A viscoplastic constitutive model for nickel-base superalloy,part 2: modeling under anisothermal conditions

In Part 2 of this study, extensive deformation tests were carried out on the nickel-base polycrystalline superalloy IN738LC under isothermal and anisothermal conditions between 450 and 950 °C. Under the isothermal conditions, the material showed almost no rate/time-dependency below 700 °C, while it showed distinct rate/time-dependency above 800 °C. Regarding the cyclic deformation, slight cyclic hardening behavior was observed when the temperature was below 700 °C and the imposed strain rate was fast, whereas in the case of the temperature above 800 °C or under slower strain rate conditions, the cyclic hardening behavior was scarcely observed. Unique inelastic behavior was observed under in-phase and out-of-phase anisothermal conditions: with an increase in the number of cycles, the stress at higher temperatures became smaller and the stress at lower temperatures became larger in absolute value although the stress range was approximately constant during the cyclic loading. In other words, the mean stress continues to evolve cycle-by-cycle in the direction of the stress at lower temperatures. Based on the experimental results, it was assumed that evolution of the variable Y that had been incorporated into a kinematic hardening rule in Part 1 of this study is active under higher temperatures and is negligible under lower temperatures. The material constants used in the constitutive equations were determined with the isothermal data, and were expressed as functions of temperature empirically. The extended viscoplastic constitutive equations were applied to the anisothermal cyclic loading as well as the monotonic tension, stress relaxation, creep and cyclic loading under the isothermal conditions. It was demonstrated that the present viscoplastic constitutive model was successful in describing the inelastic behavior of the material adequately, including the anomalous inelastic behavior observed under the anisothermal conditions, owing to the consideration of the variable Y.     

Kinematic hardening model suitable for ratchetting with steady-state

A new kinematic hardening model useful for simulating the steady-state in ratchetting is developed within the framework of the strain hardening and dynamic recovery format. The model is formulated to have two kinds of dynamic recovery terms, which operate at all times and only in a critical state, respectively. The model is examined on the basis of nonproportional experiments of Modified 9Cr–1Mo steel at 550°C and IN738LC at 850°C. The experiments include multiaxial, as well as uniaxial, ratchetting, multiaxial cyclic stress relaxation, and nonproportional cyclic straining along a butterfly-type strain path. It is shown that the model is successful in simulating the experiments, and that the model is featured by the capability of representing appropriately the steady-state in ratchetting under multiaxial and uniaxial cyclic loading.     

Ratchetting of viscoplastic material with cyclic softening,part 1: experiments on modified 9Cr–1Mo steel

Uniaxial and multiaxial ratchetting tests were conducted at temperatures between 200 and 600 °C on modified 9Cr–1Mo steel, which exhibits both viscoplastic and cyclic softening behavior. Anomalous behavior was observed in the stress-controlled uniaxial ratchetting tests; the material exhibited outstanding ratchetting in the tensile direction under zero mean stress. Under the uniaxial conditions, the ratchetting deformation significantly depended on the loading rate and hold time in addition to parameters such as the maximum stress and stress ratio. The uniaxial ratchetting was also accelerated to a great extent when cyclic deformation was given before the ratchetting tests. Under the multiaxial conditions, the ratchetting depended on the steady stress, cyclic strain range and strain rate. The ratchetting progressed faster as the steady stress or strain range became larger, or the strain rate became smaller, as expected. Monotonic compression tests were carried out to investigate the reason for the rachetting under no mean stress. Strain range change tests were also conducted to investigate the effect of strain range on the cyclic softening behavior of the material in detail.     

The viscoplastic behavior of hastelloy-X single crystal

A viscoplastic constitutive model for Hastelloy-X single crystal material is developed based on crystallographic slip theory. The constitutive model was constructed for use in a viscoplastic self-consistent model for isotropic Hastelloy-X polycrystalline material, which has been described in a recent publication. It is found that, by using the slip geometry known from the metallurgical literature, the anisotropic response can be accurately predicted. The model was verified by using tension and torsion data taken at 982°C (1800°F). The constitutive model used on each slip system is a simple unified visoplastic power law model in which weak latent interaction effects are taken into account. The drag stress evolution equations for the octahedral system are written in a hardening/recovery format in which both hardening and recovery depend on separate latent interaction effects between the octahedral crystallographic slip systems. The strain rate behavior of the single crystal material is well correlated by the constitutive model in uniaxial and torsion tests, but it is necessary to include latent information effects between the octahedral slip systems in order to obtain the best possible representation of biaxial cyclic strain rate behavior. Finally, it was observed that the single crystal exhibited dynamic strain aging at 871°C (1600°F). Similar dynamic strain aging occurs at 649°C (1200°F) in the polycrystalline version of the alloy.     

Meso-mechanical constitutive model for ratchetting of particle-reinforced metal matrix composites

Based on the Eshelby equivalent inclusion theory and Mori-Tanaka averaging method, a meso-mechanical cyclic elasto-plastic constitutive model is proposed to predict the ratchetting of particle-reinforced metal matrix composites. In the proposed model, a Hill-typed incremental formulation is used to simulate the elasto-plastic responses of the composites during cyclic loading with assumptions of elastic particle, elasto-plastic metal matrix and perfect interfacial bond between metal matrix and particles. A new nonlinear kinematic hardening rule extended from the Ohno-Abdel-Karim model [M. Abdel-Karim, N. Ohno, Kinematic hardening model suitable for ratchetting with steady-state, Int. J. Plasticity 16 (2000) 225-240] is employed to describe the ratchetting of metal matrix which dominates the ratchetting of the composites. With further assumption of spherical particles, the proposed meso-mechanical cyclic constitutive model is verified by comparing the predicted uniaxial ratchetting of SiC_P/6061Al composites with corresponding experiments obtained at room temperature [G.Z. Kang, Uniaxial time-dependent ratchetting of SiC_P/6061Al alloy composites at room and high temperature, Comp. Sci. Tech. 66 (2006) 1418-1430]. In the meantime, the effects of different tangent operators employed in the numerical implementation of the proposed model, i.e., continuum (or elasto-plastic) tangent operator C^ep and algorithmic (or consistent) one C^alg, on the predicted ratchetting are also discussed. It is concluded that the proposed model predicts the uniaxial ratchetting of SiC_P/6061Al composites at room temperature reasonably.     

A viscoplastic study of crack-tip deformation and crack growth in a nickel-based superalloy at elevated temperature

Viscoplastic crack-tip deformation behaviour in a nickel-based superalloy at elevated temperature has been studied for both stationary and growing cracks in a compact tension (CT) specimen using the finite element method. The material behaviour was described by a unified viscoplastic constitutive model with non-linear kinematic and isotropic hardening rules, and implemented in the finite element software ABAQUS via a user-defined material subroutine (UMAT). Finite element analyses for stationary cracks showed distinctive strain ratchetting behaviour near the crack tip at selected load ratios, leading to progressive accumulation of tensile strain normal to the crack-growth plane. Results also showed that low frequencies and superimposed hold periods at peak loads significantly enhanced strain accumulation at crack tip. Finite element simulation of crack growth was carried out under a constant ΔK-controlled loading condition, again ratchetting was observed ahead of the crack tip, similar to that for stationary cracks.A crack-growth criterion based on strain accumulation is proposed where a crack is assumed to grow when the accumulated strain ahead of the crack tip reaches a critical value over a characteristic distance. The criterion has been utilized in the prediction of crack-growth rates in a CT specimen at selected loading ranges, frequencies and dwell periods, and the predictions were compared with the experimental results.     

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.     

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.     

Cyclic multiaxial and shear finite deformation response of OFHC: Part I,experimental results

A series of experiments has been conducted on oxygen free high conductivity (OFHC) copper hollow cylinders under cyclic free-end torsion and biaxial tension–torsion at large strains. In addition, equations are developed to account for the finite rotation and strains in electrical resistance strain gages. In free-end cyclic torsion experiments with shear strain range equal to 23%, a significant strain in the axial direction is observed and it accumulates with a constant rate cycle by cycle. In the biaxial tension–torsion (multiaxial ratchetting) experiments, in which the primary (constant) axial stress is larger than the initial yield stress of the material, the loading conditions are varied to determine the influence of primary axial stress, cyclic shear strain range, pre-cyclic hardening and loading sequence on multiaxial ratchetting. Some important experimental features are high-lighted and recommended to help modeling efforts later.     

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.     

Constitutive modeling of strain range dependent cyclic hardening

In this paper, a new approach for constitutive modeling of strain range dependent cyclic hardening is proposed by extending the kinematic hardening model based on the critical state of dynamic recovery. It is assumed that isotropic, as well as kinematic, hardening consists of several parts, and that each part of isotropic hardening evolves when the corresponding part of kinematic hardening is in the critical state of dynamic recovery. The extended model is capable of simulating the cyclic hardening behavior in which different characteristics of cyclic hardening appear depending on strain range. The model is verified by simulating the relatively large cyclic straining tests of 304 stainless steel at ambient temperature, in which cyclic hardening does not stabilize before rupture if strain range exceeds a certain value. The model is further verified by predicting the history dependence of cyclic hardening under incremental cyclic loading and the maximum plastic strain dependence of strain hardening in cyclic tension.     

Constitutive model for uniaxial transformation ratchetting of super-elastic NiTi shape memory alloy at room temperature

The transformation ratchetting of super-elastic NiTi shape memory alloy was observed by the uniaxial stress-controlled cyclic tests [Kang, G.Z., Kan, Q.H., Qian, L.M., Liu, Y.J, 2009a. Ratchetting deformation of super-elastic and shape memory NiTi Alloys. Mech. Mater. 41, 139–153]. It is concluded that the NiTi alloy presents apparent ratchetting behaviour, and the ratchetting is collectively caused by the cyclic accumulation of residual induced-martensite and the transformation-induced plastic deformation (i.e., namely transformation ratchetting). Based on the experimental results, a cyclic constitutive model was constructed in the framework of generalized plasticity [Lubliner, J., Auricchio, F., 1996. Generalized plasticity and shape memory alloys. Int. J. Solids Struct. 33, 991–1003] to describe the transformation ratchetting of super-elastic NiTi alloy. The proposed model simultaneously accounts for the evolutions of residual induced-martensite and transformation-induced plastic strain during the stress-controlled cyclic loading by introducing an internal variable z_c, i.e., cumulated induced-martensite volume fraction. The dependence of transformation ratchetting on the applied stress levels and the phase transformation hardening behaviour of the NiTi alloy are also considered in the developed model. The anisotropic phase transformation behaviours of the alloy presented in the tension and compression cases are described by employing a Drucker–Prager-typed transformation surface. It is shown that the simulated results of transformation ratchetting obtained by the proposed model are in good agreement with the corresponding experiments, since the typical features of transformation ratchetting are reasonably captured by the proposed model.     

循环硬化材料本构模型的隐式应力积分和有限元实现

针对新发展的、能够描述循环硬化行为应变幅值依赖性的粘塑性本构模型,讨论了它的数值实现方法。首先,为了能够对材料的循环棘轮行为（Ratcheting）和循环应力松弛现象进行描述,对已有的本构模型进行了改进;然后,在改进模型的基础上,建立了一个新的、全隐式应力积分算法,进而推导了相应的一致切线刚度（Consistent Tangent Modulus）矩阵的表达式;最后,通过ABAQUS用户材料子程序UMAT将上述本构模型进行了有限元实现,并通过一些算例对一些构件的循环变形行为进行了有限元数值模拟,讨论了该类本构模型有限元实现的必要性和合理性。     

Uniaxial ratcheting of SS304 stainless steel at high temperatures: visco-plastic constitutive model

The uniaxial ratcheting of SS304 stainless steel at high temperatures (300, 600 and 700 °C) were analyzed experimentally, and described by a cyclic constitutive visco-plasticity model. The rate dependence of the material was accounted for by introducing a viscous term. The cyclic hardening and cyclic flow behavior of the material under asymmetrical stress-controlled cycling were described by the evolution rules of kinematic hardening back stress and isotropic deforming resistance. Under the isothermal condition, temperature effect was included by terms involving temperature in the evolution equations of isotropic deforming resistance. The effect of load history on ratcheting was also considered by introducing a fading memory function of the maximum inelastic strain amplitude and isotropic deformation resistance. After the material constants were determined from the experimental data, the uniaxial ratcheting of SS304 stainless steel was numerically simulated and compared with the corresponding experimental results at high temperatures. The predicted results agree well with the experimental ones.     

Cyclic ratchetting of 1070 steel under multiaxial stress states

Cyclic ratchetting behavior of 1070 steel is studied under proportional and nonproportional loading with specific emphasis on the ratchetting rate decay mechanisms for large numbers of loading cycles. Under proportional loading, where the principal stress directions are unchanged, the ratchetting evolves in the mean stress direction. Under nonproportional loading, however, the ratchetting direction is determined by the loading path and can be different from the mean stress direction. The ratchetting rate decreases with increasing loading cycles, displaying a power law relationship with the number of loading cycles. The experimental ratchetting results indicate that under cyclic loading the material exhibits a tendency toward complying with a linear hardening rule with concomitant hysteresis loop closure. Based on the fundamental framework of plasticity theory and detailed evaluation of the stress-strain behaviors, the ratchetting can be classified into two basic types; Type I, which is identifiable with proportional loading where the ratchetting is due to the different values of the plastic modulus function at the symmetric loading points with respect to the mean stress state, and Type II, which represents nonproportional loading where the ratchetting is driven by the noncoincidence of the plastic strain rate vector and the translation direction of the yield surface (backstress rate vector). The Armstrong-Frederick-based plasticity models modified by Chaboche et al. and Bower are ill-suited for describing the experimental results of both types of ratchetting. The Ohno-Wang model, which introduces a threshold concept, can account for the ratchetting rate decay of Type II ratchetting, providing results that agree with experimental observations. Modification may be needed for the Ohno-Wang model so that the model can better describe Type I ratchetting.     

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.     

Inelastic analysis of new thermal ratchetting due to a moving temperature front

Results of the joint project by the Working Group of Inelastic Analysis, Committee on High Temperature Strength, Society of Materials Science, Japan, are summarized. The purpose of this project, consisting of two parts, is to find out an appropriate inelastic constitutive equation which can simulate the new thermal ratchetting of a cylinder subjected to a traveling axial temperature distribution. In the first part of this project, uniaxial strain and stress-controlled ratchetting behavior is simulated by plural constitutive equations in order to examine the ability to reproduce basic deformation behavior of 316FR stainless steel at room temperature and 650 °C. In the second part, inelastic analyses of the thermal ratchetting of a cylinder due to a traveling temperature distribution are performed. Experimental data of ten specimens made of SUS 316 and 316FR stainless steels are prepared for evaluating the following items: traveling distance, stress level and hold time at high temperature. A comparison between experiments and corresponding inelastic analyses using eight types of constitutive models is made, and the selection of the appropriate constitutive equation is discussed.