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Summary This paper presents a model which can be used for fast landslides where coupling between solid and pore fluid plays a fundamental role. The proposed model is able to describe debris flows where the difference of velocities between solid grains and fluid is important. The approach is based on the mathematical model proposed by Zienkiewicz and Shiomi, which is similar to those of Pitman and Le and Pudasaini. The novelty of the present work is the numerical technique used, the smoothed particle hydrodynamics (SPH). We propose to use a double set of nodes for soil and water phases, the interaction between them being described by a suitable drag law. The paper presents both mathematical and numerical models, describing the main assumptions and their limitations. Then, the model is applied to (1) a simple case where shocks and expansion waves appear, (2) a dam break problem on a horizontal plane with a frictional soil phase, and (3) a debris flow which happened in Hong Kong. The main conclusions that can be drawn from the applications are: Debris flows having 2 phases with important relative mobility present a rich structure of shocks and rarefaction waves, which has to be properly modeled. Otherwise, the model will have numerical damping or dispersion. Dambreak exercises provide interesting information in simple and controlled situations. We can see how both phases move relative to each other. Real debris flows can be simulated with the proposed model, obtaining reasonable results.

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The material point method can be regarded as a meshfree extension of the finite element method. This fact has two interesting consequences. On the one hand, this puts a vast literature at our disposal. On the other hand, many of this inheritance has been adopted without questioning it. A clear example of it is the use of the Voigt algebra, which introduces an artificial break point between the formulation of the continuum problem and its discretized counterpart. In the authors' opinion, the use of the Voigt rules leads to a cumbersome formulation where the physical sense of the operations is obscured since the well-known algebra rules are lost. And with them, the intuition about how stresses and strains are related.To illustrate it, we will describe gently and meticulously the whole process to solve the nonlinear governing equations for isothermal finite strain elastodynamics, concluding with a compact set of expressions ready to be implemented effortless. In addition, the classic Newmark-algorithm has been accommodated to the local maximum-entropy material point method framework by means of an incremental formulation. K E Y W O R D SB free, finite strain, material point method, Newmark-, Voigt free * This aspect is controlled at the very beginning of the modelization, specifically during the meshing process.This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

TheBfree finite element approach is applied to the governing equations describing the consolidation process in saturated poroelastic medium with intrinsically incompressible solid and fluid phases. Under this approach, where Voigt notation is avoided, the finite element equilibrium equations and the linearization of the coupled governing equations are fully derived using tensor algebra. In order to assess theBfree approach for the consolidation equations, direct comparison with analytical solution of the response of a homogeneous and isotropic water-saturated poroelastic finite column under harmonic load is presented. The results illustrate the capability of this finite element approach of reproducing accurately the response of quasistatic phenomena in a saturated porous medium.