International Journal of Modern Engineering Research (IJMER)              www.ijmer.com         Vol.2, Issue.6, Nov-Dec. 2...
International Journal of Modern Engineering Research (IJMER)              www.ijmer.com         Vol.2, Issue.6, Nov-Dec. 2...
International Journal of Modern Engineering Research (IJMER)                 www.ijmer.com         Vol.2, Issue.6, Nov-Dec...
International Journal of Modern Engineering Research (IJMER)               www.ijmer.com         Vol.2, Issue.6, Nov-Dec. ...
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A Review on Two-Temperature Thermoelasticity

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A Review on Two-Temperature Thermoelasticity

  1. 1. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4224-4227 ISSN: 2249-6645 A Review on Two-Temperature Thermoelasticity 12 Kiran Bala 1 Research scholar in Mathematics, Singhania University, Rajasthan,India . 2 Department of Mathematics, Government College, Barwala, Panchkula, Haryana, IndiaABSTRAC: The present paper deals with the review on have an inherent paradox arising from the assumption thatthe development of the theory of two-temperature the thermal waves propagate at infinite velocity and it is athermoelasticity. The basic equations of two-temperature physically unreasonable result.thermoelasticityin context of Lord and Shulman [6] theory Generalized thermoelasticity theories have beenand Green and Naghdi[15] theories of generalized developed with the objective of removing the paradox ofthermoelasticity are reviewed. Relevant literature on two- infinite speed of heat propagation inherent in thetemperature thermoelasticity is also reviewed. conventional coupled dynamical theory of thermoelasticity in which the parabolic type heat conduction equation isKeyWords:Two-temperature, generalized thermoelasticity, based on Fouriers law of heat conduction. This newlybasic equations emerged theory which admits finite speed of heat propagation is now referred to as the hyperbolic 1. INTRODUCTION thermoelasticity theory, since the heat equation for rigid Thermoelasticity deals with the dynamical system conductor is hyperbolic-type differential equation. The firstwhose interactions with the surrounding include not only generalized theory of thermoelasticity is due to Lord andmechanical work and external work but the exchange of Shulman [6] who coupled elasticity with a way in whichheat also. Changes in temperatures causes thermal effects temperature can travel with a finite wave speed. Theon materials. Some of these thermal effects include thermal approach of Lord and Shulman [6] begins with the fullstress, strain, and deformation. Thermal deformation simply nonlinear equations but they are mainly interested inmeans that as the "thermal" energy (and temperature) of a developing a linear theory since they begin with "smallmaterial increases, so does the vibration of its strains and small temperature changes". The secondatoms/molecules and this increased vibration results in what generalization to the coupled theory is known as thecan be considered a stretching of the molecular bonds - generalized theory with two relaxation times. Muller[7]which causes the material to expand. Of course, if the introduced the theory of generalized thermoelasticity withthermal energy (and temperature) of a material decreases, two relaxation times. A more explicit version wasthenthe material will shrink or contract. Thus, thermoelasticity introduced by Green and Laws [8], Green and Lindsay [9]is based on temperature changes induced by expansion and and independently by Suhubi[10]. In this theory thecompression of the test part. Thus, the theory of temperature rates are considered among the constitutivethermoelasticity is concerned with predicting the variables. This theory also predicts finite speeds ofthermomechanical behaviour of elastic solids. It represents propagation for heat and elastic waves similar to the Lord-a generalization of both the theory of elasticity and theory Shulman theory. It differs from the latter in that Fouriersof heat conduction in solids.The theory of thermoelasticity law of heat conduction is not violated if the body underwas founded in 1838 by Duhamel [1], who derived the consideration has a centre of symmetry. Dhaliwal andequations for the strain in an elastic body with temperature Sherief[11] extended the Lord and Shulman (L-S) theorygradients. Neumann [2], obtained the same results in 1841. for an anisotropic media. Chandrasekharaiah[12]referred to However, the theory was based on independence this wave-like thermal disturbance as "second sound".of the thermal and mechanical effects. The total strain was These writers investigate the propagation of a thermal pulsedetermined by superimposing the elastic strain and the in a thermoelastic shell employing each of the linearizedthermal expansion caused by the temperature distribution equations for the three thermoelastic theories, Classical,only. The theory thus did not describe the motion associated Lord-Shulman and Green-Lindsay. Their numerical resultswith the thermal state, nor did it include the interaction typically demonstrates that Classical theory leads to abetween the strain and the temperature distributions. Hence, smooth pulse while that of Lord-Shulman is less smooththermodynamic arguments were needed, and it was showing discontinuities in derivatives. The theory of GreenThomson [3], in 1857 who first used the laws of and Lindsay [9] leads to strong pulse behaviour displayingthermodynamics to determine the stresses and strains in an distinct jumps such as to the behaviour of stainless steel runelastic body in response to varying temperatures. tanks which holds cryogenic liquids for rocket fuel at There are three types of thermoelasticity i.e. NASAs John C. Stennis Space Centre, the strong pulseuncoupled, coupled and generalized thermoelasticity. The solution is definitely of interest.theory of classical uncoupled theory of thermoelasticity Green and Naghdi[13-15] proposed three newpredicts two phenomena not compatible with physical thermoelastic theories based on entropyequality than theobservations. First, the equation of heat conduction of this usual entropy inequality. The constitutive assumption fortheory does not contain any elastic term contrary to the fact the heat flux vector are different in each theory. Thus theythat the elastic changes produce heat effects. Second, the obtained three theories which are called thermoelasticity ofheat equation is of parabolic type predicting infinite speeds type I, thermoelasticity of type II and thermoelasticity ofof propagation for heat waves. The classical uncoupled and type III. Green and Naghdi[15] postulated a new conceptcoupled thermoelastic theories of Biot[4] and Nowacki[5] ingeneralizedthermoelasticity which is called the www.ijmer.com 4224 | Page
  2. 2. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4224-4227 ISSN: 2249-6645thermoelasticity without energy dissipation. The principal temperature generalized thermoelasticity, based on thefeature of this theory is that in contrast to the classical theory of Youssef to solve boundary value problems of one-thermoelasticity, the heat flow does not involve energy dimensional finite piezoelectric rod with loading on itsdissipation. Also, the samepotential function which is boundary with different types of heating.Abbas and Youssefdefined to derive the stress tensor is used to determine the [31]analysed a finite element model of two-temperatureconstitutive equation for the entropy flux vector. In generalized magneto-thermoelasticity.Ezzat et.al.[32]addition, the theory permits the transmission of heat as studied the two-temperature theory in generalized magneto-thermal waves at finite speeds. Dhaliwal and Wang [16] thermo-viscoelasticity. Mukhopadhyay and Kumar[33]formulated the heat-flux dependent thermoelasticity theory studiedthermoelastic interactions on two-temperaturefor an elastic material with voids. This theory includes the generalized thermoelasticity in an infinite medium with aheat-flux among the constitutive variables and assumes an cylindrical cavity.Youssef[34] constructed a model of two-evolution equation for the heat-flux. Hetnarski and temperature generalized thermoelasticity for an elastic half-Ignaczack[17] examined five generalizations to the coupled space with constant elastic parameters. Ezzat and Awad[35]theory and obtained a number of important analytical derived the equations of motion and the constitutiveresults. Literature on generalized thermoelasticity is relations for the theory of micropolar generalized two-available in the books like "Thermoelastic Solids" by temperature thermoelasticity. Kaushal et al. [36]solved theSuhubi[10], "Thermoelastic Deformations" by lesan and boundary-value problem in frequency domain in the contextScalia[18], "Thermoelastic Models of Continua" by Iesan of two generalized theories of thermoelasticity (Lord and[19], "Thermoelasticity with Finite Wave Speeds" by Shulman, Green and Lindsay) by employing the HankelIgnaczak and Ostoja-Starzewski [20] etc. transform. Kumar et.al.[37]established a variational principle of convolutional type and a reciprocal principle in II.LITERATURE SURVEY the context of linear theory of two-temperature Chen and Gurtin[21] and Chen et al. [22-23] have generalized thermoelasticity, for a homogeneous andformulated a theory of heat conduction in deformable isotropic body. Kumar and Mukhopadhyay[38]bodies, which depends on two distinct temperatures, the investigated the propagation of harmonic plane waves inconductive temperature  and the thermodynamic elastic media in the context of the linear theory of two-temperature T. The two-temperature theory involves a temperature-generalized thermoelasticity. Youssef [39]material parameter a* > 0. The limit a*→ 0 implies that → solved a problem of thermoelastic interactions in an elasticT and hence classical theory can be recovered from two- infinite medium with cylindrical cavity thermally shockedtemperature theory. The two-temperature model has been at its bounding surface and subjected to moving heat sourcewidely used to predict the electron and phonon temperature with constant velocity. Youssef and El-Bary[40]studieddistributions in ultrashort laser processing of metals. For two-temperature generalized thermoelasticity with variabletime-independent situations, the difference between these thermal conductivity.Awad[41] write a note on the spatialtwo temperatures is proportional to the heat supply, and in decay estimates in non-classical linear thermoelastic semi-the absence of any heat supply, the two temperatures are cylindrical bounded domains. El-Karamany[42] presentedidentical. For time-dependent problems, however, and for two-temperature theory in linearwave propagation problems in particular, the two micropolarthermoviscoelastic anisotropic solid.El-temperatures are in general different, regardless of the Karamany and Ezzat [43] introduced the two generalpresence of a heat supply. The two temperatures T and models of fractional heat conduction law for non-and the strain are found to have representations in the homogeneous anisotropic elastic solid, obtained theform of a travelling wave plus a response, which occurs constitutive equations for the two-temperature fractionalinstantaneously throughout the body. Warren and Chen [24] thermoelasticity theory, proved uniqueness and reciprocalinvestigated the wave propagation in the two-temperature theorems and established the convolutional variationaltheory of thermoelasticity. Following Boley and Tolins, principle.El-Karamany and Ezzat[44] gave the constitutive[25], they studied the wave propagation in the two- laws for two-temperature Green–Naghdi theories andtemperature theory of coupled thermoelasticity. proved that the two-temperature thermoelasticity theory Youssef [26] developed a new theory of admits dissipation of energy and the theory of elasticitygeneralized thermoelasticity by taking into account the without energy dissipation is valid only when the two-theory of heat conduction in deformable bodies, which temperatures coincide.Ezzat and El-Karamany[45] studieddepends on two distinct temperatures, the conductive the two-temperature theory in generalized magneto-temperature and the thermodynamic temperature where the thermoelasticity with two relaxation times. Ezzat and El-difference between these two temperatures is proportional Karamany[46] constructed fractional order heat conductionto the heat supply. Youssef and Al-Harby[27] studied the law in magneto-thermoelasticity involving twostate-space approach of two-temperature generalized temperatures.Miglani and Kaushal [47] studied theaxi-thermoelasticity of infinite body with a spherical cavity symmetric deformation in generalized thermoelasticity withsubjected to different types of thermal loading . Youssef two temperatures.Mukhopadhyayet. al.[48] presented theand Al-Lehaibi[28]applied the state space techniques of theory of two-temperature thermoelasticity with two phase-two-temperature generalized thermoelasticityto one- lags. Singh and Bijarnia[49] studied the propagation ofdimensional problemof half-space subjected to thermal plane waves in anisotropic two-temperature generalizedshock and traction free. Youssef [29] studied two- thermoelasticity.Youssef [50] presented a theory of two-dimensional problem of a two-temperature generalized temperature thermoelasticity without energy dissipation.thermoelastic half-space subjected to ramp type heating. Banik and Kanoria [51] studied the effects of three-phase-Youssef andBassiouny[30]used the theory of two- lag on two-temperature generalized thermoelasticity for www.ijmer.com 4225 | Page
  3. 3. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4224-4227 ISSN: 2249-6645infinite medium with spherical cavity.Bijarnia and Singh conductive temperature and satisfying the relation − T =[52] studied the propagation of plane waves in an a* ,ii , where a* > 0, is the two-temperature parameter andanisotropic generalized thermoelastic solid with diffusion. K, K* are material characterstic constants.Ezzat et al. [53] introduced both modified Ohms andFouriers laws to the equations of the linear theory of IV. ACKNOWLEDGEMENTSmagneto-thermo-viscoelasticity involving two-temperature Author is highly grateful to Dr.Baljeet Singh,theory, allowing the second sound effects obtained the exact Assistant Professor, Post Graduate Government College,formulas of temperature, displacements, stresses, electric Sector 11, Chandigarh for his valuable guidance whilefield, magnetic field and current density.Singh and Bala[54] preparing this review.studied the reflection of P and SV waves from the freesurface of a two-temperature thermoelastic solid half-space. REFERENCES [1] J. M. C. Duhamel,., Me’moiresur’lecalcul des actions III. THE BASIC EQUATIONS OF TWO- mol’eculairesd’evelopp’ees par les changements de TEMPERATURE GENERALIZED temp‘eraturedans les corps solides,M’emoirs par Divers THERMOELASTICITY Savans (Acad. Sci. Paris), 5, 1838, 440-498. [2] K. E. 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  4. 4. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4224-4227 ISSN: 2249-6645[20] J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity temperature thermoelasticity, International Journal of with finite wavespeeds, (Oxford University Press, 2009). Engineering Science, 48,2010, 128-139.[21] P.J Chen and M. E. Gurtin, On a theory of heat conduction [39] H.M. Youssef, Two-temperature generalized thermoelastic involving twotemperatures, Z. Angew. Math. Phys. 19 in finite mediumwith cylindrical cavity subjected to moving (1968) 614-627. heat source, Archive of Applied[22] P.JChen., M. E. Gurtin and W. O. Williams, A note on non- Mechanics, 80 ,2010, 1213-1224. simple heatconduction, Z. Angew. Math. Phys. 19, 1968, [40] H.MYoussef and A. A. El-Bary, Two-Temperature 969-970. Generalized Thermoelasticity with Variable Thermal[23] P.J Chen., M. E. Gurtin and W. O. 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