Michel Bornert

An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals

Variational approaches for nonlinear elasticity show that Hill’s incremental formulation for the prediction of the overall behaviour of heterogeneous materials yields estimates which are too stiff and may even violate rigorous bounds. This paper aims at proposing an alternative ‘affine’ formulation, based on a linear thermoelastic comparison medium, which could yield softer estimates. It is first described for nonlinear elasticity and specified by making use of Hashin–Shtrikman estimates for the linear comparison composite; the associated affine self-consistent predictions are satisfactorily compared with incremental and tangent ones for power-law creeping polycrystals. Comparison is then made with the second-order procedure (Ponte Castaneda, P., 1996. Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids, 44 (6), 827–862) and some limitations of the affine method are pointed out; explicit comparisons between different procedures are performed for isotropic, two-phase materials. Finally, the affine formulation is extended to history-dependent behaviours; application to the self-consistent modelling of the elastoplastic behaviour of polycrystals shows that it offers an improved alternative to Hill’s incremental formulation.

Experimental characterization of the local strain field in a heterogeneous elastoplastic material

A new technique, which allows to characterize the local strain field over a domain representative of the microstructure of a heterogeneous material, is described. It is based on scanning electron microscopy, microelectrolithography, image analysis and in situ tensile tests. The in-plane components of the local strain field are characterized by their averages per phase and their distribution functions. The results are accurate for global strains between 5 and 15%. It is also possible to get contour plots of these components of the local strain field over the considered domain. The obtained strain maps give a powerful qualitative information on the strain localization modes during the deformation. This technique has basically been developed for two-phase elastoplastic materials, namely iron/silver and iron/copper blends, submitted to uniaxial tensile tests; it could also be used for polycrystals or other composite materials and for other mechanical tests.

Morphologically representative pattern-based bounding in elasticity

Abstract A general theory for the homogenization of heterogeneous linear elastic materials that relies on the concept of “morphologically representative pattern” is given. It allows the derivation of rigorous bounds for the effective behaviour of the Voigt-Reuss-type, which apply to any distribution of patterns, or of the Hashin-Shtrikman-type, which are restricted to materials whose pattern distributions are isotropic. Particular anisotropic distributions of patterns can also be considered: Hashin-Shtrikman-type bounds for anisotropic media are then generated. The resolution of the homogenization problem leads to a complex composite inclusion problem with no analytical solution in the general case. Here it is solved by a numerical procedure based on the finite element method. As an example of possible application, this procedure is used to derive new bounds for matrix-inclusion composites with cubic symmetry as well as for transversely isotropic materials.

Experimental and numerical characterisation of in-plane deformation in two-phase materials

The aim of the present work consists in the comparison of in-plane strain fields with out-of-plane displacements in micro-areas of an Ag/Ni-composite after a macroscopic compressive deformation of 8.6%. The in-plane deformations in an Ag/Ni-composite have been analysed experimentally with a high resolution object grating technique and numerically using the finite element method. The out-of-plane displacements were measured with an atomic force microscope (AFM). The development of local strain fields in micro-areas at the surface of an Ag/Ni-composite was simulated numerically using the FE-method in plane strain condition. A real cut-out of the microstructure served as input for the calculation. The out-of-plane displacements determined by AFM measurements were used further to correct the in-plane values of strains evaluated by the object grating technique. The roughness on the surface of the sample was characterised by fractal dimensions and compared with the in-plane strains in the same micro-region.

Second-order estimates for the effective behaviour of viscoplastic polycrystalline materials

Abstract This paper is concerned with the application of the “second-order” nonlinear homogenisation procedure (Ponte Castaneda, J. Mech. Phys. Solids 44 (6) (1996) 827) to generate estimates of the self-consistent type for the effective behaviour of fcc and hcp viscoplastic polycrystals. The method has the distinctive property that it leads to estimates that are exact to second-order in the heterogeneity contrast, and which are expected to be more accurate, particularly when compared to rigorous bounds, than those resulting from earlier homogenisation schemes such as the Hill “incremental” method or its “total” formulation (Hutchinson) for pure power-law viscous materials. Special attention is paid to large grain anisotropy leading to correspondingly large heterogeneity contrast, and to highly nonlinear behaviour. Comparisons are also carried out with estimates derived from other more recent homogenisation schemes such as the “tangent” and “affine” methods. The results, illustrated for zirconium- and ice-type polycrystals, show that the second-order procedure offers the potential for significantly improved results, at least relative to the Hill incremental formulation.

Micromechanical Applications of Digital Image Correlation Techniques

Digital image correlation (DIC) is applied to images obtained with a scanning electron microscope (SEM) in order to determine the local strain field in a heterogeneous material on a representative domain of the microstructure, millimetric in size, the local contrast being provided by a microgrid deposited on the surface of the sample, with a typical 5μm step. The specific difficulties encountered when applying this classical technique for CCD images to SEM images are discussed. They are related to geometric images defects, high noise levels or imperfect scanning or to the non optimal local contrast, which are all potential sources of loss of accuracy. Several procedures to correct such imperfections are presented. The investigation of the residual strain field in a ferritic-martensitic steel after 1% total strain under uniaxial traction is then shortly presented.

Micro-scale experimental investigation of the swelling anisotropy of the Callovo-Oxfordian argillaceous rock

Abstract An experimental study of the swelling anisotropy of the Callovo-Oxfordian argillaceous rock under hydration is presented. The investigation, which combines environmental scanning electron microscopy (ESEM) and digital image correlation techniques, has been carried out at the micrometric scale of the composite microstructure of the rock. Specimens were hydrated in the ESEM over a wide range of relative humidity and observations conducted on two planes: plane 1 parallel to the bedding plane, and plane 2 perpendicular to it. The observations reveal that the local swelling (which can be quantified at a local gauge length of about 5 μm) is strongly anisotropic in both planes. The global swelling, measured over areas of about 500 μm in width, is also clearly anisotropic in plane 2 (with major swelling direction normal to the bedding plane), but not in plane 1. The global isotropy in plane 1 arises from the uniform distribution of the orientation of anisotropic local strains, while the anisotropic swelling in plane 2 is due to a preferred local orientation.

New bounds and estimates for porous media with rigid perfectly plastic matrix

Abstract We derive new rigorous bounds and self-consistent estimates for the effective yield surface of porous media with a rigid perfectly plastic matrix and a microstructure similar to Hashins composite spheres assemblage. These results arise from a homogenisation technique that combines a pattern-based modelling for linear composite materials and a variational formulation for nonlinear media. To cite this article: N. Bilger et al., C. R. Mecanique 330 (2002) 127–132.

Second–order estimates of the self–consistent type for viscoplastic polycrystals

The ‘second–order’ homogenization procedure of Ponte Castañeda is used to propose new estimates of the self–consistent type for the effective behaviour of viscoplastic polycrystals. This is accomplished by means of appropriately generated estimates of the self–consistent type for the relevant ‘linear thermoelastic comparison composite’, in the homogenization procedure. The resulting nonlinear self–consistent estimates are the only estimates of their type to be exact to second order in the heterogeneity contrast, which, for polycrystals, is determined by the grain anisotropy. In addition, they satisfy the recent bounds of Kohn and Little for two–dimensional power–law polycrystals, which are known to be significantly sharper than the Taylor bound at large grain anisotropy. These two features combined, suggest that the new self–consistent estimates, obtained from the second–order procedure, may be the most accurate to date. Direct comparison with other self–consistent estimates, including the classical incremental and secant estimates, for the special case of power–law creep, appear to corroborate this observation.

On the choice of parameters in the phase field method for simulating crack initiation with experimental validation

The phase field method is a versatile simulation framework for studying initiation and propagation of complex crack networks without dependence to the finite element mesh. In this paper, we discuss the influence of parameters in the method and provide experimental validations of crack initiation and propagation in plaster specimens. More specifically, we show by theoretical and experimental analyses that the regularization length should be interpreted as a material parameter, and identified experimentally as it. Qualitative and quantitative comparisons between numerical predictions and experimental data are provided. We show that the phase field method can predict accurately crack initiation and propagation in plaster specimens in compression with respect to experiments, when the material parameters, including the characteristic length are identified by other simple experimental tests.