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International audienceFor more than a decade, numerical methods for periodic elasticity, based on the fast Fourier transform, have been used successfully as alternatives to more conventional (fem, bem) numerical techniques for composites. These methods are based on the direct, point-wise, discretization of the Lippmann-Schwinger equation, and a subsequent truncation of underlying Fourier series required for the use of the fast Fourier transform. The basic FFT scheme is very attractive, because of its simplicity of implementation and use. However, it cannot handle pores or rigid inclusions, for which a specific (and significantly more involved) treatment is required. In the present paper, we propose a new FFT-based scheme which is as simple as the basic scheme, while remaining valid for infinite contrasts. Since we adopted an energy principle as an alternative to the Lippmann-Schwinger equation, our scheme is derived within a variational framework. As a by-product, it provides an energetically consistent rule for the homogenization of boundary voxels, a question which has been pending since the introduction of Fourier-based methods

To cite this version:Sébastien Brisard, Luc Dormieux. Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2012, 217-220, pp.197-212 AbstractWe report on the mathematical analysis of two different, FFT-based, numerical schemes for the homogenization of composite media within the framework of linear elasticity: the basic scheme of Suquet (1994, 1998), and the energy-based scheme of Brisard and Dormieux (2010). Casting these two schemes as Galerkin approximations of the same variational problem allows us to assert their well-posedness and convergence. More importantly, we extend in this work their domains of application, by relieving some stringent conditions on the reference material which were previously thought necessary. The origins of the flaws of each scheme are identified, and a third scheme is proposed, which seems to combine the strengths of the basic and energy-based schemes, while leaving out their weaknesses. Finally, a rule is proposed for handling heterogeneous pixels/voxels, a situation frequently met when images of real materials are used as input to these schemes.

With the recent increase in underground urban development as well as for transportation, tunnels need to he driven in increasingly dif&ult soil conditions. In most cases the ground itself is not stable and face stability is achieved by applying fluid pressure to the tunnel front. The question of determining the retaining pressure to he used has received special consideration in the past hecause of the concern for safety during construction, and also hecause of the damage that could he caused to surface structures by the failure of a shallow tunnel. The problem is three-dimensional and can he studied by using the limit state design method. Solutions are available for the case of a circular tunnel in purely cohesive ground, but very little is known of the face stability of a tunnel driven in sandy soils. This Paper approaches this latter problem from the point of view of limit analysis. Both safety against face collapse and blow-out are considered. Three upper hound solutions are derived from the consideration of mechanisms based on the motion of rigid conical blocks. The results are compared with lower hound solutions published in a previous article. A failure criterion is proposed for the tunnel face in the general case of a cohesive and frictional soil, and charts are provided to allow a bracketed estimate of the required retaining pressure. A comparison with centrifuge laboratory tests shows close agreement between the theoretical upper hound solutions and the face pressures at collapse measured experimentally.

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