Corrigendum to “Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites” [Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 197–212]

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Corrigendum to “Combining Galerkin approximation

techniques with the principle of Hashin and Shtrikman

to derive a new FFT-based numerical method for the

homogenization of composites” [Comput. Methods

Appl. Mech. Engrg. 217–220 (2012) 197–212]

Sébastien Brisard, Luc Dormieux

To cite this version:

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Corrigendum to “Combining Galerkin approximation techniques with the principle of

Hashin and Shtrikman to derive a new FFT-based numerical method for the

homogenization of composites” [Comput. Methods Appl. Mech. Engrg. 217-220 (2012)

197–212]

S. Brisarda,∗_{, L. Dormieux}a

a_{Universit´e Paris-Est, Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTAR, F-77455 Marne-la-Vall´ee}

Assumption 1 in our original paper can be replaced with the following, less stringent assumption.

Assumption 1. There existsλ > 0 such that at any point x ∈ Ω, the eigenvalues of[C(x) − C0] are greater than λ in absolute

value.

Proof of Theorem 4 requires that the local stiffness be bounded from below and above. Therefore, Assumption 2 must be altered as follows

Assumption 2. There existsκmax> κmin> 0 and µmax> µmin >

0 such that at any point x ∈Ω

κmin≤κ(x) ≤ κmax, µmin≤µ(x) ≤ µmax.

Then, the end of the proof of Theorem 4 (starting from “Taking advantage of the isotropy”) must be modified as follows. Proof of Theorem 4. [. . . ] Taking advantage of the isotropy of both local and reference materials, the above volume averages can be expanded |Ω| τ+: (C − C0)−1: τ+= Z κ(x)>κ0 kτhyd_(x)k2 d[κ(x) − κ0] dΩ +Z µ(x)>µ0 kτdev_(x)k2 2µ(x) − µ 0 d Ω, and |Ω| τ−_{: S} 0: (S − S0)−1: S0: τ−= Z κ(x)<κ0 κ(x) kτhyd_(x)k2 dκ0[κ0−κ(x)] dΩ +Z µ(x)<µ0 µ(x) kτdev_(x)k2 2µ0µ0−µ(x) dΩ. From Assumption 2, we first have

|Ω| τ+: (C − C0)−1: τ+≥ R κ(x)>κ0 kτhyd_(x)k2_d_Ω d(κmax−κ0) + R µ(x)>µ0kτ dev_(x)k2_d_Ω 2 (µmax−µ0) , ∗ Corresponding author.

Email addresses: [email protected](S. Brisard), [email protected](L. Dormieux) then |Ω| τ−_{: S} 0: (S − S0)−1: S0: τ−≥ κmin R κ(x)<κ0kτ hyd_(x)k2_d_Ω dκ0(κ0−κmin) +µmin R µ(x)<µ0kτ dev_(x)k2_d_Ω 2µ0(µ0−µmin) . from which the following bound results

a(τ, $) ≥ α |Ω| Z Ω h kτhyd (x)k2+ kτdev(x)k2idΩ = αkτk2_V, (49) with1 α = minn [d (κmax−κ0)]−1, 2 (µmax−µ0) −1

κmin[dκ0(κ0−κmin)]−1, µmin2µ0(µ0−µmin) −1_o.

The proof of the first statement is complete, since k$k_V= kτk_V,

and α > 0.

1_{In this definition of α, it is assumed that κ}

min< κ0< κmaxand µmin< µ0<

µmax. The proof remains valid if any of these inequalities are not verified. For

example, if µmin≥µ0, then, from Assumption 2, µ(x) ≥ µ0at any point x ∈Ω.

In other words, the integral over the set of points x ∈Ω such that µ(x) < µ0is

null. Then Eq. (49) still holds, provided that α is defined as follows α = minn

[d (κmax−κ0)]−1, 2 (µmax−µ0)−1, κmin[dκ0(κ0−κmin)]−1o.