Circularly polarized wave propagation in a class of bodies defined by a new class of implicit constitutive relations
In this paper, we show that circularly polarized transverse stress waves, standing shear stress waves, and oscillatory shear stress waves can propagate in a new class of viscoelastic solid bodies which are a subclass of bodies described by implicit constitutive theories. The class of models that is...
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| Published in | Zeitschrift für angewandte Mathematik und Physik Vol. 65; no. 5; pp. 1003 - 1010 |
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| Format | Journal Article |
| Language | English |
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| Abstract | In this paper, we show that circularly polarized transverse stress waves, standing shear stress waves, and oscillatory shear stress waves can propagate in a new class of viscoelastic solid bodies which are a subclass of bodies described by implicit constitutive theories. The class of models that is being considered includes as sub-classes, the classical Kelvin–Voigt model, the new models introduced by Rajagopal wherein the Cauchy–Green tensor is a non-linear function of the stress, and the Navier–Stokes fluid model. The solutions established in this paper are generalizations of solutions that have been established within the context of nonlinear elasticity by Carroll, and Destrade and Saccomandi, to the new class of elastic and viscoelastic bodies that are being considered. |
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| AbstractList | In this paper, we show that circularly polarized transverse stress waves, standing shear stress waves, and oscillatory shear stress waves can propagate in a new class of viscoelastic solid bodies which are a subclass of bodies described by implicit constitutive theories. The class of models that is being considered includes as sub-classes, the classical Kelvin–Voigt model, the new models introduced by Rajagopal wherein the Cauchy–Green tensor is a non-linear function of the stress, and the Navier–Stokes fluid model. The solutions established in this paper are generalizations of solutions that have been established within the context of nonlinear elasticity by Carroll, and Destrade and Saccomandi, to the new class of elastic and viscoelastic bodies that are being considered. |
| Author | Rajagopal, K. R. Saccomandi, Giuseppe |
| Author_xml | – sequence: 1 givenname: K. R. surname: Rajagopal fullname: Rajagopal, K. R. organization: Department of Mechanical Engineering, Texas A & M University – sequence: 2 givenname: Giuseppe surname: Saccomandi fullname: Saccomandi, Giuseppe email: giuseppe.saccomandi@unipg.it organization: Dipartimento di Ingegneria Industriale, Università degli Studi di Perugia |
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| Cites_doi | 10.1016/j.wavemoti.2007.07.002 10.1002/andp.18922831210 10.1016/j.ijengsci.2010.06.013 10.1007/3-540-29089-3 10.1007/BF01602111 10.1023/A:1026062615145 10.1098/rspa.1950.0035 10.1007/BF00130471 10.1016/j.wavemoti.2014.02.004 10.1017/S0022112005008025 10.1007/s00033-006-6084-5 10.1103/PhysRevE.72.016620 10.1098/rspl.1865.0052 10.1093/qjmam/30.2.223 10.1016/j.amc.2013.03.049 10.1007/BF01453713 |
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| Keywords | Viscoelasticity Implicit elastic models Shear waves Circularly polarized waves Secondary 74B99 Primary 74J30 |
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| References | Kannan, K., Rajagopal K.R., Saccomandi G.: Unsteady motions of a new class of elastic solids. submitted (2013) CarrollM.M.Oscillatory shearing of nonlinearly elastic solidsZ. Angew Math. Phys.197425838810.1007/BF016021110291.73027 VoigtW.Ueber inner Reiburg fester Koerper, insbesondere der MetalleAnnalen der Physik189228367169310.1002/andp.18922831210 BoltzmannL.Sitzungsber KaiserlichAkad. Wiss (Wien) Math. Naturwiss Classe187470II275 MaxwellJ.C.On the dynamical theory of gasesPhilos. Trans. R. Soc. Lond. A18661572678 OldroydJ.G.On the formulation of rheological equations of stateProc. R. Soc. Lond. A1950200235913519210.1098/rspa.1950.0035 ThompsonW.On the elasticity and viscosity of metalsProc. R. Soc. Lond. A186514289297 RajagopalK.R.On implicit constitutive theoriesAppl. Math.200348279319199437810.1023/A:10260626151451099.74009 BoillatG.La propagation des ondes1965ParisGauthier-Villars0151.45104 CarrollM.M.Some results on finite amplitude elastic wavesActa Mechanica1967316718110.1007/BF01453713 MalekJ.PrusaV.RajagopalK.R.Generalizations of a Navier–Stokes fluid from a new perspectiveInt. J. Eng. Sci.20104819071924277875210.1016/j.ijengsci.2010.06.0131231.76073 CarrollM.M.Finite amplitude standing waves in compressible elastic solidsJ. Elast.1978832332810.1007/BF00130471 CarrollM.M.Plane circular shearing of incompressible fluids and solidsQ. J. Mech. Appl. Math.19763022323410.1093/qjmam/30.2.223 DestradeM.SaccomandiG.On finite amplitude elastic waves propagating in compressible solidsPhys. Rev. E2005720016620217839110.1103/PhysRevE.72.016620 Burgers, J.M.: Mechanical considerations-model systems. phenomenological theories of relaxation and viscosity. In: First Report on Viscosity and Plasticity, 2nd ed. Nordeman Publishing Company Inc., New York. Prepared by the committee of viscosity of the Academy of Sciences at Amsterdam (1939) DafermosC.Hyperbolic Conservation Laws in Continuum Physics20052HeidelbergSpringer10.1007/3-540-29089-31078.35001 NarayananA.RajagopalK.R.Unsteady flows of a class of novel generalizations of a Navier–Stokes fluidAppl. Math. Comput.201321999359946305571110.1016/j.amc.2013.03.049 Saccomandi, G.: On the mathematical structure of the determining equations for transverse waves in nonlinear elasticity. submitted (2013) DestradeM.SaccomandiG.Nonlinear transverse waves in deformed dispersive solidsWave Motion200845325336245005110.1016/j.wavemoti.2007.07.0021231.74244 SpencerA.J.M.EringenC.Theory of invariantsContinuum Physics1971New YorkAcademic Press239353 RajagopalK.R.On implicit constitutive theories for fluidsJ. Fluid Mech.2006550243249226398410.1017/S00221120050080251097.76009 RajagopalK.R.The elasticity of elasticityZAMP20075830931710.1007/s00033-006-6084-51113.74006 C. Dafermos (362_CR8) 2005 K.R. Rajagopal (362_CR16) 2003; 48 K.R. Rajagopal (362_CR18) 2007; 58 362_CR3 A.J.M. Spencer (362_CR20) 1971 362_CR11 M. Destrade (362_CR10) 2008; 45 M. Destrade (362_CR9) 2005; 72 K.R. Rajagopal (362_CR17) 2006; 550 M.M. Carroll (362_CR4) 1967; 3 W. Voigt (362_CR22) 1892; 283 L. Boltzmann (362_CR2) 1874; 70 M.M. Carroll (362_CR6) 1976; 30 M.M. Carroll (362_CR7) 1978; 8 J.G. Oldroyd (362_CR15) 1950; 200 A. Narayanan (362_CR14) 2013; 219 M.M. Carroll (362_CR5) 1974; 25 J.C. Maxwell (362_CR12) 1866; 157 J. Malek (362_CR13) 2010; 48 W. Thompson (362_CR21) 1865; 14 G. Boillat (362_CR1) 1965 362_CR19 |
| References_xml | – reference: MalekJ.PrusaV.RajagopalK.R.Generalizations of a Navier–Stokes fluid from a new perspectiveInt. J. Eng. Sci.20104819071924277875210.1016/j.ijengsci.2010.06.0131231.76073 – reference: OldroydJ.G.On the formulation of rheological equations of stateProc. R. Soc. Lond. A1950200235913519210.1098/rspa.1950.0035 – reference: BoillatG.La propagation des ondes1965ParisGauthier-Villars0151.45104 – reference: CarrollM.M.Oscillatory shearing of nonlinearly elastic solidsZ. Angew Math. Phys.197425838810.1007/BF016021110291.73027 – reference: RajagopalK.R.On implicit constitutive theories for fluidsJ. Fluid Mech.2006550243249226398410.1017/S00221120050080251097.76009 – reference: MaxwellJ.C.On the dynamical theory of gasesPhilos. Trans. R. Soc. Lond. A18661572678 – reference: CarrollM.M.Plane circular shearing of incompressible fluids and solidsQ. J. Mech. Appl. Math.19763022323410.1093/qjmam/30.2.223 – reference: DestradeM.SaccomandiG.Nonlinear transverse waves in deformed dispersive solidsWave Motion200845325336245005110.1016/j.wavemoti.2007.07.0021231.74244 – reference: Saccomandi, G.: On the mathematical structure of the determining equations for transverse waves in nonlinear elasticity. submitted (2013) – reference: DestradeM.SaccomandiG.On finite amplitude elastic waves propagating in compressible solidsPhys. Rev. E2005720016620217839110.1103/PhysRevE.72.016620 – reference: VoigtW.Ueber inner Reiburg fester Koerper, insbesondere der MetalleAnnalen der Physik189228367169310.1002/andp.18922831210 – reference: ThompsonW.On the elasticity and viscosity of metalsProc. R. Soc. Lond. A186514289297 – reference: CarrollM.M.Finite amplitude standing waves in compressible elastic solidsJ. Elast.1978832332810.1007/BF00130471 – reference: Burgers, J.M.: Mechanical considerations-model systems. phenomenological theories of relaxation and viscosity. In: First Report on Viscosity and Plasticity, 2nd ed. Nordeman Publishing Company Inc., New York. Prepared by the committee of viscosity of the Academy of Sciences at Amsterdam (1939) – reference: Kannan, K., Rajagopal K.R., Saccomandi G.: Unsteady motions of a new class of elastic solids. submitted (2013) – reference: SpencerA.J.M.EringenC.Theory of invariantsContinuum Physics1971New YorkAcademic Press239353 – reference: BoltzmannL.Sitzungsber KaiserlichAkad. Wiss (Wien) Math. Naturwiss Classe187470II275 – reference: RajagopalK.R.On implicit constitutive theoriesAppl. Math.200348279319199437810.1023/A:10260626151451099.74009 – reference: NarayananA.RajagopalK.R.Unsteady flows of a class of novel generalizations of a Navier–Stokes fluidAppl. Math. Comput.201321999359946305571110.1016/j.amc.2013.03.049 – reference: CarrollM.M.Some results on finite amplitude elastic wavesActa Mechanica1967316718110.1007/BF01453713 – reference: DafermosC.Hyperbolic Conservation Laws in Continuum Physics20052HeidelbergSpringer10.1007/3-540-29089-31078.35001 – reference: RajagopalK.R.The elasticity of elasticityZAMP20075830931710.1007/s00033-006-6084-51113.74006 – volume: 45 start-page: 325 year: 2008 ident: 362_CR10 publication-title: Wave Motion doi: 10.1016/j.wavemoti.2007.07.002 – volume: 283 start-page: 671 year: 1892 ident: 362_CR22 publication-title: Annalen der Physik doi: 10.1002/andp.18922831210 – volume: 48 start-page: 1907 year: 2010 ident: 362_CR13 publication-title: Int. J. Eng. Sci. doi: 10.1016/j.ijengsci.2010.06.013 – ident: 362_CR19 – volume-title: La propagation des ondes year: 1965 ident: 362_CR1 – volume-title: Hyperbolic Conservation Laws in Continuum Physics year: 2005 ident: 362_CR8 doi: 10.1007/3-540-29089-3 – volume: 25 start-page: 83 year: 1974 ident: 362_CR5 publication-title: Z. Angew Math. Phys. doi: 10.1007/BF01602111 – volume: 157 start-page: 26 year: 1866 ident: 362_CR12 publication-title: Philos. Trans. R. Soc. Lond. A – volume: 48 start-page: 279 year: 2003 ident: 362_CR16 publication-title: Appl. Math. doi: 10.1023/A:1026062615145 – ident: 362_CR3 – volume: 200 start-page: 23 year: 1950 ident: 362_CR15 publication-title: Proc. R. Soc. Lond. A doi: 10.1098/rspa.1950.0035 – start-page: 239 volume-title: Continuum Physics year: 1971 ident: 362_CR20 – volume: 8 start-page: 323 year: 1978 ident: 362_CR7 publication-title: J. Elast. doi: 10.1007/BF00130471 – ident: 362_CR11 doi: 10.1016/j.wavemoti.2014.02.004 – volume: 550 start-page: 243 year: 2006 ident: 362_CR17 publication-title: J. Fluid Mech. doi: 10.1017/S0022112005008025 – volume: 58 start-page: 309 year: 2007 ident: 362_CR18 publication-title: ZAMP doi: 10.1007/s00033-006-6084-5 – volume: 72 start-page: 0016620 year: 2005 ident: 362_CR9 publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.72.016620 – volume: 14 start-page: 289 year: 1865 ident: 362_CR21 publication-title: Proc. R. Soc. Lond. A doi: 10.1098/rspl.1865.0052 – volume: 70 start-page: 275 issue: II year: 1874 ident: 362_CR2 publication-title: Akad. Wiss (Wien) Math. Naturwiss Classe – volume: 30 start-page: 223 year: 1976 ident: 362_CR6 publication-title: Q. J. Mech. Appl. Math. doi: 10.1093/qjmam/30.2.223 – volume: 219 start-page: 9935 year: 2013 ident: 362_CR14 publication-title: Appl. Math. Comput. doi: 10.1016/j.amc.2013.03.049 – volume: 3 start-page: 167 year: 1967 ident: 362_CR4 publication-title: Acta Mechanica doi: 10.1007/BF01453713 |
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| Title | Circularly polarized wave propagation in a class of bodies defined by a new class of implicit constitutive relations |
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