A mathematical framework for finite strain elastoplastic consolidation Part 1: Balance laws, variational formulation, and linearization

Abstract

A mathematical formulation for finite strain elasto plastic consolidation of fully saturated soil media is presented. Strong and weak forms of the boundary-value problem are derived using both the material and spatial descriptions. The algorithmic treatment of finite strain elastoplasticity for the solid phase is based on multiplicative decomposition and is coupled with the algorithm for fluid flow via the Kirchhoff pore water pressure. Balance laws are written for the soil-water mixture following the motion of the soil matrix alone. It is shown that the motion of the fluid phase only affects the Jacobian of the solid phase motion, and therefore can be characterized completely by the motion of the soil matrix. Furthermore, it is shown from energy balance consideration that the effective, or intergranular, stress is the appropriate measure of stress for describing the constitutive response of the soil skeleton since it absorbs all the strain energy generated in the saturated soil-water mixture. Finally, it is shown that the mathematical model is amenable to consistent linearization, and that explicit expressions for the consistent tangent operators can be derived for use in numerical solutions such as those based on the finite element method.

Hydro-mechanical modelling of the excavation damaged zone around an underground excavation at Mont Terri Rock Laboratory

“…In order to solve this non-linear problem, a Newton-Raphson scheme is proposed to find a new solution of displacement and pressure fields, for which equilibrium is met. The idea is to define a linear auxiliary problem deriving from the continuum one (instead of the discretized one as it is more usually done) similar to the work of Borja and Alarcon [29]. This approach gives the same results as standard FEM procedure but makes the linearization easier, especially for coupled problem in large strain formulation.…”

On boundary condition in tunnels under partial saturation

“…In order to solve numerically in finite element code the field equations using this new boundary condition, a linear auxiliary problem can be defined following the ideas of Borja & Alarcon [1995] and the field of unknowns is obtained through a Newton-Raphson scheme. The linear auxiliary problem is discretized using the finite element methodology [Zienkiewicz & Taylor, 2000].…”

Hydro mechanical numerical modelling of geotechnical problems using local second gradient models

“…Similarly to the work of Borja (Borja and Alarcon 1995), we chose in this paper to get the linearized equation for the continuum problem instead of for the discretized one as it is more usually done. More details on the computation can be seen in a forthcoming paper (Collin, Chambon and Charlier 2004).…”

“…The reader interested by numerical computations of multiphasic materials can read for instance the book of Lewis and Schrefler (Lewis and Schrefler 2000). An other important reference dealing with hyper elasto plastic modelling is the paper written by Borja and Alarcon (Borja and Alarcon 1995).…”