On the Ohno–Wang kinematic hardening rules for multiaxial ratcheting modeling of medium carbon steel

This paper evaluates the performance of four Ohno–Wang type constitutive models in predicting ratcheting responses of medium carbon steel S45C for a set of axial/torsional loading paths. Suggestions are also made for further modification. The four models are the Ohno–Wang model, the McDowell model, the Jiang–Sehitoglu model and the AbdelKarim–Ohno model. It is shown that the Ohno–Wang model and the McDowell model overestimate the multiaxial ratcheting. Whereas, the Jiang–Sehitoglu model yields good predictions for most loading conditions used in this study with an appropriate modification of the dynamic recovery term. The AbdelKarim–Ohno model gives acceptable predictions for all considered multiaxial conditions when used with an evolution function for μ_i, but gives poor predictions of uniaxial ratcheting if the parameter μ_i is determined from a multiaxial ratcheting response. A new modified Ohno–Wang hardening rule is proposed for better adaptability under diverse situations by multiplying a factor to the dynamic recovery term, which is dependent on noncoaxiality of the plastic strain rate and back stress. This new model predicts ratcheting strain reasonably well for the test cases.     

An advancement in cyclic plasticity modeling for multiaxial ratcheting simulation

In a search for a constitutive model for ratcheting simulations, the models by Chaboche, Ohno–Wang, McDowell, Jiang–Sehitoglu, Voyiadjis–Basuroychowdhury and AbdelKarim–Ohno are evaluated against a set of uniaxial and biaxial ratcheting responses. With the assumption of invariant shape of the yield surface during plastic loading, the ratcheting simulations for uniaxial loading are primarily a function of the plastic modulus calculation, whereas the simulations for multiaxial loading are sensitive to the kinematic hardening rule of a model. This characteristic of the above mentioned models is elaborated in this paper. It is demonstrated that if all parameters of the kinematic hardening rule are determined from uniaxial responses only, these parameters primarily enable a better plastic modulus calculation. However, in this case the role of the kinematic hardening rule in representing the ratcheting responses for multiaxial loading is under-appreciated. This realization motivated many researchers to incorporate multiaxial load dependent terms or parameters into the kinematic hardening rule. This paper evaluates some of these modified rules and finds that none is general enough to simulate the ratcheting responses consistently for the experiments considered. A modified kinematic hardening rule is proposed using the idea of Delobelle and his co-workers in the framework of the Chaboche model. This new rule introduces only one multiaxial load dependent parameter to the Chaboche model, but performs the best in simulating all the ratcheting responses considered.     

Simulation of ratcheting strain to a high number of cycles under biaxial loading

The Ohno–Wang kinematic hardening rule is modified to incorporate the Burlet–Cailletaud radial evanescence term for an improved simulation of the ratcheting behavior. The Delobelle parameter δ^′ is implemented in the modified model to compromise shakedown of the Burlet–Cailletaud hardening rule and over-prediction of the Ohno–Wang model. An evolution equation is proposed for δ^′ to simulate the ratcheting strain over an extended domain of cycles. Ratcheting tests were conducted on S45C steel under four types of nonproportional axial–torsional loading. The new model is found to yield reasonably accurate predictions of ratcheting strain to a much higher number of cycles compared with other studies.     

Kinematic hardening rules in uncoupled modeling for multiaxial ratcheting simulation

An earlier paper by the authors evaluated the performance of several coupled models in simulating a series of uniaxial and biaxial ratcheting responses. This paper evaluates the performance of various kinematic hardening rules in an uncoupled model for the same set of ratcheting responses. A modified version of the Dafalias–Popov uncoupled model has been demonstrated to perform well for uniaxial ratcheting simulation. However, its performance in multiaxial ratcheting simulation is significantly influenced by the kinematic hardening rules employed in the model. Performances of eight different kinematic hardening rules, when engaged with the modified Dafalias–Popov model, are evaluated against a series of rate-independent multiaxial ratcheting responses of cyclically stabilized carbon steels. The kinematic hardening rules proposed by Armstrong–Frederick, Voyiadjis–Sivakumar, Phillips, Tseng–Lee, Kaneko, Xia–Ellyin, Chaboche and Ohno–Wang are examined. The Armstrong–Frederick rule performs reasonably for one type of the biaxial ratcheting response, but fails in others. The Voyiadjis–Sivakumar rule and its constituents, the Phillips and the Tseng–Lee rules, can not simulate the biaxial ratcheting responses. The Kaneko rule, composed of the Ziegler and the prestress directions, and the Xia–Ellyin rule, composed of the Ziegler and Mroz directions, also fail to simulate the biaxial ratcheting responses. The Chaboche rule, with three decomposed Armstrong–Frederick rules, performs the best for the whole set of ratcheting responses. The Ohno–Wang rule performs well for the data set, except for one biaxial response where it predicts shakedown with subsequent reversal of ratcheting.     

Visco-plastic constitutive modeling on Ohno-Wang kinematic hardening rule for uniaxial ratcheting behavior of Z2CND18.12N steel

Experimental results of monotonic uniaxial tensile tests at different strain rates and the reversed strain cycling test showed the characteristics of rate-dependence and cyclic hardening of Z2CND18.12N austenitic stainless steel at room temperature, respectively. Based on the Ohno-Wang kinematic hardening rule, a visco-plastic constitutive model incorporated with isotropic hardening was developed to describe the uniaxial ratcheting behavior of Z2CND18.12N steel under various stress-controlled loading conditions. Predicted results of the developed model agreed better with experimental results when the ratcheting strain level became higher, but the developed model overestimated the ratcheting deformation in other cases. A modified model was proposed to improve the prediction accuracy. In the modified model, the parameter m_i of the Ohno-Wang kinematic hardening rule was developed to evolve with the accumulated plastic strain. Simulation results of the modified model proved much better agreement with experiments.     

Influence of non-proportional loading on ratcheting responses and simulations by two recent cyclic plasticity models

Aubin and her coworkers conducted a unique set of experiments demonstrating the influence of loading non-proportionality on ratcheting responses of duplex stainless steel. In order to further explore their new observation, a set of experiments was conducted on stainless steel (SS) 304L under various biaxial stress-controlled non-proportional histories. This new set of data reiterated Aubin and her coworkers’ observation and illustrated many new responses critical to model development and validation. Two recent and different classes of cyclic plasticity models, the modified Chaboche model proposed by Bari and Hassan and the version of the multi-mechanism model proposed by Taleb and Cailletaud, are evaluated in terms of their simulations of the SS304L non-proportional ratcheting responses. A modeling scheme for non-proportional ratcheting responses using the kinematic hardening rule parameters in addition to the conventionally used isotropic hardening rule parameter (yield surface size change) in the modified Chaboche model is evaluated. Strengths and weaknesses of the models in simulating the non-proportional ratcheting responses are identified. Further improvements of these models needed for improving the non-proportional ratcheting simulations are suggested in the paper.     

Uniaxial ratcheting and fatigue failure of tempered 42CrMo steel: Damage evolution and damage-coupled visco-plastic constitutive model

Uniaxial ratcheting and fatigue failure of tempered 42CrMo steel were observed by the tests under the uniaxial stress-controlled cyclic loading with non-zero mean stress [G.Z. Kang, Y.J. Liu, Mater. Sci. Eng. A 472 (2008) 258–268]. Based on the obtained experimental results, the evolution features of whole-life ratcheting behavior and low-cycle fatigue (LCF) damage of the material were discussed first. Then, in the framework of unified visco-plasticity and continuum damage mechanics, a damage-coupled visco-plastic cyclic constitutive model was proposed to simulate the whole-life ratcheting and predict the fatigue failure life of the material presented in the uniaxial stress cycling with non-zero mean stress. In the proposed model, the damage was divided into two parts, i.e., elastic damage and plastic damage, which were described by the evolution equations with the same form but different constants, since the maximum applied stresses in most of loading cases were lower than the nominal yielding strength of the material. The ratcheting of the material was still described by employing a nonlinear kinematic hardening rule based on the Abdel-Karim–Ohno combined kinematic hardening model [M. Abdel Karim, N. Ohno, Int. J. Plast. 16 (2000) 225–240] but extended by considering the effect of damage. The maximum strain criterion combined with an elastic damage threshold was employed to determine the failure life of the material caused by two different failure modes, i.e., fatigue failure (caused by low-cycle fatigue due to plastic shakedown) and ductile failure (caused by large ratcheting strain). The simulated whole-life ratcheting behavior and predicted failure life of tempered 42CrMo steel are in a fairly good agreement with the experimental ones.     

316L不锈钢随动强化模型改进研究

本文对316L不锈钢进行了单轴与多轴非比例路径下的应力控制棘轮试验,考察了应力幅值、平均应力和加载历程对棘轮特性的影响。同时进行了应变控制循环试验以研究材料的应力松弛特性。试验结果表明轴向棘轮效应在对称剪切荷载下效果明显,同时棘轮应变随应力幅值和平均应力的增加而增加。研究了Chen-Jiao随动强化模型与Jiang-Sehitoglu随动强化模型采用的单轴与多轴参数对背应力分量增量方向的影响,将Chen-Jiao模型中的多轴系数替换为界面饱和率,并在此基础上引入新的参数对塑性模量系数进行修正,计算结果表明修正后的模型能提升应力控制下多轴棘轮的预测精度,并能很好的预测应力松弛现象,表明了新模型的正确性与有效性。     

Reconcile the perfectly elastoplastic model to simulate the cyclic behavior and ratcheting

Perfectly elastoplastic constitutive model is modified through a smoothing factor introduced by Liu [Liu, C.-S., 2003. Smoothing elastoplastic stress–strain curves obtained by a critical modification of conventional models. Int. J. Solids Struct. 40, 2121–2145]. The new model allows plasticity to happen in a non-zero-measure yield volume in stress space, rather than that of conventional zero-measure yield surface, and within the yield volume the plastic modulus is varying continuously. It endows a specific strain-hardening rule of flow stress and is able to describe the phenomena of strain hardening, cyclic hardening, the Bauschinger effect, mean-stress relaxation, strain ratcheting, out-of-phase hardening, as well as erasure-of-memory. In order to suppress the over prediction of ratcheting we consider a scalar function of smoothing factor, which can simulate the saturation behavior of uniaxial/multiaxial strain ratcheting. These effects are demonstrated through numerical examples. The existence of stress equilibrium point and limiting surface is a natural result without requiring an extra design. Moreover, the non-linear constitutive equations can be converted into a linear system for augmented stress in the Minkowski space, of which the symmetry group is a proper orthochronous Lorentz group SO_o(5, 1). The augmented stress is a time-like vector, moving on hyperboloids inside the cone. When taking the Prager kinematic hardening rule into account we can simulate some cyclic behaviors of SAE 4340 and grade 60 steels within a certain accuracy through the use of only three material constants and a fixed smoothing factor. To simulate the ratcheting behaviors of SS304 stainless steel we allow the smoothing factor to be an exponential decaying function of λ.     

Anatomy of coupled constitutive models for ratcheting simulation

This paper critically evaluates the performance of five constitutive models in predicting ratcheting responses of carbon steel for a broad set of uniaxial and biaxial loading histories. The models proposed by Prager, Armstrong and Frederick, Chaboche, Ohno-Wang and Guionnet are examined. Reasons for success and failure in simulating ratcheting by these models are elaborated. The bilinear Prager and the nonlinear Armstrong-Frederick models are found to be inadequate in simulating ratcheting responses. The Chaboche and Ohno-Wang models perform quite well in predicting uniaxial ratcheting responses; however, they consistently overpredict the biaxial ratcheting responses. The Guionnet model simulates one set of biaxial ratcheting responses very well, but fails to simulate uniaxial and other biaxial ratcheting responses. Similar to many earlier studies, this study also indicates a strong influence of the kinematic hardening rule or backstress direction on multiaxial ratcheting simulation. Incorporation of parameters dependent on multiaxial ratcheting responses, while dormant for uniaxial responses, into Chaboche-type kinematic hardening rules may be conducive to improve their multiaxial ratcheting simulations. The uncoupling of the kinematic hardening rule from the plastic modulus calculation is another potentially viable alternative. The best option to achieve a robust model for ratcheting simulations seems to be the incorporation of yield surface shape change (formative hardening) in the cyclic plasticity model.     

Cyclic multiaxial and shear finite deformation responses of OFHC Cu. Part II: An extension to the KHL model and simulations

In this part, the Khan–Huang–Liang (KHL) constitutive model was extended to account for kinematic hardening characteristic behavior of materials. The extended model is then generalized and used to simulate experimental response of oxygen free high conductivity (OFHC) copper under cyclic shear straining and biaxial tension–torsion (multiaxial ratchetting) experiments presented in Part I (Khan et al., 2007). In addition, a new modification for the non-linear kinematic hardening rule of Karim–Ohno (Abdel-Karim and Ohno, 2000) is proposed to simulate multiaxial ratchetting behaviors. Although, the kinematic hardening contributes the most to the response, it is shown that, the loading rate effect, and a coupled isotropic and kinematic hardening effect should also be considered while simulating the multiaxial ratchetting behavior of OFHC copper. Furthermore, the newly modified kinematic hardening rules is able to fairly well simulate the multiaxial ratchetting experiments under different loading conditions, irrespective of the value of applied axial tensile stress, shear strain amplitude, pre-cyclic hardening and/or loading sequence.     

内压循环下薄壁圆筒的环向棘轮应变预测

对单轴和比例加载下的棘轮应变进行预测目前仍有较大困难。本文考虑将移动极限面加入材料双面本构模型中，对原提出的叠加型随动强化律进行了修正，可以对较多循环数的1Crl8Ni9Ti不锈钢薄壁圆管的环向棘轮应变做出预测。     

Evaluation of cyclic plasticity models in ratcheting simulation of straight pipes under cyclic bending and steady internal pressure

This paper evaluates seven cyclic plasticity models for structural ratcheting response simulations. The models evaluated are bilinear (Prager), multilinear (Besseling), Chaboche, Ohno–Wang, Abdel Karim–Ohno, modified Chaboche (Bari and Hassan) and modified Ohno–Wang (Chen and Jiao). The first three models are already available in the ANSYS finite element package, whereas the last four were implemented into ANSYS for this study. Experimental responses of straight steel pipes under cyclic bending with symmetric end rotation history and steady internal pressure were recorded for the model evaluation study. It is demonstrated that when the model parameters are determined from the material response data, none of the models evaluated perform satisfactorily in simulating the straight pipe diameter change and circumferential strain ratcheting responses. A detailed parameter sensitivity study with the modified Chaboche model was conducted to identify the parameters that influence the ratcheting simulations and to determine the ranges of the parameter values over which a genetic algorithm can search for refinement of these values. The refined parameter values improved the simulations of straight pipe ratcheting responses, but the simulations still are not acceptable. Further, improvement in cyclic plasticity modeling and incorporation of structural features, like residual stresses and anisotropy of materials in the analysis will be essential for advancement of low-cycle fatigue response simulations of structures.     

Uniaxial time-dependent ratchetting: Visco-plastic model and finite element application

Based on a visco-plastic model, a time-dependent formulation was introduced. The model can describe as time-dependent ratchetting in term of revising the Abdel-Karim–Ohno nonlinear kinematic hardening rule [M. Abdel-Karim, N. Ohno, Kinematic hardening model suitable for ratchetting with steady-state, Int. J. Plast. 16 (2000) 225–240] by a static recovery term. It is shown that the simulated results are in good agreement with the corresponding experiment results of SS304 stainless steel [G.Z. Kang, Q.H. Kan, J. Zhang, Time-dependent ratchetting experiments of SS304 stainless steel, Int. J. Plast. 22 (2006) 858–894]. Then, the proposed model with static recovery term was implemented into the finite element package. Based on the radial return method and backward Euler’s integration, a new implicit stress integration algorithm was proposed, and a new expression of consistent tangent modulus was derived. Finally, the reasonability of such implementation was verified by some numerical samples.     

Uniaxial and non-proportionally multiaxial ratcheting of SS304 stainless steel at room temperature: experiments and simulations

The uniaxial and non-proportionally multiaxial ratcheting behaviors of SS304 stainless steel at room temperature were initially researched by experiment and then were theoretically described by a cyclic constitutive model in the framework of unified visco-plasticity. The effects of cyclic stress amplitude, mean stress, and their histories on the ratcheting were experimentally investigated under uniaxial and different multiaxial loading paths. The shapes of non-proportional loading paths were linear, circular, elliptical and rhombic, respectively. In the constitutive model, the rate-dependent behavior of the material was reflected by a viscous term; the cyclic flow and cyclic hardening behaviors of the material under asymmetrical stress-controlled cycling were reflected by the evolution rules of kinematic hardening back stress and isotropic deforming resistance, respectively. The effect of loading history on the ratcheting was also considered by introducing two fading memorization functions for maximum inelastic strain amplitude and isotropic deformation resistance, respectively, into the constitutive model. The effect of multiaxial loading path on the ratcheting was reflected by a non-proportional factor defined in this work. The predicting ability of the developed model was proved to be good by comparing the simulations with corresponding experiments.     

Ratchetting of viscoplastic material with cyclic softening,part 2: application of constitutive models

Simulation capability on ratchetting of modified 9Cr–1Mo steel at 550 °C was discussed using several constitutive models in the present paper. It was revealed that the authors' previous model, which uses an Armstrong–Frederick kinematic hardening rule, has a strong tendency to overestimate both uniaxial and multiaxial ratchetting of the material. On the contrary, the Ohno–Wang (OW) I model tended to underestimate the uniaxial and multiaxial ratchetting. The OW II and III models predicted the uniaxial and multiaxial ratchetting with better accuracy. Regarding the uniaxial ratchetting under the zero mean stress condition described in part 1 of this study, none of the constitutive models was able to simulate it even qualitatively. On the basis of the OW I model, a constitutive model incorporating a tension–compression asymmetry was proposed to predict the ratchetting behavior under the zero mean stress condition. The simulation capability of the proposed model was discussed in comparison with that of the other constitutive models.     

Constitutive and damage model for 63Sn37Pb solder under uniaxial and torsional cyclic loading

A constitutive model with Ohno–Wang kinematic hardening rule is developed and employed to simulate the isothermal cyclic behavior of Sn–Pb solder under uniaxial and torsional loading. An implicit constitutive integration scheme is presented for inelastic flow of solder. Then a modified low cycle fatigue life prediction model is put forward in which the sum of maximum shear strain range and normal strain range based on the critical plane concept is adopted to replace the uniaxial strain range used by Stolkarts et al. [Stolkarts, V., Keer, L.M., Fine, M.E., 1999. Damage evolution governed by microcrack nucleation with application to the fatigue of 63Sn–37Pb solder. J. Mech. Phys. Solids 47, 2451–2468]. Comparison of the experimental results and simulation verifies that the stress strain hysteresis loops and peak stress decline curve of solder can be reasonably modeled over a wide range of loading conditions with implement of damage coupled constitutive model, and the lifetime estimations of 63Sn37Pb solder based on the assumption of microcrack nucleation governed damage is effective to provide a conservative prediction.     

Computational modeling of inelastic large ratcheting strains

A framework for phenomenological hyperelasto-plasticity with initial anisotropy, kinematic hardening as well as anisotropic damage is presented in [Menzel et al., Int. J. Plasticity (2004), in press]. In this contribution, we exploit and extend this framework to include several back-stresses in order to capture the ratcheting response of polycrystalline metals subjected to cyclic stress with non-zero mid-value. The evolution equations for kinematic hardening resemble a linear combination of the multiple-Armstrong–Frederick and the Burlet–Cailletaud models, which are extended to the large strain setting. The capability of the model to capture various phenomenological characteristics, in particular multi-axial ratcheting, is illustrated by numerical examples. Comparisons with uni-axial and bi-axial experimental ratcheting results for carbon steel are given. Finally, the finite element analysis of a simplified railway turnout component subjected to cyclic loading is presented.