# (0, 1, 2, 4) Interpolation by G -splines

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**Example text**

Hill’s condition (Hill [[79]], 1952) dictates the size requirements on the RVE. The classical argument is as follows. For any perfectly bonded heterogeneous body, in the absence of body forces, two physically important loading states satisfy Hill’s condition. 2) (2) pure tractions in the form: t|∂Ω = L · n ⇒ σ where E and L are constant strain and stress tensors, respectively. Clearly, for Hill’s conditions to be satisﬁed within a macroscopic body under nonuniform external loading, the sample must be large enough to possess small boundary ﬁeld ﬂuctuations relative to its size.

E. to make the material data reliable, the sample must be small enough that it can be considered as a material point with respect to the size of the domain under analysis, but large enough to be a statistically representative sample of the microstructure, see also Fig. 3. 1 Testing procedures 47 If the eﬀective response is assumed isotropic then only one test loading (instead of usually six), containing non-zero dilatational ( tr3σ and tr3 ) and def def deviatoric components (σ = σ − trσ I and = − tr I), is necessary to de3 3 termine the eﬀective bulk and shear moduli: def 3κ∗ = tr σ 3 Ω tr 3 Ω and def 2µ∗ = σ Ω Ω : σ : Ω Ω .

Therefore, once either C or IE∗ are known, the other can be determined. In the case of isotropy we may write def Cκ = 1 κ2 κ∗ − κ1 v2 κ∗ κ2 − κ1 and def Cµ = 1 µ2 µ∗ − µ1 . 22) Clearly, the microstress ﬁelds are minimally distorted when C κ = C µ = 1. Remark: There has been no approximation yet. The “burden” in the computations has shifted to the determination of C. Classical methods approximate C. For example, the simplest approximation is C = I, which is the Voigt approximation, IE∗ = (IE1 + v2 (IE2 − IE1 ) : I) or for the Reuss approximation ˆ = I.