Geodesic
On determination of the stress state of an elastic-porous medium
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 2, pp. 204-215

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The model of elastic porous solid medium for simulation of fluid penetration in an elastic porous body has been investigated. Similar processes can occur as a result of exposure of, for example, hard coal seams or deep solid type concrete, glass, etc. to fluid under high pressure. Assuming that an elastic body is a bundle of capillaries, a linear relationship between the filtration coefficient and the first invariant of the stress tensor has been revealed. Therefore, the problem of filtration through a deformable elastic medium has been reduced to two tasks: finding the stress tensor and solving the problem of filtration with the known filtration coefficient. In the general case, both tasks are complicated for analytical studies, but they can be solved numerically, for example, by the method of finite element analysis. The problem is significantly simplified for infinitely long cylindrical bodies. In the present work, the simplest mathematical model of plane elastic stress state in the transverse weightness field within both Hooke's law and Darcy's law with a constant filtration coefficient has been considered. In this case, the elastic deformation is described by the Airy biharmonic function, while filtration by the harmonic function. Based on Green's integral formula, integral relations combined into a single system have been found for the desired functions. The numerical solution has been performed using the method of boundary elements, with the help of which the problem has been reduced to a system of linear equations. Using a round tube, comparative analysis of the numerical and analytical solutions has been carried out. Numerical values have been obtained for the required parameters of an elliptical tube.
Keywords: elastic porous medium, strain, pressure, harmonic equation, biharmonic equation, numerical methods.
Mots-clés : filtration 
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     author = {E. A. Mikishanina and A. G. Terentiev},
     title = {On determination of the stress state of an elastic-porous medium},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {204--215},
     publisher = {mathdoc},
     volume = {159},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2017_159_2_a5/}
}
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E. A. Mikishanina; A. G. Terentiev. On determination of the stress state of an elastic-porous medium. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 2, pp. 204-215. http://geodesic.mathdoc.fr/item/UZKU_2017_159_2_a5/