DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2014.19.2335 DYNAMICAL SYSTEMS SERIES B Volume 19, Number 7, September 2014 pp. 2335–2352 ON THE THEORY OF VISCOELASTICITY FOR MATERIALS WITH DOUBLE POROSITY Merab Svanadze Institute for Fundamental and Interdisciplinary Mathematics Research Ilia State University K. Cholokashvili Ave., 3/5, 0162, Tbilisi, Georgia Abstract. In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with double porosity is presented and the basic partial differential equations are derived. The system of these equations is based on the equations of motion, conservation of fluid mass, the effective stress concept and Darcy’s law for materials with double porosity. This theory is a straightforward gen- eralization of the earlier proposed dynamical theory of elasticity for materials with double porosity. The fundamental solution of the system of equations of steady vibrations is constructed by elementary functions and its basic prop- erties are established. Finally, the properties of plane harmonic waves are studied. The results obtained from this study can be summarized as follows: through a Kelvin-Voigt material with double porosity three longitudinal and two transverse plane harmonic attenuated waves propagate. 1. Introduction. The concept of porous media is used in many areas of applied science (e.g., biology, biophysics, biomechanics) and engineering. There are a num- ber of theories which describe mechanical properties of porous materials. The gen- eral 3D theory of consolidation for materials with single porosity was formulated in [7]. One important generalization of this theory that has been studied extensively started with the work [3], where a fissured porous medium is modeled as two com- pletely overlapping flow regions: one representing the porous matrix, and the other the fissure network. The theory of consolidation for elastic materials with double porosity was presented in [6, 34, 55]. This theory unifies the earlier proposed models of Barenblatt for porous media with double porosity [3] and Biot for porous media with single porosity [7]. With regard to the Aifantis’ quasi-static theory, the cross-coupled terms were included in the equations of fluid mass conservation in [35, 36]. The phenomenolog- ical equations of the quasi-static theory for double porosity media were established and the method to determine the relevant coefficients was presented in [4, 5, 39]. The governing system of equations for an anisotropic material with double porosity was obtained in [56]. In the governing equations of the above mentioned theories of poroelasticity the inertial term was neglected and quasi-static problems were investigated (see [4 - 7, 34 - 36, 39, 56]). On the other hand, the inertial effect plays a pivotal role in the investigation of various problems of vibrations and wave propagation through double porosity media. Therefore, it is important to study a full dynamic model 2010 Mathematics Subject Classification. Primary: 74D05, 74F10; Secondary: 74A60, 74J05. Key words and phrases. Viscoelasticity, double porosity, Kelvin-Voigt material. 2335