Type Heat Conduction Research Articles

Time decay of viscoelastic plates with type II heat conduction

In this work, we consider the thermomechanical problem determined by a viscoelastic plate coupled with heat conduction of type II. This problem differs significantly from what can be found in the literature since we assume that the dissipation mechanism is mechanical while the heat conduction mechanism is conservative. We first prove the existence of a semigroup of contractions defining the solutions in a suitable Hilbert space. Exponential stability of the solutions is also proved by means of a known characterization of the exponentially stable semigroups. The lack of differentiability of the semigroup is also proved. Finally, we use energy arguments to show that the only solution that can be zero after a finite time is the null solution.

Laser-Induced Thermoelastic Response in an Isotropic Medium Having Variable Material Moduli

Thermal displacement and stress distribution are analysed in the domain of plane-strain geometry of an isotropic thermoelastic medium having variable material moduli. In this context of the analysis, laser induced heat sources are planted on the reference plane. To analyse the thermal responses due to the injected heat sources into the medium, hyperbolic type heat conduction model with three phase-lags incorporated. Analytical solutions are expressed in-terms of Laplace–Fourier integral transform domain and the physical behaviour of displacement and stresses are depicted through a discretised form of inverse-integral transformation technique. Finally, parameterized characteristic analysis are presented graphically and the salient features of thermal displacements are highlighted.

Stability of a linear thermoelastic Bresse system with second sound under new conditions on the coefficients

AbstractIn this paper, we discuss the stability of a linear one‐dimensional thermoelastic Bresse system, where the coupling is given through the first component of the Bresse model with the heat conduction of second sound type. We state the well‐posedness and show the polynomial stability of the system, where the decay rate depends on the smoothness of initial data. Moreover, we prove the non exponential and the exponential decay depending on new conditions on the parameters of the system. The proof is based on a combination of the energy method and the frequency domain approach.

Non-exponential stability to a Timoshenko system with heat conduction and Kelvin–Voigt damping

We consider a Timoshenko system with partially Kelvin–Voigt damping, and with Cattaneo/Fourier type heat conduction. When the Timoshenko system with only heat conduction or only partially Kelvin–Voigt damping, some authors showed the non-exponential stability. In this paper, we add the Kelvin–Voigt damping in the thermoelastic Timoshenko system, and prove that the system is also not exponentially stable in cases of Fourier and Cattaneo type heat conduction, whether the wave speeds are equal or not. Additionally, we prove the semigroup decays with the optimal rate t−12.

On the decay of MGT-Viscoelastic plate with heat conduction of Cattaneo type in bounded and unbounded domains

In this paper, we consider a viscoelastic plate equation of Moore-Gibson-Thompson viscoelastic plate with heat conduction of Cattaneo type. We investigate the system in bounded and unbounded domains, seeking exponential stability in bounded domains and polynomial decay rates for the Cauchy problem. It turns out that our system is exponentially stable in the bounded domain, while the system has regularity loss in the Cauchy problem. Our results are proved in the sub-critical case $ K>0. $

Exponential stability for a flexible structure with Fourier's type heat conduction and distributed delay

Exponential stability for a flexible structure with Fourier's type heat conduction and distributed delay

A thermodynamic analysis of an enhanced theory of heat conduction model: Extended influence of finite strain and heat flux

An Eulerian form of thermodynamically consistent generalized thermoelastic model capable of accounting for thermal signals is proposed. The heat flux is assumed to consist of both energetic and dissipative components. The constitutive relations for the stress, entropy and the heat fluxes are derived in the spatial coordinate system. Later on, a domain of dependence inequality for the proposed hyperbolic type heat conduction model is proved. The linearised form of this improved thermoelasticity theory of finite deformation is employed to study the thermoelastic interactions due a continuous source of heat in isotropic elastic solids. This newly developed model is also applied in a thermoelastic saturated porous medium. In order to obtain the exact analytical expressions of the field functions, appropriate integral transformations are employed in a convenient way.

Extensional and flexural modes of Rayleigh–Lamb wave in an orthotropic thermoelastic layer lying over a viscoelastic half-space

In this article the characteristics of the extensional and flexural modes, propagating in a thermoelastic orthotropic layer lying over a viscoelastic half-space, are analyzed. The complete analysis is carried out in the framework of a thermodynamically consistent hyperbolic type heat conduction model without energy dissipation. The normal-mode-analysis is adopted and a general form of dispersive equation is derived for an anisotropic thermoelastic layered medium. A prominent distinction with the isotropic elastic solids is observed in the symmetric as well as anti-symmetric modes of dispersion curves. In turn, such deformation reshapes the wave propagation while the deformation stiffening changes significantly the phase velocities of the wave till the acoustic radiation stresses are balanced by elastic stresses in the current configuration of the hyperelastic medium.

The Cauchy Problem for Thermoelastic Plates with Two Temperatures

We consider the decay rates of solutions to thermoelastic systems in materials where, in contrast to classical thermoelastic models for Kirchhoff type plates, two temperatures are involved, related by an elliptic equation. The arising initial value problems deal with systems of partial differential equations involving Schrödinger like equations, hyperbolic and elliptic equations. Depending on the model – with Fourier or with Cattaneo type heat conduction – we obtain polynomial decay rates without or with regularity loss. This way we obtain another example where the loss of regularity in the Cauchy problem corresponds to the loss of exponential stability in bounded domains. The well-posedness is done using semigroup theory in appropriate space reflecting the different regularity compared to the classical single temperature case, and the (optimal) decay estimates are obtained with sophisticated pointwise estimates in Fourier space.

Stability of abstract thermoelastic systems with inertial terms

We investigate coupled systems of thermoelastic type in a general abstract form both modeling Fourier and Cattaneo type heat conduction. In particular we take into account a possible inertial term. A complete picture of the regions of exponential stability resp. non-exponential stability for the arising parameters (two from the type of thermoelastic system, one from the inertial term) is given. The regions of loss of exponential stability, while moving from the Fourier to the Cattaneo law, are thus clearly recognized and interestingly large. The polynomial stability in regions of non-exponential stability is also characterized.

Telegraph equation: two types of harmonic waves, a discontinuity wave, and a spectral finite element

The telegraph equation $$\tau \partial ^2 u/\partial t^2 + \partial u/ \partial t = \tau c^2 \partial ^2 u / \partial x^2$$ arises in studies of waves in dissipative media with a damping coefficient $$1/\tau $$ , or from a Maxwell–Cattaneo type heat conduction with a relaxation time $$\tau $$ . To elucidate basic properties of this equation, two harmonic wave solutions are compared: (1) temporally attenuated and spatially periodic (TASP) and (2) spatially attenuated and temporally periodic (SATP). The phase velocities of both waves are equal to the energy velocities and less than the group velocities. The phase velocities of the two waves are different, and less than c, but both naturally lead to a speed c for the propagation of discontinuities. The two harmonic wave solutions are suitable for different initial-boundary value problems: TASP for those with space periodicity and SATP for those with time periodicity. The asymptotic behaviors of the harmonic wave solutions when the telegraph equation transitions into a nondissipative wave equation or into a parabolic diffusion equation are presented. Only the SATP waves survive when the equation turns parabolic. The spectral finite element method is formulated for 1d Maxwell–Cattaneo heat conduction based on the SATP wave solutions. The element thermal conductivity matrix is reduced to that for a conventional (nonspectral) finite element when the frequency tends to zero.

Component level modelling of heat transfer during vapour phase soldering with finite difference ADI approach

In this paper, the complex heat transfer process of the vapour phase soldering has been investigated on the level of electronic components. VPS is gaining increased attention lately, and the process needs alternative approaches in modelling, compared to conventional soldering processes in electronics mass manufacturing. Component level modelling was not studied deeply in the literature before, so our focus pointed to heat transfer on large size surface mounted electronic components. Applying the Fourier type heat conduction equation, a detailed 3D thermal model of a polyester capacitor on a printed circuit board was implemented, based on X-ray images of an actual assembly. Our model incorporates inner geometry, material inhomogeneity, composite materials and anisotropic thermal conductivity as important thermal features. Transient heating was calculated with Finite Difference Method combining an alternating direction implicit (ADI) approach, using averaged heat transfer coefficient. Validation measurements were performed in our experimental VPS system. The measured data show good agreement with the calculation results and points to possible application for use in advanced engineering and manufacturing environment.

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

Continuum mechanics with dislocations, with the Cattaneo type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov type system of the first order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov type formulation brings the mathematical rigor (the well-posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).

Enhancing boiling and condensation co-existing heat transfer in a small and closed space by heat-conduction bridges

This paper aims at enhancing the boiling and condensation co-existing phase change heat transfer process inside a small and closed space by heat-conduction bridges. The working fluid is deionized water and the copper heat-conduction bridges are machined directly on the boiling surface and connect the boiling and the condensation surface. Three different heat-conduction bridges that are all rectangular pillars of different sizes and layouts are tested. Experiments were carried out to study the influences of working fluid filling ratio, the cross sectional area size of the heat-conduction bridge and the gap between the heat-conduction bridges (the bridge gap) on boiling and condensation co-existing heat transfer inside the small and closed space. Experimental results show that the heat-conduction bridges can greatly enhance the boiling and condensation co-existing heat transfer process. The optimum filling ratio is 50% for 3 type heat-conduction bridges. The boiling, condensation and overall heat transfer coefficient of the heat-conduction bridges whose cross-sectional area is 2mm×2mm and bridge gap is 1mm (HCB-C) is the largest among the three heat-conduction bridges.

Guyer-Krumhansl–type heat conduction at room temperature

Results of heat pulse experiments in various artificial and natural materials are reported in this paper. The experiments are performed at room temperature with macroscopic samples. It is shown that temperature evolution does not follow Fourier's law but is well explained by the Guyer-Krumhansl equation. The observations confirm the ability of non-equilibrium thermodynamics to formulate universal constitutive relations for thermomechanical processes.

Stability for thermoelastic plates with two temperatures

We investigate the well-posedness, the exponential stability, or the lack thereof, of thermoelastic systems in materials where, in contrast to classical thermoelastic models for Kirchhoff type plates, two temperatures are involved, related by an elliptic equation. The arising initial boundary value problems for different boundary conditions deal with systems of partial differential equations involving Schrodinger like equations, hyperbolic and elliptic equations, which have a different character compared to the classical one with the usual single temperature. Depending on the model -with Fourier or with Cattaneo type heat conduction -we obtain exponential resp. non-exponential stability, thus providing another examples where the change from Fourier's to Cattaneo's law leads to a loss of exponential stability.

Decay property of the Timoshenko–Cattaneo system

We study the Timoshenko system with Cattaneo’s type heat conduction in the one-dimensional whole space. We investigate the dissipative structure of the system and derive the optimal [Formula: see text] decay estimate of the solution in a general situation. Our decay estimate is based on the detailed pointwise estimate of the solution in the Fourier space. We observe that the decay property of our Timoshenko–Cattaneo system is of the regularity-loss type. This decay property is a little different from that of the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647–667]) in the low frequency region. However, in the high frequency region, it is just the same as that of the Timoshenko–Fourier system (see [N. Mori and S. Kawashima, Decay property for the Timoshenko system with Fourier’s type heat conduction, J. Hyperbolic Differential Equations 11 (2014) 135–157]) or the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647–667]), although the stability number is different. Finally, we study the decay property of the Timoshenko system with the thermal effect of memory-type by reducing it to the Timoshenko–Cattaneo system.

Exponential decay in nonsimple thermoelasticity of type III

This paper deals with the model proposed for nonsimple materials with heat conduction of type III. We analyze first the general system of equations, determine the behavior of its solutions with respect to the time, and show that the semigroup associated with the system is not analytic. Two limiting cases of the model are studied later. Copyright © 2015 John Wiley & Sons, Ltd.

On the time-dependent Cattaneo law in space dimension one

We consider the one-dimensional wave equationεutt-uxx+[1+εf′(u)]ut+f(u)=hwhere ε=ε(t) is a decreasing function vanishing at infinity, providing a model for heat conduction of Cattaneo type with thermal resistance decreasing in time. Within the theory of processes on time-dependent spaces, we prove the existence of an invariant time-dependent attractor, which converges in a suitable sense to the attractor of the classical Fourier equationut-uxx+f(u)=hformally arising in the limit t→∞.

DECAY PROPERTY FOR THE TIMOSHENKO SYSTEM WITH FOURIER'S TYPE HEAT CONDUCTION

We study the Timoshenko system with Fourier's type heat conduction in the one-dimensional (whole) space. We observe that the dissipative structure of the system is of the regularity-loss type, which is somewhat different from that of the dissipative Timoshenko system studied earlier by Ide–Haramoto–Kawashima. Moreover, we establish optimal L2decay estimates for general solutions. The proof is based on detailed pointwise estimates of solutions in the Fourier space. Also, we introuce here a refinement of the energy method employed by Ide–Haramoto–Kawashima for the dissipative Timoshenko system, which leads us to an improvement on their energy method.