New formulation of nonlinear kinematic hardening model, Part I: A Dirac delta function approach
Introduction
Elastic-plastic analysis of mechanical bodies is an important aspect of design and structural integrity assessment in a range of industries. The loading conditions considered in the analysis can vary significantly between applications, from relatively simple monotonic loading to complex load histories with multiple load and unload stages. The phenomenological approach to plasticity modelling which is based on macroscopic observations of material behaviour and the concept of a yield surface in the stress space is a convenient way of structure behaviour prediction. The yield surface changes size and moves according to isotropic and kinematic hardening rules respectively, where, in simplest formulation, the plastic deformation is defined in terms of linear functions (Hill, 1950; Prager, 1949). These plasticity models are relatively simple to implement but are limited in their ability to represent real plastic behaviour, particularly under cyclic loading conditions.
To describe cyclic plasticity more realistically, isotropic and kinematic hardening models are modified either through a multisurface approach (Dafalias and Popov, 1975; Mróz, 1967) or by introduction of nonlinear hardening functions. In the latter approach, a saturation function implemented through the Voce rule is used in isotropic hardening and a dynamic recovery term implemented through the Armstrong and Frederick (1966) rule (A-F rule) is used in kinematic hardening. These models are able to better represent experimentally observed plastic deformation under complex load cycles than the linear models. The A-F model was generalized by Chaboche (1979) by decomposing the back stress into several components. This model is widely used in practice, incorporated in several commercial Finite Element Method programs and is the basis of many other plasticity models incorporating various mathematical formulations for the dynamic recovery term. These types of model have been shown to give good agreement with experimental observation of ratcheting rate and stabilized stress-strain states for cyclic softening and hardening. However, Döring et al. (2003), Xu et al. (2016) and Zhu et al. (2017) have shown that in many cyclic loading situations A-F models may not accurately capture the form of the evolving stress-strain curve, particularly during transitions at load reversals.
Several plasticity models that reasonably accurately represent the form of cyclic plasticity curves, including transition regions have been proposed. Döring et al. (2003) extended the Jiang and Sehitoglu (1996) model with A-F type of kinematic hardening to allow it to simulate transient behaviour with reasonable accuracy for a wide strain range. The cyclic plasticity modelling framework of Zhu et al. (2014), Zhu et al. (2017) and Zhu and Poh (2016) incorporates a new kinematic hardening rule in which the back stress is decomposed into long-range, middle-range and short range components with different nonlinear features. Each component consists of a special type of nonlinear kinematic hardening rule with a linear hardening term and a dynamic recovery term, resulting in a logarithmic function after integration for a monotonic loading step. The logarithmic function shows better prediction of the general stress-strain curve shape than the exponential function from the A-F rule, but still with certain deviations from experimental results. The authors attribute these to the fact that experimental results were fitted to only 3 components of back stress decomposition and the discrepancy is, in general, present in all Chaboche back stress decomposition models. The representation of the stress-strain curve can be improved by increasing the number of Chaboche back stress decompositions; however this can lead to highly complicated constitutive modelling and a complex procedure in the material specification process. To eliminate the issue of the numerous back stresses in Chaboche model frameworks, a phase mixture approach (Eisenträger et al., 2018a, 2018b; Naumenko et al., 2011; Naumenko and Gariboldi, 2014) is applied for the unified description of the material behaviour to minimize the number of material parameters. The phase mixture model originates from materials science, i.e., hardening and softening behaviour is simulated based on an iso-strain composite with soft and hard constituents. It is assumed that the alloy is made of soft subgrains surrounded by hard boundaries, while the volume fraction of the hard constituent is closely related to the microstructure (e.g., mean subgrain size) and decreases toward a saturation value to model softening (Naumenko et al., 2011). To simplify parameter identification based on microstructural observations, a back stress and a softening variable are introduced as internal variables through a continuum mechanics approach (Naumenko et al., 2011; Naumenko and Gariboldi, 2014).
Many structural steels exhibit Lüders-type of yielding, in which a pronounce plateau occurs after yield such that plastic deformation takes place without increasing loading up to a point of subsequent strain hardening. This phenomenon has been studied experimentally for different materials, temperatures and loading conditions by Ballarin et al. (2009), Elliot et al. (2004), Wang and Huang (2017) and Zhang and Jiang (2005b). The physical background of the phenomenon of the plateau together with advanced theoretical modelling by models of crystal plasticity and strain gradient plasticity are given by Hallai and Kyriakides (2013), Mazière et al. (2017) and Yoshida et al. (2008). To solve a wide range of engineering problems, concerning structural plastic behaviour of different components, Goto et al. (1998), Shen et al. (1995) and Ucak and Tsopelas (2011, 2012) included the yielding plateau into the phenomenological plasticity framework. Bounding surface and strain amplitude parameters dividing the loading into regions of plateau and strain hardening are introduced, allowing representation of loading and unloading from the plateau region with different hardening responses.
Nonlinear nature of unloading stress-strain curves after plastic pre-straining have been experimentally observed by Lee et al. (2017a), Lee et al. (2017b), Lee et al. (2013), Mendiguren et al. (2015), Yang et al. (2004), Yu (2009) and Zajkani and Hajbarati (2017), who have reported reduction in measured unloading chord modulus of up to 40% of the initial value for different metals. This effect is usually neglected in modelling cyclic plasticity but is significant in engineering applications such as sheet metal forming, where spring-back due to elastic recovery occurs. Several mathematical models based on empirical relationships between Young's modulus and accumulated plastic pre-strain have been developed by Chen et al. (2016), Luo and Ghosh (2003), Yoshida et al. (2002), Zang et al. (2006) and Zavattieri et al. (2009). A review of the physical background of the phenomenon of nonlinearity of the unloading stress-strain curve slope is given by Chen et al. (2016). Explanation mechanisms, such as second order nonlinear elasticity via atomic bond stretching, twinning/detwinning, textural changes and material damage are excluded from consideration due to inconsistency with many sets of experimental results. The continuum elastic-plastic response due to internal and residual stresses and pile-up dislocation motion are established as two main mechanisms behind the phenomenon of elastic modulus reduction during accumulation of plastic strain.
Materials such as rocks and soils exhibit strong dependence of plastic flow on hydrostatic pressure. To incorporate this effect in a material model, all three stress invariants are usually considered in the constitutive equation (Lai et al., 2016; Shen et al., 2017; Smith et al., 2015; Sun et al., 2018). Some metals show a less pronounced dependence on hydrostatic pressure (Spitzig and Richmond, 1984) and the effect is introduced into yield criteria in several studies (Wilson, 2001; Brünig et al., 2000, 2008; Brünig and Gerke, 2011; Cazacu and Barlat, 2004; Bai and Wierzbicki, 2008; Mirone and Corallo, 2010). These indicate that stress triaxiality is important in predicting the onset of yield, as well as damage and fracture, in the metals considered. Hydrostatic pressure component is also present during the uniaxial tension-compression stress state. This effect on the form of cyclic stress-strain curves is observed in experimental curves from Halama et al. (2017), Kang et al. (2003), Kowalewski et al. (2014), Taleb and Cailletaud (2010), Voyiadjis et al. (2012) and Wang et al. (2015) articles, where tension-compression plasticity tests showed a larger hardening effect for compression dominated loading. Voyiadjis et al. (2012) demonstrated that improved representation of this effect can be achieved by including the first stress invariant in the isotropic hardening constants.
The cyclic plasticity modelling concept proposed here accounts for the various plasticity phenomena identified above in a unified mathematical framework based on a nonlinear kinematic hardening rule with non-saturation effect and without back stress decomposition. The deformation process, including the transition effect between different stress-strain loops, is determined by a framework of Dirac delta functions and introduction of an additional stress surface. This approach can simulate stepwise dependence for different internal variables and is shown to be predictive in modelling transition of initial monotonic curves to subsequent cyclic plasticity curves. The effect of differences between tensile and compression stress-strain curves is represented by an incremental form of hardening slope dependency on the first stress invariant. The ability of the model to determine initiation of plasticity at the actual yield point means that unloading curve nonlinearity is included and the model naturally accounts for variation in Young's modulus during plastic strain accumulation without use of additional features to account for this phenomenon. The cyclic hardening/softening effects and ratcheting are considered in a companion paper (Part II). Experimental testing of S355J2 low carbon structural steel together with experimental results for other metals from the literature shows that the new model can accurately represent plasticity loop branches for different loading stages of loading.
Section snippets
Experimental setup
Tension-compression tests of S355J2 low carbon structural steel with chemical composition given in Table 1 were performed on a Zwick/Roell Amsler Z250 Material Testing Machine with capacity of 250 kN. The test specimen geometry and dimensions are shown in Fig. 1. Both strain and force control were used, with total strain rate of 5·10−4 s−1 and stress rate of 1 MPa/s for both monotonic and cyclic loading. The strain was measured by a 10 mm gauge length extensometer. Preliminary tests were
Modelling background
The proposed nonlinear kinematic hardening model is based on observations made during the experimental investigation of S355J2 low carbon steel. It will be shown that the general form of the rule is valid for other metals, with particular forms of the evolution variables required for different materials.
The model is based on the small strain assumption with strain decomposition on elastic and plastic parts:
The elastic part of the strain tensor is described with the Hooke's law for
Determination of cyclic yield strength and isotropic hardening rule
To determine the evolution rule for expansion of the yield surface (5) and a mathematical form for the isotropic hardening variable, the cyclic yield strength of the material was determined. A crucial factor in the yield strength determination is accurate measurement of the yield point. Abdel-Karim (2011) discussed various methods of yield stress determination, which significantly affect the choice of the yield surface and hardening rules. The S355J2 steel exhibits discontinuous yield under
Stress-strain curve shape modelling
Experimental measurement of the yield stress during different loading conditions determined the shape of the yield surface and the isotropic hardening rule for its change in size. The size of the elastic domain remains unchanged during plastic deformation for the investigated S355J2 steel. Subsequently, the stress-strain curve shape and variation between curves during the deformation process are uniquely described by the kinematic hardening rule, which defines translation of the yield surface.
Shift of the stress-strain curves
Although the proposed kinematic hardening rule describes the shape of a single deformation stress-strain curve precisely, its direct application to all deformation curves at the same time will deviate from experimental results. This is due to the shift of stress-strain curves that occurs with reversal of load path. To determine the effect load reversals on the plastic deformation history, it is convenient to investigate the measured stress-strain curves for S355J2 steel from a random loading
Dirac delta functions framework
Developing a new mathematical framework for describing cyclic plasticity phenomena is required for a certain types of internal variables which behave as constants during certain intervals of loading. An example of such a variable is previously accumulated plastic strain, introduced by Voyiadjis et al. (2012) to better describe stress-strain curve prediction. The concept of this variable is that previously accumulated plastic strain remains constant during a step of active loading and changes to
Developing of the evolution equation for shift of back stress reference tensor
Investigation of the deformation curve from Fig. 8 suggests that there are two main reasons for changing the reference point of the stress-strain curves. The first case is change of the loading direction such as from Curve 2 to Curve 3 and the second case is transition between two curves without changing loading path such as from Curve 3 to Curve 1. It is assumed that the shift point keeps constant during a step of loading and changes its value only when Case 1 or Case 2 occurs.
Non-proportional hardening
The proposed model of kinematic hardening in the form (19) evolves back stress only in the direction of plastic strain. This behaviour has been observed experimentally but only for proportional loading. Lamba and Sidebottom (1978) demonstrated that better agreement with experimental results of non-proportional loading is achieved when back stress can evolve in a different direction according to the Mróz (1967) kinematic hardening model. In A-F models, the directionality is provided by the
Transition from the initial stress-strain curve
For many metals the shape of initial stress-strain curve is different from the subsequent curves of cyclic loading. Fig. 2 demonstrates a typical stress-strain response for carbon steels where the initial stress-strain curve has a plateau. It is seen that initial monotonic stress-strain curve is different from other curves and cannot be approximated by the same hardening law as other curves directly.
The occurrence of a plateau during monotonic loading is known as the Luder's band phenomenon,
Isotropic hardening compensation
Materials usually exhibit combined isotropic and kinematic hardening such that the yield surface can move and change its size. It has been established experimentally here that in the case of S355J2 steel the larger part of the deformation process is described by kinematic movement of the yield surface and the smaller part of isotropic hardening occurs when the initial size of the yield surface is decreased, as shown in Fig. 3d. To fully associate the uniaxial stress-strain curve shape with the
Elastic-plastic transition during unloading
Fig. 20a and Fig. 20b shows the results of stress-strain slope modulus measurement during unloading stages, with reduction of up to 20% of initial value of the chord modulus. There are several modelling concepts to take into account variations of the unloading curve modulus during plastic deformation. They can be divided into two main groups of linear and nonlinear modeling of elastic modulus. With linear modelling, Young's modulus is represented as a chord modulus, which is a linear slope
Effect of hydrostatic pressure
It is observed in many metals that cyclic plasticity deformation curves have different hardening slopes for tensile and compression loading (Voyiadjis et al., 2012). Such difference is significant for notched samples with high hydrostatic pressure. As hydrostatic pressure is present during the uniaxial tension-compression stress state, its effect on stress-strain curves is also expected. Fig. 21 shows the uniaxial monotonic stress-strain curves obtained for both tensile and compression tests
Conclusions
This paper proposes a new continuum model of cyclic plasticity in metals, capable of accurate representation of complex cyclic stress-strain curves. The theoretical development is based on experimental results from cyclic tension-compression tests of S355J2 low carbon steel and experimental stress-strain curves from the literature. The review of the literature on phenomenological cyclic plasticity models with A-F kinematic hardening rules indicates certain restrictions in modelling cyclic
Acknowledgments
This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Currie grant agreement No 643159.
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