ON A REFINED HEAT CONDUCTION THEORY FOR MICROPERIODIC LAYERED SOLIDS

Abstract

A refined averaged theory of a rigid heat conductor with a microperiodic structure is used to solve a one-dimensional initial boundary value problem of heat conduction in a periodically layered plate with a large number of homogeneous isotropic layers. In such a theory, the temperature θ = θ(x,t) (0 ≤ x ≤ L, t ≥ 0)is approximated by θ(x,t) = θ₀(x,t) + η(x)θ₁(x,t) where θ₀(x,t) is a temperature-corrector and η = η(x) is a prescribed microshape function; and the functions θ₀ = θ₀(x,t) and θ₁ = theta;₁(x,t) are to be found by solving an initial-boundary value problem described by a system of linear partial differential equations with averaged coefficients subject to suitable initial and boundary conditions. A uniqueness theorem for the averaged problem is proved and two closed-form solutions for a periodically layered semispace are obtained. One of the two solutions represents the temperature field in the layered semispace due to a sudden heating of the boundary plane, while the other stands for the temperature field in the layered semispace produced by laser surface heating. Numerical examples are included.     

SAINT-VENANT'S PRINCIPLE FOR A MICROPERIODIC COMPOSITE THERMOELASTIC SEMISPACE: THE DYNAMICAL REFINED AVERAGE THEORY

A Saint-Venant's principle associated with a one-dimensional dynamic coupled thermoelastic effective modulus theory (EMT) for a microperiodic layered semispace was presented in J. Thermal Stresses, vol. 23, pp. 1-14, 2000. It was shown there that a thermoelastic energy associated with a solution to an initial boundary value problem of the theory decays exponentially as a distance x from the thermomechanical load region goes to infinity and that its decay length depends on the time t, an effective velocity d c 1 * ¢ , an effective time unit d T* ¢ , and an effective thermoelastic coupling parameter d k * ¢ . In the present article the Saint-Venant's principle is extended to include a refined averaged theory (RAT) for a microperiodic layered thermoelastic semispace in which a microstructural length is taken into account (see [IFTR Report #25, pp. 1-158, 1995]). It is shown that for such an extended theory, a similar exponential decay estimate for a thermoelastic energy holds true. In the refined estimate the thermoelastic energy depends on a number of microstructural parameters while its decay length is independent of these parameters; and the decay length for small (large) times is comparable to that of a pure thermal (elastic) energy for a rigid (elastic) semispace for every time t > 0 .     

AXISYMMETRIC TEMPERATURE DISTRIBUTIONS ON A TRANSVERSELY ISOTROPIC SEMI-INFINITE SOLID

An exact static solution for the axisymmetric boundary value problem of a transversely isotropic semispace subjected to a point heat source is constructed by similarity transformations. The closed-form expressions for the temperature and components of displacements and stresses are derived. In the particular cases of uniform and parabolic-type temperature distributions on a circular area of the surface, the expressions for the displacements and stresses at a distance z beneath the surface are determined. The Mathematica software is used, and the numerical results are presented on graphs depicting the spatial variation of the displacements and stresses in a semispace of cobalt material.     

An investigation on thermoelastic interactions under an exact heat conduction model with a delay term

The present work is concerned with a very recently proposed heat conduction model: an exact heat conduction model with a single delay term. A generalized thermoelasticity theory was proposed by Roy Choudhuri based on the heat conduction law with three-phase-lag effects for the purpose of considering the delayed response in time due to the microstructural interactions in the heat transport mechanism. However, the model defines an ill-posed problem in Hadamard sense. Quintanilla has recently proposed to reformulate this heat conduction model as an alternative heat conduction theory with a single delay term and subsequently, Leseduarte and Quintanilla investigated the spatial behavior of the solutions for this theory and they extended the results to a thermoelasticity theory by considering the Taylor series approximation of the equation of heat conduction with one delay term. In the present work, we consider the thermoelasticity theory based on this newly proposed heat conduction model and investigate a problem of thermoelastic interactions. State-space approach is used to formulate the problem and the formulation is then applied to a problem of an isotropic elastic half-space with its plane boundary subjected to sudden increase in temperature and zero stress. The integral transform method is applied to obtain the solution of the problem. A detailed analysis of analytical results is provided by finding the short-time approximated solutions of different field variables analytically and comparing the results of the present model with the corresponding results reported for other existing theories. An attempt has also been made to illustrate the problem and numerical values of field variables are obtained for a particular material. Results are analyzed with different graphs. To the best of the author\textquoteright s knowledge, this thermoelastic model is not yet investigated by any researcher in this direction.     

Mixed nonstationary problem of heat conduction for a microperiodic two-layered half-space

The paper deals with a mixed nonstationary problem of heat conduction for a microperiodic two-layered half-space. The body is assumed to be initially at zero temperature, the boundary plane is kept at a known temperature for time 0 ≤ t ≤ t₀ and is insulated for t > t₀. The half-space is composed of periodically repeated two-layered laminae with layering parallel to the boundary. The problem is solved within the framework of homogenized model with microlocal parameters [Cz. Woźniak, A nonstandard method of modelling of thermoelastic periodic composites, International Journal of Engineering Science 25, (1987), pp. 483–499, S.J. Matysiak Cz. Woźniak, On the modelling of heat conduction problem in laminated bodies, Acta Mechanica, 65, (1986), pp. 223–238.]. The influence of thermal and geometric properties of the composite components on the temperature distribution for some special case of the boundary condition is analyzed.     

Numerical Solution of Periodic Heat Transfer in an Anisotropic Cylinder Subject to Asymmetric Heat Flux

Abstract

The problem of heat conduction in a two-dimensional anisotropic cylinder subject to asymmetric and periodic heat flux distribution on the outer wall is solved numerically. The dimensional analysis of the problem reveals that the heat conduction is a function of five nondimensional parameters: nondimensional frequency (α), cylinder outer to inner radius ratio (R₂), Biot number (Bi), orthotropicity factor (K₁), and anisotropicity factor (K_r1). A systematic study of the effect of each parameter is carried out over the influential range for each parameter. The results show that, depending on the combination of these parameters, the magnitude and/or phase of heat conduction in an anisotropic cylinder can be significantly different from those of an orthotropic and isotropic cylinder when subjected to the same externally imposed heat flux distribution.     

A Global Heat Compliance Measure Based Topology Optimization for the Transient Heat Conduction Problem

For topology optimization with transient loads, heat compliance varies with transient heat analysis. The peak value of the transient heat compliance should be minimized. Thus, this article proposes a global heat compliance measure to handle this kind of topology optimization for the transient heat conduction problem. The optimization model is then constructed by the global heat compliance measure. The finite-element, equivalent static loads, and continuum shape based sensitivity analyses are derived using the adjoint variable method. Through case studies, the effectiveness of the proposed global heat compliance measure for the transient heat conduction topology optimization is validated.     

PLANE HARMONIC WAVES IN A MICROPERIODIC LAYERED THERMOELASTIC SOLID REVISITED

In previous work a one-dimensional averaged theory of thermoelastic waves in a microperiodic layered infinite solid was proposed in which an eighth-order-in-time partial differential equation involving a high intrinsic mechanical frequency Ω_M and a high intrinsic thermal frequency Ω_T is a central equation. Also, the existence of two harmonic thermoelastic waves of a given frequency ω propagating in a positive direction normal to the layering was previously established when Ω_M → ∞ and Ω_T < ∞, or Ω_M < ∞ and Ω_T → ∞. The existence of two harmonic thermoelastic waves of a given frequency ω propagating in the positive direction normal to the layering is proved when Ω_M < ∞ and Ω_T < ∞. Also, a closed form of the associated velocities and attenuation coefficients is obtained. Numerical results illustrating propagation of the two waves in a particular microperiodic layered thermoelastic solid are included.     

On the Theory of Two-Temperature Thermoelasticity with Two Phase-Lags

The present paper is concerned with the theory of two temperature thermoelasticity with two phase-lags in which the theory of heat conduction in deformable bodies depends on two distinct temperatures – the conductive temperature and the thermodynamic temperature. A generalized heat conduction law with dual-phase-lag effects was proposed by Tzou (1995) for the purpose of considering the delayed response in times due to the microstructural interactions in the heat transport mechanism. Recently, Quintanilla (2008) has proposed to combine this constitutive equation with a two temperature heat conduction theory and has proved that a dual-phase-lag theory with two temperatures is a well-posed problem. In the present work we consider the basic equations concerning this dual-phase lag theory of two temperature thermoelasticity and make an attempt to establish some important theorems in this context. A uniqueness theorem has been established for a homogeneous and isotropic body. An alternative characterization of mixed boundary initial value problem is formulated and a variational principle as well as reciprocal principle have been established.     

PLANE HARMONIC WAVES IN A MICROPERIODIC LAYERED INFINITE THERMOELASTIC SOLID

A one-dimensional dynamical coupled refined averaged thermoelasticity for a microperiodic composite is used to study harmonic waves propagating in a layered infinite solid. In such a theory the waves are governed by an eighth-order-in-time partial differential equation in which the two intrinsic frequencies are present: a mechanical frequency Ω and a thermal frequency α; both these frequencies are high if the layering period l is small. The existence of harmonic waves of a given frequency ω propagating in a positive direction normal to the layering is established when (i) Ω→∞, α<∞ or (ii) Ω<∞, α→∞. It is shown that in each of the two cases there are two plane harmonic thermoelastic waves of a given frequency ω that are dispersive and attenuated. Also, a closed form of velocities and attenuation coefficients for the two waves are obtained. The velocities and attenuation coefficients, treated as the functions of ω, are illustrated graphically for a unit cell made of the two homogeneous isotropic thermoelastic layers: a zirconium oxide [Zr O₂] and a titanium alloy [Ti–6Al–4V].     

An integral equation approach to diffusion

A method of solution of transient diffusion, e.g. heat conduction, problems in homogeneous and isotropic media with internal sources and arbitrary (including nonlinear) boundary conditions and initial conditions is proposed. The method is based on the reduction of the problem to one only involving surface values of temperature and/or heat flux in the form of an integral equation through the introduction of fundamental solutions and the use of Green's theorem. The integral equation is solved numerically for a specific example.     

On thermoelastic problem of a thermosensitive functionally graded rectangular plate with instantaneous point heat source

The present article attempts to investigate transient thermoelastic problem of a functionally graded thin rectangular plate with thermosensitive and spatial variable dependent material properties. All the material properties are assumed to be isotropic. Kirchhoff’s variable transformation is used to deal with the nonlinearity of the three dimensional heat conduction equation and that equation is solved by the integral transform method considering an instantaneous point heat source. The thermoelastic behavior of the rectangular plate is studied together with the thermal deflection and thermally induced resultant moments. Numerical computations are carried out for ceramic-metal-based functionally graded material, in which alumina is selected as ceramic and nickel as metal. The results are illustrated graphically.     

Non-Fourier effects on transient temperature resulting from periodic on-off heat flux

The transient temperatures resulting from a periodic on-off heat flux boundary condition have many applications, including, among others, the sintering of catalysts frequently found during coke burnoff, and the use of laser pulses for annealing of semiconductors. In such situations, the duration of the pulses is so small (i.e. picosecond-nanosecond) that the classical heat diffusion phenomenon breaks down and the wave nature of energy propagation characterized by the hyperbolic heat conduction equation governs the temperature distribution in the medium. In this work, an explicit analytic solution is presented for a linear transient heat conduction problem in a semi-infinite medium subjected to a periodic on-off type heat flux at the boundary x = 0 by solving the hyperbolic heat conduction equation. The non-linear case allowing for the added effect of surface radiation into an external ambient is studied numerically.     

On the Harmonic Vibrations in Linear Thermoelasticity Without Energy Dissipation

In the present paper we consider a prismatic cylinder occupied by an anisotropic homogeneous compressible linear thermoelastic material that is subject to zero body force and heat supply and zero displacement and thermal displacement on the lateral boundary. The motion is induced by a harmonic time–dependent displacement and thermal displacement specified pointwise over the base. We establish some spatial estimates for an appropriate cross–sectional measure associated with the amplitude of the corresponding harmonic vibration in the high frequency range that describe how the amplitude evolves with respect to the axial distance to the excited base. In fact, in the high frequency range we establish certain algebraical estimates that predict decay or growth slower than that for the low frequency range.     

Assessment of Thermal Properties via Nanosecond Thermoreflectance Method

Nanosecond time-domain thermoreflectance (ns-TDTR) is an all optical method of determining independently a variety of thermal parameters of both homogeneous and layered materials. Despite its relative experimental simplicity, the sensitivity of the temperature decay (measured by the transient reflectivity signal) to the relevant thermal properties has yet to be fully characterized. In principle, it is possible to simultaneously extract multiple thermal parameters from a single measurement. In practice, however, changes to several of these parameters may result in experimentally indistinguishable variations to the transient reflectivity signal. In this work, we focus on investigating thermal properties of bulk material and the contact resistance between the thin-film coating that is needed for the ns-TDTR method and the bulk substrate. To extract multiple properties from one temperature decay trace, we divide the data into temporal sub-regions known to be influenced to different degrees by each individual thermal parameter and iteratively fit with a 1-D heat conduction model to independently determine the contact resistance and cross-plane thermal conductivity.     

On the Spatial Behavior in Thermoelasticity Without Energy Dissipation

In the present paper we consider a prismatic cylinder occupied by an anisotropic homogeneous compressible linear thermoelastic material that is subject to zero body force and heat supply and zero displacement and thermal displacement on the lateral boundary. The motion is induced by a harmonic time-dependent displacement-thermal displacement specified pointwise over the base. We establish some spatial estimates for some appropriate cross-sectional measures associated with the transient and steady state solutions that describe how they evolves with respect to the axial distance to the excited base. For the transient solutions a domain of influence is observed inside which a measure of the solution decays exponentially with respect to the axial variable, while outside of the domain the solution vanishes. For the harmonic in time vibrations we establish spatial estimates describing how the amplitude of vibration evolves with respect to the axial variable, provided the frequency is lower than a critical value. All results are established for the thermoelastic materials whose constitutive coefficients satisfy strong ellipticity conditions.     

Relativistic moving heat source

The relativistic heat conduction (RHCE) model is particularly important in the analysis of processes involving moving heat sources (MHS) at speeds or frequencies comparable with those of heat propagation in the medium. This paper establishes a unified framework for solving heat conduction problems using the RHCE model. It offers “Fundamental Solutions” in one, two, and three spatial dimensions, for the transient response due to an instantaneous point MHS. Moreover, it presents the transient response due to a continuous point MHS, the quasi-steady response due a periodic point MHS, as well as guidelines for solving the RHCE equation under various loading and boundary conditions.     

The effect of prandtl number on conjugate heat transfer from rectangular fins

The problem of conjugate heat transfer from rectangular fins is treated numerically. Results are provided for the three Prandtl numbers (three fluids) 0.021, 0.7 and 5.0 and convection - conduction parameters meters in the range of 0 ≤ √Re k_fL/k_sb = CCP ≤ 5. The results indicated great effects of both CCP and Pr. Comparisons with the simple conventional fin theory show that concerning the fin efficiency, the simple theory yields acceptable results while the temperature variation and local heat flux distributions are not correctly predicted.     

Estimation of a Location- and Time-Dependent High-Magnitude Heat Flux in a Heat Conduction Problem Using the Kalman Filter and the Approximation Error Model

This paper aims to estimate a location- and time-dependent high-magnitude heat flux in a heat conduction problem. The heat flux is applied on a small region of a surface of a flat plate, while transient temperature measurements are taken on the opposite surface. This inverse problem is solved using the Kalman filter and a reduced forward model, obtained by simplifications of a three-dimensional and nonlinear heat conduction problem. To deal with the modeling errors of this reduced model, the Approximation Error Model is used. The results show that excellent estimates can be obtained at feasible computational times.