PSI - Issue 13

Marco Schmidt et al. / Procedia Structural Integrity 13 (2018) 91–96 M. Schmidt et al. / Structural Integrity Procedia 00 (2018) 000–000

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have been proposed to study shear mechanisms in critical regions and thus achieving stress triaxialities around zero, see Bao and Wierzbicki (2004), Bru¨nig et al. (2008) and Driemeier et al. (2010). Unfortunately this type of specimens tends to rotations within the shear region leading to rather tension dominated stress states. Consequently, biaxial test specimens were developed, (Makinde et al. (1992), Kuwabara (2007), Gerke et al. (2017)) to cover a wide range of stress triaxialities with one specimen geometry. Due to the geometry of the test specimens shear tension as well as shear compression loads should be possible within a biaxial test setup. The presented paper outlines the basic concept of a phenomenological anisotropic damage and failure model. Biaxial experiments and corresponding numerical simulations reflect the influence of negative stress triaxialities on the damage behavior. The experimental evaluation is carried out by digital image correlation for the analysis of the strain fields and by scanning electron microscope for the investigations of the fracture surface. In conclusion, a cut-o ff value for negative stress triaxialities below which no damage occurs is discussed, see also Bru¨nig et al. (2018). The anisotropic damage and failure model for ductile metals is based on the additive decomposition of the strain rate tensor into an elastic, a plastic and a damage part (Bru¨nig (2003)). For this purpose, damaged and fictitious undamaged configurations are introduced, which are coupled by damage tensors. The plastic material behavior is characterized by a stress-state-dependent yield condition, which indicates the onset of plastic flow, and by a stress-state-dependent flow rule. A similar approach is used to model the damage behavior. The damage condition f da = α I 1 + β J 2 − σ = 0 (1) characterizes onset of damage. It is formulated in terms of the first and second deviatoric invariant of the Kirchho ff stress tensor, I 1 and J 2 , and the damage threshold σ . The parameters α and β in Eq. (1) depend on the stress triaxiality (2) where σ m = I 1 / 3 means the hydrostatic stress and σ eq = √ 3 J 2 represents the von Mises equivalent stress as well as on the Lode parameter with the principal stress components T 1 ≥ T 2 ≥ T 3 . The parameters α and β are based on numerical calculations of di ff erent single-pore models (Bru¨nig et al. (2013)) as well as on results of biaxial experiments and associated numerical simulations (Bru¨nig et al. (2016)). In particular, the parameter α is taken to be α ( η ) = − 0 . 15 for η cut < η < 0 0 . 33 for η > 0 , (4) where η cut is the cut-o ff value for negative stress triaxialities below which no damage occurs. The parameter β ( η, ω ) = β 0 ( η, ω = 0) + β ω ( ω ) ≥ 0 (5) with β 0 ( η ) = − 1 . 28 η + 0 . 85 (6) and β ω ( ω ) = − 0 . 017 ω 3 − 0 . 065 ω 2 − 0 . 078 ω (7) is a non-negative function depending on the stress triaxiality (Eq. (2)) and the Lode parameter (Eq. (3)). Besides the damage condition, the damage rule ˙ H da = ˙ µ ¯ α 1 √ 3 1 + ¯ β N + ¯ δ M (8) 2. Continuum damage and fracture model η = σ m σ eq = I 1 3 √ 3 J 2 ω = 2 T 2 − T 1 − T 3 T 1 − T 3 (3)