# American Institute of Mathematical Sciences

September  2014, 19(7): 2027-2038. doi: 10.3934/dcdsb.2014.19.2027

## Spatial behavior in the vibrating thermoviscoelastic porous materials

 1 Department of Mathematics, Al. I. Cuza University, 700506, Iaşi, Romania

Received  April 2013 Revised  July 2013 Published  August 2014

In this paper we study the spatial behavior of the amplitude of the steady-state vibrations in a thermoviscoelastic porous beam. Here we take into account the effects of the viscoelastic and thermal dissipation energies upon the corresponding harmonic vibrations in a right cylinder made of a thermoviscoelastic porous isotropic material. In fact, we prove that the positiveness of the viscoelastic and thermal dissipation energies are sufficient for characterizing the spatial decay and growth properties of the harmonic vibrations in a cylinder.
Citation: Stan Chiriţă. Spatial behavior in the vibrating thermoviscoelastic porous materials. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2027-2038. doi: 10.3934/dcdsb.2014.19.2027
##### References:
 [1] S. Chiriţă, Spatial decay estimates for solutions describing harmonic vibrations in a thermoelastic cylinder, Journal of Thermal Stresses, 18 (1995), 421-436. doi: 10.1080/01495739508946311. [2] S. Chiriţă and A. Scalia, On the spatial and temporal behaviour in linear thermoelasticity of materials with voids, Journal of Thermal Stresses, 24 (2001), 433-455. doi: 10.1080/01495730151126096. [3] S. Chiriţă, C. Galeş and I. D. Ghiba, On spatial behavior of the harmonic vibrations in Kelvin-Voigt materials, Journal of Elasticity, 93 (2008), 81-92. doi: 10.1007/s10659-008-9167-z. [4] S. Chiriţă and C. D'Apice, On Saint-Venant's principle in a poroelastic arch-like region, Mathematical Methods in the Applied Sciences, 33 (2010), 1743-1754. doi: 10.1002/mma.1294. [5] S. Chiriţă and I. D. Ghiba, Strong ellipticity and progressive waves in elastic materials with voids, Proceedings of the Royal Society A, 466 (2010), 439-458. [6] M. Ciarletta and A. Scalia, On uniqueness and reciprocity in linear thermoelasticity of materials with voids, Journal of Elasticity, 32 (1993), 1-17. doi: 10.1007/BF00042245. [7] M. Ciarletta and B. Straughan, Thermo-poroacoustic acceleration waves in elastic materials with voids, Journal of Mathematical Analysis and Applications, 333 (2007), 142-150. doi: 10.1016/j.jmaa.2006.09.014. [8] M. Ciarletta, M. Svanadze and L. Buonanno, Plane waves and vibrations in the theory of micropolar thermoelasticity for materials with voids, European Journal of Mechanics - A/Solids, 28 (2009), 897-903. doi: 10.1016/j.euromechsol.2009.03.008. [9] S. C. Cowin and J. W. Nunziato, Linear theory of elastic materials with voids, Journal of Elasticity, 13 (1983), 125-147. [10] C. D'Apice and S. Chiriţă, Spatial behaviour in a Mindlin-type thermoelastic plate, Quarterly of Applied Mathematics, 61 (2003), 783-796. [11] C. D'Apice and S. Chiriţă, On Saint-Venant's principle for a linear poroelastic material in plane strain, Journal of Mathematical Analysis and Applications, 363 (2010), 454-467. doi: 10.1016/j.jmaa.2009.09.032. [12] J. N. Flavin and R. J. Knops, Some spatial decay estimates in continuum dynamics, Journal of Elasticity, 17 (1987), 249-264. doi: 10.1007/BF00049455. [13] C. Galeş, On spatial behavior of harmonic vibrations in viscoelastic Reissner-Mindlin plates, International Journal of Solids and Structures, 48 (2011), 243-248. [14] C. Galeş and S. Chiriţă, On spatial behavior in linear viscoelasticity, Quarterly of Applied Mathematics, 67 (2009), 707-723. [15] D. Ieşan, Some theorems in the theory of elastic materials with voids, Journal of Elasticity, 15 (1985), 215-224. doi: 10.1007/BF00041994. [16] D. Ieşan, A theory of thermoelastic material with voids, Acta Mechanica, 60 (1986), 67-89. [17] D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic, Dordrecht, 2004. doi: 10.1007/978-1-4020-2310-1. [18] D. Ieşan, On a theory of thermoviscoelastic materials with voids, Journal of Elasticity, 104 (2011), 369-384. doi: 10.1007/s10659-010-9300-7. [19] D. Ieşan and L. Nappa, Thermal stresses in plane strain of porous elastic solids, Meccanica, 39 (2004), 125-138. doi: 10.1023/B:MECC.0000005118.15612.01. [20] A. Scalia, A. Pompei and S. Chiriţă, On the behavior of steady time-harmonic oscillations in thermoelastic materials with voids, Journal of Thermal Stresses, 27 (2004), 209-226. doi: 10.1080/01495730490264330. [21] K. Sharma and P. Kumar, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, Journal of Thermal Stresses, 36 (2013), 94-111. doi: 10.1080/01495739.2012.720545. [22] B. Straughan, Stability and Wave Motion in Porous Media, Series in applied mathematical sciences, 165, Springer-Verlag, Berlin-Heidelberg-New York, 2008. [23] S. K. Tomar, J. Bhagwan and H. Steeb, Time harmonic waves in thermo-viscoelastic material with voids, Journal of Vibration and Control, 20 (2014), 1119-1136. doi: 10.1177/1077546312470479.

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##### References:
 [1] S. Chiriţă, Spatial decay estimates for solutions describing harmonic vibrations in a thermoelastic cylinder, Journal of Thermal Stresses, 18 (1995), 421-436. doi: 10.1080/01495739508946311. [2] S. Chiriţă and A. Scalia, On the spatial and temporal behaviour in linear thermoelasticity of materials with voids, Journal of Thermal Stresses, 24 (2001), 433-455. doi: 10.1080/01495730151126096. [3] S. Chiriţă, C. Galeş and I. D. Ghiba, On spatial behavior of the harmonic vibrations in Kelvin-Voigt materials, Journal of Elasticity, 93 (2008), 81-92. doi: 10.1007/s10659-008-9167-z. [4] S. Chiriţă and C. D'Apice, On Saint-Venant's principle in a poroelastic arch-like region, Mathematical Methods in the Applied Sciences, 33 (2010), 1743-1754. doi: 10.1002/mma.1294. [5] S. Chiriţă and I. D. Ghiba, Strong ellipticity and progressive waves in elastic materials with voids, Proceedings of the Royal Society A, 466 (2010), 439-458. [6] M. Ciarletta and A. Scalia, On uniqueness and reciprocity in linear thermoelasticity of materials with voids, Journal of Elasticity, 32 (1993), 1-17. doi: 10.1007/BF00042245. [7] M. Ciarletta and B. Straughan, Thermo-poroacoustic acceleration waves in elastic materials with voids, Journal of Mathematical Analysis and Applications, 333 (2007), 142-150. doi: 10.1016/j.jmaa.2006.09.014. [8] M. Ciarletta, M. Svanadze and L. Buonanno, Plane waves and vibrations in the theory of micropolar thermoelasticity for materials with voids, European Journal of Mechanics - A/Solids, 28 (2009), 897-903. doi: 10.1016/j.euromechsol.2009.03.008. [9] S. C. Cowin and J. W. Nunziato, Linear theory of elastic materials with voids, Journal of Elasticity, 13 (1983), 125-147. [10] C. D'Apice and S. Chiriţă, Spatial behaviour in a Mindlin-type thermoelastic plate, Quarterly of Applied Mathematics, 61 (2003), 783-796. [11] C. D'Apice and S. Chiriţă, On Saint-Venant's principle for a linear poroelastic material in plane strain, Journal of Mathematical Analysis and Applications, 363 (2010), 454-467. doi: 10.1016/j.jmaa.2009.09.032. [12] J. N. Flavin and R. J. Knops, Some spatial decay estimates in continuum dynamics, Journal of Elasticity, 17 (1987), 249-264. doi: 10.1007/BF00049455. [13] C. Galeş, On spatial behavior of harmonic vibrations in viscoelastic Reissner-Mindlin plates, International Journal of Solids and Structures, 48 (2011), 243-248. [14] C. Galeş and S. Chiriţă, On spatial behavior in linear viscoelasticity, Quarterly of Applied Mathematics, 67 (2009), 707-723. [15] D. Ieşan, Some theorems in the theory of elastic materials with voids, Journal of Elasticity, 15 (1985), 215-224. doi: 10.1007/BF00041994. [16] D. Ieşan, A theory of thermoelastic material with voids, Acta Mechanica, 60 (1986), 67-89. [17] D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic, Dordrecht, 2004. doi: 10.1007/978-1-4020-2310-1. [18] D. Ieşan, On a theory of thermoviscoelastic materials with voids, Journal of Elasticity, 104 (2011), 369-384. doi: 10.1007/s10659-010-9300-7. [19] D. Ieşan and L. Nappa, Thermal stresses in plane strain of porous elastic solids, Meccanica, 39 (2004), 125-138. doi: 10.1023/B:MECC.0000005118.15612.01. [20] A. Scalia, A. Pompei and S. Chiriţă, On the behavior of steady time-harmonic oscillations in thermoelastic materials with voids, Journal of Thermal Stresses, 27 (2004), 209-226. doi: 10.1080/01495730490264330. [21] K. Sharma and P. Kumar, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, Journal of Thermal Stresses, 36 (2013), 94-111. doi: 10.1080/01495739.2012.720545. [22] B. Straughan, Stability and Wave Motion in Porous Media, Series in applied mathematical sciences, 165, Springer-Verlag, Berlin-Heidelberg-New York, 2008. [23] S. K. Tomar, J. Bhagwan and H. Steeb, Time harmonic waves in thermo-viscoelastic material with voids, Journal of Vibration and Control, 20 (2014), 1119-1136. doi: 10.1177/1077546312470479.
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