On the decay of exponential type for the solutions in a dipolar elastic body

References

• Gurtin ME. The linear theory of elasticity. In: Truesdell C, editor. Handbuch der Physik. Vol. VIa/2. Berlin/Heidelberg/New York: Springer-Verlag; 1972. p. 1–296.
   Google Scholar
• Edelstein WS. A spatial decay estimate for the heat equation. Z Angew Math Phys. 1969;20:900–905. doi: 10.1007/BF01592299
   Web of Science ®Google Scholar
• Knowles JK. On Saint-Venant's principle in the two-dimensional linear theory of elasticity. Arch Ration Mech Anal. 1996;21:1–22. doi: 10.1007/BF00253046
   Google Scholar
• Toupin RA. Saint-Venant's principle. Arch Ration Mech Anal. 1965;18:83–96. doi: 10.1007/BF00282253
   Google Scholar
• Boley B. Some observations on Saint-Venant's principle. Proceedings of the Third U.S. National Congress of Applied Mechanics. New York: ASME; 1958. p. 259–264.
   Google Scholar
• Horgan CO. Recent developments concerning Saint-Venant's principle: an update. Appl Mech Rev. 1989;42(11):295–303. doi: 10.1115/1.3152414
   Google Scholar
• Karp D, Durban D. Saint-Venant's principle in dynamics of structure. Appl Mech Rev. 2011;64(2):020801-1–020801-20. doi: 10.1115/1.4004930
   Web of Science ®Google Scholar
• Knowles JK. On the spatial decay of solutions of the heat equation. Z Angew Math Phys. 1971;22:1050–1056. doi: 10.1007/BF01590873
   Web of Science ®Google Scholar
• Horgan CO, Payne LE, Wheeler LT. Spatial decay estimates in transient heat conduction. Quart Appl Math. 1984;42:119–127. doi: 10.1090/qam/736512
   Web of Science ®Google Scholar
• Ericksen JL. Uniformity in shells. Arch Ration Mech Anal. 1970;37:73–84. doi: 10.1007/BF00281663
   Web of Science ®Google Scholar
• Chirita S. On the spatial decay estimates in certain time-dependent problems of continuum mechanics. Arch Mech. 1995;47:755–771.
   Google Scholar
• Nunziato JW. On the spatial decay of solutions in the nonlinear theory of heat conduction. J Math Anal Appl. 1974;48:687–698. doi: 10.1016/0022-247X(74)90143-7
   Web of Science ®Google Scholar
• Chirita S. Spatial decay estimates for solutions describing harmonic vibrations in a thermoelastic cylinder. J Therm Stresses. 1995;18:421–436. doi: 10.1080/01495739508946311
   Web of Science ®Google Scholar
• Iesan D, Quintanilla R. Decay estimates and energy bounds for porous elastic cylinders. Z Angew Math Phys. 1995;46:268–281. doi: 10.1007/BF00944757
   Web of Science ®Google Scholar
• Flavin JN, Knops RJ. Some spatial decay estimates in continuum dynamics. J Elasticity. 1987;17:249–264. doi: 10.1007/BF00049455
   Web of Science ®Google Scholar
• Flavin JN, Knops RJ, Payne LE. Energy bounds in dynamical problems for a semi-infinite elastic beam. In: Eason G, Ogden RW, editors. Elasticity: mathematical methods and applications. Chichester: Ellis-Horwood; 1990. p. 101–111.
   Google Scholar
• Eringen AC. Theory of thermo-microstretch elastic solids. Int J Eng Sci. 1990;28:1291–1301. doi: 10.1016/0020-7225(90)90076-U
   Web of Science ®Google Scholar
• Eringen AC. Microcontinuum field theory. Vol. I. Foundations and solids. New York: Springer-Verlag; 1999.
   Google Scholar
• Mindlin RD. Micro-structure in linear elasticity. Arch Ration Mech Anal. 1964;16:51–78. doi: 10.1007/BF00248490
   Web of Science ®Google Scholar
• Green AE, Rivlin RS. Multipolar continuum mechanics. Arch Ration Mech Anal. 1964;17:113–147. doi: 10.1007/BF00253051
   Web of Science ®Google Scholar
• Fried E, Gurtin ME. Thermomechanics of the interface between a body and its environment. Contin Mech Thermodyn. 2007;19(5):253–271. doi: 10.1007/s00161-007-0053-x
   Web of Science ®Google Scholar
• Marin M, Agarwal RP, Mahmoud SR. Nonsimple material problems addressed by the Lagrange's identity. Boundary Value Probl. 2013;2013:1–14, Art. No. 135. doi: 10.1186/1687-2770-2013-1
   Google Scholar
• Marin M, Ellahi R, Chirila A. On solutions of saint-venant's problem for elastic dipolar bodies with voids. Carpathian J Math. 2017;33(2):219–232.
   Web of Science ®Google Scholar
• Marin M. Lagrange identity method for microstretch thermoelastic materials. J Math Anal Appl. 2010;363(1):275–286. doi: 10.1016/j.jmaa.2009.08.045
   Web of Science ®Google Scholar
• Marin M, Vlase S, Ellahi R, et al. On the partition of energies for the backward in time problem of thermoelastic materials with a dipolar structure. Symmetry (Basel). 2019;11(7):1–16, Art. No. 863.
   Google Scholar
• Marin M, Nicaise S. Existence and stability results for thermoelastic dipolar bodies with double porosity. Contin Mech Thermodyn. 2016;28(6):1645–1657. doi: 10.1007/s00161-016-0503-4
   Web of Science ®Google Scholar
• Marin M, Florea O. On temporal behaviour of solutions in thermoelasticity of porous micropolar bodies. An St Univ Ovidius Constanta. 2014;22(1):169–188.
   Google Scholar
• Abbas IA, Othman MIA. Generalized thermoelasticity of thermal shock problem in an isotropic hollow cylinder and temperature dependent elastic moduli. Chin Phys B. 2012;21(1):14601–14606. doi: 10.1088/1674-1056/21/1/014601
   Web of Science ®Google Scholar
• Othman MIA, Abbas IA. Generalized thermoelasticity of thermal shock problem in a non-homogeneous isotropic hollow cylinder with energy dissipation. Int J Thermophys. 2012;33(5):913–923. doi: 10.1007/s10765-012-1202-4
   Web of Science ®Google Scholar
• Othman MIA, Abbas IA. Effect of rotation on a magneto-thermoelastic hollow cylinder with energy dissipation using finite element method. J Comput Theor Nanosci. 2015;12(9):2399–2404. doi: 10.1166/jctn.2015.4039
   Google Scholar
• Seadawy AR, Iqbal M, Lu D. Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. J Taibah Univ Sci. 2019;13(1):1060–1072. doi: 10.1080/16583655.2019.1680170
   Web of Science ®Google Scholar
• Ahmad H, Seadawy AR, Khan TA, et al. Analytic approximate solutions for some nonlinear parabolic dynamical wave equations. J Taibah Univ Sci. 2020;14(1):346–358. doi: 10.1080/16583655.2020.1741943
   Web of Science ®Google Scholar
• Mehrabadi MM, Cowin SC, Horgan CO. Strain energy density bounds for linear anisotropic elastic materials. J Elasticity. 1993;30:191–196. doi: 10.1007/BF00041853
   Web of Science ®Google Scholar
• Iesan D, Quintanilla R. On Saint-Venant's principle for microstretch elastic bodies. Int J Eng Sci. 1997;35(14):1277–1290. doi: 10.1016/S0020-7225(97)00049-9
   Web of Science ®Google Scholar
• Choudhuri SKR. On a thermoelastic three-phase-lag model. J Thermal Stress. 2007;30(3):231–238. doi: 10.1080/01495730601130919
   Web of Science ®Google Scholar
• Marin M, Öchsner A. Complements of higher mathematics. Cham: Springer; 2018.
   Google Scholar