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.     

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.     

Kinematic hardening rules with critical state of dynamic recovery,part I: formulation and basic features for ratchetting behavior

Kinematic hardening rules formulated in a hardening/dynamic recovery format are examined for simulating rachetting behavior. These rules, characterized by decomposition of the kinematic hardening variable into components, are based on the assumption that each component has a critical state for its dynamic recovery to be activated fully. Discussing their basic features, the authors show that they can predict much less accumulation of uniaxial and multiaxial ratchetting strains than the Armstrong and Frederick rule. Comparisons with multilayer and multisurface models are made also, resulting in a finding that the simple one in the present rules is similar to the multilayer model with total strain rate replaced by inelastic (or plastic) strain rate. Part II of this work deals with applications to experiments.     

Modified kinematic hardening rule for multiaxial ratcheting prediction

A modified kinematic hardening rule is proposed in which one biaxial loading dependent parameter δ′ connecting the radial evanescence term [(α:n)ndp] in the Burlet–Cailletaud model with the dynamic recovery term of Ohno–Wang kinematic hardening rule is introduced into the framework of the Ohno–Wang model. Compared with multiaxial ratcheting experimental data obtained on 1Cr18Ni9Ti stainless steel in the paper and CS1026 steel conducted by Hassan et al. [Int. J. Plasticity 8 (1992) 117], simulation results by modified model are quite well in all loading paths. The simulations of initial nonlinear part in ratcheting curves can be improved greatly while the evolutional parameter δ′ related to plastic strain accumulation is added into the modified 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.     

Effect of the rate dependence of nonlinear kinematic hardening rule on relaxation behavior

Relaxation experiments for metallic materials and solid polymers have exhibited nonlinear dependence of stress relaxation on prior loading rate; the relaxed stress associated with the fastest prior strain rate has the smallest magnitude at the end of the same relaxation periods. Modeling capability for the basic feature of relaxation behavior is qualitatively investigated in the context of unified state variable theory. Unified constitutive models are categorized into three general classes according to the rate dependence of kinematic hardening rule, which defines the evolution of the back (equilibrium) stress and is the major difference among constitutive models. The first class of models adopts the nonlinear kinematic hardening rule proposed by Armstrong and Frederick. In this class, the back stress appears to be rate-independent under loading and subsequent relaxation conditions. In the second class of models, a stress rate term is incorporated into the Armstrong–Frederick rule and the back stress then becomes rate-dependent during relaxation condition even though it remains rate-independent under loading condition. The final class proposed here includes a new nonlinear kinematic hardening rule that causes the back stress to be rate-dependent all the time. It is shown that the apparent rate dependence of the back stress during relaxation enables constitutive models to predict the influence of prior loading rate on relaxation behavior.     

新型铝锡硅合金高温塑性变形流变应力的研究

采用高温等温压缩变形方法,在温度为373-673K范围和应变速率为0.001-1.0s^-1范围内,测定了新型Al-10Sn-4Si合金的流变应力曲线,结果表明,该合金为正应变速率敏感材料并且具有稳态流变特征;稳态流变应力随变形速率的增加而增大,随变形温度的升高而降低,通过回归分析,建立了该合金高温塑性变形时稳态流变应力的半经验方程,这种稳态流变特征与动态回复、动态再结晶及局部晶界粘滞性流动行为有关,受热激活过程控制。     

A crystal plasticity model that incorporates stresses and strains due to slip gradients

This work is concerned with incorporating the kinematic and stress effects of excess dislocations in a constitutive model for the elastoplastic behavior of crystalline materials. The foundation of the model is a three term multiplicative decomposition of the deformation gradient in which the two classical terms of plastic and elastic deformation are included along with an additional term for long range strain due to the collective effects of excess dislocations. The long range strain is obtained from an assumed density of Volterra edge dislocations and is directly related to gradients in slip. A new material parameter emerges which is the size the region about a continuum point that contributes to long range strains.Using Hookean elasticity, the stress at a point is linearly related to the sum of the elastic plus the long range strain fields. However, the driving force for slip is postulated to be due only to the elastic stress so that the long range stress is a back stress in the constitutive relationship for plastic deformation. A consistent balance of the total deformation rate with the three proposed mechanisms of deformation leads to a set of differential equations that can be solved for the elastic stress, rotation and pressure which then implicitly defines the material state and equilibrium stress. Results from the simulation of a tapered tensile specimen demonstrate that the constitutive model exhibits isotropic and kinematic type hardening effects as well as changes in the pattern of plastic deformation and necking when compared to a material without slip gradient effects.     

Crack tip asymptotic character of anti-plane stress and strain rate for linear fractional constitutive relations

Linear-fractional strain rate and stress relations are used to simulate materials undergoing steady state creep. The crack tip asymptotic character of the stress and strain rate field is obtained in exact and approximate form. In the limit as the radial distance emanating from the crack tip approaches zero, the stress field corresponds to that for an ideal plastic material while the exact and approximate solutions tend to coincide. Discussed is the nonhomogeneous singular character of the strain rate field that possess different orders of singularities in a circular region around the crack tip.     

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.     

Creep,plasticity, and fatigue of single crystal superalloy

Single crystal components in gas turbine engines are subject to such extreme temperatures and stresses that life prediction becomes highly inaccurate resulting in components that can only be shown to meet their requirements through experience. Reliable life prediction methodologies are required both for design and life management. In order to address this issue we have developed a thermo-viscoplastic constitutive model for single crystal materials. Our incremental large strain formulation additively decomposes the inelastic strain rate into components along the octahedral and cubic slip planes. We have developed a crystallographic-based creep constitutive model able to predict sigmoidal creep behavior of Ni base superalloys. Inelastic shear rate along each slip system is expressed as a sum of a time dependent creep component and a rate independent plastic component. We develop a new robust, computationally efficient rate-independent crystal plasticity approach and combined it with creep flow rule calibrated for Ni-based superalloys. The transient variation of each of the inelastic components includes a back stress for kinematic hardening and latent hardening parameters to account for the stress evolution with inelastic strain as well as the evolution for dislocation densities. The complete formulation accurately predicts both monotonic and cyclic tests at different crystallographic orientations for constant and variable temperature conditions (low cycle fatigue (LCF) and thermo-mechanical fatigue (TMF) tests). Based on the test and modeling results we formulate a new life prediction criterion suitable for both LCF and TMF conditions.     

不同加载状态下TA2钛合金绝热剪切破坏响应特性

一般认为绝热剪切现象在宏观上表现为材料动态本构失稳,即热软化大于应变硬化.本文采用帽型受迫剪切试样研究TA2钛合金的动态力学特性和本构失稳过程.首先对剪切区加载应力状态进行理论和数值分析,通过合理设计帽型试样,剪切区变形可近似按剪切状态处理;结合二维数字图像相关法(two-dimensional digital image correlation,DIC-2D)直接测试试样剪切区应变演化,给出帽型受迫剪切实验的等效应力-应变响应曲线.进一步,利用Hopkinson压杆对TA2钛合金开展动态压缩及帽型剪切对比试验研究,比较压缩、剪切试验得到的等效应力-应变曲线,采用"冻结"试样方法分析试样中绝热剪切局域化演化过程,探讨不同加载状态下TA2钛合金的绝热剪切破坏现象及其动态力学响应特性.实验结果表明,在塑性变形初始阶段,动态压缩及剪切加载下的等效应力-应变曲线符合较好,但随塑性损伤发展及绝热剪切带形成,两者出现分离,表明损伤及绝热剪切演化过程与应力状态相关.剪切试样实验得到的本构"软化"特性能够反映绝热剪切带起始、破坏演化过程的力学响应特性,而在动态压缩实验中,即使试样中已出现双锥形的绝热剪切带及局部裂纹分布,其表观等效应力-应变曲线并不出现软化特征,动态压缩实验无法得到关于绝热剪切起始、发展以及破坏的本构软化响应特性.     

A non-equilibrium thermodynamic geometric structure for thermoviscoplasticity with maximum dissipation

A maximum dissipation principle induces a class of viscoplastic evolution equations within a non-linear, non-equilibrium thermodynamic structure which extends Gibbs thermostatics. The evolution of a relaxation process is determined through non-linear affinities, which generalize the linear Onsager construct, along with the maximum dissipation principle and the long term states. The relaxation modulus is the norm of the plastic strain rate. Forced processes, in which the control variables change with time, are assumed to be relaxation processes at each instant. This class of thermoviscoplastic models includes the three-dimensional Freed–Chaboche–Walker model in which the internal state variables are the back stress, the drag strength and the elastic yield strength for creep initiation.     

A simple plasticity model for prediction of non-coaxial flow of sand

A bounding surface plasticity model for non-coaxiality, another aspect of anisotropic behavior of sands under rotation of principal stress axes; is developed in the critical state framework. Numerous experimental evidences exist that corroborate dependence of plastic shear strain rate direction on inherent fabric anisotropy. At first, general expressions for plastic strain rate with respect to possible emerge of non-coaxial flow are obtained. Consequently, using an anisotropy state parameter that is specially developed for this model and accounts for the interaction between imposed loading and soil fabric; effect of anisotropy on plastic flow direction is taken into account. Besides, novel circumstances are proposed for plastic modulus and dilatancy under rotation of principal stress axes. Finally, it is shown that the model is able to simulate successfully the non-coaxial behavior of sands subjected to principal stress axes rotation.     

径向惯性对薄壁圆管中弹塑性复合应力波传播的影响

弹塑性压扭复合应力波在薄壁管中的传播特性,已得到较为深入的研究,但为得到简单波解,大部分研究忽略了薄壁圆管中与径向惯性有关的周向应力σθ的影响。该文采用便于动态数值方法应用的增量型弹塑性本构关系,应用有限差分数值方法,计算了考虑径向惯性效应的弹塑性薄壁管中复合应力波的演化规律和传播特性,并与无径向惯性效应的计算结果作了对比,结果表明薄壁管中的径向惯性效应对弹塑性复合应力的传播有较大的影响。     

Investigation on the plastic work-heat conversion coefficient of 7075-T651 aluminum alloy during an impact process based on infrared temperature measurement technology

The plastic work-heat conversion coefficient is one key parameter for studying the work-heat conversion under dynamic deformation of materials. To explore this coefficient of 7075-T651 aluminum alloy under dynamic compression, dynamic compression experiments using the Hopkinson bar under four groups of strain rates were conducted, and the temperature signals were measured by constructing a transient infrared temperature measurement system. According to stress versus strain data as well as the corresponding temperature data obtained through the experiments, the influences of the strain and the strain rate on the coefficient of plastic work converted to heat were analyzed.The experimental results show that the coefficient of plastic work converted to heat of 7075-T651 aluminum alloy is not a constant at the range of 0.85–1 and is closely related to the strain and the strain rate. The change of internal structure of material under high strain rate reduces its energy storage capacity, and makes almost all plastic work convert into heat.     

What is behind the plastic strain rate?

The plastic strain rate plays a central role in macroscopic models on elasto-viscoplasticity. In order to discuss the concept behind this quantity, we propose, first, a kinetic toy model to describe the dynamics of sliding layers representative of plastic deformation of single crystalline metals. The dynamic variable is given by the distribution function of relative strains between adjacent layers, and the plastic strain rate emerges as the average hopping rate between energy wells. We demonstrate the behavior of this model under different deformations and how it captures the elastic-to-plastic transition. Second, the kinetic toy model is reduced to a closed evolution equation for the average of the relative strain, allowing us to make a direct link to macroscopic theories. It is shown that the constitutive relation for the plastic strain rate does not only depend on the stress, but also on the macroscopic applied deformation rate, contrary to common practice.