On numerical analyses in the presence of unstable saturated porous materials

The numerical analysis of the dynamic evolution problem concerning an elastic–plastic saturated porous media in the presence of softening (or non-associativity) is considered in the framework of the Biot formulation extended to take into account plastic phenomena. The nite step boundary value problem, obtained by discretization in time of the continuous initial boundary value problem, is studied and the issue of its ill-posedness is particularly addressed. The conditions for the loss of ellipticity are established for the linearized problem solved at each iteration when using the Newton–Raphson scheme. In particular, the roles of the algorithmic properties on this loss of ellipticity are derived in details. The integration scheme of the balance of mass equation plays a major role and it is shown that the uid ow (Darcy’s law) does indeed introduce a length scale but in addition to being dependent on the integration time step, it is found to be insu cient for regularization. To illustrate and corroborate the obtained results, a one-dimensional example (exhibiting all the features of the three-dimensional situation) is considered and the corresponding linearized nite step problem is solved in closed form