On a micro-macro transition for poroelastic Biot's model and corresponding Gassmann-type relations
- Wilmanski, Krzysztof
2010 Mathematics Subject Classification
- 74F10 74L10 74Q15 74E30
- Micro-macro transitions, mechanics of poroelastic materials, Biot's model
In the paper we consider a micro-macro transition for a linear thermodynamical model of poroelastic media which yields the Biot's model. We investigate a two-component poroelastic linear model in which a constitutive dependence on the porosity gradient is incorporated and this is compared with the classical Biot's model without added mass effects. We analyze three Gedankenexperiments: jacketed undrained, jacketed drained and unjacketed and derive a generalization of classical Gassmann relations between macroscopic material parameters and microscopic compressibility moduli of the solid, and of the fluid. Dependence on the porosity is particularly exposed due to its importance in acoustic applications of the model.
In particular we show that Gassmann relations follow as one of two physically justified solutions of the full set of micro-macro compatibility relations. In this solution the coupling to the porosity gradient is absent. Simultaneously, we demonstrate the second solution which lies near the Gassmann results but admits the coupling. In both models couplings are weak enough to admit, within the class of problems of acoustic wave analysis, an approximation by a "simple mixture" model in which coupling of stresses is fully neglected.