Kinematic hardening rules for modeling uniaxial and multiaxial ratcheting

Ratcheting, which is the strain accumulation observed under the unsymmetrical stress controlled loading and non-proportional loadings, is modeled using the simplified viscoplasticity theory based on overstress (VBO). The influences of kinematic hardening laws on the uniaxial and multiaxial non-proportional ratcheting behavior of CS 1026 carbon steel have been investigated. The following kinematic hardening rules have been considered: the classical kinematic hardening rule, the kinematic hardening rules introduced by Armstrong–Frederick, Burlet–Cailletaud and the modified Burlet–Cailletaud. The investigated loading conditions include uniaxial stress controlled test with non-zero mean stress, and axial strain controlled cyclic test of thin-walled tubular specimen in the presence of constant pressure. Numerical results are compared with the experimental data obtained by Hassan and Kyriakides [Hassan T, Kyriakides S. Ratcheting in cyclic plasticity, part I: uniaxial behavior. Int J Plast 1992;8:91–116] and Hassan et al. [Hassan T, Corona E, Kyriakides S. Ratcheting in cyclic plasticity, part I: multiaxial behavior. Int J Plast 1992;8:117–146]. It is observed that all investigated kinematic hardening rules do not improve ratcheting behavior under multiaxial loading, but over-prediction still exists.     

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.     

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.     

Stabilized cyclic stress-strain calculation for metals under multiaxial loading

A simple plasticity model for modeling the stabilized cyclic stress-strain responses is developed to consider the effect of non-proportional additional hardening. In the proposed model, the plastic modulus for uniaxial loading is extended to multiaxial loading by introducing the non-proportionality factor and the additional hardening coefficient. The two introduced factors take into account the effects of non-proportional additional hardening, not only on the shape of the loading path, but also on the material and its microstructure. And then, the basic Armstrong-Frederick nonlinear hardening rule is modified to model the evolution of the back stress. The consistency condition is enforced to obtain the relationship between the back stress and plastic modulus. The proposed model requires only six material constants for estimating the stabilized responses. Comparisons between the test results (30CrNiMo8HH steel, SA 333 Gr.6 steel, and 1 %CrMoV steel) and model predictions show that the proposed model predicts relatively accurate stress responses under both proportional and non-proportional loading paths.     

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.     

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.     

Experimental and numerical analyses of mean stress relaxation in cold expanded plate of Al‐alloy 2024‐T3 in double shear lap joints

In this paper, the mean stress relaxation behavior of simple Al‐alloy 2024‐T3 specimens and also the mean stress relaxation around the hole of cold expanded specimen are studied. The analyses are performed through the combination of the nonlinear isotropic hardening and Chaboche nonlinear kinematic hardening model accompanied by the results of experimental tests. The strain‐controlled axial tests are performed at two different strain amplitudes, while the stress‐controlled tests of cold expanded specimens are performed for three different imposed load amplitudes. The constitutive equations of the hardening model are coded as a UMAT subroutine in FORTRAN programming language and implemented in the commercial finite element code of ABAQUS. The accuracy of the hardening model has been proved in two steps: first by simulations of mean stress relaxation during the uniaxial strain‐controlled cyclic tests and second by simulation of strain ratcheting during the stress‐controlled cyclic loading. The stress and strain distributions after cold expansion process are examined as well as the mean stress relaxation due to cyclic loading. The results show the influences of imposed stress amplitude on increasing mean stress relaxation and also the effect of cold expansion level on reducing the mean stress relaxation.     

Springback investigation of anisotropic aluminum alloy sheet with a mixed hardening rule and Barlat yield criteria in sheet metal forming

A mixed hardening model has been implemented based on Lemaitre and Chaboche non-linear kinematic hardening theory to consider cyclic behavior and the Bauschinger effect. The Chaboche isotropic hardening theory is incorporated into the non-linear kinematic hardening model to introduce a surface of nonhardening in the plastic strain space. The bending and reverse bending case study has verified the effectiveness of the mixed hardening model by comparison with the proposed experiment results. Barlat’89 yielding criterion is adopted for it does not has any limitation while Hill’s non-quadratic yield criterion is for the case that the principal axes of anisotropy coincides with principal stress direction. The Backward–Euler return mapping algorithm was applied to calculate the stress and strain increment. The mixed hardening model is implemented using ABAQUS user subroutine (UMAT). The comparisons with linear kinematic hardening model and isotropic hardening model in NUMISHEET’93 benchmark show that the mixed hardening model coupled with Barlat’89 yield criteria can well reflect stress and strain distributions and give a more favorable springback angle prediction.     

The choice of cyclic plasticity models in fatigue life assessment of 304 and 1045 steel alloys based on the critical plane-energy fatigue damage approach

The present study intends to examine various cyclic plasticity models in fatigue assessment of 304 and 1045 steels based on the critical plane-energy damage approach developed earlier. Cyclic plasticity models of linear hardening, nonlinear, multi-surface, and two-surface were chosen to study fatigue damage and life of materials under proportional and non-proportional loading conditions. The effect of additional hardening induced due to non-proportional loading in 1045 steel and particularly in 304 steel was further evaluated as different constitutive models were employed. In the present study, the plasticity models were calibrated by the equivalent cyclic stress–strain curves. The merits of the models were then investigated to assess materials deformation under proportional and non-proportional loading conditions. Under non-proportional loading, the cyclic plasticity models were found to be highly dependent upon the employed hardening rule as well as the materials properties/coefficients.The stress and strain components calculated through constitutive laws were then used as input parameters to evaluate fatigue damage and assess the fatigue life of materials based on the critical plane-energy approach.The calculated values of stress components based on constitutive laws resulted in a good agreement with those of experimentally obtained under various loading paths of proportional and non-proportional conditions in 1045 steels. In 304 steel, the calculated stress components were however found in good agreement when plasticity models were employed for proportional loading conditions. Under non-proportional loading, the application of the multi-surface plasticity model in conjunction with the fatigue damage approach resulted in more reasonable results as compared with other plasticity models. This can be attributed to the motion of the yield surface in deviatoric stress space in the multi-surface model encountering additional hardening effect through estimated higher stress values under non-proportional loading conditions.Predicted fatigue lives based on the critical plane-energy damage approach showed such range of agreements as ±1.05–±3.0 factors in 1045 and 304 steels as compared with experimental life data when various constitutive plasticity models were employed.     

LOW-CYCLE FATIGUE UNDER NON-PROPORTIONAL LOADING

A series of strain-controlled, low-cycle fatigue experiments have been conducted on 42CrMo steel under various loading paths including circular, square, cruciform, and rectangular paths. Present experiments have shown that there is additional hardening under non-proportional cyclic loading. Non-proportional cyclic additional hardening also results in a shorter life for multiaxial low cycle fatigue. A non-proportionality measure of strain path based on both a physical basis and macromechanical phenomena is proposed. The loading path effect on additional hardening is also described well. Low-cycle fatigue damage accumulation and the evolution process under non-proportional loading is analysed via the Continuum Damage Mechanics Model of Chaboche. A non-proportinality measure is introduced in the damage evolution equation and a modified Coffin-Manson type formula is derived. A novel fatigue life prediction approach based on the critical-plane concept of Brown and Miller is proposed.     

Uniaxial and non-proportionally multiaxial ratcheting of U71Mn rail steel: experiments and simulations

The ratcheting and strain cyclic characteristics of U71Mn rail steel were experimentally researched under uniaxial and non-proportionally multiaxial cyclic loading at room temperature. The effects of cyclic strain, stress and their histories on strain cyclic characteristics and ratcheting were studied, respectively. It is shown that: U71Mn rail steel exhibits a cyclic stabilization and non-memorization for previous loading history under strain cycling; however, the ratcheting of the material depends greatly not only on the current values of mean stress and stress amplitude, but also on their histories; the non-proportionality of multiaxial loading path only causes a negligible additional hardening for the material. Based on the Ohno–Wang non-linear kinematic hardening model [Int. J. Plast. 9 (1993) 375, 391], the uniaxial and multiaxial ratcheting behaviours of the material were simulated by a visco-plastic constitutive model. The simulated results are in good consistence with the experimental ones.     

A nonuniform hardening plasticity model for concrete materials

For the most part, the developments of constitutive models for for the concrete by the theory of plasticity have in the past been made to scarch for a suitable failure surface. The initial yield surface is usually assumed to have the same shape as the failure surface but with a reduced size. The subsequent loading surfaces are then obtained by the uniform expansion of the initial one. This approach is found generally inadequate in predicting the deformational behavior of concrete for a wide range of loading conditions.

The present non-uniform hardening plasticity model adopts the most sophisticated failure model of Willam-Warnke or Hsieh-Ting-Chen as the bounding surface; assume an initial yield surface with a shape that is different from the failure surface; proposes a nonuniform hardening rule for the subsequent loading surfaces with a hydrostatic pressure and Lode-angle dependent plasticity modulus; and utilizes a nonassociated flow rule for a general formulation.

The work-hardening stress-strain behaviors of concrete based on the present model are found in good agreement with experimental results involving a wide range of stress states and different types of concrete material. The important features of inelastic behavior of concrete, including brittle failure in tension; ductile behavior in compression; hydrostatic sensitivities; and volumetric dilation under compressive loadings can all be represented by this improved constitutive model.     

The role of a loading surface in viscoplasticity theory

Summary A multiaxial theory of viscoplasticity is further generalized to incorporate an isotropic loading surface and an anisotropic hardening rule for the unsymmetric cyclic distortion of the yield surface. The loading surface is defined as the smallest surface in stress space containing all the previous loading points; and the subsequent elastic region is assumed to be defined jointly by a yield surface and an isotropic surface which is contained by the loading surface and expands with a rate depending on the location of the loading point and the quasistatic loading point. The theory is applied to the modeling of stress-strain response and yield surface deformation of 1100-O aluminum subject to non-proportional loading in tension-torsion space.With 8 Figures     

A robust kinematic hardening rule for cyclic plasticity with ratchetting effects

Summary A robust kinematic hardening rule is proposed which appropriately blends the deviatoric stress rate rule and the Tseng-Lee rule in order to satisfy both the experimental observations made by Phillips et al. [1]–[5] and the nesting of the yield surface to the limit surface. The work presented in Part I is confined to the theoretical formulation of kinematic hardening rule with limited correlation to experimental results. A more general expression for the plastic modulus is proposed. The expressions proposed by McDowell and by Dafalias can be obtained as a special case of the proposed expression. An additional parameter is introduced that reflects the dependence of the plastic modulus on the angle between the deviatoric stress rate tensor and the direction of the limit backstress relative to the yield backstress.     

A mixed hardening rule coupled with Hill48’ yielding function to predict the springback of sheet U-bending

A mixed hardening model has been used to model the Bauschinger effect. This hardening model is based on Lemaitre and Chaboche nonlinear kinematic hardening theory to consider cyclic behavior and the Bauschinger effect. Hill’48 yielding criterion is used because of the general stress state and relative ease of formulation. The backward Euler return mapping algorithm is applied to calculate the stress and strain increment. The mixed hardening model is implemented based on UMAT subroutine of FEA code ABAQUS. The NUMISHEET’93 benchmark shows that the mixed hardening model coupled with anisotropic yield criteria can give a favorable springback angle prediction.     

高强轨道钢双轴压-扭棘轮行为的实验和模拟

在室温下对高强轨道钢进行了单轴和非比例双轴压-扭循环变形行为的实验,讨论了不同加载路径对轨道钢棘轮变形行为的影响。结果显示:该轨道钢呈现出明显的循环软化效应和压缩方向的棘轮行为,且棘轮行为的演化表现出强烈的加载路径相关性;在椭圆路径下,棘轮应变较其他四种路径更小。进而建立了基于Abdel Karim-Ohno非线性随动硬化律的非比例多轴循环棘轮本构模型,并通过在随动硬化和各向同性软化律中引入非比例因子来考虑非比例路径对双轴压-扭棘轮行为的影响。实验结果和模拟结果的对比表明:该本构模型能够较好地模拟高强度轨道钢的非比例双轴压-扭棘轮行为。     

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。