Accounting for a Distribution of Morphologies and Orientations on Stresses Analysis by X-Ray and Neutron Diffraction: Normalized Self-Consistent Modeling

Abstract

Abstract. The historical Eshelby-Kröner self-consistent model is only valid in the case when grains can be assumed similar to ellipsoids aligned preferentially along a same direction into the polycrystal. In this work, distributions of crystallites morphologies and geometrical orientations were accounted for, owing to the so-called normalized self-consistent model, in order to satisfy Hill's averages principles. Different nonlinear εϕψ-vs.-sin 2 ψ distributions were predicted in elasticity, even in the absence of crystallographic texture, in the case when several morphologies and geometrical orientations coexist within the same polycrystal. IntroductionPolycrystals with a morphological texture have macroscopic anisotropic properties in the absence of crystallographic texture. Many studies have been conducted to characterize the mechanical properties of such materials. The classical Eshelby-Kröner self-consistent model enables to take into account non-spherical grains when certain conditions are fulfilled [1][2][3]. According to these restrictions, only an ideal morphological texture can be taken into account through (identical) ellipsoidal inclusions whose principal axes are aligned along specific directions in the specimen. Such a microstructure is not generally observed in a real specimen; several morphologies or several grains with different morphological orientations could coexist within the same crystalline material. It is the case, for example, of titanium alloy Ti-17 polycrystal which consists of needle-shaped α crystallites mixed to slightly equiaxed prior β-grains [4]. Another example could be chosen among composite materials, made of epoxy resin and carbon-epoxy reinforcing strips, exhibiting an inplane distribution on the morphologies [5], the microstructure of which was successfully modeled through disc-shaped inclusions.It was recently numerically shown [5] that in such a situation, the Eshelby-Kröner self-consistent model does not simultaneously fulfil anymore both the so-called "Hill's average relations over the mechanical states", historically established in [6]. Therefore, this theoretical approach fails to represent a morphological texture featuring various grain shapes or a relative disorientation of the morphologies coexisting. The main interest of the present contribution is to show the possible influence of this morphological texture in the context of stress analysis by diffraction, in elasticity, using a normalized elastic self-consistent model. Then, we will be interested in the εϕψ-vs.-sin 2 ψ diagrams. If transversely isotropic grains are randomly oriented, the effective property of the polycrystal will be approximately isotropic, with values depending on the volume fraction and aspect ratio of the inclusions [7]. A random morphologic texture can then be neglected without consequence, within the context of modelling the multi-scale behaviour of heterogeneous materials consisting of quasi-isotropic elementary inclusions. This was demonstrated, at least numerically in [4]. Therefore, spec...

On the choice of homogenization method to achieve effective mechanical properties of composites reinforced by ellipsoidal and spherical particles

“…To overcome all these several theoretical difficulties, Li [39] proposed an effective-medium-field micromechanics approximation using normalized strain localization tensor. Under this approximation, the strain localization tensor can be written, in the inclusion's frame of reference Rl attached to its main axes, as [39,40]:…”

“…The effective material properties obtained using the numerical FFTb homogenization techniques were compared with three different analytical methods: the normalized self-consistent scheme (NSC) [39,40], the Mori-Tanaka model (MT) [16] and Lielens' model [18]. The influence of the inclusions morphology on the accuracy of these homogenization techniques to predict the mechanical properties of reinforced composites was investigated.…”

On the Influence of the Inclusions’ Morphology on the Accuracy of the Prediction of Reinforced Composites Mechanical Behaviour

“…The strainlocalization tensor A i in this relation differs from one model to another. In the normalized selfconsistent [3,4], Mori-Tanaka [5] and Lielens' [6] models, the strain-localization tensors are expressed respectively as follows:…”

“…In this work, the effective material properties obtained using the numerical FFTb homogenization techniques were compared to three different analytical methods: the normalized selfconsistent scheme (NSC) [3,4], the Mori-Tanaka model (MT) [5] and Lielens' model [6]. The aim of this paper is to exhibit the influence of the inclusions morphology on the accuracy of these homogenization techniques to predict the mechanical properties of reinforced composites consisting not only for spherical or ellipsoidal inclusions but also for a mixture of spherical and ellipsoidal inclusions.…”