Pierre Gilormini

Variational self-consistent estimates for cubic viscoplastic polycrystals: the effects of grain anisotropy and shape

Abstract A fundamental problem in mechanics of materials is the computation of the macroscopic response of polycrystalline aggregates from the properties of their constituent single-crystal grains and the microstructure . In this work, the nonlinear homogenization method of deBotton and Ponte Castaneda (1995) [deBotton, G., Ponte Castaneda, P., 1995. Variational estimates for the creep behavior of polycrystals. Proc. R. Soc. Lond. A 448, 121–142] is used to compute “variational” self-consistent estimates for the effective behavior of various types of cubic viscoplastic polycrystals. In contrast with earlier estimates of the self-consistent type, such as those arising from the “incremental” and “tangent” schemes, the new estimates are found to satisfy all known bounds, even in the strongly nonlinear, rate-insensitive limit, and to exhibit a more realistic scaling law at large grain anisotropy. Also, unlike the Taylor and Reuss estimates, they are able to account for grain shape in a rigorous statistical sense. For these reasons, they can be shown to be significantly more accurate than earlier estimates. Thus, for example, the new self-consistent estimates can be less than half the corresponding Taylor estimates for ionic polycrystals with highly anisotropic “flat” grains.

Variational self-consistent estimates for texture evolution in viscoplastic polycrystals

Abstract The variational self-consistent (V-SC) method is presented for simulating texture evolution in viscoplastic polycrystals undergoing large deformation. The theory is based on a recent non-linear homogenization procedure which makes use of estimates of the self-consistent type for the instantaneous response of a suitably chosen “linear comparison polycrystal” to generate corresponding estimates for the non-linear viscoplastic polycrystal. Making use of consistent estimates for the average strain rate and spin in the grains of the polycrystal, and of standard kinematical arguments, evolution equations are derived for the crystallographic and morphological textures. Applications to titanium polycrystals under uniaxial tension, compression and plane-strain compression performed at 750 °C are presented. The texture predictions and macroscopic stress–strain responses are compared with those given by earlier models, and with finite element (FEM) simulations and experimental measurements taken from the literature. Better overall agreement with FEM simulations is obtained for the V-SC estimate than for the other models. However, the comparison with the experimental results is not as good, especially for plane-strain compression, where additional effects due to friction cannot be ruled out.

Accurate estimates for the creep behavior of hexagonal polycrystals

Abstract A recently proposed “variational” self-consistent model is used to estimate the macroscopic response of various types of HCP viscoplastic polycrystals, including titanium, ice, and zirconium. Unlike the “incremental” and “tangent” models, which lead to predictions that tend to the Taylor upper and Reuss lower bounds, respectively, in the strongly nonlinear, rate-insensitive limit, the new model leads to estimates that tend to neither bound and which are somewhat intermediate between the two. The new estimates are quite possibly the most accurate to date, especially for polycrystals with highly anisotropic grains. In addition, the new self-consistent model is capable of handling the commonly occurring situation of different rate-sensitivity exponents for different slip families. Sample calculations for zirconium show a clear transition—from the high to the low rate-sensitivity exponent—in the response curve for the polycrystal, but only if at least four independent systems are available for each slip family.

Variational self-consistent estimates for viscoplastic polycrystals with highly anisotropic grains

Abstract A recently established variational procedure is used to compute self-consistent estimates for viscoplastic polycrystals with highly anisotropic ionic and HCP grains. For large values of the grain anisotropy M , the predictions for the macroscopic flow stress are found to scale as M γ , where γ depends on the number (less than 5) of independent slip systems that are available for plastic flow, but not on the strain-rate sensitivity. The predictions of other nonlinear extensions of the self-consistent scheme are inconsistent with this scaling law, suggesting that they may be less accurate than the variational estimates proposed here.

A modified affine theory for the overall properties of nonlinear composites

Abstract The expressions of the intraphase second moments of stress and strain in heterogeneous linear thermoelastic materials are derived; they are used to quantify the intraphase heterogeneity predicted by the ‘classical affine’ formulation for the homogenization of nonlinear composites. By analogy with the ‘modified secant’ approach, a ‘modified affine’ formulation that accounts for intraphase heterogeneity is proposed.

Finite element solution of the problem of a spherical inhomogeneity in an infinite power-law viscous matrix

Abstract The problem of a spherical inclusion surrounded by an unbounded matrix loaded at infinity plays a prominent role in the mechanics of materials, and is addressed in this paper for power-law viscous materials via the finite element method. With respect to a previous analysis, the results of which are further developed, the influence of the necessarily finite size of the mesh is carefully analyzed (by using two types of boundary conditions, among other tests) and two finite element codes are compared. In addition to the localization rules for the strain rate and stress deviator in the inclusion, the analysis also covers the influence of a dilute concentration of deformable spheres in a power-law matrix, which generalizes previous studies on cavities or rigid inclusions. All the results are fitted by curves, and this is expected to assist in further practical applications.

A similarity between the classical and modified secant extensions of the self-consistent model

Abstract The first and second moments of the strain (or stress) fields predicted by the classical and modified secant extensions of the self-consistent model in isotropic mixtures of isotropic incompressible power-law phases are shown to be identical if the same macroscopic strain (or stress) is applied. This proves directly that the modified extension does predict a softer overall behavior for such composites.

Realizable compressibility and conductivity in isotropic two-phase composites

Abstract It is shown in a very simple manner how any bulk modulus and any conductivity of isotropic two-phase composites located between the Hashin and Shtrikman bounds can be realized exactly by suitable microstructures. The latter are assemblages of two types of composite spheres.

Inclusion élastoplastique en surface

Resume A laide des elements finis, nous avons etudie lecoulement cyclique dun grain elastoplastique en surface favorablement oriente, en monoglissement, pour des aciers ferritiques. Nous avons ainsi pu expliquer partiellement certaines observations classiques comme laugmentation des deformations plastiques ou la limitation du nombre de systemes de glissement actives. Toutefois, ces calculs qui ne prennent pas en compte un gradient de proprietes au voisinage de la surface semblent sous-estimer leffet observe experimentalement.

Homogenization-Based Predictions for Texture Evolution in Halite

The “variational” homogenization method developed by deBotton and Ponte Castaneda [2] is used here to predict texture development in halite polycrystals at room and high temperatures accounting for hardening and grain shape changes. The new predictions are compared with those of the Taylor and “tangent” model of Molinari et al. [5] for uniaxial tension and compression. The predictions of the “variational” model are found to be intermediate between the Taylor and “tangent” predictions, although not too different from either, as a consequence of the relatively high isotropy of the halite single crystal grains.