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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ScienceDirect Comput. Methods Appl. Mech. Engrg. 283 (2015) 1587–1588 www.elsevier.com/locate/cma

Corrigendum

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. Brisard ∗ , L. Dormieux Universit´e Paris-Est, Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTAR, F-77455 Marne-la-Vall´ee, France Available online 7 November 2014

Assumption 1 in our original paper can be replaced with the following, less stringent assumption. Assumption 1. There exists λ > 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 exist κmax > 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   ∥τ hyd (x)∥2 ∥τ dev (x)∥2 −1 + + |Ω | τ : (C − C0 ) : τ = dΩ + d Ω, κ(x)>κ0 d [κ(x) − κ0 ] µ(x)>µ0 2 [µ(x) − µ0 ] and |Ω | τ − : S0 : (S − S0 )−1 : S0 : τ − =

 κ(x)<κ0

κ(x) ∥τ hyd (x)∥2 dΩ + dκ0 [κ0 − κ(x)]

 µ(x)<µ0

DOI of original article: http://dx.doi.org/10.1016/j.cma.2012.01.003. ∗ Corresponding author.

E-mail addresses: [email protected] (S. Brisard), [email protected] (L. Dormieux). http://dx.doi.org/10.1016/j.cma.2014.10.021 c 2014 Elsevier B.V. All rights reserved. 0045-7825/⃝

µ(x) ∥τ dev (x)∥2 d Ω. 2µ0 [µ0 − µ(x)]

1588

S. Brisard, L. Dormieux / Comput. Methods Appl. Mech. Engrg. 283 (2015) 1587–1588

From Assumption 2, we first have  |Ω | τ + : (C − C0 )

−1

: τ+ ≥

κ(x)>κ0

∥τ hyd (x)∥2 d Ω

d (κmax − κ0 )

 +

µ(x)>µ0

∥τ dev (x)∥2 d Ω

2 (µmax − µ0 )

,

then |Ω | τ − : S0 : (S − S0 )

−1

: S0 : τ − ≥

κmin



κ(x)<κ0

∥τ hyd (x)∥2 d Ω

dκ0 (κ0 − κmin )

+

µmin



µ(x)<µ0

∥τ dev (x)∥2 d Ω

2µ0 (µ0 − µmin )

from which the following bound results    α ∥τ hyd (x)∥2 + ∥τ dev (x)∥2 d Ω = α∥τ ∥2V , a (τ , ϖ ) ≥ |Ω | Ω

(49)

with1   α = min [d (κmax − κ0 )]−1 , [2 (µmax − µ0 )]−1 , κmin [dκ0 (κ0 − κmin )]−1 , µmin [2µ0 (µ0 − µmin )]−1 . The proof of the first statement is complete, since ∥ϖ ∥V = ∥τ ∥V , and α > 0.



1 In this definition of α, it is assumed that κ min < κ0 < κmax and µmin < µ0 < µmax . The proof remains valid if any of these inequalities is not verified. For example, if µmin ≥ µ0 , then, from Assumption 2, µ(x) ≥ µ0 at any point x ∈ Ω . In other words, the integral over the set of points x ∈ Ω such that µ(x) < µ0 is null. Then Eq. (49) still holds, provided that α is defined as follows  −1  −1  −1  α = min d (κmax − κ0 ) , 2 (µmax − µ0 ) , κmin dκ0 (κ0 − κmin ) .