On extensional waves in a poroelastic cylinder within the framework of viscosity-extended Biot theory: the case of traction-free open-pore cylindrical surface

Abstract

SUMMARY For a fluid saturated porous cylinder, in which the saturating fluid is allowed to move freely across the traction-free cylindrical boundary surface, the dispersion relation for extensional deformation, established over four decades ago by Gardner within the framework of Biot's theory, yield a trivial and two non-trivial roots. One of the non-trivial roots corresponds to the coupled Biot fast-compressional and shear waves and it is the poroelastic analogue of the extensional wave of the elasticity theory. The nature of the second non-trivial root is not fully understood yet. Also there is the suggestion that the scope of applicability of Gardner's first non-trivial root might be limited. Apart from these matters, since the Biot theory is also incapable of accounting for the viscous bulk- and shear-relaxations that occur within the saturating Newtonian fluid because its fluid stress tensor is devoid of the viscous stress tensor part, therefore, we re-examined this poroelastic extensional problem with viscosity-extended Biot constitutive relations. The erstwhile trivial third root of the Biot–Gardner framework is not non-vanishing in this viscosity-extended Biot framework. In the frequency regime below the Biot relaxation frequency, the other two non-trivial roots in this framework are the same as those obtained by Gardner within the classical Biot framework. We find there is nothing dubious about the formula of Gardner's first root. The scenario in which it is not expected to hold true, in fact the Biot theory itself breaks down. We find the Gardner's second root to be essentially the Biot slow-compressional wave, the out-of-phase/differential compressional motion of the two phases. The new non-vanishing third root is akin to the slow-shear process, the out-of-phase/differential shear motion of the two phases, and a process analogue to the Biot slow-compressional wave.