Analysis of fractional-order thermoelasticity with a heat source and three relaxation times

The evolution of residual stresses resulting from cooling an adhesive bond configuration on its lateral surfaces at a constant rate through the glass transition of the polymer are considered. A nonlinear, viscoelastic (free-volume) model serves for the thermoviscoelastic characterization of the polymer. The simultaneous solution to the heat diffusion and the transient thermoviscoelatic problems are addressed. Both an infinite (one-dimensional) and a finite (two-dimensional) domain are studied. A "critical" cooling time exists, in the present case on the order of a few seconds, which separates the control of the solidification process according to whether the relaxation or thermal diffusion time scale governs. The short time "quenching process," i.e., when the time scale is governed by thermal diffusion, leads to essentially constant residual stresses. Slower cooling increasingly invokes the time and rate sensitive properties of the polymer and leads to monotically decreasing residual stresses with longer cooling times. To reduce residual stresses by a factor of two from their maximal values requires cooling times on the order of one or two days. These results are not drastically altered by changes in the thicknesses of the bond components. Apart from singular behavior of the stress components in the two-dimensionally finite domain "quenching" has the effect of producing significantly different stress distributions (including stress "spikes") than slow or thermoelastic analyses would suggest. This observation is attributed to the interaction of the bending response of the metal components early in the cooling history under the high thermal gradients, which deformations are then partially frozen in during the subsequent cooling of the polymer. Implications of these results for systems possessing geometric and material differences subjected to various thermal cooling ranges are also discussed. The results demonstrate the importance of knowing the bulk relaxation or creep spectrum for the polymer. In the second part of the thesis the effect of the residual stresses on fracture behavior of an adhesive bond are addressed within the context of linear fracture mechanics for dissimilar materials. The crack faces are found to be in contact at the fractured end during the (residually stress) unloading process. A significantly error results if this contact zone is not taken into account. The combined effect of the mechanical loads and the residual stresses on the energy release rate is also studied. The total energy release rate from the combined effect is not necessarily higher or equal to the sum of the individual contribution from external loads and from residual stresses separately.

Thermodynamic and Gasdynamic Aspects of a Boiling Liquid Expanding Vapour Explosion

Casting is considered as a major manufacturing process. Thermal analysis of a solidifying medium is of great importance for appropriate design of casting processes. The conventional governing equation of a solidifying medium is based on the Fourier heat conduction law, which does not account for the phase-lag between the heat flux and the temperature gradient. In this paper, the concept of phase-lag during the phenomenon of solidification is investigated. This concept is considered by utilization of the hyperbolic heat conduction equation, known generally as the Maxwell–Cattaneo relation. In this way, the effect of finite heat wave speed on the thermal behavior of a solidifying medium is studied. In this context, some numerical example problems are analyzed with the meshless radial point interpolation method. The effect of the relaxation time on the thermal behavior of the solidifying medium is investigated. Moreover, the results of Fourier and non-Fourier heat conduction equations are compared. It is observed that based on the specific solidification process and the amount of relaxation time, the results of the Fourier and non-Fourier conduction laws can be quite different. The most prominent effect of the relaxation time is to alter the initiation of the solidification at each point.

Nonlinear transient heat transfer and thermoelastic stress analyses of a thick-walled FGM cylinder with temperature-dependent materials are performed by using the Hermitian transfinite element method. Temperature-dependency of the material properties has not been taken into account in transient thermoelastic analysis, so far. Due to the mentioned dependency, the resulting governing FEM equations of transient heat transfer are highly nonlinear. Furthermore, in all finite element analysis performed so far in the field, Lagrangian elements have been used. To avoid an artificial local heat source at the mutual boundaries of the elements, Hermitian elements are used instead in the present research. Another novelty of the present paper is simultaneous use of the transfinite element method and updating technique. Time variations of the temperature, displacements, and stresses are obtained through a numerical Laplace inversion. Finally, results obtained considering the temperature-dependency of the material properties are compared with those derived based on temperature independency assumption. Furthermore, the temperature distribution and the radial and circumferential stresses are investigated versus time, geometrical parameters and index of power law. Results reveal that the temperature-dependency effect is significant.

State space approach to thermoelasticity with two relaxation times

State space approach to generalized thermo-viscoelasticity with two relaxation times

A boundary element method based on the Laplace technique is developed for transient coupled thermoelasticity problems with relaxation times in a two-dimensional finite domain. The dynamic thermoelastic model of Green and Lindsay (Green, A. E., and Lindsay, K. E., Thermoelasticity, Journal of Elasticity, Vol. 2, No. 1, 1972, pp. 1-7) and Lord and Shulman (Lord, H. W., and Shulman, Y., A Generalized Dynamic Theory of Thermoelasticity, Journal of the Mechanics and Physics of Solids, Vol. 15, 1967, pp. 299-309) are selected. The Laplace transform method is applied in the time domain, and the resulting equations in the transformed field are discretized using a boundary element method. The nodal dimensionless temperature and displacements in the transformed domain are inverted to obtain the actual physical quantities, using the numerical inversion of the Laplace transform method. The concern is with thermoelastic waves detection, propagation, and reflection in a finite domain that have not been reported on in the past. Comparison is made with other solutions, and coupling and relaxation time effects in stress, displacement, and temperature distribution are investigated. Details of the formulation and numerical implementation are presented.

Mathematically, the well-known heat conduction equation is a parabolic partial differential equation, which defines the temperature at a point as proportional to the difference in the average of the surrounding temperatures. Marin indicated that the associated equation assumes that heat propagates at an infinite speed, which does not satisfy Einstein's special theory of relativity. Later, Landau discovered the existence of heat waves in liquid Helium II by observing the anomalous thermal conductivity at certain temperatures. Recently, it was reported that heat waves also occur in carbon nanotubes. To eliminate such physical inconsistencies, researchers have attempted to improve the associated equation in various ways, such as by introducing a relaxation time. In this study, we focused on the heat conduction equation, in which two relaxation times were introduced by Tzou, and attempted to couple it with the dynamic thermoelastic equation so that heat and stress waves could be generated and propagated simultaneously, but at different speeds. As a simple example, a one-dimensional bar problem was investigated and solved numerically using the Laplace transform technique. The results showed that in the conventional heat conduction equation, the temperature was diffusely distributed from the heating point in the depth direction, whereas the spike-shaped compressive stress propagated at a constant speed. However, in the heat conduction equation with relaxation times, the temperature distribution has discontinuities propagating at a constant speed, confirming that the heat wave can be simulated. However, the compressive stress results showed that the thermoelastic coupling effect increased the period of stress occurrence and reduced the peak stress.

Purpose The purpose of this paper is to frame a dual-phase-lag model using the fractional theory of thermoelasticity with relaxation time. The generalized Fourier law of heat conduction based upon Tzou model that includes temperature gradient, the thermal displacement and two different translations of heat flux vector and temperature gradient has been used to formulate the heat conduction model. The microstructural interactions and corresponding thermal changes have been studied due to the involvement of relaxation time and delay time translations. This results in achieving the finite speed of thermal wave. Classical coupled and generalized thermoelasticity theories are recovered by considering the various special cases for different order of fractional derivatives and two different translations under consideration. Design/methodology/approach The work presented in this manuscript proposes a dual-phase-lag mathematical model of a thick circular plate in a finite cylindrical domain subjected to axis-symmetric heat flux. The model has been designed in the context of fractional thermoelasticity by considering two successive terms in Taylor’s series expansion of fractional Fourier law of heat conduction in the two different translations of heat flux vector and temperature gradient. The analytical results have been obtained in Laplace transform domain by transforming the original problem into eigenvalue problem using Hankel and Laplace transforms. The numerical inversions of Laplace transforms have been achieved using the Gaver−Stehfast algorithm, and convergence criterion has been discussed. For illustrative purpose, the dual-phase-lag model proposed in this manuscript has been applied to a periodically varying heat source. The numerical results have been depicted graphically and compared with classical, fractional and generalized thermoelasticity for various fractional orders under consideration. Findings The microstructural interactions and corresponding thermal changes have been studied due to the involvement of relaxation time and delay time translations. This results in achieving the finite speed of thermal wave. Classical coupled and generalized thermoelasticity theories are recovered by considering the various special cases for different order of fractional derivatives and two different translations under consideration. This model has been applied to study the thermal effects in a thick circular plate subjected to a periodically varying heat source. Practical implications A dual-phase-lag model can effectively be incorporated to study the transient heat conduction problems for an exponentially decaying pulse boundary heat flux and/or for a short-pulse boundary heat flux in long solid tubes and cylinders. This model is also applicable to study the various effects of the thermal lag ratio and the shift time. These dual-phase-lag models are also practically applicable in the problems of modeling of nanoscale heat transport problems of semiconductor devices and accordingly semiconductors can be classified as per their ability of heat conduction. Originality/value To the authors’ knowledge, no one has discussed fractional thermoelastic dual-phase-lag problem associated with relaxation time in a finite cylindrical domain for a thick circular plate subjected to an axis-symmetric heat source. This is the latest and novel contribution to the field of thermal mechanics.

PurposeThe purpose of this paper is to study the transient waves caused by a line heat source with a stable internal heat source inside isotropic homogenous thermoelastic perfectly conducting half‐space permeate into a uniform magnetic field. The formulation is applied under three theories of generalized thermoelasticity Lord‐Shulman (L‐S) theory with one relaxation time, Green‐Lindsay (G‐L) theory with two relaxation times, as well as the classical dynamical coupled theory. The problem is reduced to the solution of three differential equations by introducing the elastic and thermoelastic potentials.Design/methodology/approachThe normal mode analysis is used to obtain the expressions. Numerical results are given and illustrated graphically. Comparisons are made with the results predicted by the three theories in the presence and absence of magnetic field and the internal heat source.FindingsThe results are graphically described for the medium of copper. We can conclude that the magnetic field has a great effect on the displacement components and this effect produces the same trend under the three theories. The results show that the relaxation times have salient effect to the distribution of displacement at small values of time.Originality/valueThe present theoretical results may provide interesting information for experimental scientists /researchers/seismologist working on this subject.

In this work, a non-Fourier heat conduction model considering the effect of heat source is presented and applied into pulsed laser deposition (PLD) technique to study target temperature evolvement when the target is radiated by high-power (≥1013 W/m2) short-pulse (in the order of picosecond) laser. Under this kind of irradiation conditions, the finite propagation speed of heat wave must be taken into account. The temperature profiles obtained from our model are compared with those obtained from Fourier conduction model and non-Fourier model without heat source term. The effects of heat source and relaxation time on non-Fourier temperature evolution are emphatically studied. It is found that the effect of non-Fourier heat conduction mainly shows in that it takes the target temperature certain time to start to increase and the increasing of temperature with time is faster obviously than that of Fourier heat conduction case. As for the heat source, it plays an important role that makes surface temperature ascending quicker than the case without heat source. Results also demonstrate that with the increasing of relaxation time, the Non-Fourier effect becomes remarkable more and more. Meanwhile, the corresponding physical mechanisms are investigated in detail.

Contact problems always exist in mechanical systems such as roller bearings, railway wheels, grinding wheels, etc. In this study, a steady-state thermoelastic analysis is considered for a cylindrical sliding contact which can be used to model many systems. For the contact problems, heat is generated from friction between the cylinder and the semi-infinite plane, which is assumed to be rigid and insulated.Rather than using a presumed Hertzian pressure distribution, numerical solutions are obtained by using a simply-discretized method and functional regulation. As predicted, the results indicate that the maximum contact pressure and temperature are generated at the undeformed contact point.

In this thesis the sensitivity of a modelled tropical cyclone to sea surface temperature (SST) and Convective Available Potential Energy (CAPE) is investigated. In Chapter 1 previous studies on this topic are discussed and an overview is given of our current knowledge of the effects of latent and sensible heat fluxes from the ocean on tropical cylcones. In Chapter 2 the experiments that make up this study are descibed in some detail. They have been carried out to compare the importance of SST to that of CAPE on tropical cyclone intensity. In order to achieve this, three experiments were designed, a control experiment, an experiment with increased SST and finally an experiment with increased SST, but CAPE equal to that of the control run. The simulations were carried out with the CSIRO Division of Atmospheric Research Limited area model using the Betts Miller cumulus parameterisation scheme. Both the model and this scheme are described in Chapter 3. The tropical cyclone used in this study is Tropical Cyclone Connie, which occurred in the Timor Sea from 15 to 23 January. The track and central pressure of the control simulation are compared with the observations and other simulations of the same storm. The simulation in this study compares very well to the other simulations and given its coarse resolution of 76 x 76 km, produces an acceptable simulation of cyclone Connie, given its coarse resolution of 76 x 76 km. When comparing the three experiments to each other it is found that the storm with increased SST and increased CAPE produces the strongest tropical cyclone. Finally, the sensitivity of the Betts Miller scheme to a model parameter called the relaxation time is discussed briefly. The relaxation time simulates the lag between the large scale forcing and the convective response. It was found that a decrease in relaxation time leads to faster convective adjustment and deeper tropical cyclones. It is concluded in Chapter 5 that SST is not as obviously tied to tropical cyclone intensity as was previously thought. Evaporation from the surface was found to play a very important role, but the importance of CAPE is still unclear. For that reason it will be necessary to perform more simulations with other convective schemes that respond directly to CAPE and are independent of external parameters that have not been verified against observations.

The effect of relaxation times is studied on plane waves propagating through semiconductor half-space medium by using the eigen value approach. The bounding surface of the half-space is subjected to a heat flux with an exponentially decaying pulse and taken to be traction free. Solution of the field variables are obtained in the form of series for a general semiconductor medium. For numerical values, Silicon is considered as a semiconducting material. The results are represented graphically to assess the influences of the thermal relaxations times on the plasma, thermal, and elastic waves.

In order to consider the effect of micro-structural interactions in the fast transient process of heat transport, a modified bio-heat transfer equation was developed based on the dual-phase-lag model. This work would use the corresponding equation to analyze the thermal response in a perfused tissue subjected to the spatial heating, the heat flux decays exponentially with the distance from the heating surface. In analysis process, the surface convection effect is taken into account. The modified discretization scheme based on the Laplace transform is extended to solve the non-Fourier bio-heat equation with boundary conditions. The effects of perfusion rate and relaxation times on the behavior of bioheat transfer are discussed as well as the boundary effect.