TIME DOMAIN FORMULATION OF THE BIOT POROELASTIC THEORY USING FRACTIONAL CALCULUS

Here an integro-differential formulation for time domain poroelasticity is derived and included into a finite element framework for structural dynamics. Further, an algorithm for the time integration of the spatially discretized finite element equations is developed. The Biot poroelastic theory (formulated in the frequency domain) is taken as a starting point. In this theory, the dissipative force is proportional to the relative motion between the solid phase and the viscous fluid in the pores with a constant of proportionality, drag force coefficient, that is frequency dependent. The dissipative force is described in the time domain by a convolution term with a kernel that is the Fourier inverse of the drag force coefficient. As the exact analytical inversion of the drag force coefficient seems highly intractable, we have combined an asymptotic expansion for high frequencies with a low frequency correction term, yielding an analytically invertible drag force coefficient. The convolution with the fourier inverse of the asymptotic expansion, can be identified as a semi-derivative.