An updated version of the multimechanism model for cyclic plasticity
Introduction
Fatigue is one of the most important phenomena encountered in mechanical structures for different industries like nuclear, automotive, aeronautics. An accurate prediction of this phenomenon is often closely related to safety in addition to economic aspects. It results from cyclic loading applied to the material. Along the past decade, many studies show the complexity of the cyclic elasto-visco-plastic behavior: cyclic softening/hardening depending not only on strain amplitude (Jiang and Zhang, 2008) but also on the loading path (Taleb and Hauet, 2009); effect of non-proportional loading (Aubin et al., 2003a, Aubin et al., 2003b, Aubin and Degallaix, 2006, Bocher et al., 2001, Hassan et al., 2008, Kang et al., 2001, Kang et al., 2002a, Kang et al., 2004, Kang et al., 2005, Kang and Gao, 2004, Khan et al., 2007, Portier et al., 2000), effect of plastic strain memory, distortion of the yield surface under non-proportional loading (Aubin et al., 2003a, Aubin et al., 2003b, Aubin and Degallaix, 2006, Feigenbaum and Dafalias, 2007, Lissenden and Lei, 2004, Vincent et al., 2002, Vincent et al., 2004) and so on. The macroscopic observations are sometimes explained through microstructural analyses in few papers (Bocher et al., 2001, Feaugas and Gaudin, 2004, Gaudin and Feaugas, 2004, Taleb and Hauet, 2009). Furthermore, the complexity of the cyclic behavior is increasing at high temperature (Kang et al., 2005, Kang and Gao, 2004).
Two main classes of phenomenological constitutive models are developed during the last decade: Chaboche’s type models (Abdel-Karim and Ohno, 2000, Bari and Hassan, 2000, Bari and Hassan, 2001, Bari and Hassan, 2002, Chen and Kim, 2003, Chen and Jiao, 2004, Kang et al., 2002b, Kobayashi and Ohno, 2002, Mayama et al., 2007, Ohno and Abdel-Karim, 2000, Yaguchi and Takahashi, 2005, Yoshida, 2000, Yoshida and Uemori, 2002) and the multimechanism model type (Sai and Cailletaud, 2007, Taleb et al., 2006, Wolff and Taleb, 2008). Two recent versions of these two model types were recently evaluated by comparison of 304L stainless steel non-proportional ratcheting responses, and weaknesses were exhibited. It was a motivation for further experimental and model developments. One will find an interesting review of different constitutive theories in Chaboche (2008).
A more recent version of the Chaboche’s model is proposed in (Krishna et al., 2009). In addition to the strain memory dependence introduced in the model following (Chaboche et al., 1979), this new version takes into account the non-proportionality effect through Tanaka Model (1994). The capabilities of the model have been demonstrated by simulating a number of proportional and non-proportional cyclic plasticity tests on a ferritic material and an austenitic (304L) stainless steel. However, the time dependence behavior cannot be described by the new constitutive equations despite the large number of the parameters (more than 40). The time dependence is essential for the 304L SS even at room temperature.
In order to better understand the behavior under non-proportional loading paths and capture the physical phenomena responsible of the macroscopic observations, we have performed a multiscale experimental investigations about the cyclic behavior of the 304L SS (Taleb and Hauet, 2009). The effect of the non-proportionality may be shown by the cross effect which represents the over strengthening observed when a proportional cyclic loading in a given direction is followed by a subsequent proportional cyclic loading in another direction (Cailletaud et al., 1984, Benallal and Marquis, 1987, Benallal et al., 1989). This cross effect is revisited in (Taleb and Hauet, 2009) through new strain as well as stress controlled tests. Thus, the additional cyclic hardening observed for these paths has been explained by the high density of defects generated: multiple slip systems, intersecting stacking faults and twins, formation of heterogeneous dislocation structures. Furthermore, it is shown that martensite nucleation takes place at the intersections between micro-shear bands or twin faults. The material anisotropy induced by the loading path has been evaluated by means of magnetic measurements; the martensitic needles and platelets seem to develop in the axial direction.
In the light of the experimental data base performed in Taleb and Hauet (2009), the modeling of the isotropic hardening by the multimechanism model (Taleb et al., 2006) is improved here. The capabilities of the model have been demonstrated considering the cyclic plasticity responses of Taleb and Hauet (2009) and Hassan et al. (2008) in addition to a new non-proportional ratcheting test.
The next section is devoted to the presentation of the main constitutive equations of the multimechanism model. The details about the identification of the material parameters are given in the third section while the fourth one demonstrates the capabilities of the model in simulating cyclic stress and strain controlled tests under proportional and non-proportional loading paths. Some concluding remarks are given in the last section.
Section snippets
The “old” version
The first version of the multimechanism model called 2M1C (two mechanisms, one criterion) has been proposed by Cailletaud and Sai (1995). This section shows the operational version of the 2M1C model without detailing the full equation set that can be found in the original paper. A mechanism is defined by a local stress, a kinematic variable, and a flow rule. Two mechanisms are combined in one loading function f to define one criterion.
The inelastic strain is composed of two components
Procedure
In addition to A1 = A2 = 1, we choose N = 2 and C11 = E where E is the Young modulus. We have then to identify “only” 17 parameters in total: the initial yield stress, 8 parameters for isotropic hardening, 6 for kinematic hardening and 2 for time dependent effects.
The identification process has been conducted in four main steps evaluating the parameters related to the following phenomena: isotropic hardening for proportional loading paths, time dependent effects, isotropic hardening under
Simulation of tests not used in the identification process
The objective here is to evaluate the capabilities of the multimechanism model established above. One can remark that in the identification process used in Section 3, and except the creep test, only cyclic strain controlled tests have been used. Our challenge now is to keep the same set of parameters listed in Table 1 and to simulate other tests performed using specimens of the same material but not necessarily machined from the same bunch. In addition to strain controlled test, we will
Concluding remarks
In this paper we have proposed an updated version of the multimechanism model in which the strain memory and the non-proportional effects are included in the isotropic hardening variable. The latter are taken into account through a simplified version of Benallal–Marquis model, which is chosen for its simplicity despite certain drawbacks. The latter appear in the simulations of two specific conditions: (i) creep tests and; (ii) cross hardening tests where the angle between the deviatoric stress
Acknowledgements
The authors acknowledge the financial support of the “Région Haute Normandie” and the European Community, through the FEDER program, for acquisition of the equipment used in the experimental study.
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