Propagation of Thermo-Elastic Waves at Several Typical Interfaces Based on the Theory of Dipolar Gradient Elasticity
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摘要
The reflection and transmission properties of thermo-elastic waves at five possible interfaces between two different strain gradient thermo-elastic solids are investigated based on the generalized thermo-elastic theory without energy dissipation(the GN theory). First, the function of free energy density is postulated and the constitutive relations are defined. Then,the temperature field and the displacement field are obtained from the motion equation in the form of displacement and the thermal transport equation without energy dissipation in the strain gradient thermo-elastic solid. Finally, the five types of thermo-elastic interfacial conditions are used to calculate the amplitude ratios of the reflection and transmission waves with respect to the incident wave. Further, the reflection and transmission coefficients in terms of energy flux ratio are calculated and the numerical results are validated by the energy conservation along the normal direction. It is found that there are five types of dispersive waves, namely the coupled longitudinal wave(the CP wave), the coupled thermal wave(the CT wave), the shear wave, and two evanescent waves(the coupled SP wave and SS wave), that become the surface waves at an interface. The mechanical interfacial conditions mainly influence the coupled CP waves, SV waves, and surface waves, while the thermal interfacial conditions mainly influence the coupled CT waves.
The reflection and transmission properties of thermo-elastic waves at five possible interfaces between two different strain gradient thermo-elastic solids are investigated based on the generalized thermo-elastic theory without energy dissipation(the GN theory). First, the function of free energy density is postulated and the constitutive relations are defined. Then,the temperature field and the displacement field are obtained from the motion equation in the form of displacement and the thermal transport equation without energy dissipation in the strain gradient thermo-elastic solid. Finally, the five types of thermo-elastic interfacial conditions are used to calculate the amplitude ratios of the reflection and transmission waves with respect to the incident wave. Further, the reflection and transmission coefficients in terms of energy flux ratio are calculated and the numerical results are validated by the energy conservation along the normal direction. It is found that there are five types of dispersive waves, namely the coupled longitudinal wave(the CP wave), the coupled thermal wave(the CT wave), the shear wave, and two evanescent waves(the coupled SP wave and SS wave), that become the surface waves at an interface. The mechanical interfacial conditions mainly influence the coupled CP waves, SV waves, and surface waves, while the thermal interfacial conditions mainly influence the coupled CT waves.
The reflection and transmission properties of thermo-elastic waves at five possible interfaces between two different strain gradient thermo-elastic solids are investigated based on the generalized thermo-elastic theory without energy dissipation(the GN theory). First, the function of free energy density is postulated and the constitutive relations are defined. Then,the temperature field and the displacement field are obtained from the motion equation in the form of displacement and the thermal transport equation without energy dissipation in the strain gradient thermo-elastic solid. Finally, the five types of thermo-elastic interfacial conditions are used to calculate the amplitude ratios of the reflection and transmission waves with respect to the incident wave. Further, the reflection and transmission coefficients in terms of energy flux ratio are calculated and the numerical results are validated by the energy conservation along the normal direction. It is found that there are five types of dispersive waves, namely the coupled longitudinal wave(the CP wave), the coupled thermal wave(the CT wave), the shear wave, and two evanescent waves(the coupled SP wave and SS wave), that become the surface waves at an interface. The mechanical interfacial conditions mainly influence the coupled CP waves, SV waves, and surface waves, while the thermal interfacial conditions mainly influence the coupled CT waves.
引文
[1]Toupin RA.Elastic materials with couple stresses.Arch Ration Mech Anal.1962;11:385-414.
[2]Mindlin RD,Tiersten HF.Effects of couple stress in linear elasticity.Arch Ration Mech Anal.1962;11:415-48.
[3]Eringen AC.Mechanics of micromorphic materials.In:Gortler H,editors.Proceedings pf the XI,international congress of applied mechanics.New York:Springer(1964)
[4]Eringen AC.Linear theory of micropolar elasticity.J Math Mech.1966;15:909-24.
[5]Eringen AC.Nonlocal continuum field theories.Berlin:Springer;2001.
[6]Mindlin RD.Micro-structure in linear elasticity.Arch Ration Mech Anal.1964;16:51-78.
[7]Georgiadis HG.The mode III crack problem in microstructured solids governed by dipolar gradient elasticity:static and dynamic analysis.ASME J Appl Mech.2003;70:517-30.
[8]Georgiadis HG,Vardoulakis I,Velgaki EG.Dispersive rayleigh-wave propagation in microstructured solids characterized by dipolar gradient elasticity.J Elast.2004;74:17-45.
[9]Gourgiotis PA,Georgiadis HG,Neocleous I.On the reflection of waves in half-spaces of microstructured materials governed by dipolar gradient elasticity.Wave Motion.2013;50:437-55.
[10]Li YQ,Wei PJ.Reflection and transmission of plane waves at the interface between two different dipolar gradient elastic half-spaces.Int J Solids Struct.2015;56-57:194-208.
[11]Li YQ,Wei PJ,et al.Reflection and transmission of elastic waves at the interface between two gradientelastic solids with surface energy.Eur J Mech A Solid.2015;52:54-71.
[12]Li YQ,Wei PJ,et al.Band gaps of elastic waves in 1-D phononic crystal with dipolar gradient elasticity.Acta Mech.2016;227:1005-23.
[13]Green AE,Lindsay KA.Thermalelasticity.J Elast.1972;2:1-7.
[14]Green AE,Naghdi PM.On thermodynamics and the nature of the second law.Proc R Soc Lond A.1977;357:253-70.
[15]Lord HW,Shulman YA.Generalized dynamical theory of thermoelasticity.J Mech Phys Solids.1967;15:299-309.
[16]Chandrasekharaiah DS.Hyperbolic thermoelasticity:a review of recent literature.Appl Mech Rev.1998;51(12):705-30.
[17]Hetnarski RB,Ignaczak J.Generalized thermoelasticity.J Therm Stress.1999;22:451-76.
[18]Green AE,Naghdi PM.Thermoelasticity without energy dissipation.J Elast.1993;31:189-208.
[2]Mindlin RD,Tiersten HF.Effects of couple stress in linear elasticity.Arch Ration Mech Anal.1962;11:415-48.
[3]Eringen AC.Mechanics of micromorphic materials.In:Gortler H,editors.Proceedings pf the XI,international congress of applied mechanics.New York:Springer(1964)
[4]Eringen AC.Linear theory of micropolar elasticity.J Math Mech.1966;15:909-24.
[5]Eringen AC.Nonlocal continuum field theories.Berlin:Springer;2001.
[6]Mindlin RD.Micro-structure in linear elasticity.Arch Ration Mech Anal.1964;16:51-78.
[7]Georgiadis HG.The mode III crack problem in microstructured solids governed by dipolar gradient elasticity:static and dynamic analysis.ASME J Appl Mech.2003;70:517-30.
[8]Georgiadis HG,Vardoulakis I,Velgaki EG.Dispersive rayleigh-wave propagation in microstructured solids characterized by dipolar gradient elasticity.J Elast.2004;74:17-45.
[9]Gourgiotis PA,Georgiadis HG,Neocleous I.On the reflection of waves in half-spaces of microstructured materials governed by dipolar gradient elasticity.Wave Motion.2013;50:437-55.
[10]Li YQ,Wei PJ.Reflection and transmission of plane waves at the interface between two different dipolar gradient elastic half-spaces.Int J Solids Struct.2015;56-57:194-208.
[11]Li YQ,Wei PJ,et al.Reflection and transmission of elastic waves at the interface between two gradientelastic solids with surface energy.Eur J Mech A Solid.2015;52:54-71.
[12]Li YQ,Wei PJ,et al.Band gaps of elastic waves in 1-D phononic crystal with dipolar gradient elasticity.Acta Mech.2016;227:1005-23.
[13]Green AE,Lindsay KA.Thermalelasticity.J Elast.1972;2:1-7.
[14]Green AE,Naghdi PM.On thermodynamics and the nature of the second law.Proc R Soc Lond A.1977;357:253-70.
[15]Lord HW,Shulman YA.Generalized dynamical theory of thermoelasticity.J Mech Phys Solids.1967;15:299-309.
[16]Chandrasekharaiah DS.Hyperbolic thermoelasticity:a review of recent literature.Appl Mech Rev.1998;51(12):705-30.
[17]Hetnarski RB,Ignaczak J.Generalized thermoelasticity.J Therm Stress.1999;22:451-76.
[18]Green AE,Naghdi PM.Thermoelasticity without energy dissipation.J Elast.1993;31:189-208.