PSI - Issue 28

R. Moreira et al. / Procedia Structural Integrity 28 (2020) 943–949

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R. Moreira et al. / Structural Integrity Procedia 00 (2019) 000–000

1. Introduction The reduction of pollution and fuel consumption is an important goal to the transportation industry. The weight reduction of vehicles has a strong effect on the reduction of greenhouse gas emissions and on the fuel consumption. In this sense, magnesium alloys tend to replace steels and aluminum alloys in order to go further in the structural weight reduction. There are a few works in the literature regarding multiaxial stress-strain relations for magnesium alloys, but all of them stated that the magnesium alloys cyclic behaviour is quite different from the one found in steels or even in aluminum alloys. One evidence of this difference is the hardening asymmetry found in these alloys, also their yield stresses at compression and tension are different with a distinct mechanical behaviour, Yu et al. (2011), Xiong et al. (2012), Reis et al. (2013), Albinmousa and Jahed (2014). Commercial finite element packages do not have intrinsic cyclic plasticity models to modulate such type of materials. They assume that the yield stress in tension and compression are equal in absolute value for the same strain amplitude, which is true for steels but not for magnesium alloys. At present state of the art, it is not possible to accurately estimate the cyclic stress-strain relation of magnesium alloys using the available constitutive continuum plasticity models. In this sense, a new phenomenological approach called Hypo-strain (HYPS) has been developed to capture the asymmetric behaviour of magnesium alloys, Anes et al. (2015), (2018) and (2019). One way to accurately describe the unusual cyclic behaviour of magnesium alloys according to the loading using a commercial finite element packages is to implement an external subroutine to update the material cyclic response. Therefore, the main objective of this work is to implement the phenomenological Hypo-strain approach on UMAT subroutine to run in Abaqus/Standard 6.14. The results show that the phenomenological model implemented in a subroutine UMAT are in agreement with the experimental data. 2. Theoretical Background 2.1. Armstrong-Frederick Plasticity Model In general, continuum plasticity models have three main parts to model the material response under plastic deformation, namely: the yield function, the flow rule and the hardening rule. Armstrong-Frederick model is a nonlinear kinematic model, the yield function F is given through Eq. (1): � � � 3 2 � � �: � � � � � � (1) Where is the deviatoric stress, is the back stress and is the yield stress. The kinematic hardening is governed through the back-stress tensor that can be calculated by Eq. (2): � 2 3 � (2) Where and are material parameters, is the increment of the back stress, � is the increment of plastic strain. The quantity is the increment of the accumulated plastic strain. 2.2. Phenomenological Hypo-strain Model From the experiments, it was found that AZ31B-F magnesium alloy hysteresis loop can be approximated with a very good accuracy using a third-degree polynomial function for any value of total strain. In order to obtain these functions, it is considered six specific points on a hysteresis loop, as shown in the figure 1. For the tension branch, the polynomial function is obtained using the experimental data at points 4, 5, 6 and 1. Similarly, the polynomial function