Keywords: post-liquefaction, Newtonian model, Bingham viscoplastic model, regularisation, variational principle, finite element method, augmented Lagrangian method.

The approach exposed in this paper makes it possible to simulate the evolution of liquefied soil embankments subject to gravitational forces induced by seismic loading. Newtonian as well as Bingham viscoplastic models have been successively adopted to capture the rheological behaviour of the liquefied soil. The numerical strategy is based on the establishment of a variational principle for the velocity field defined on the current configuration of the liquefied soil mass combined with a step- by-step time integration procedure making it possible to follow the geometry changes between two successive configurations. Several numerical schemes based on this approach implemented in a finite element code are developed, and numerical illustrations are presented.

It is firstly considered that liquefied soil behaves like a classical incompressible Newtonian fluid (zero shear strength), and hence, the minimum principle governing its evolution is equivalent to the minimum principle of the potential energy used in linear elasticity [1]. A Lagrangian finite element method is used to solve this problem.

Then, the incompressible Bingham viscoplastic model characterised by two parameters relating to the undrained shear strength and viscosity coefficient, which represents the real behaviour of the liquefied soil more accurately, is adopted. In order to circumvent the numerical difficulties associated with the non-linearity and the non-differentiability of some terms deriving from the constitutive behaviour law, an approximation to the true Bingham model by means of regularisation technique is introduced, using the Papanastasiou's approximation [2]. The numerical treatment by means of an iterative algorithm is described.

Then a direct implementation of the unregularised Bingham model, which involves variational inequalities that are briefly described and requires special numerical treatment, is finally developed, A generalised Uzawa's algorithm using the so-called Fortin-Glowinski decomposition-co-ordination method by augmented Lagrangian, which appears to be numerically efficient, is presented [3].

The performances of the numerical schemes described for solving the problem, are finally compared on the illustrative example of a liquefied soil embankment.

1

J. Salençon, "Handbook of Continuum Mechanics", Springer, 2000.

2

T.C. Papanastasiou, "Flow of materials with yield", Journal of Rheology, 31, 385-404, 1987. doi:10.1122/1.549926

3

M. Fortin, R. Glowinski, "Méthodes de Lagrangien augmenté: applications à la résolution numérique de problèmes aux limites", Dunod, 1982.

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