This review article is a continuation of a previous article by the author, Thermoelasticity with second sound: A review, which appeared in this journal in March, 1986 (Appl Mech Rev39 (3) 355-376). Here, attention is focused on papers published during the past 10-12 years. Contributions to the theory of thermoelasticity with thermal relaxation and the temperature-rate dependent thermoelasticity theory are reviewed. The recently developed theory of thermoelasticity without energy dissipation is described, and its characteristic features highlighted. A glance is made at the new thermoelasticity theory which includes the so-called dual-phase-lag effects. There are 338 references

Chandrasekharaiah, D.S.

Jordan, P.M.

Puri, P.

ePrints@Bangalore University

Downloaded FromComments on the articles‘‘Hyperbolic thermoelasticity:A review of recent literature’’Chandrasekharaiah DS, 1998,Appl Mech Rev 5112, 705–729and ‘‘Thermoelasticity withsecond sound: A review’’Chandrasekharaiah DS, 1986,Appl Mech Rev 393, 355–376P PuriDepartment of Mathematics, University of New Orleans,New Orleans LA 70148; ppuri@math.uno.eduPM Jordan*Code 7181, Naval Research Laboratory,Stennis Space Center, MS 39529;pjordan@nrlssc.navy.milIn the articles Thermoelasticity with second sound: A review@1# and Hyperbolic thermoelasticity: A review of recent lit-erature @2#, Chandrasekharaiah has presented an in-depthlook at nonconventional ~a.k.a. generalized or non-Fourier!theories of thermoelasticity. The motivation driving the for-mulation of these theories is the desire to overcome the infi-nite propagation speed of thermal signals predicted by con-ventional thermoelasticity ~CTE!, the so-called ‘‘paradox ofheat conduction.’’In @1#, two of these nonconventional theories are exam-ined in the context of the Danilovskaya problem ~DP!. ~Inthe DP, the homogeneous and isotropic thermoelastic half-space x.0, under a stress free boundary condition ~BC! atx50, is subjected to a Heaviside, or step, temperature BC attime t501.) The first he refers to as extended thermoelas-ticity ~ETE! and the second as temperature-rate dependentthermoelasticity ~TRDTE!. In both ETE and TRDTE, theparabolic diffusion equation of CTE is replaced with a hy-perbolic heat transport equation. As a result, both theoriespredict thermal waves ~ie, second sound! propagating withfinite speeds.In ETE, a single relaxation time t.0 appears and secondsound propagates with speed vT5Ak/t , where k is used hereto denote the thermal diffusivity. It is noted that ETE reducesto CTE in the limit t→0. TRDTE was presented in 1972 byGreen and Lindsay @3#. This theory involves the two relax-ation times a0 and a, where a>a0.0 and, in the case ofhomogeneous and isotropic materials, reduces to CTE in thelimit a→0. ~While it has been postulated that a0 is actuallynon-negative, it must be noted that TRDTE admits secondsound only when a0.0 @1#.! An important aspect of TRDTEis that Fourier’s heat law is not violated in materials thathave a center of symmetry at each point @2,3#.Appl Mech Rev vol 56, no 4, July 2003 45: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 02/11/2Although it was not pointed out in @1#, Chandrasekharaiahdid note in @2# several physically unrealistic results associ-ated with TRDTE, in particular the fact that the displacementsuffers jump discontinuities in the presence of a step tem-perature BC. A natural question that arises is why was thisproblem with TRDTE not reported in @1#, especially sincethe author of that paper derived parts of the small-time solu-tion to the DP for a TRDTE medium? ~See @2# and the ref-erences therein for a discussion of the problems withTRDTE.!The intent of the present Letter is the following: i) Showthat the small-time expression given in @1# for the normalstress corresponding to the DP for a TRDTE medium is in-correct; ii) Show how this erroneous expression could havelead to the aforementioned shortcoming of TRDTE beingmissed in @1#; and iii) Give for the record the correct small-time expressions for the normal stress, displacement, andstrain corresponding to the DP for a TRDTE medium. Lastly,all quantities below are dimensionless, unless stated other-wise the same notation employed in @1# is used here, and thereader is referred to @1# for the definition of all undefinedsymbols.In the Laplace transform domain, the normal stress isgiven by ~see Eq. ~5.53! of @1#!s¯~x ,s !5T0s~11as !$e2n1x2e2n2x%n122n22 , (1)where s is the transform parameter. For large s , it can beshown thatn1,2’s/V1,2* 1r1,2 ,1n122n22 ’1s2AM 01N0s3M 03/2 212s4M 03/2 S ~11e!22 3N02M 0 D ,(2)where V1,2* (V2*.V1*) and M 0 are positive constants and V2*denotes the speed of the second sound ~ie, thermal! wave.Using Eqs. ~2!, the large-s expression for s¯(x ,s) is found tobes¯~x ,s !’T0~11as !AM 0F1s1N0s2M 0212s3M 03S ~11e!22 3N02M 0 D G$exp@2~r11s/V1*!x#2exp@2~r21s/V2*!x#%. (3)Expanding and rearranging Eq. ~3! into increasing powers of1/s , and then truncating all terms after 1/s2 so as to match@1#, givess¯~x ,s !’T0AM 0Fa1 1s~11aR0!1S0s2G$exp@2~r11s/V1*!x#2exp@2~r21s/V2*!x#%, (4)*Corresponding author© 2003 American Society of Mechanical Engineers1016 Terms of Use: http://www.asme.org/about-asme/terms-of-useAppl Mech Rev vol 56, no 4, July 2003 Discussion 452Downlwhere R05N0 /M 0 and S05(2R02aM 021(11e)213aR02)/2. ~The quantity S0 does not appear in @1#, it isintroduced here for convenience.! Inverting Eq. ~4!, thesmall-time expression for the normal stress is found to bes~x ,t !’T0AM 0(j512~21 ! j11$ad~ t2x/V j*!1H~ t2x/V j*!3@~11aR0!1~ t2x/V j*!S0#%e2r jx, (5)where H() is the Heaviside unit step function and d()denotes the Dirac delta function. ~The notation used here forH() is slightly different than that of @1#.! Equation ~5! is thecorrect form of Eq. ~5.58! in @1#. Comparing the former withthe latter it is clear why the latter is incorrect; the contribu-tion of the term as , which is part of the quantity (11as) inthe numerator of Eq. ~5.53! of @1#, is missing in the inverse.Indeed, no term with coefficient a that appears in Eq. ~5! ispresent in Eq. ~5.58! of @1#. In particular, Eq. ~5.58! of @1#does not contain the two delta function terms that it should.A second consequence of these missing terms is that theexpressions given in Eq. ~5.61! of @1# for s1,2* , whereT0s1,2* [@s12s2#x5x1,2*[s(x1,2* ,t10)2s(x1,2* ,t20) de-note the amplitudes of the jumps in s across the wavefrontsx5x1,2* , are also incorrect. Specifically, since the ~correct!expression for s(x ,t) exhibits two propagating delta func-tions, us1,2* u5‘ in the sense of @4#.From the Laplace transforms of Eqs. ~5.49! and ~5.51! in@1#, it can be shown that u¯5s22(]s¯/]x), where u¯ is theimage of the x-component of the displacement vector in theLaplace transform domain. Consequently, using Eq. ~1!, itfollows thatu¯~x ,s !52T0~11as !$n1e2n1x2n2e2n2x%s~n122n22!. (6)Again using the approximations given in Eqs. ~2!, the large-s expansion of u¯ turns out to beoaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.orgu¯~x ,s !’T0AM 0(j512~21 ! jF1s S aV j*D11s2 S ar j1 aR011V j* D Gexp@2~r j1s/V j*!x# .(7)On inverting Eq. ~7!, the small-time solution for u is foundto beu~x ,t !’T0AM 0(j512~21 ! jF aV j* 1~ t2x/V j*!S ar j1aR011V j*D Ge2r jxH~ t2x/V j*!. (8)From Eq. ~8!, it is clear that u always admits two propagat-ing jump discontinuities, the amplitudes of which are givenbyu1,2* 57S aV1,2* AM 0D e2r1,2x1,2* . (9)In Eq. ~9!, T0u1,2* [@u12u2#x5x1,2* denote the amplitudes ofthe jumps in u across x5x1,2* . ~The quantities u1,2* are intro-duced here in a manner consistent with the notation conven-tion of @1#.!While not given in @1#, the small-time relation for thestrain will be given here for completeness. To this end, Eq.~6! is differentiated with respect to x , re-expressed using theidentities n1,22 5$s(11e)1s2L06(n122n22)%/2 and the ap-proximations given in Eqs. ~2!, and then inverted to yield thesmall-time strain relation]u]x~x ,t !’~11e!2 E0ts~x ,t8!dt81L02 s~x ,t !1T02 (j512e2r jx$H~ t2x/V j*!1ad~ t2x/V j*!%.(10)In Eq. ~10!, s(x ,) denotes the right-hand side of Eq. ~5!.Figure 1 shows a comparison of the inverse of Eq. ~6!with the small-time solution given in Eq. ~8!. The inverse ofEq. ~6! was computed numerically using Tzou’s Riemannsum inversion algorithm ~TRSIA! @5# and the values of thematerial parameters were obtained from Table II of @1#. Asshown in Fig. 1, Eq. ~8! is a very good/excellent approxima-tion to u for t&0.05. In addition, the two propagating jumpsare clearly visible, with uu1*u.uu2*u, and it is noted that x1,2*are the elastic ~trailing! and thermal ~leading! wavefronts,respectively.It must be pointed out that the presence of propagatingjumps in u violates the continuity of displacements require-ment @@6#, p. 142#, and thus indicates that TRDTE is incon-sistent with the continuum theory of matter under a step~actually any discontinuous! temperature BC ~see @2# and thereferences therein!. These jumps, which occur in both theFig. 1 u vs. x for a050.05, a50.1, e50.0356, and T051. Solid:Inverse of Eq. ~6! computed using TRSIA with 1000 terms. Broken:Small-time solution given in Eq. ~8!./ on 02/11/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use453 Discussion Appl Mech Rev vol 56, no 4, July 2003Downloaded Fromcoupled (e.0) and uncoupled (e50) cases, vanish only inthe limit a→0. ~For a treatment of the uncoupled, spheri-cally symmetric case for a shell, see @7#.!Finally, it should be mentioned that an error similar to theone corrected here, in which all d() and d8() terms aremissing from the Laplace inverse, occurs in the expressionfor the strain ~ie, Eq. ~48!! in @8#. ~It is of interest to note thathad the correct expression for the strain been obtained in @8#,the drawbacks with TRDTE could have been uncovered in1980.! However, while Eq. ~5.58! of @1# is incorrect, and thiserror appears to have directly resulted in the primary physi-cally objectionable feature of TRDTE being overlooked in@1# as well as to the mistaken claim ~@1#, p 371! that theTRDTE expression for s(x ,t) reduces to its ETE counterpart~ie, Eq. ~4.39! of @1#! when a5a05t , Chandrasekharaiah’stwo articles @1,2# nevertheless provide an excellent review ofthe literature on nonconventional thermoelasticity and con-tain a wealth of information on the subject.PM Jordan was supported by CORE/ONR/NRL funding~PE 602435N!.REFERENCES@1# Chandrasekharaiah DS ~1986!, Thermoelasticity with second sound:A review, Appl. Mech. Rev., 39~3!, 355–376.@2# Chandrasekharaiah DS ~1998!, Hyperbolic thermoelasticity: A reviewof recent literature, Appl. Mech. Rev., 51~12!, 705–729.@3# Green AE and Lindsay KA ~1972!, Thermoelasticity, J. Elast., 2, 1–7.@4# Boley BA and Hetnarski RB ~1968!, Propagation of discontinuities incoupled thermo-elastic problems, ASME J. Appl. Mech., 35, 489–494.@5# Tzou DY ~1997!, Macro- to Microscale Heat Transfer: The LaggingBehavior, Taylor and Francis, Washington DC.@6# Achenbach JD ~1973!, Wave Propagation in Elastic Solids, North-Holland, Amsterdam.@7# Jordan PM and Puri P ~2001!, Thermal stresses in a spherical shellunder three thermo-elastic models, J. Therm. Stresses, 24, 47–70.@8# Chandrasekharaiah DS ~1980!, On generalised thermoelastic wavepropagation, Proc of Indian Acad Sci (Math Sci) 89, 43–52.: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 02/11/2Author’s Response to ‘‘Commentson the articles ‘Hyperbolicthermoelasticity: A review of recentliterature’ ChandrasekharaiahDS, 1998, Appl Mech Rev 5112,705–729 and ‘Thermoelasticitywith second sound: Areview’ Chandrasekharaiah DS,1986, Appl Mech Rev 393,355–376’’DS ChandrasekharaiahDepartment of Mathematics, Bangalore University,Bangalore 560001, IndiaThe authors of the Letter to the Editor have reinvestigatedthe Danilovskaya’s problem ~DP! in the contextof TRDTE. They have suggested some correction termsto the expressions ~5.58! and ~5.61! of @1#. They have alsoobtained small-time solutions for the displacement and strainfields, which were not reported in @1#. Their analysis bringsout the fact that some physically unrealistic features ofTRDTE ~summarized in @2#! can be seen in DP aswell.But for the unnecessarily aggressive language, repetitivestatements, and undue length, the letter makes a useful con-tribution to the literature on TRDTE. I am thankful to theauthors of the letter for correcting a couple of mathematicalexpressions which I had derived some 20 years back.I am moved by their overall opinion on my two ReviewArticles @1,2#.016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Comments on the articles "Hyperbolic thermoelasticity: A review of recent literature" (Chandrasekharaiah DS, 1998, appl mech rev 51(12), 705-729) and "Thermoelasticity with second sound: a review" (Chandrasekharaiah DS, 1986, appl mech rev 39(3), 355-376)

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Comments on the articles "Hyperbolic thermoelasticity: A review of recent literature" (Chandrasekharaiah DS, 1998, appl mech rev 51(12), 705-729) and "Thermoelasticity with second sound: a review" (Chandrasekharaiah DS, 1986, appl mech rev 39(3), 355-376)

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