Abstract
Linear theory of nonlocal elastic material with double porosity structure is developed within the context of Eringen’s theory of nonlocal elasticity. Energy density function is constructed from the basic variables, and then, constitutive relations are derived, which are used to develop the field equations for an isotropic homogeneous nonlocal elastic material with double porosity. It is found that there may exist four basic plane waves in an unbounded medium consisting of three sets of coupled dilatational waves and an independent transverse wave. The major impact of the presence of nonlocality in the medium is that all the four propagating plane waves face cut-off frequencies. The coupled dilatational waves are dispersive and attenuating in nature, while the transverse wave is dispersive and non-attenuating in nature below their respective cut-off frequencies and beyond which they disappear. It is also noticed that coupled waves are affected by the presence of voids, while the transverse wave is independent of the presence of voids in the medium. In the case of non-Voigt model, the coupled dilatational waves face critical frequencies in the low-frequency range. The effect of nonlocality and voids is shown graphically on the dispersion curve of the plane waves for a particular model.
Similar content being viewed by others
References
Cowin, S.C., Nunziato, J.W.: Linear elastic materials with voids. J. Elast. 13(2), 125–147 (1983)
Nunziato, J.W., Cowin, S.C.: A non-linear theory of elastic materials with voids. Arch. Ration. Mech. Anal. 72(2), 175–201 (1979)
Puri, P., Cowin, S.C.: Plane waves in linear elastic materials with voids. J. Elast. 15(2), 167–183 (1985)
Ieşan, D.: A theory of thermoelastic materials with voids. Acta Mech. 60(1–2), 67–89 (1986)
Ieşan, D., Quintanilla, R.: On a theory of thermoelastic materials with a double porosity structure. J. Therm. Stresses 37(9), 1017–1036 (2014)
Svanadze, M.: Plane waves, uniqueness theorems and existence of eigen frequencies in the theory of rigid bodies with a double porosity structure. In: Albers, B., Kuczma, M. (eds.) Continuous Media with Microstructure 2, pp. 287–306. Springer International Publishing, Switzerland (2016)
Svanadze, M.: Steady vibration problems in the theory of elasticity for materials with double voids. Acta Mech. 229(4), 1517–1536 (2018)
Singh, D., Kumar, D., Tomar, S.K.: Plane harmonic waves in a thermoelastic solid with double porosity. Math. Mech. Solids 25(4), 869–886 (2020)
Kröner, E.: Elasticity theory of material with long range cohesive forces. Int. J. Solid Struct. 3, 731–742 (1967)
Edelen, D.G.B., Laws, N.: On the thermodynamics of system with nonlocality. Arch. Ration. Mech. Anal. 43(1), 24–35 (1971)
Edelen, D.G.B., Green, A.E., Laws, N.: Nonlocal continuum mechanics. Arch. Ration. Mech. Anal. 43(1), 36–44 (1971)
Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)
Eringen, A.C.: Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10, 425–435 (1972)
Eringen, A.C.: Nonlocal Continuum Field Theories. Springer-Verlag, New York (2002)
Altan, S.B.: Uniqueness in the linear theory of nonlocal elasticity. Bull. Tech. Univ. Istanb. 37, 373–385 (1984)
Altan, S.B.: Uniqueness of initial-boundary value problems in nonlocal elasticity. Int. J. Solid. Struct. 25(11), 1271–1278 (1989)
Chirita, S.: On some boundary value problems in nonlocal elasticity. In. Amale Stinfice ale Universitatii “AL. I. CUZA" din Iasi Tomul. 22, 2(1976) https://doi.org/10.1080/17455030.2020.1721612
Shaat, M., Ghavanloo, E., Fazelzadeh, S.A.: Review on nonlocal continuum mechanics: physics, material applicability, and mathematics. Mech. Mater. 150, 103587 (2020)
Kaur, G.: Wave Propagation in Nonlocal Elastic Solid with Voids. Panjab University, Chandigarh (2019).. (Thesis)
Eringen, A.C.: On Rayleigh surface waves with small wavelengths. Lett. Appl. Eng. Sci. 1, 11–17 (1973)
Eringen, A.C.: Plane waves in a nonlocal micropolar elasticity. Int. J. Eng. Sci. 22(8–10), 1113–1121 (1984)
Khurana, A., Tomar, S.K.: Wave propagation in nonlocal microstretch solid. Appl. Math. Model. 40(11–12), 5858–5875 (2016)
Khurana, A., Tomar, S.K.: Rayleigh-type waves in nonlocal micropolar elastic solid half-space. Ultrasonics 73, 162–168 (2017)
Khurana, A., Tomar, S.K.: Waves at interface of dissimilar nonlocal micropolar elastic half-spaces. Mech. Adv. Mat. Struct. 26(10), 825–833 (2019)
Gopalakrishnan, S., Narendar, S.: Wave Propagation in Nanostructures. Springer International Publishing, Switzerland (2013)
Singh, D., Kaur, G., Tomar, S.K.: Waves in nonlocal elastic solid with voids. J. Elasticity 128(1), 85–114 (2017)
Bachher, M., Sarkar, N.: Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer. Waves Rand. Compl. Med. 29(4), 595–613 (2019)
Sarkar, N., Tomar, S.K.: Plane waves in nonlocal thermoelastic solid with voids. J. Therm. Stresses 42(5), 580–606 (2019)
Kumar, S., Tomar, S.K.: Plane waves in nonlocal micropolar thermoelastic material with voids. J. Therm. Stresses 43(11), 1355–1378 (2020)
Kaur, G., Singh, D., Tomar, S.K.: Rayleigh-type wave in a nonlocal elastic solid with voids. Eur. J. Mech. A Solids 71, 134–151 (2018)
Singh, B.: Rayleigh-type surface waves in a nonlocal thermoelastic solid half-space with voids. Waves Rand. Compl. Med. 1–12, (2020)
Ciarletta, M., Ieşan, D.: Non-classical Elastic Solids. Pitman Research Notes in Mathematics Series, pp. 239–301. Longman Scientific & Technical, London (1993)
Lakes, R.S.: Physical meaning of elastic constants in Cosserat, void, and microstretch elasticity. J. Mech. Mat. Struct. 11(3), 217–229 (2016)
Borcherdt, R.D.: Viscoelastic Waves in Layered Media. Cambridge University Press, Cambridge (2009)
Acknowledgements
Authors are thankful to DST, New Delhi, and JSPS for providing funds under DST-JSPS project sanctioned to Sushil K. Tomar and Sohichi Hirose through Grant No. DST/INT/JSPS/P-322/2020 and JPJSBP-120207707.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kumar, D., Singh, D., Tomar, S.K. et al. Waves in nonlocal elastic material with double porosity. Arch Appl Mech 91, 4797–4815 (2021). https://doi.org/10.1007/s00419-021-02035-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-021-02035-8