PSI - Issue 8

Giorgio De Pasquale et al. / Procedia Structural Integrity 8 (2018) 75–82 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

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1 = 2 ; 2 = ; 3 = 2

(2.1)

(a)

(b)

Fig. 3. a) Octet-truss RVE, b) RVE geometry and dimensions.

The main concept at the basis of this homogenization scheme is the equivalence between the strain energy of the lattice RVE and that of the equivalent homogeneous material replacing it. Because of this assumption, it is possible to estimate the components of the stiffness matrix [C] of the equivalent homogeneous material at the upper scale. Furthermore, taking into account the periodical nature of the lattice a set of prescribed periodic boundary conditions (PBCs) must be imposed to the RVE. At each time, these PBCs are applied in order to obtain only one component of the strain field different from zero: this strain field represents on the one hand the “exact solution” for the homogeneous medium and on the other hand the average strain field for the RVE of the lattice structure.

̅̅̅ = 1 ∫ = 0 , ℎ , = 1, … , 3 .

(2.2)

The constitutive law for the equivalent homogenous material (at the macroscopic scale) can be defined as (Voigt’s notation) :

{ ̅} = [C]{ ̅} .

In Eq. (2.2) { ̅} and { ̅} are the stresses and strains of the equivalent homogeneous continuum, respectively. The stiffness matrix components are determined, column by column, by solving six linear static analyses. In each static analysis the PBCs are imposed in such a way that only one component of the average strain field of the RVE is non null and equal to the imposed (arbitrary) strain. The six linear static analyses are implemented in ANSYS® environment and the PBCs are imposed automatically between couples of opposite nodes (belonging to opposite faces of the RVE, respectively). The set of PBCs for each analysis has been parametrized and coded within a dedicated macro (implemented according to the ANSYS parametric design language). Finally, the stiffness matrix components [C] are estimated (Eq. 2.4): In Eq. (2.4) i corresponds to the actual static case ( = 1, … ,6) . The elastic properties of the material can be derived from the material's compliance matrix [S] as follows (here below the example of an orthotropic material is shown): (2.3) C αi = ̅̅̅ = 1 ∫ ( 1 , 2 , 3 ) ℎ = 1, … , 6 (2.4)