PSI - Issue 71

Akash S.S. et al. / Procedia Structural Integrity 71 (2025) 180–187

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(1) This material, which appears homogeneous macroscopically, is characterized using macro-stress and macro-strain, which are obtained by averaging the stress and strain tensors over the volume of the SVE as expressed in eqn.1. σ The equivalence between the actual heterogeneous composite medium and the homogeneous medium represented by the average stresses and strains and the e ff ective elastic constants needs to be examined. For this purpose, the SVEis subjected to appropriate boundary tractions , or boundary displacements that would produce uniform stresses ( ̅̅̅̅ ) and strains ( ϵ ̅̅̅̅ )) in a homogeneous medium. The total strain energy U stored within a volume V of the e ff ective medium is: U = 1 2 σ ij̅̅̅̅ ϵ ij̅̅̅ V The strain energy U’ stored in the heterogeneous SVE of volume V is: ′ = 1 2 ∫ = 1 2 ∫ ( − ̅̅̅ + ̅̅̅ ) (2) ̅̅̅̅ = 1 ∫σ ( , , ) ϵ ̅̅̅ = 1 ∫ϵ ( , , ) =0 ⇒ ′ − = 1 2 ∫ [ ( − ̅ )] Using Gauss theorem, the volume integral in eqn.5 can be converted into a surface integral. Thus ′ − = 1 2 ∫ ( − ̅ ) where n is the unit outward normal and S is the surface. On the surface: = ̅ ⇒ ′ − =0 (7) The spatially-averaged stress and strain quantities defined in equations 1 and 2 thus ensure equivalence in strain energy between the equivalent homogeneous material and the original heterogeneous material. So, they can be used for calculating the effective material properties of the SVE using the following equations. (5) (6) (3) (4) = 1 2 ∫ ( − ̅̅̅ ) + 12 ̅̅̅ ∫ = 1 2 ∫ ( − ̅ ) + 1 2 ̅̅̅̅ ̅̅̅̅ Subtracting eqn.2 from eqn.3 yields: ′ − = 1 2 ∫ The equilibrium equation implies that ( − ̅̅̅ )