PSI - Issue 42

Michael Brünig et al. / Procedia Structural Integrity 42 (2022) 1137–1144 M. Bru¨nig et al. / Structural Integrity Procedia 00 (2019) 000–000

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of corresponding numerical tools are important aspects to predict deformation behavior, damage mechanisms and fracture modes of ductile metals in engineering applications. It has been observed in several experiments that damage and fracture processes on the micro-level are a ff ected by stress states and by loading histories. Therefore, development of appropriate constitutive models has to be based on detailed analysis of the stress-state-dependent and loading-path-dependent phenomena. In this context, results of tests with di ff erent specimens have been published during the last decades. For example, evolution of stress-state dependent damage and fracture mechanisms have been studied by uniaxial experiments and corresponding numerical simulations with unnotched and di ff erently notched specimens Bao and Wierzbicki (2004); Bonora et al. (2005); Bru¨nig et al. (2008); Driemeier et al. (2010); Gao et al. (2010); Roth and Mohr (2016). However, these tests with uniaxially loaded specimens only cover a small range of stress triaxialities and, therefore, experiments with biaxially loaded cruciform specimens have been proposed taking into account proportional and non-proportional loading paths Bru¨nig et al. (2015, 2018); Chow and Lu (1992); Green et al. (2004); Kuwalinski et al. (2011); Kuwabara (2007). The experimental and numerical results reveal that formation of damage and fracture mechanisms are remarkably influenced by the loading history and, thus, must be examined in further detail. In this context, new experiments and corresponding numerical simulations with the biaxially loaded X0-specimen Gerke et al. (2017) cut from thin aluminum alloy sheets undergoing di ff erent non-proportional paths have been discussed Bru¨nig et al. (2019). In these tests remarkably di ff erent damage and fracture mechanisms compared to proportional loading paths have been detected. Thus, further experimental and numerical investigations with steel sheets have been performed and are presented in this paper. The analysis is based on a continuum damage model here with special focus on di ff erent stress states and loading histories. In these tests digital image correlation (DIC) technique is used to monitor evolution of displacements in three directions and to compute strain fields in critical parts of the specimens. Corresponding numerical analysis of the experiments deliver stress fields leading to di ff erent damage mechanisms and fracture processes. In the present paper new experiments and corresponding numerical simulations with specimens cut from steel sheets are presented to investigate the influence of proportional and non-proportional loading paths on damage and fracture mechanisms on the micro-scale. An anisotropic continuum damage model is discussed based on criteria cor responding to di ff erent stress-state-dependent damage and failure mechanisms on the micro-scale. Experiments with biaxially loaded specimens and corresponding numerical simulations have been performed. Results for proportional and corresponding non-proportional loading histories are presented. During the experiments strain fields in critical re gions of the specimens are monitored by digital image correlation technique. Numerical simulations detect evolution of di ff erent stress parameters and reveal stress distributions which lead to di ff erent failure modes. The results demon strate the e ffi ciency of the experimental program covering a wide range of stress states, the accuracy of the proposed continuum model as well as the influence of loading history on damage and fracture behavior in steel sheets. defining elastic, plastic and damage strain rate tensors. Plastic behavior is governed by the yield condition f pl = a ¯ I 1 + ¯ J 2 − c = 0 (1) where ¯ I 1 and ¯ J 2 represent the first and second deviatoric invariants of the e ff ective Kirchho ff stress tensor T¯ with respect to the undamaged configurations, a is the hydrostatic stress sensitivity parameter and c denotes the current yield stress. The isochoric e ff ective plastic strain rate ˙¯ H pl = ˙ γ N¯ (2) characterizes the formation of plastic strains where N¯ = 1 / ( 2 ¯ J 2 ) dev T¯ is the normalized e ff ective deviatoric stress tensor and γ means the equivalent plastic strain measure. In addition, the damage condition f da = α I 1 + β J 2 − σ = 0 (3) 2. Continuum damage model The continuum damage model proposed by Bru¨nig (2003a) is used to analyze damage and fracture behavior in ductile metals. Introduction of damaged and fictitious undamaged configurations leads to the kinematic approach