PSI - Issue 5

Danuta Miedzińska et al. / Procedia Structural Integrity 5 (2017) 484– 491 Danuta Miedzińska / Structural Integrity Procedia 00 (2017) 000 – 000

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Fig. 1. Characteristic dimensions of RVE.

The accurate mathematical condition which has to fulfilled to decide that the material segment is representative for total volume was given by Hill (1963) . The Hill’s condition can be derived as f ollows. In mechanics the effective elastic constants can be defined as a proportionality factors of volume averaged stress and strain tensors: 〈 〉 = 〈 〉 (2) The effective elastic constants can be also defined as a proportionality factors of volume averaged strain energy and strain tensors: 〈 〉 = 1 2 〈 〉〈 〉 = 1 2 〈 〉〈 〉 (3) But 〈 〉 is defined as volume averaged strain local energy, which can be assumed as: 〈 〉 ≡ 1 2 〈 〉 = 1 2 〈 〉〈 〉 + 1 2 〈 ′ 〉〈 ′ 〉 (4) where ′ i ′ are the fluctuations of averaged stress and strain appropriately. Putting (2) and (3) to (4) we get: ( − )〈 〉〈 〉 = 〈 ′ 〉〈 ′ 〉 (5) On the base of Equation (5) it can be noticed that effective material constants defined in mechanical and energetic way can be similar only when fluctuations of averaged stress and strain disappear: 〈 ′ 〉〈 ′ 〉 = 0 (6) On the base of (4) this condition can be written as: 〈 〉 = 〈 〉〈 〉 (7) To achieve the Hill’s condition the external stress acting on RVE cannot depend on microstructure. In the other words, the RVE should have such dimension that its effective elastic constants do not depend on homogenous boundary conditions. Many real materials do not fulfill the Hill’s condition (e.g. some composites or fractured materials) But for materials with a great number of microcracks the application of mechanical (2) or energetic (3) way for material properties description allows to use the Hill’s condition. In such cases the damaged material is treated as homogenous one, and the influence of cracks is consider by damage parameter usage. In the presented study the numerical analyses using Finite Element Method were applied for selection of RVE for raster 2D and 3D models with pores random distribution.