The effect of the rate sensitivity

In this subsection, we present results for HCP polycrystals with a fixed value of the grain anisotropy, M1 =10 and M2 → ∞, but with different rate sensitivities m=1/n. Moreover, the c/_a ratio of the HCP crystal is taken to be 1_.633. Due to the ab-

sence of the pyramidal slips (τC → ∞), only four linearly independent slip systems are

available for each grain. Thus, strong interactions between grains with different orien- tations are expected to occur in order to accommodate general plastic deformations. Predicting accurately the macroscopic response of these materials is a challenging problem that may serve to discriminate among different nonlinear homogenization approaches. Fig. 5.1(a) shows plots for the FOSO estimates of the effective flow stress ̃σ0, normalized by the reference flow stress τ_B, as a function of the rate sen-

sitivity m =1/_n. The corresponding VAR upper bounds of the SC type (Nebozhyn et al., 2001), POSO estimates of the SC type (Liu and Ponte Casta˜neda, 2004b), tangent estimates (Lebensohn et al., 1993), as well as the Reuss lower bounds are also included for comparison. Note that the Taylor upper bounds tend to infinity in this case, due to the lack of five linearly independent slip systems. Note further that two different versions of the POSO estimates are presented: (i) the energy (E) version derived from expression (5.24) in solid lines, and (ii) the constitutive (C) version derived from expression (5.25) in dashed lines. We observe from Fig. 5.1(a)

0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 m HCP ~ σ τB0 POSO(E) POSO(C) VAR FOSO Reuss 1 2 / 10 / B A C A M M τ τ τ τ = = = = ∞ Tangent (a) 1 1.5 2 2.5 0 0.2 0.4 _m 0.6 0.8 1 HCP POSO FOSO VAR FOSO VAR ( e) e SDσ σ ( e) e SD D D 1 2 / 10 / B A C A M M τ τ τ τ = = = = ∞ (b)

Figure 5.1: The effective flow stress and field fluctuations for untextured, HCP poly- crystals with isotropic two-point statistics (w1 =w2 =1), and with contrast param- eters M1 = τB/τA = 10 and M2 = τC/τA → ∞, for uniaxial tension, as functions of

the rate sensitivity m=1/_n. Plots are shown for (a) the effective flow stress normal-

ized by the reference flow stress τB, and (b) the overall standard deviations of the

equivalent stress SD(_σe), and of the equivalent strain rate SD(_De), normalized by

the macroscopic equivalent stress σe, and the macroscopic equivalent strain rateDe,

respectively.

(m=1), they deviate from each other for smaller values of_m, with the tangent esti-

mates tending to the Reuss lower bounds as m →0. On the other hand, the FOSO estimates, as well as both versions of the POSO estimates, lie within the VAR upper bounds and Reuss lower bounds for all values of m. In particular, both the FOSO and POSO estimates decrease monotonically with decreasing values of m, with the FOSO results lying somewhat above the POSO results, while the opposite is true for the VAR bounds, which increase monotonically with decreasing values of m.

Figure5.1(b)displays the corresponding plots for the FOSO estimates of the over- all standard deviations of the equivalent stress and equivalent strain rate, SD(_σe)and

SD(D_e), normalized by the macroscopic equivalent stress σ_e and strain rate D_e, re-

spectively. The corresponding VAR and POSO results are also shown for comparison. It can be seen from this figure that, for both the stress and strain rate fluctuations, the FOSO and VAR estimates stay fairly close to each other for all values of m, where the stress fluctuations increase monotonically with decreasing values of m,

while the corresponding strain rate fluctuations remain almost a constant. (Note that the similarity of the FOSO and VAR estimates for the overall field fluctuations is coincidental and not a general result, as will be seen below.) By contrast, the corresponding POSO estimates exhibit a qualitatively different behavior, where as

m decreases the stress fluctuation varies non-monotonically, while the correspond- ing strain rate fluctuation increases significantly, exceeding the stress fluctuations for

m ≲ 0.6. The above observation would first appear to be in contradiction with the observations made in Fig. 5.1(a), where the FOSO estimates stay closer to the POSO estimates than to the VAR estimates. This may be explained in terms of the fact that the POSO estimates shown in Fig. 5.1(b) (and all the POSO estimates below) make direct use of the field fluctuations in the LCC to estimate the corresponding quantities in the actual nonlinear polycrystals, although the field fluctuations in the nonlinear composite and in the LCC are known (Idiart and Ponte Casta˜neda,2007c) to be different for the POSO estimates, due to the lack of full stationarity. In fact, this reference suggests that additional terms involving difficult-to-compute numerical derivatives are required—in addition to the field fluctuations in the LCC—to recover the field fluctuations in the actual nonlinear composite. However, this requires rather heavy numerical computations and is not pursued here for simplicity. On the other hand, both the FOSO and VAR estimates are fully stationary, so that the field fluctu- ations in the LCC are entirely consistent with those in the nonlinear polycrystals. As already mentioned, this is a remarkable advantage of the new FOSO estimates, which can yield as byproducts estimates of the field fluctuations in the actual nonlinear composites, without additional computational cost.