Keywords: homogenisation, finite elements, periodic composites, cellular geo-composites.

Cellular geo-composites (i.e. GEOWEB and similar systems) have a wide area of application in modern geotechnical engineering. Retaining walls, embankments, road sub-soil strengthening are their typical application. Any attempt to numerical modelling of a geo-structure including a cellular geo-composite requires evaluation of its mechanical properties. As in the most of practical cases the macro-scale of the structure is large enough compared to the scale of a single cell of the geo-composite, averaged (homogenised) mechanical properties are sought. It is difficult to obtain them on the basis of direct measurements, while properties of the geo-composite constituents i.e. high density poliethylene bands and natural soil or artificial fills can be easily evaluated experimentally.

Cellular geo-composites are to high extent periodic, with known and easily controllable morphology of a typical cell, which makes them a perfect example of a periodic composite media with a three-dimensional micro-structure.

The constitutive models of composite components, namely fill, are in general nonlinear (here, elasto-plastic models are involved) and frictional contact phenomena between the band and the fill are to be taken into account. The aforementioned aspects cause a finite element based homogenisation approach to be a natural tool for evaluating geo-composite properties.

In the presented homogenisation procedure, see [1], macro stress-strain constitutive relationships of the general incremental form d are obtained. Accompanying tangent stiffness matrices of the composite relating stress and strain increments as: dd are numerically evaluated, as well. Moreover the method give the insight into deformation, stress/strain decomposition on the micro-level explaining macroscopically observed effects. The method is based on the finite element formulation of a micro-level analysis of the representative cell of the composite controlled by selected macroscopic stress or strain components. The unknown field are displacements , describing perturbation of the deformation, constrained by periodicity conditions. In the case of stress control macroscopic strains emerge as additional global unknowns.

The BVP are set in the 3 or 2 dimensional domains of the representative cell as shown in the Figure 60.1. Formulation accepts any, in general non-linear, constitutive model. Incremental solution of resulting equations, relates macroscopic stress with its strain counterparts and allows to trace paths in a stress-strain space, which represent constitutive behaviour of equivalent media. Static condensation of the extended set of the incremental FE equations written at the equilibrium state for given load step yields tangent module matrix of the equivalent media.

At the element level, the homogenisation procedure is applied not only to standard continuum element but also to membrane and contact interface elements which are present in the model of the geo-composite cell. Formulas are given for the stiffness and the residual forces, used in the micro-level analysis of representative cell of the composite.

Presented results, obtained with a custom version of Z_SOIL.PC code [2], include examples of paths in macro-stress-strain space, homogenised stiffness module histories and their relevance to constituent's data. The comparison is given between simplified (but numerically efficient) formulation of a problem as a generalized plane strain case (Figure 60.1b) and the full 3D one (Figure 60.1a).

1

Urbanski A. Unified, finite element based approach to the problem of homogenisation of structural members with periodic microstructure, Proceedings of ECCM, Munchen 1999.

2

Z_SOIL.PC. Users manual, ZACE Services 2001.

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