On the equations governing the motion of an anisotropic poroelastic material

We address Biot's equations governing the motion of an anisotropic fluid-saturated poroelastic material with certain properties. First, we investigate the uniqueness in solutions of the three-dimensional governing equations for the regular region of the poroelastic material and enumerate the conditions sufficient for the uniqueness. Next, by applying Hamilton's principle to the motion of the region, we obtain a variational principle that generates only the Biot–Newton equations and the associated natural boundary conditions. Then, by extending the variational principle for the region with an internal fixed surface of discontinuity through Legendre's transformation, we derive a six-field variational principle that operates on all the poroelastic field variables. The variational principle leads, as its Euler–Lagrange equations, to all the governing equations, including the jump conditions but the initial conditions, as a generalized version of the Hellinger–Reissner variational principle. Moreover, we consider the free vibrations of the region, and we discuss some basic properties of eigenvalues and present a variational formulation by Rayleigh's quotient. This work provides a standard tool with the features of variational principles when numerically solving the governing equations in heterogeneous media with finite element methods, treating the free vibrations and consistently deriving some one-dimensional/two-dimensional equations of the poroelastic region.