Mechanical field fluctuations in polycrystals estimated by homogenization techniques

Abstract

The fluctuation of mechanical fields arising in polycrystals is investigated. These materials are viewed as composites of the Hashin-Shtrikman type with a large number of anisotropic phases and a 'granular' topology. We show that the estimation of the intra-phase stress and strain (rate) second moments comes down to the resolution of a linear system of equations. Applied to a linear viscous face-centred cubic (FCC) polycrystal, it is observed that significant local slip rates are estimated even when the corresponding Schmid factor vanishes, due to the intergranular interactions. For the application to viscoplastic polycrystals, the secant and affine nonlinear extensions of the self-consistent scheme are compared. At large stress sensitivity (n = 30), it is observed that the secant linearization leads to almost uniform slip rates for all slip systems in every phase, whereas the affine approach predicts a much larger spread. Furthermore, there is no one-to-one relation between the phase-average stress (or strain rate) and the corresponding second moment. It is emphasized that intra-phase strain-rate heterogeneities should be accounted for when dealing with microstructure evolution.

An enhanced affine formulation and the corresponding numerical algorithms for the mean-field homogenization of elasto-viscoplastic composites

“…One well known limitation of such modeling is the absence of non linear response under hydrostatic stress (see Weng, 1992, 1993 for a discussion on this issue in elasto-plasticity). The definition of better reference moduli is still an active research subject in the literature, an interesting approach being the use of the second-order moment of strain in order to better take into account the non-uniformity of the strain field (see Hu (1996), Brenner et al (2004), González et al (2004) and Ponte Castañ eda (2002) for a variational approach).…”

A second-moment incremental formulation for the mean-field homogenization of elasto-plastic composites

“…C el r can then be computed analytically. Otherwise, if the inclusions are not spheroids or E is computed with an anisotropic matrix's modulus tensor, then the partial derivatives can be approximated numerically, as in Brenner et al (2004).…”

Elastoviscoplastic micromechanical modeling of the transient creep of ice

“…The SC scheme is therefore especially appropriate for polycrystals. The expression of the stress localization tensor in a general context of anisotropy is given, e.g., by Brenner et al [2004].…”

“…It must be made clear, however, that long-range interactions associated with spatial correlations between dislocation avalanches are not considered owing to the simplicity of the adopted local constitutive relation (5). For purely elastic and purely viscoplastic regime, the standard deviations (8) can be calculated by an appropriate differentiation of the corresponding effective potential [Brenner et al, 2004].…”