The poroviscoelastodynamic solution to Mandel's problem

Mandel's problem of quasi-static poroelasticity describes axial compression of a rectangular sample from a porous, fluid-saturated, and elastic material while allowing the pore fluid to drain from the lateral boundaries. This paper reports the closed-form analytical solution to an extension of this problem that considers dynamic response of a porous, fluid-saturated, and viscoelastic specimen with similar geometry and boundary conditions to harmonic excitation. Biot's theory of poroelastodynamics, along with frequency-domain formulation of the viscoelastic constitutive behavior, are used for this purpose. Results indicate that a quasi-static approximation of the dynamic Mandel's problem could generate significantly deviant results compared to the presented dynamic solution, even at frequencies that are substantially smaller than the first natural frequency of the specimen. In this aspect, the solution delineates the existence of four timescale categories pertaining to viscous energy dissipation within the solid and fluid phases of the tested material, as well as the specimen natural frequencies and the loading excitation frequency. Relative order of the described timescales is shown to shape the various forms that the sample frequency response may take. These variations are demonstrated via three case studies involving the brain tissue, asphalt mix, and shale. The presented poroviscoelastodynamic solution verifies previous findings from experimental studies suggesting that a disentangled exhibition of the poroelastic and viscoelastic energy dissipation mechanisms may be captured by the loss angle curves of a frequency sweep test on poroviscoelastic materials.