Paul Germain (1920-2009)

International audienceNotice nécrologique de Paul Germai

Convergence of iterative methods based on Neumann series for composite materials: theory and practice

Iterative Fast Fourier Transform methods are useful for calculating the fields in composite materials and their macroscopic response. By iterating back and forth until convergence, the differential constraints are satisfied in Fourier space, and the constitutive law in real space. The methods correspond to series expansions of appropriate operators and to series expansions for the effective tensor as a function of the component moduli. It is shown that the singularity structure of this function can shed much light on the convergence properties of the iterative Fast Fourier Transform methods. We look at a model example of a square array of conducting square inclusions for which there is an exact formula for the effective conductivity (Obnosov). Theoretically some of the methods converge when the inclusions have zero or even negative conductivity. However, the numerics do not always confirm this extended range of convergence and show that accuracy is lost after relatively few iterations. There is little point in iterating beyond this. Accuracy improves when the grid size is reduced, showing that the discrepancy is linked to the discretization. Finally, it is shown that none of the three iterative schemes investigated over-performs the others for all possible microstructures and all contrasts.Comment: 41 pages, 14 figures, 1 tabl

Incremental variational principles with application to the homogenization of nonlinear dissipative composites

International audienceThis study is devoted to the overall response of nonlinear composites composed of phases which have a partly reversible and partly irreversible behavior, typically elasto-viscoplastic constituents. Upon use of an implicit time-discretization scheme, the evolution equations describing the constitutive behavior of the phases are reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a nonuniform transformation field. Two different techniques for approximating the nonuniform eigenstrains by piecewise uniform eigenstrains and for linearizing the nonlinear thermoelastic problem will be presented

On the effective behavior of nonlinear inelastic composites: II A second-order procedure

International audienceA new method for determining the overall behavior of composite materials comprising nonlinear viscoelastic and elasto-viscoplastic constituents is presented. Part I of this work showed that upon use of an implicit time-discretization scheme, the evolution equations describing the constitutive behavior of the phases can be reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a transformation strain which is a nonuniform field (not even uniform within the phases). In part I of this paper the nonlinearity was handled using a variational (or secant) technique. In this second part of the study, a proper modification of the second-order procedure of Ponte Castañeda is proposed and leads to replacing, at each time-step, the actual nonlinear viscoelastic composite by a linear viscoelastic one. The linearized problem is even further simplified by using an “effective internal variable” in each individual phase. The resulting predictions are in good agreement with exact results and improve on the predictions of the secant model proposed in part I of this paper

On the influence of local fluctuations in volume fraction of constituents on the effective properties of nonlinear composites. Application to porous materials

International audienceComposite materials often exhibit local fluctuations in the volume fraction of their individual constituents. This paper studies the influence of such small fluctuations on the effective properties of composites. A general asymptotic expansion of these properties in terms of powers of the amplitude of the fluctuations is given first. Then, this general result is applied to porous materials. As is well-known, the effective yield surface of ductile voided materials is accurately described by Gurson's criterion. Suitable extensions for viscoplastic solids have also been proposed. The question addressed in the present study pertains to nonuniform distributions of voids in a typical volume element or in other words to the presence of matrix-rich and pore-rich zones in the material. It is shown numerically and analytically that such deviations from a uniform distribution result in a weakening of the macroscopic carrying capacity of the material

On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles.

International audienceA new method for determining the overall behavior of composite materials comprised of nonlinear inelastic constituents is presented. Upon use of an implicit time-discretization scheme, the evolution equations describing the constitutive behavior of the phases can be reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a transformation strain which is a nonuniform field (not even uniform within the phases). In this first part of the study the variational technique of Ponte Castaneda is used to approximate the nonuniform eigenstrains by piecewise uniform eigenstrains and to linearize the nonlinear thermoelastic problem. The resulting problem is amenable to simpler calculations and analytical results for appropriate microstructures can be obtained. The accuracy of the proposed scheme is assessed by comparison of the method with exact results

Model-reduction in micromechanics of polycrystalline materials

International audienceThe present study is devoted to the generalization of the Nonuniform Transformation Field Analysis (NTFA), a model-reduction approach introduced by the authors. First, the local fields of internal variables are decomposed on a reduced basis of modes. Second, the effective (average) dissipation potential of the phases is replaced by accurate approximations. The reduced evolution equations of the models, in other words the homogenized constitutive relations, can be entirely expressed explicitly in terms of quantities which are pre-computed " off-line ". The example of creep of polycrystalline ice is used to assess the accuracy of the models. Their predictions, both the overall response and the local response, are shown to be in good agreement with full-field simulations with a significant speed-up

Porous materials with two populations of voids under internal pressure: II. Growth and coalescence of voids

International audienceThis study is devoted to the mechanical behavior of polycrystalline materials with two populations of voids, small spherical voids located inside the grains and larger spheroidal voids located at the grain boundaries. In part I of the work, instantaneous effective stress-strain relations were derived for fixed microstructure. In this second part, the evolution of the microstructure is addressed. Differential equations governing the evolution of the microstructural parameters in terms of the applied loading are derived and their integration in time is discussed. Void growth results in a global softening of the stress-strain response of the material. A simple model for the prediction of void coalescence is proposed which can serve to predict the overall ductility of polycrystalline porous materials under the combined action of thermal dilatation and internal pressure in the voids

Model reduction for micromechanics of materials

International audienceUpscaling (also known as homogenization, or change of scales) is a common practice in multiscale problems when the scales are well separated. In linear elasticity the structure of the homogenized constitutive relations is strictly preserved in the change of scales. In other words, a composite made from linear elastic constituents behaves macroscopically as a linear elastic homogeneous solid whose effective properties can be computed once for all by solving a finite number of unit-cell problems. Unfortunately there is no exact scale-decoupling in multiscale nonlinear problems which would allow to solve only a few unit-cell problems once for all and then use them subsequently at the larger scale. The surviving coupling between the scales has led to the development of FEM2 methods which are accurate but come at a formidable cost. Another approach, involving a certain degree of approximation, is to investigate the response of representative volume elements along specific loading paths and to use them to calibrate macroscopic phenomenological models. Unfortunately most of the huge body of information generated by these micromechanical analyses is lost, or discarded, since very often only the overall response is used. In order to derive less arbitrary constitutive relations and to reduce the formidable cost of full-field simulations, mean-field theories based on approximations of the local fields have been proposed. The crudest approximations are contradicted by experimental evidence, but recent improvements accounting for instance for intraphase field fluctuations have been proposed. This is typically the case of the Non Uniform Transformation Field Analysis ([1][2]). A new version of this model will be proposed in this talk, with the aim of preserving the underlying variational structure of the constitutive relations (similar objective in [3] [4]).[1] J.C. Michel, P. Suquet, Int. J. Solids Structures 40, 6937-6955 (2003). [2] J.C. Michel, P. Suquet, in F. Aliabadi and U. Galvanetto (eds) Multiscale Modelling in Solid Mechanics -- Computational Approaches, Imperial College Press, 159-206 (2009). [3] F. Fritzen, M. Leuschner, Computer Methods in Applied Mechanics and Engineering 260, 143–154 (2013). [4] S. Lall, P. Krysl, J. E. Marsden, Physica D 184, 304–318 (2003)

Ductile damage of porous materials with two populations of voids

International audienceThis note presents an upper bound and an estimate for the yield yield function of a material with a rigid ideally plastic matrix and two scale cavities. The results are compared to numerical simulations. The laws of evolution of the two porosities are determined