Impact of the grain size distribution on the yield stress of heterogeneous materials

Abstract

When the mean grain size is larger than ∼100 nm, it is well accepted for different heterogeneous materials that the macroscopic yield stress follows the so-called Hall–Petch relation. However, in this classic formalism, only the mean grain size is considered in a semi-phenomenological way and the fact that the grains form a population of stochastic nature with different sizes and shapes is not stated. Moreover, an efficient homogenization procedure leading to the aforementioned grain size dependent behaviour from the individual properties of grains has not yet been reported in the literature mainly due to a lack of statistical description. In this paper, a recently developed self-consistent scheme making use of the “translated fields” technique for elastic–viscoplastic materials is used as micro–macro scale transition. The representative volume element is composed of grains supposed to be spherical and randomly distributed with a grain size distribution following a log-normal statistical function. The viscoplastic strain rate of the grains depends on their individual size. Intragranular plastic anisotropy and strain hardening are not considered in order to focus this work on grain size heterogeneities only. Numerical results with unimodal log-normal distributions firstly display that the overall yield stress depends not only on the mean grain size but also on the dispersion of the distribution. A decrease of the yield stress with an increase of the dispersion occurs and is more important when the mean grain size is on the order of the μm. Secondly, prediction of the evolution of the internal structure indicates an increase of the internal stresses when the dispersion is increased. Lastly, numerical results concerning bimodal grain size distributions, considered as mixtures of unimodal log-normal distributions, are discussed.

Introduction

The determination of the behaviour of heterogeneous materials with complex microstructures constitutes a challenge in the design of advanced materials and the modelling of their effective behaviour during their processing. Volume fractions, crystallographic and morphologic orientations but also sizes and shapes of the grains are constitutive elements of the microstructure of heterogeneous materials (metals, ceramic, intermetallics, etc.) which have to be tailored.

In the last decades, the Hall–Petch relation (Hall, 1951, Petch, 1953) linking the yield strength to the mean grain size has been found to match quantitatively the grain size effect on the behaviour of polycrystalline aggregates (metals, intermetallics, ceramics, etc.). Originally, this relation which claims that the tensile yield stress of several polycrystalline steels scaled with the inverse square root of their mean grain size has been attributed by Hall and Petch to the development of intragranular dislocation pileups having a slip length proportional to the mean grain size. Indeed, to justify their experimental results, they used the theoretical model of an equilibrated pileup of straight dislocations developed by Eshelby et al. (1951) to derive the aforementioned relation. Improvement of the pileup model was given later by Cottrell (1958). Other authors (Conrad, 1963, Thompson et al., 1973) have also derived the Hall–Petch behaviour by measuring through TEM observations dislocation density within grains and by determining the increase of the plastic flow stress with the square root of dislocation density. It has been also found that grain boundaries are common sources of dislocations (Li, 1963, Hirth, 1972, Sutton and Baluffi, 1995, Fu et al., 2001, Biner and Morris, 2003, Espinosa et al., 2005). A complete description of the different theoretical explanations considering grain size effect is reported in Meyers and Chawla (1984). Anyway, the hypothesis of pileups to explain the Hall–Petch behaviour for the yield stress with a classical −0.5 exponent (inverse square root of mean grain size dependence) is still widely used by metallurgists (Armstrong et al., 1962, Armstrong, 1970, Hansen and Ralph, 1982, Narutani and Takamura, 1991, Nieh and Wadsworth, 1991) except for very large grained materials for which a − 1 exponent (inverse mean grain size dependence) is rather found (Hirth, 1972, Dollar and Gorczyca, 1982, Zonghoa et al., 1995). Furthermore, it has been shown that the Hall–Petch behaviour continues to be valid until a very fine grain regime (on the order of 100 nm) following Masumura et al., 1998, Khan et al., 2000, Khan et al., 2006 who explored a lot of experimental data.

As pointed out by Weng (1983), the grain size dependent behaviour has been often expressed in terms of macroscopic stress, without accounting for the polycrystalline nature of the representative volume element (RVE). Indeed, plastic deformation is inhomogeneous due to heterogeneous behaviour related to plastic anisotropy (Schmid factors) and to the physical mechanisms responsible for grain size effects. Thus, Weng (1983) considered a Hall–Petch type equation with a single valued grain size at the scale of the slip systems and used the Berveiller–Zaoui’s model (Berveiller and Zaoui, 1979) to derive the homogenized behaviour of copper polycrystals which leaded also to a Hall–Petch type equation. More recently, non-local dislocation mechanics models using the concept of geometrically necessary dislocations (GNDs) have been developed (Acharya and Beaudoin, 2000, Cheong et al., 2005) and predict pretty well the strain-hardening rate dependence on grain size after the yield point making use of the evolution of GNDs (or equivalently the lattice incompatibility). Another attempt to capture grain size effects were developed by Forest et al. (2000) making use of a heterogeneous Cosserat medium. Again, only the mean grain size is considered in all these recent developments.

Since the distribution of grain size in heterogeneous materials provides heterogeneity, it appears fundamental to get an accurate description of the effect of grain size on the local interactions and behaviours, and also, a relevant mathematical description of the grain size statistics. As shown in Fig. 1 which deals with an electron back-scattering diffraction (EBSD) map of a polycrystalline sample of zirconium-α (LETAM, 2005), grains display significantly different sizes and the mean grain size is about 5.5 μm. Advanced homogenization procedures developed these last decades such that the self-consistent procedure did not focused on the effect of grain size distribution on the behaviour of polycrystalline materials. As a matter of fact, micromechanical models assuming spherical grains generally suffer from a lack of statistical description about grain diameters which are stochastic internal parameters of the microstructure inherent to the processing route (prior rolling, annealing, etc.).

Until now, contributions focused on the role of grain size distribution on mechanical behaviour of heterogeneous materials (and particularly their yield stress) are not numerous. Ghosh and Raj (1981) developed an analytical model to investigate the effect of varying simple bimodal and multimodal grain diameter distributions (triangular spreads around 2 or 3 discrete grain sizes assuming spherical grains) on creep of polycrystalline metals with a constrained uniform strain rate in all grains (Taylor, 1938). Kurzydlowski (1990) examined the role of some log-normal distributions of the grain volumes in polycrystals. He stated not only the importance of the mean grain size but also the width of the grain size range. However, only a few continuous distributions are used and the macroscopic plastic flow stress is directly assimilated to the mean stress using a simple averaging rule of mixture on the distributed grain volumes. A more recent one is due to Masumura et al. (1998) who accounted for a grain size distribution with the simultaneous presence of Hall–Petch strengthening mechanism for “large” grains and further mechanisms (like vacancy transport in the grain boundary region, i.e., the Coble creep mechanism (Coble, 1963)) operating for “very fine” grains (with sizes lower than a critical value on the order of several nm). From a quite simple weighted average over grain volumes in the distribution similar to Kurzydlowski, they found a good agreement with experimental data for several fine grained polycrystalline materials.

The objective of the present paper is to study, in a systematic statistical way, grain size effects on the macroscopic yield stress of heterogeneous materials assuming a given grain size distribution with higher moments than the mean grain size. Especially, the role of the grain size dispersion is underscored. To achieve this objective, the present modelling includes three combined pillars depicted in Part 2: (i) a given statistical grain size distribution (possibly multimodal) characterizing the heterogeneity of the aggregate associated with local grain size effect; (ii) some adequate constitutive relations at the scale of the grains including grain size as characteristic internal length scale parameter; (iii) a relevant scale transition scheme embodied here by a new self-consistent model based on the “translated fields” technique developed for heterogeneous elastic–viscoplastic materials. Thus, intragranular plastic anisotropy and strain hardening are voluntary not taken into account since correlations between grain size and lattice orientation (crystallographic texture) need to be quantified for a complete description of couplings between texture and grain size. Furthermore, the case of nano-grained materials is not treated here (we only consider grain sizes larger than 0.1 μm) since grain boundaries may then occupy a non-negligible volume fraction within the material and are willing to be regarded as a new phase (see e.g., Carsley et al., 1995, Kim et al., 2001, Jiang and Weng, 2004, Capolungo et al., 2005, Warner et al., 2006). Part 3 displays quantitative numerical results for common metals with grain sizes log-normally distributed (unimodal distributions) in terms of macroscopic yield stress, local plastic strains, second-order internal stresses and stored energy. Secondly, calculations have been also performed on bimodal grain size distributions to study more complex microstructures.