PSI - Issue 13

Moritz Zistl et al. / Procedia Structural Integrity 13 (2018) 57–62 M. Zistl et al. / Structural Integrity Procedia 00 (2018) 000–000

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2

to analyze the formation of strain fields in critical notched regions of the specimen where damage and fracture are expected to localize. Fracture mechanisms on the micro-scale are visualized by scanning electron microscope analysis of fracture surfaces.

2. Continuum damage model

Large inelastic deformations and the evolution of damage in ductile metals can be modeled by the continuum framework presented by Bru¨nig (2003). It introduces damaged and fictitious undamaged configurations. The thermo dynamically consistent framework is based on kinematic description of damage leading to the definition of damage strain tensors. The e ff ective undamaged configurations are considered to formulate equations modeling elastic-plastic behavior of the undamaged matrix material. The onset of damage is described by the damage criterion

1 + β J 2 − σ = 0 .

f da = α I

(1)

It is formulated in terms of the stress invariants I 1 = tr T and J 2 = 1 / 2 dev T · dev T of the Kirchho ff stress tensor T as well as in the damage threshold σ . The damage mode parameters α and β correspond to di ff erent stress-state dependent damage mechanisms on the micro-level and depend on the stress triaxiality and the Lode parameter (Bru¨nig et al. (2013)). In addition, the evolution of macroscopic deformations of the material caused by stress-state-dependent damage processes acting on the micro-scale are described by the damage strain rate tensor ˙ H da = ˙ µ ¯ α 1 √ 3 1 + ¯ β N . (2) Here N = 1 / ( 2 √ J 2 ) dev ˜ T is the normalized deviatoric stress tensor. The stress-state-dependent parameters ¯ α and ¯ β are also chosen to be functions of to the stress triaxiality and the Lode parameter, see Bru¨nig et al. (2013) for further details. These stress-state-dependent parameters ( ¯ α and ¯ β ) are kinematic variables describing the portion of volumetric and isochoric damage-based deformations and ˙ µ represents the equivalent damage strain rate measure characterizing the amount of increase in damage. The damage rule (2) takes into account a volumetric part ( ¯ α 1 / √ 3 1 ) corresponding to isotropic growth of voids as well as a deviatoric part ( ¯ β N ) corresponding to anisotropic evolution of micro-shear cracks. Therefore, both basic damage mechanisms acting on the micro-level can be modeled by the proposed damage rule (2). The experimental program has been developed by Gerke et al. (2017) to propose two-dimensional tests revealing the e ff ect of stress state on inelastic behavior in ductile metals. The experiments are performed using the biaxial test machine type LFM-BIAX 20 kN shown in Fig. 1. The specimens are loaded by four individually driven cylinders with loading up to ± 20 kN. The cruciform X0 specimen (Fig. 2) is characterized by crosswise arranged notches with a central opening. The investigated material is the aluminum alloy AlSiMgMn EN AW-6082. Specimens are taken from sheets with 4 mm thickness. In the center of this specimen four notches in thickness direction have been milled (b) which will lead to high stresses and the localization of irreversible deformations in these regions. The notched parts have the length of 6 mm and their reduced thickness is 2 mm whereas the notch radii are 3 mm in plane (c) and 2 mm in thickness (d) directions, respectively. The loading conditions are shown in Fig. 3. The specimen will be separately loaded in two directions with F 1 and F 2 . First results with this specimen have been presented by Gerke et al. (2018). There for di ff erent loading ratios damage and fracture mechanisms caused by proportional and one non-proportional loading path are considered and the results are compared and discussed. 3. Biaxial experiments with the X0-specimen