Abstract
In the present paper, the linear theory of elasticity for materials with double voids is considered. Some basic properties of plane harmonic waves of this theory are established. The internal boundary value problems (BVPs) of steady vibrations are formulated. On the basis of Green’s tensors, the internal BVPs are reduced to the equivalent Fredholm’s integral equations of the second kind with symmetrical kernel. The existence of eigenfrequencies of the internal BVPs of steady vibrations is proved. Finally, the formula of the asymptotic distribution of these eigenfrequencies is obtained.
Similar content being viewed by others
References
Achenbach, J.D.: Wave Propagation in Elastic Solids. American Elsevier Publishing Company, Inc, New York (1975)
Arendt, W., Nittka, R., Peter, W., Steiner, F.: Weyl’s law: spectral properties of the Laplacian in mathematics and physics. In: Arendt, W., Schleich, W.P. (eds.) Mathematical Analysis of Evolution, Information, and Complexity, pp. 1–77. WILEY- VCH Verlag GmbH & Co. KGaA, Weinheim (2009)
Bai, M., Elsworth, D., Roegiers, J.C.: Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs. Water Resour. Res. 29, 1621–1633 (1993)
Barenblatt, G.I., Zheltov, I.P., Kochina, I.N.: Basic concept in the theory of seepage of homogeneous liquids in fissured rocks (strata). J. Appl. Math. Mech. 24, 1286–1303 (1960)
Berryman, J.G., Wang, H.F.: Elastic wave propagation and attenuation in a double porosity dual-permeability medium. Int. J. Rock Mech. Min. Sci. 37, 63–78 (2000)
Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)
de Boer, R.: Theory of Porous Media: Highlights in the Historical Development and Current State. Springer, Berlin (2000)
Burchuladze, T.V., Gegelia, T.G.: The Development of the Potential Methods in the Elasticity Theory. Metsniereba, Tbilisi (1985)
Carleman, T.: Über die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen. Ber. der Sächs. Akad. d. Wiss. Leipzig 88, 119–132 (1936)
Ciarletta, M., Passarella, F., Svanadze, M.: Plane waves and uniqueness theorems in the coupled linear theory of elasticity for solids with double porosity. J. Elast. 114, 55–68 (2014)
Cowin, S.C.: Bone poroelasticity. J. Biomech. 32, 217–238 (1999)
Cowin, S.C., Cardoso, L.: Blood and interstitial flow in the hierarchical pores pace architecture of bone tissue. J. Biomech. 48, 842–854 (2015)
Cowin, S.C., Nunziato, J.W.: Linear elastic materials with voids. J. Elast. 13, 125–147 (1983)
Gegelia, T., Jentsch, L.: Potential methods in continuum mechanics. Georgian Math. J. 1, 599–640 (1994)
Gelet, R., Loret, B., Khalili, N.: Borehole stability analysis in a thermoporoelastic dual-porosity medium. Int. J. Rock Mech. Min. Sci. 50, 65–76 (2012)
Gentile, M., Straughan, B.: Acceleration waves in nonlinear double porosity elasticity. Int. J. Eng. Sci. 73, 10–16 (2013)
Ieşan, D.: Method of potentials in elastostatics of solids with double porosity. Int. J. Eng. Sci. 88, 118–127 (2015)
Ieşan, D., Quintanilla, R.: On a theory of thermoelastic materials with a double porosity structure. J. Therm. Stress. 37, 1017–1036 (2014)
Khalili, N., Selvadurai, A.P.S.: A fully coupled constitutive model for thermo-hydro-mechanical analysis in elastic media with double porosity. Geophys. Res. Lett. 30, 2268 (2003)
Kupradze, V.D.: Potential Methods in the Theory of Elasticity. Israel Program Sci. Transl, Jerusalem (1965)
Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O., Burchuladze, T.V.: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland, Amsterdam, New York, Oxford (1979)
Nunziato, J.W., Cowin, S.C.: A non-linear theory of elastic materials with voids. Arch. Ration. Mech. Anal. 72, 175–201 (1979)
Pleijel, A.: Propriétés asymptotiques des fonctions et valeurs propres de certains problems de vibrations. Arkiv för Math. Astr. och Fysik 27A, 1–100 (1940)
Rohan, E., Naili, S., Cimrman, R., Lemaire, T.: Multiscale modeling of a fluid saturated medium with double porosity: relevance to the compact bone. J. Mech. Phys. Solids 60, 857–881 (2012)
Scarpetta, E., Svanadze, M.: Uniqueness theorems in the quasi-static theory of thermoelasticity for solids with double porosity. J. Elast. 120, 67–86 (2015)
Scarpetta, E., Svanadze, M., Zampoli, V.: Fundamental solutions in the theory of thermoelasticity for solids with double porosity. J. Therm. Stress. 37, 727–748 (2014)
Straughan, B.: Stability and Wave Motion in Porous Media. Springer, New York (2008)
Straughan, B.: Stability and uniqueness in double porosity elasticity. Int. J. Eng. Sci. 65, 1–8 (2013)
Straughan, B.: Convection with Local Thermal Non-equilibrium and Microfluidic Effects. Springer, Berlin (2015)
Svanadze, M.: Asymptotic distribution of eigenfunctions and eigenvalues of the boundary value problems of linear theory of elastic mixtures. Georgian Math. J. 3, 177–200 (1996)
Svanadze, M.: Plane waves and boundary value problems in the theory of elasticity for solids with double porosity. Acta Appl. Math. 122, 461–471 (2012)
Svanadze, M.: Uniqueness theorems in the theory of thermoelasticity for solids with double porosity. Meccanica 49, 2099–2108 (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, Basel (2016)
Weyl, H.: Über die asymptotische Verteilung der Eigenwerte. Nachr. Ges. Wiss. Göttingen 1911, 110–117 (1911)
Weyl, H.: Über die Abhängigkeit der Eigenschwingungen einer Membran von deren Begrenzung. J. Reine Angew. Math. 141, 1–11 (1912)
Weyl, H.: Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten elastischen Kórpers. Rend. Circolo Mat. Palermo 39, 1–49 (1915)
Wilson, R.K., Aifantis, E.C.: On the theory of consolidation with double porosity. Int. J. Eng. Sci. 20, 1009–1035 (1982)
Zhao, Y., Chen, M.: Fully coupled dual-porosity model for anisotropic formations. Int. J. Rock Mech. Min. Sci. 43, 1128–1133 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Svanadze, M. Steady vibration problems in the theory of elasticity for materials with double voids. Acta Mech 229, 1517–1536 (2018). https://doi.org/10.1007/s00707-017-2077-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-017-2077-z