PSI - Issue 42

Michael Brünig et al. / Procedia Structural Integrity 42 (2022) 1137–1144

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M. Bru¨nig et al. / Structural Integrity Procedia 00 (2019) 000–000

3

characterizes onset and continuation of damage. For isotropic elastic and plastic behavior the damage condition (3) can be expressed in terms of the first and second deviatoric stress invariants I 1 and J 2 of the Kirchho ff stress tensor with respect to the damaged configurations and σ is the material toughness to micro-defect propagation. In Eq. (3) the parameters α and β are damage mode variables considering various damage processes on the micro-scale. These parameters are taken to be functions of the stress triaxiality (4) (ratio of the mean stress σ m = I 1 / 3 and the von Mises equivalent stress σ eq = √ 3 J 2 ) as well as of the Lode parameter (5) written in terms of the principal values T 1 , T 2 and T 3 of the Kirchho ff stress tensor T . For high stress triaxialities damage is mainly caused by isotropic growth of voids and their coalescence whereas for nearly zero or small negative ones evolution of micro-shear-cracks will lead to damage. For moderate positive stress triaxialities both basic mech anisms act simultaneously while no damage has been observed in ductile metals for high negative stress triaxialities. Accumulation of these damage processes will then lead to final fracture. Furthermore, the damage strain rate tensor ˙ H da = ˙ µ ¯ α 1 √ 3 1 + ¯ β √ 2 N (6) models formation of irreversible strains caused by damage where µ is the equivalent damage strain measure and N = 1 / ( √ 2 J 2 ) dev ˜ T represents the stress related deviatoric tensor. The stress-state-dependent parameters ¯ α and ¯ β are kinematic variables describing the portion of volumetric and isochoric damage-induced strain rates. These variables also represent the dependence on di ff erent damage and fracture processes on the micro-scale. η = σ m σ eq = I 1 3 √ 3 J 2 ω = 2 T 2 − T 1 − T 3 T 1 − T 3 with T 1 ≥ T 2 ≥ T 3 The experiments are carried out with the biaxial test machine shown in Fig. 1(a). Loading is realized by four electro mechanically driven cylinders with forces up to ± 20 kN located in perpendicular axes. The specimens are fixed in the heads of the cylinders with clamped boundary conditions. During the tests three-dimensional displacement fields in selected regions of the cruciform specimens are recorded by digital image correlation (DIC) technique. The stereo setting contains eight 6.0 Mpx cameras with lighting system shown in Fig. 1(b), four cameras on the top and four on the bottom side of the specimen allowing detailed analysis of 3D strain distribution. The analyzed ductile metal is the steel X5CrNi18-10 (EN 10088-3) and specimens are cut out from thin sheets with 4 mm thickness. The biaxially loaded H-specimen shown in Fig. 2(a) has a central opening and four notched regions. This is the section where strain fields will be analyzed by DIC. The specimens dimensions are 240 mm in each axis and the depth of the notches is 1 mm reducing the thickness in these regions from 4 mm to 2 mm at the thinnest points, see Fig. 2(b) and (c). During the experiments the specimen is biaxially loaded by F 1 and F 2 and the displacements between the red points, u 1 . 1 and u 1 . 2 in 1-direction and u 2 . 1 and u 2 . 2 in 2-direction, shown in Fig. 2(d) are monitored by DIC. This leads to the relative displacements ∆ u re f . 1 = u 1 . 1 − u 1 . 2 and ∆ u re f . 2 = u 2 . 1 − u 2 . 2 , respectively, taken into account in the load-displacement curves. The investigated loading paths are shown in Fig. 2(e). In the proportional one (P 1 /+ 1) the specimen is biaxially loaded by F 1 : F 2 = 1 : 1 with the load maximum F 1 = F 2 = 11.0 kN. In the non-proportional case (NP 1 /+ 1) the specimen is first loaded by F 1 only up to F 1 = 9.5 kN (NP 1 / 0 switch), then F 1 is kept constant and in axis 2 the additional load F 2 increases up to F 2 = 9.5 kN and in the final step the specimen is proportionally loaded up to final fracture at F 1 = F 2 = 13.0 kN and 15.0 kN (NP 1 /+ 1 end), respectively. 3.2. Numerical aspects 3. Experiments and corresponding numerical simulations 3.1. Experimental equipment and specimen

The numerical simulations have been performed using the finite element program ANSYS enhanced by a user defined material subroutine taking into account the proposed continuum damage model. Integration of the constitutive