Particle swarm optimization procedure in determining parameters in Chaboche kinematic hardening model to assess ratcheting under uniaxial and biaxial loading cycles

As parameters in Chaboche model are difficult to be determined from experimental data, a single objective particle swarm optimization procedure was employed to obtain them. Hysteresis loop and uniaxial and biaxial ratcheting simulations were conducted to validate the determined models. Chaboche models determined by particle swarm optimization give more accurate simulation of ratcheting compared with the model determined by trial and error method. Chaboche models containing different backstress components were studied. Models determined considering uniaxial ratcheting can only predict uniaxial ratcheting precisely, while giving very bad simulation of biaxial ratcheting. The linear hardening rule in the N3L1 model clearly decreases the rate of the accumulation of ratcheting strain, and the N3L1 model gives the best simulations. For biaxial ratcheting, the fourth backstress component can decrease the rate of the accumulation apparently, while it has a little influence on prediction of uniaxial ratcheting.     

一种预测复合材料棘轮行为的循环塑性-损伤模型

构建循环塑性本构模型并揭示其微观机制,目前仍然是复合材料力学研究富有挑战性的课题。本文提出了一种循环塑性-损伤模型,用以预测在循环载荷作用下纤维增强复合材料的应力-应变响应。该模型是在作者前期提出的描述非线性滞后行为的弹塑性本构模型的基础上的进一步扩展。它可以预测加载时的非线性响应、卸载和重加载时的迟滞行为及大量循环下的棘轮现象。作为基准问题验证,将Kawai等的实验数据与本文模型的数值预测进行了比较。结果表明,该模型能够模拟碳纤维/环氧树脂单向复合材料在偏轴循环加载下的棘轮行为。      

Visco-plastic constitutive model considering non-proportional hardening at elevated temperature under multiaxial loading

Based on the Chaboche unified visco-plastic constitutive model, a visco-plastic constitutive model which can take into account non-proportional hardening was proposed for nickel-base alloy at elevated temperature under multiaxial loading. In the proposed multiaxial visco-plastic constitutive model, the non-proportional hardening is considered as a change of kinematic hardening property. The kinematic hardening parameters which can evolve exponentially with the cumulative plastic strain and rotation factor resulted from loading path were proposed to take into account the non-proportional hardening at high temperature under non-proportional loading. Experimental verification showed that the proposed model can accurately predict the peak stresses and the hysteresis loop compared with the original model during the tension-torsion loading process for nickel-base alloy at elevated temperature under non-proportional loading.     

Cyclic loading of beams based on the Chaboche model

The time independent non-unified version of the Chaboche constitutive model for the cyclic loading which includes the kinematic and isotropic hardening is discussed in detail. The performance of the Chaboche constitutive model in predicting ratcheting response of CS1026 steel for a broad set of mechanical uniaxial and biaxial loading histories is considered. A numerical iterative method is used to calculate the stresses and strains in beams due to cyclic loading. The reported experimental data of the stainless steel, available in the literature, is used for the verification of the results. It is concluded that the Chaboche model performs quite well in predicting the uniaxial ratcheting or shakedown responses. In addition, imposing the isotropic hardening effect to the constitutive equations results to lower ratcheting rate at initial cycles. While the kinematic hardening effect remains the major factor in prediction of the ratcheting response.     

Modeling of uniaxial and biaxial ratcheting behavior of 1026 Carbon steel using the simplified Viscoplasticity Theory Based on Overstress (VBO)

Summary.  Hassan and Kyriakides [1], Hassan et al. [2], and Corona et al. [3] performed uniaxial and biaxial ratcheting experiments on heat-treated 1026 Carbon steel. The loading histories performed with uniaxial and tubular specimens were selected to simulate those encountered in nuclear reactor vessels. The stress-strain diagram of 1026 Carbon steel was used to determine the material constants in a simplified version of VBO. Small rate dependence was allowed. The model represents some nonlinear rate dependence, kinematic hardening and cyclically neutral behavior. The set of material constants determined only from uniaxial tests was used throughout the paper. Numerical experiments included are: (i) uniaxial stress-controlled cycling with various mean stresses, (ii) strain-controlled axial tests with tubular specimens under constant and variable internal pressure, and (iii) examination and variation of certain material constants of the VBO model that can influence ratcheting. Very good agreement with the experiments is found for the uniaxial case. However the ratchet strain accumulation during biaxial cycling is over-predicted by VBO and other constitutive models. Received November 5, 2001; revised June 12, 2002 Published online: January 16, 2003     

An inelastic constitutive model of blood vessels

Summary The present paper is concerned with the formulation of a constitutive model describing the hysteresis loops of stress-strain relations of blood vessels under cyclic loading conditions. It is assumed that the hysteresis loop is composed of elastic deformation and viscoplastic deformation. Hence the total strain is expressed as the sum of the elastic part and the inelastic part on the basis of a finite deformation theory. Then the elastic part is established by postulating a strain energy function of an exponential type, while the inelastic part is formulated by modifying the nonlinear kinematic hardening rule in the viscoplastic model proposed by Chaboche et al. A comparison of the numerical result with the literature shows that the present model can describe the hysteresis loop qualitatively.     

Numerical and experimental study of ratcheting in cold expanded plate of Al‐alloy 2024‐T3 in double shear lap joints

In this paper strain ratcheting in cold expanded flat plate of Al‐alloy 2024‐T3 in double shear lap joints was studied both experimentally and numerically. In the experimental part, two types of symmetric strain‐controlled and asymmetric stress‐controlled cyclic tests were performed. Also, the cold expanded double shear lap joints subjected to cyclic stress‐controlled tests. The required parameters for simulating the cyclic plastic behaviour of Al‐alloy 2024‐T3 were obtained on the basis of the experimental responses. In the numerical part, a combination of nonlinear isotropic and nonlinear kinematic hardening model (Chaboche) was implemented in the commercial finite element code of ABAQUS, using the subroutine UMAT written in FORTRAN. The results of simulations give an accurate prediction of ratcheting for all types of loading. The obtained results show that increasing the mean stress increases the strain ratcheting. It is clearly shown that the cold expansion process decreases the magnitude of strain ratcheting remarkably compared with “as drilled” specimens and the decrease is bigger for larger cold expansion sizes. Also, it is shown that the middle plane has the highest amount of ratcheting compared to the pin entrance plane and exit plane of the plate hole.     

Constitutive and damage modelling of H11 subjected to low‐cycle fatigue at high temperature

Hot‐work tool steel H11 is extensively applied in extrusion industries as extrusion tools. The understanding of its mechanical properties and damage evolution as well as failure is crucial for its implementation. In this paper, a finite element (FE) model employing Chaboche unified constitutive model and ductile damage rule is proposed to simulate the mechanical responses of H11 subjected to low‐cycle fatigue (LCF). Accumulated inelastic hysteresis energy is adopted to demonstrate the impact on damage initiation and evolution rules. A series of tension and LCF experiments were conducted to investigate H11's mechanical properties and its deterioration processes. In addition, to deeply understand the deformation and damage mechanism, scanning electron microscope (SEM) investigations were performed on the fracture section of gauge‐length part of the specimen after failure. Furthermore, the parameters in both constitutive model and damage rule are identified based on experimental data. The comparison of the hysteresis loop of the first cycle and stable cycle with different strain amplitudes demonstrates that the Chaboche constitutive model provides high precision to predict the evolution of mechanical properties. Based on the reliable achieved constitutive model, LCF behaviour prediction with damage rule was executed successfully using FE model and gains a good agreement with the experiments. It is believed that the proposed FE method lays the foundation of structure analysis and rapid design optimization in further applications.     

Isotropic‐kinematic hardening framework to assess ratcheting response of steel samples undergoing asymmetric loading cycles

The present study intends to characterize ratcheting response of several steel alloys subject to asymmetric loading cycles through coupling the Ahmadzadeh‐Varvani kinematic hardening rule with isotropic hardening rules of Lee and Zavrel, Chaboche, and Kang. The Ahmadzadeh‐Varvani kinematic hardening rule was developed to address ratcheting progress over asymmetric stress cycles with relatively a simple framework and less number of coefficients. Inclusion of isotropic hardening rules to the framework improved ratcheting response of materials mainly over the first stage of ratcheting. Lee and Zavrel model (ISO‐I) developed an exponential function to account for accumulated plastic strain as yield surface is expanded over stage I and early stage II of ratcheting. Isotropic models by Chaboche (ISO‐II) and Kang (ISO‐III) encountered yield surface evolution in the framework by introducing an internal variable that takes into account the prior maximum plastic strain range. The choice of isotropic hardening model coupled to the kinematic hardening model is highly influenced by material softening/hardening response.     

Steady ratcheting strains accumulation in varying temperature fatigue tests of PMMA

The evolutions of ratcheting strains of polymethyl methacrylate (PMMA) at different temperatures and stress levels were experimentally investigated. A steady ratcheting strain growth region with a constant rate was observed in all specimens, which occupied significant part of total fatigue failure life. Experimental results also showed that the steady ratcheting growth rate varied with applied temperatures and loading. In this paper, theory of thermally activated process for glassy polymers was used to describe the plastic deformations during the cycle. Based on the correlations between ratcheting strains per cycle and hysteresis loop energy, a new ratcheting strains accumulative model for polymer materials was developed, which quantificationally elucidated the effects of temperature, loading frequency, mean stress and stress amplitude on the accumulative rate of ratcheting strains. Comparing the predications from the proposed model with experimental ratcheting strain data of PMMA, it was found that the model could describe the steady ratcheting strain accumulative behaviors under arbitrary temperatures and loading conditions exactly.     

Simulation of low cycle fatigue stress‐strain response in 316LN stainless steel using non‐linear isotropic kinematic hardening model—A comparison of different approaches

Cyclic stress‐strain response of 316LN stainless steel subjected to low cycle fatigue at strain amplitude of ±0.4% and at 873 K is simulated using finite element analysis with non‐linear isotropic‐kinematic hardening Chaboche model. Four different approaches have been used in simulating cyclic stress response and hysteresis loops: 3 based on Chaboche model‐parameters and the fourth on direct experimental data (stabilized loop and cyclic stress‐strain curve [CSSC]). Among them, simulations performed with direct experimental data have not yielded expected initial cyclic response. The source of data used for evaluation of kinematic‐hardening (KH) parameters determined the extent of closeness between experimental results and Chaboche‐model predictions. KH parameters determined from first‐cycle loop and modified‐CSSC predicted the overall stress‐strain response (from initial to stabilized condition) with reasonable fit, compared with other approaches. All 4 approaches though predicted stabilized response, simulations based on “KH‐parameters from stabilized‐cycle” accurately described stabilized response with coefficient of determination (r²) 0.995.     

GH4169合金高温力学行为本构建模及参数识别

针对涡轮盘用GH4169合金开展了高温下单调拉伸、对称循环及非对称循环的实验工作,结果表明,该材料具有比较明显的循环软化和平均应力松弛特性.采用带Ohno/Wang修正的Chaboche粘塑性理论本构方程,对其表现出的复杂力学现象进行本构建模,介绍了Levenberg-Marquadt非线性优化算法,结合材料实验数据并通过该算法识别了本构方程参数,将本构方程通过用户子程序嵌入到有限元软件ABAQUS中,对GH4169合金的上述实验现象进行了数值模拟,计算曲线与实验曲线取得了较好的一致性.     

Predicting damage and failure under low cycle fatigue in a 9Cr steel

Experimental data have been generated and finite element models developed to examine the low cycle fatigue (LCF) life of a 9Cr (FB2) steel. A novel approach, employing a local ductile damage initiation and failure model, using the hysteresis total stress–strain energy concept combined with element removal, has been employed to predict the failure in the experimental tests. The 9Cr steel was found to exhibit both cyclic softening and nonlinear kinematic hardening behaviour. The finite element analysis of the material's cyclic loading was based on a nonlinear kinematic hardening criterion using the Chaboche constitutive equations. The models’ parameters were calibrated using the experimental test data available. The cyclic softening model in conjunction with the progressive damage evolution model successfully predicted the deformation behaviour and failure times of the experimental tests for the 9Cr steels performed.     

Cyclic uniaxial and multiaxial loading with yield surface distortion consideration on prediction of ratcheting

In this study, the yield surface distortion was incorporated in the cyclic plasticity modeling as well as its center movement. The combination of Chaboche’s model and the yield surface distortion model of Baltov was used in a set of uniaxial and multiaxial loadings. The variation of the stress amplitude and the mean stress and different multiaxial loadings such as tension-torsion tests were studied. It was shown that the consideration of the distortion of the yield surface via the distortion parameter and its sign in modeling has an important effect on the plastic strain increment determination and so on the ratcheting rate. The combined model was applied to the experimental results. It was shown that the combination of the nonlinear kinematic hardening model of Chaboche and the yield surface distortion leads to a good estimation of the ratcheting strain increment in different uniaxial and multiaxial tests.     

Stress–strain time-dependent behavior of A356.0 aluminum alloy subjected to cyclic thermal and mechanical loadings

This article presents the cyclic behavior of the A356.0 aluminum alloy under low-cycle fatigue (or isothermal) and thermo-mechanical fatigue loadings. Since the thermo-mechanical fatigue (TMF) test is time consuming and has high costs in comparison to low-cycle fatigue (LCF) tests, the purpose of this research is to use LCF test results to predict the TMF behavior of the material. A time-independent model, considering the combined nonlinear isotropic/kinematic hardening law, was used to predict the TMF behavior of the material. Material constants of this model were calibrated based on room-temperature and high-temperature low-cycle fatigue tests. The nonlinear isotropic/kinematic hardening law could accurately estimate the stress–strain hysteresis loop for the LCF condition; however, for the out-of-phase TMF, the condition could not predict properly the stress value due to the strain rate effect. Therefore, a two-layer visco-plastic model and also the Johnson–Cook law were applied to improve the estimation of the stress–strain hysteresis loop. Related finite element results based on the two-layer visco-plastic model demonstrated a good agreement with experimental TMF data of the A356.0 alloy.     

双相型不锈钢S22053循环本构关系研究

为研究国产双相型不锈钢S22053在循环荷载下材料的力学性能和本构关系,该文采用S22053热轧钢板加工成母材试件,进行单调和循环加载试验,得到14种加载制度下的应力-应变曲线。基于单调加载试验曲线,分析了单调荷载下的材料性能;利用等应变幅加载试验曲线拟合了Chaboche混合强化参数,并用有限元软件ABAQUS对试验进行模拟计算,对比、验证了拟合效果。结果表明:双相型不锈钢S22053延性较好,没有明显的屈服平台和屈服点,比例极限较低;循环骨架曲线可以采用Ramberg-Osgood模型进行拟合;采用不同分量模型标定的3组混合强化参数均能较好的模拟材料的循环受力特征,其中三背应力分量模型(N2L1)拟合效果最好。研究结果可用于分析计算双相型不锈钢结构在地震作用下的受力性能。     

Ratcheting progress at notch root of 1045 steel samples over asymmetric loading cycles: Experiments and analyses

The present study examines ratcheting response of steel samples with various notch diameters through conducting several cyclic tests. Ratcheting strain values were measured through strain gauges mounted at different distances from the notch root. Local ratcheting at the notch region was highly influenced by notch diameter, notch shape, distance from the notch root, and magnitude of the nominal mean/amplitude of loading cycles. Nominal force‐controlled cycles were kept below the yield point and the Neuber's rule accommodated for the maximum/minimum local stress components along those local strains measured through the strain gauges at the notch region. Plastic strains at the vicinity of notch root over loading cycles were further accumulated by means of the Chaboche hardening model. The local ratcheting strain while progressed at the notch root plastic zone over loading cycles resulted in mean stress relaxation controlled by the model.     

McDowell模型改进及1070钢棘轮效应的预测

利用McDowell 模型对1070 钢比例和非比例循环加载条件下的棘轮效应进行了预测,通过对McDowell模型的修正,为McDowell 模型中的单轴参数Ai引入了与塑性应变累积相关的演化方程,改进后的McDowell 模型能精确的预测具有拉压平均应力的单轴棘轮效应,具有平均应力的拉扭比例加载,常轴向应力的扭转循环,非比例循环载荷以及多重步骤加载条件下单轴平均应力变化的棘轮效应,改进的模型对较大循环数的棘轮效应也能给予较好的描述。     

考虑强度和刚度退化及捏拢效应的钢筋混凝土桥墩滞回模型及其参数识别

为了有效地模拟强震作用下桥梁结构的非线性地震反应,需要确定钢筋混凝土（RC）桥墩非线性力-位移滞回关系。基于Bouc-Wen模型,给出了一种考虑RC桥墩强度和刚度退化以及捏拢效应的改进非线性滞回模型,确定了影响该滞回模型的控制参数,并在Matlab/Simulink里实现对RC桥墩各种破坏模式非线性滞回性能预测。同时,通过对1/3比尺的RC桥墩分别在压弯、扭转和弯剪作用下的往复荷载试验研究,试验得到的力-位移关系滞回曲线与本文提出的改进滞回模型预测结果吻合较好,验证了本文提出改进模型的正确性和有效性。并采用汶川地震波对绵州市回澜立交桥匝道桥的RC桥墩的滞后性能进行了数值模拟,分析结果和桥墩的震害一致,证实了采用本文提出的改进滞回模型可以较准确的预测RC桥墩在强震作用下的非线性滞后性能。在此基础上,运用无迹卡尔曼（UKF）方法对Bouc-Wen模型进行参数估计,模拟结果和误差分析表明,该方法能够精确的估计出改进的Bouc-Wen模型的参数。