The prediction of the physical and mechanical properties of porous media has been one of hot topics of continuum mechanics for more than 100 years. The presentation of mathematical models of multi-porosity continua and the intensive research of the problems of these models arise by extensive use of porous materials into civil and geotechnical engineering, hydrology, technology, etc. In addition, in recent years, very important area of application for multiple porosity elasticity is biomechanics.The mathematical models of triple-porosity materials represent a new possibility for the study of important problems of applied sciences. Nowadays, triple porosity mathematical models for solids with hierarchical macro-, meso-, and microporosity structure are mostly based on one of the following two phenomena: (i) the concept of Darcy's classical law and (ii) the concept of volume fraction.Moreover, many of the engineering problems have coupled physical nature and in almost all cases of practical interest, we encounter various coupled processes in porous media. Therefore, it is required to consider several coupled mechanical concepts simultaneously in the models of such materials.Recently, Svanadze (2020) introduced the linear models of elasticity and thermoelasticity for single and double-porosity materials in which the coupled phenomenon of the concepts of Darcy's law and the volume fraction of pore network is considered. The basic BVPs of quasi-static and steady vibrations of these models are studied by several authors.In the present talk, the coupled linear model of elastic triple-porosity materials is considered in which the coupled phenomenon of the concepts of Darcy's law and the volume fractions of three levels of pores (macro-, meso- and micropores) is proposed. The basic internal and external boundary value problems (BVPs) of steady vibrations are investigated. Indeed, the fundamental solution of the system of steady vibration equations is constructed explicitly by means of elementary functions and its basic properties are presented. The radiation conditions are established and Green's identities are obtained. The uniqueness theorems for the regular (classical) solutions of the BVPs are proved. The surface (single-layer and double-layer) and volume potentials are constructed and the basic properties of these potentials are given. The determinants of symbolic matrices of the singular integral operators are calculated explicitly. Then, the BVPs are reduced to the always solvable singular integral equations for which Fredholm’s theorems are valid. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.Acknowledgments.&nbsp;This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) [Project # FR-19-4790].

Steady Vibration Problems in the Coupled Theory of Elasticity for Triple Porosity Materials

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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