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Mechanical field fluctuations in polycrystals estimated by homogenization techniques. (English) Zbl 1104.74054
The authors consider an elastic material built with composites of Hashin-Shtrikman type. The purpose of the paper is to compute the mechanical fields inside the polycrystal using homogenization techniques. The material is supposed to be linear with a constitutive relation of the kind $$\varepsilon (x)=m(x):\sigma (x)+\varepsilon _{0}(x)$$, where $$m(x)$$ (resp. $$\varepsilon _{0}(x)$$) is the compliance (resp. the stress-free thermal strain field). The material being the aggregate of $$N$$ homogeneous phases, $$m$$ (resp. $$\varepsilon _{0}$$) is the superposition of local homogeneous fields $$m^{r}$$ (resp. $$\varepsilon _{0}^{r}$$), through characteristic functions $$\chi ^{r}$$. The material is submitted to a uniform stress field $$\Sigma$$ on its outer surface $$\partial \Omega$$. The local stress field is given through $$\sigma (x)=B(x):\Sigma +\sigma _{res}(x)$$, where $$B$$ (resp. $$\sigma _{res}$$) is the localization tensor (resp. the residual stress field coming from the incompatibility of the thermal strain field). The authors first quote from the book [M. Bornert (ed.), T. Bretheau (ed.) and P. Gilormini (ed.), Homogenization in material mechanics. II: Nonlinear behavior and open problems. Paris: Hermès Science Publications (2001; Zbl 1067.74502)] analytic expressions of the average of the fields over each phase, through the introduction of the inverse $$\widetilde{M}=E:M^{\ast }$$ of the constraint tensor introduced by R. Hill [J. Mech. Phys. Solids 13, 89-101 (1965; Zbl 0127.15302)]. They then prove that $$M^{\ast }=M$$, $$M$$ being the effective compliance, and compute the intra-phase fluctuations of these tensors. The authors then compute the derivatives of the Eshelby tensor $$S^{E}=P:M^{-1}$$, where $$P$$ is a microstructural tensor introduced by A.G. Khatchaturyan [Sov. Phys. Solid State 8, 2163-2168]. This tensor $$P$$ and its derivatives have to be computed through a numerical scheme designed for the computation of integrals and for the resolution of an implicit equation. In the main part of the paper, the authors specialize these computations in the cases of a linear viscous or of a viscoplastic face-centred cubic polycrystal, with a large number of phase ($$n=30$$). They present numerical computations for these fields, which proves that the methodology gives better results than the ones which have been previously obtained. In the linear case, they indeed prove that local slip rates can be estimated with this methodology, even if the Schmid factor vanishes. In the viscoplastic case, they compare the secant linearization and the affine approach, with the previously obtained results, on the phase-average slip rate.

##### MSC:
 74Q05 Homogenization in equilibrium problems of solid mechanics 74E15 Crystalline structure 74E30 Composite and mixture properties
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