-dimensional inverse steady-state heat conduction problem. The aim of this study is to estimate the thermal conductivity, the heat transfer coefficient, and the heat flux in irregular bodies (both separately and simultaneously) using a two-dimensional inverse analysis. The numerical procedure consists of an elliptic grid generation technique to generate a mesh over the irregular body and solve for the heat conduction equation. This article describes a novel sensitivity analysis scheme to compute the sensitivity of the temperatures to variation of the thermal conductivity, the heat transfer coefficient, and the heat flux. This sensitivity analysis scheme allows for the solution of inverse problem without requiring solution of adjoint equation even for a large number of unknown variables. The conjugate gradient method (CGM) is used to minimize the difference between the computed temperature on part of the boundary and the simulated measured temperature distribution. The obtained results reveal that the proposed algorithm is very accurate and efficient.     

A new local meshless method for steady-state heat conduction in heterogeneous materials

In this paper a truly meshless method based on the integral form of energy equation is presented to study the steady-state heat conduction in the anisotropic and heterogeneous materials. The presented meshless method is based on the satisfaction of the integral form of energy balance equation for each sub-particle (sub-domain) inside the material. Moving least square (MLS) approximation is used for approximation of the field variable over the randomly located nodes inside the domain. In the absence of heat generation, the domain integration is eliminated from the formulation of presented method and the computational efforts are reduced substantially with respect to the conventional MLPG method. A direct method is presented for treatment of material discontinuity at the heterogeneous material in the presented meshless method. As a practical problem the heat conduction in fibrous composite material is studied and the steady-state heat conduction in unidirectional fiber–matrix composites is investigated. The solution domain includes a small area of the composite system called representative volume element (RVE). Comparison of numerical results shows that the presented meshless method is simple, effective, accurate and less costly method for micromechanical analysis of heat conduction in heterogeneous materials.     

Stress intensity factors for curvilinear cracks loaded under anti-plane strain (mode III) conditions

A general solution to the elastic and thermoelastic problems with a rigid circular-arc inclusion is presented. The proposed analysis is based upon the complex variable theory dealing with sectionally holomorphic functions which is reduced to the solution of the Hilbert problem. It is indicated that both the stress and thermal stress fields near the inclusion tip possess a square-root singularity similar to that for the corresponding crack problem. In analogy to the stress intensity factors defined for crack problem, stress singularity coefficients are introduced in this paper to characterize the near tip fields. Complete stress fields and the corresponding stress singularity coefficients as the circular-arc inclusion are under uniform remote load, concentrated force and uniform heat flux are given. Failure initiation of an infinite plate embedded with a rigid arc inclusion under different loading conditions is also discussed.     

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.     

The Hyperbolic Heat Conduction Equation in an Anisotropic Material

In this work, the phase-lag concept in the wave theory of heat conduction is extended to describe the thermal behavior of an anisotropic material. This is achieved by assuming that there are phase lags of different magnitudes between each component of the heat flux vector and the summation of temperature gradients in all directions of the orthogonal coordinate system. Also, expressions are provided to specify the locations of the principal coordinate axes, the principal thermal conductivities, and the principal thermal relaxation times.     

Modeling of smart composites on account of actuation,thermal conductivity and hygroscopic absorption

The asymptotic homogenization models for smart composite materials are derived and effective elastic, actuation, thermal expansion and hygroscopic expansion coefficients for smart structures are obtained. The actuation coefficients characterize the intrinsic transducer nature of active smart materials that can be used to induce strains and stresses in a coordinated fashion. Examples of such actuators employed with smart composite material systems are derived from piezoelectric, magnetostrictive, and some other materials. The pertinent mathematical framework is that of asymptotic homogenization. The objective is to transform a general anisotropic composite material with a regular array of reinforcements and/or actuators into a simpler one that is characterized by some effective coefficients; it is implicit, of course, that the physical problem based on these homogenized coefficients should give predictions differing as little as possible from those of the original problem. The effectiveness of the derived models is illustrated by means of two- and three-dimensional examples.     

Temperature Stresses and Displacements in a Multilayer Plate with Nonlinear Conditions of Heat Exchange

We propose a method for the determination of the stress-strain state formed in a multilayer plate under the conditions of radiation and convection heating with regard for the temperature and time dependences of the coefficients of heat exchange. The problem of heat conduction is reduced to the solution of a system of Volterra-type nonlinear integral equations of the second kind for the values of temperature on the outer surfaces of the plate. The integral equations are deduced with the help of the Green function of the nonstationary problem of heat conduction for the multilayer plate with constant coefficients of heat exchange and solved by using linear splines. The problem of thermoelasticity is solved in stresses. The boundary conditions on the cylindrical surface are satisfied in the integral form. The results of the corresponding numerical calculations are presented.