The strain path dependence of multiaxial cyclic hardening behaviour

The small-strain, isotropic deformation theory is used in incremental form to model an additional cyclic hardening for any arbitrary loading path. The theory is of the unified type and does not employ yield or loading/ unloading criteria. The scalar-valued functions involved in the tensorial constitutive equations as well a growth law for these functions are identified based on idea of the equivalent state. Definitions of equivalent stress and equivalent strain have been developed to correlate step by step loading programmes taking the history of deformation into account. Use is made of the total work increment together with an interpolation method for tensor functions to generalize the simple state to a multiaxial behaviour in the strain space for a given strain increment. For the demonstration of model capability, the numerical simulation is undertaken on cyclic nonproportional paths in two-dimensional axial-shear strain space. The results are verified for stainless steel and brass by comparison with the material response experimentally obtained in the stress space.     

Cumulative damage model based on equivalent fatigue under multiaxial thermomechanical random loading

A new creep–fatigue damage cumulative model is proposed under multiaxial thermomechanical random loading, in which the damage at high temperature can be divided into the pure fatigue damage and the equivalent fatigue damage from creep. During the damage accumulation process, the elementary percentage of the equivalent fatigue damage increment is proportional to that of the creep damage increment, and the creep damage is converted to the equivalent fatigue damage. Moreover, combined with a multiaxial cyclic counting method, a life prediction method is developed based on the proposed creep–fatigue damage cumulative model. In the developed life prediction method, the effects of nonproportional hardening on the fatigue and creep damages are considered, and the influence of mean stress on damage is also taken into account. The thermomechanical fatigue experimental data for thin‐walled tubular specimen of superalloy GH4169 under multiaxial constant amplitude and variable amplitude loadings were used to verify the proposed model. The results showed that the proposed method can obtain satisfactory life prediction results.     

Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading

An analytical study is carried out on the elastic–plastic stress and strain distributions and on the shape of the plastic zone ahead of parabolic notches under antiplane shear loading and small scale yielding. The material is thought of as obeying an elastic-perfectly-plastic or a strain hardening law. When the notch root radius becomes zero, the analytical frame matches the solutions for the crack case due to Hult–McClintock (elastic-perfectly-plastic material) and Rice (strain hardening material). The analytical frame provides an explicit link between the plastic stress and the elastic stress at the notch tip. Neuber’solution for blunt notches under antiplane shear is also obtained and the conditions under which such a solution is valid are discussed in detail by using elastic and plastic notch stress intensity factors. Finally, revisiting Glinka and Molski’s equivalent strain energy density (ESED), these factors are used also to give, under antiplane shear loading, the increment of the strain energy at the notch tip with respect to the linear elastic case.     

Automatic consistency integrations for the stress update of J2 elastoplastic rate constitutive equations with combined hardening

A novel method is introduced to study numerical integrations of J₂ elastoplastic rate constitutive equations with general combined hardening. The basic idea is to transform the usual time rate constitutive equations into those with reference to the equivalent plastic strain. By virtue of tensorial matrix operations, we show that these transformed equations may be converted to a linear differential system governing the shifted stress and the plastic multiplier. From this system, we derive explicit integrations for the shifted stress and then for the back stress and the Cauchy stress. We demonstrate that these results are accurate up to within a third order term of the equivalent plastic strain increment. In particular, for pure kinematic hardening, we show that the integrations obtained can achieve automatic enforcement of both the plastic consistency condition and the loading condition, thus bypassing the numerical treatment of the latter two. Furthermore, we explain that, with the new algorithm for the stress update, the continuum tangent moduli may be used to ensure a quadratic rate of convergency in Newton's iteration scheme for the balance equation. Numerical examples suggest that the new algorithm may be more accurate and efficient than the widely used return algorithm. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

A note on the combined isotropic–kinematic workhardening rule

The Ziegler modification of the Prager hardening rule has been used in conjunction with a combined isotropic–kinematic workhardening rule in order to predict the Bauschinger effect during cyclic loadings. There has been some controversy as to whether the above workhardening rule predicts a stress increment which is consistent with the yield condition. The purpose of this communication is to establish the combined hardening rule as a consistent statement in order to clarify its correctness.     

Studies on the fracture mechanics parameters of weldment with mechanical heterogeneity

The fully plastic solutions of welded centre-cracked strip for plane stress problem were carefully investigated with the fully plastic finite element method. It was introduced for assessing the fracture mechanics parameters of weldment with mechanical heterogeneity that there existed an equivalent yielding stress and equivalent strain hardening exponent in the vicinity of crack tip keeping the assessment of fracture mechanics parameters of weldment in the same way as the homogeneous material. The equivalent yielding stress and equivalent strain hardening exponent of various matched weldment were computed and the effect of weld metal width were calculated and discussed on equivalent yielding stress and equivalent strain hardening exponent near crack tip. The engineering approach was given for estimating the fracture mechanics parameters of weldment with mechanical heterogeneity in elastic-plastic range.     

Drucker-Prager弹塑性本构关系积分:考虑非关联流动与各向同性硬化

考虑非关联流动法则以及各向同性硬化条件,采用广义中点法(Generalized Midpoint Method,GMM)进行Drucker-Prager(DP)弹塑性本构关系数值积分,给出调整后最终应力的解析解.GMM 属于隐式算法,具有良好的计算精度与数值稳定性;最近点投影法(Closest Point Project Method,CPPM)是其特例,具有一阶精度并且无条件稳定.DP 塑性势函数的特殊性质导致上述GMM 解由初始应力状态与应变增量显式确定,无需迭代求解,因此计算效率大幅提高,同时避免了迭代过程的收敛性问题.数值算例证明:当加载偏离角度较大时,GMM(ξ=1/2)的计算精度高于CPPM,可适应更大的加载步长;而对于比例加载,任意GMM 等同于精确解,采用CPPM 可获得最高的计算效率.推导了满足DP 屈服准则厚壁圆筒的弹塑性理论解,对比验证算法精度.采用非关联流动各向同性线性硬化DP 材料模拟厚壁圆筒变形局部化效应.     

Dynamic precipitation during cyclic deformation of an underaged Al-Cu alloy

The cyclic deformation behavior of Al-4Cu alloy containing shear-resistant particles was investigated systematically as a function of precipitate state. Pronounced cyclic hardening was observed in the under aged Al-4Cu-0.05Sn (wt.%) alloy strained under various imposed plastic strain amplitudes at room temperature. Such cyclic hardening is absent from the longer aging treatments. Microstructural characterization reveals that the pronounced cyclic hardening of the under aged alloy is due to the dynamic precipitation of GP zones. The dynamic precipitation occurs during all the cyclic loading process and only at the peak stress, where the hardening increment from dynamic precipitation saturates, does strain localization occur which is soon followed by failure of the material. The dynamic precipitation of GP zones has a positive effect on the low cycle fatigue performance of this alloy, and can significantly elevate the strength of this alloy without loss in ductility. Experiments performed to test the dependence of the cyclic hardening on plastic strain amplitude and strain-rate illustrate a relatively strain-rate independent and strain amplitude dependent behavior. Such kinetic behavior is approximately consistent with that expected if the GP zone formation is controlled by the vacancies production process during plastic deformation.     

AN ANISOTROPIC STRESS RESULTANT CONSTITUTIVE LAW FOR SHEET METAL FORMING

Based on Hill's quadratic orthotropic yield function, a yield function in the stress resultant space is approximated in quadratic form for sheets with planar isotropy and normal anisotropy. An equivalent stress resultant is defined and the equivalent work-conjugate generalized plastic strain rate is then derived. A power-law hardening rule between the equivalent stress resultant and generalized plastic strain is obtained under proportional straining conditions. A hemispherical punch stretching operation and a plane-strain draw operation are simulated by finite element methods based on the stress resultant constitutive law. The results of these finite element simulations are in good agreement with those using the through-the-thickness integration method. The results also indicate that the computational time of the simulations based on the stress resultant constitutive law is much shorter than that based on the through-the-thickness integration method.     

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.     

A survey on multiaxial fatigue damage parameters under non‐proportional loadings

In this paper, several multiaxial fatigue damage parameters taking into account nonproportional additional hardening are reviewed. According to the way nonproportional additional hardening is considered in the model, the damage parameters are classified into 2 categories: (1) equivalent damage parameters and (2) direct damage parameters. The equivalent damage parameters usually define a nonproportional coefficient to consider nonproportional additional cyclic hardening, and make a combination of this nonproportional coefficient with stress and/or strain quantities to calculate the equivalent damage parameters. In contrast, the direct damage parameters are directly estimated from the stress and strain quantities of interest. The accuracy of 4 multiaxial fatigue damage parameters in predicting fatigue lifetime is checked against about 150 groups of experimental data for 10 different metallic materials under multiaxial fatigue loading. The results revealed that both Itoh's model, one of equivalent damage parameters, and Susmel's model, which belong to direct damage parameters, could provide a better correlation with the experimental results than others assessed in this paper. So direct damage parameters are not better than the equivalent damage parameters in predicting fatigue lifetime.     

Study on the dislocation sub-structures of Al–Mg–Si alloys fatigued under non-proportional loadings

In this article, the fatigue dislocation sub-structures of Al–Mg–Si alloys were studied under the same equivalent stress amplitude of 93 MPa with multi-axial non-proportional loading paths, which were circle, ellipse, rectangle, and square loading paths. After fatigue test, the fatigue dislocation sub-structures of the failure specimens were also observed under different equivalent stress amplitudes, which were 47, 70, 90, 140, and 163 MPa with the transmission electron microscopy method. The results indicate that fatigue lives of the alloy are different under different loading paths with the same equivalent stress amplitude, and the fatigue life decreases with the increase in equivalent stress amplitude from 47 to 163 MPa. Cyclic additional hardening is found above medium effective stress amplitude of 93 MPa. The dislocation sub-structures strongly depend on equivalent stress amplitudes and loading paths. The dislocation sub-structures of parallel slip band, slight small dislocation cell (or labyrinth), and remarkable labyrinth feature are found under the equivalent stress amplitudes of 47, 93, and 140 MPa, respectively. The movability of the dislocation and the stress concentration degree are closely related to fatigue life and cyclic hardening of the alloy.     

Implementation of cyclic plasticity models based on a general form of kinematic hardening

This paper deals with implementation of cyclic plastic constitutive models in which a general form of strain hardening and dynamic recovery is employed to represent the multilinear, as well as non‐linear, evolution of back stress. First, in order to incorporate such a general form of kinematic hardening in finite element methods, successive substitution and its convergence are discussed for implicitly integrating stress; moreover, a new expression of consistent tangent modulus is derived by introducing a set of fourth‐rank constitutive parameters into discretized kinematic hardening. Then, the constitutive parameters introduced are specified in three cases of the general form of kinematic hardening; the three cases have distinct capabilities of simulating ratcheting and cyclic stress relaxation. Numerical examples are given to verify the convergence in successive substitution and the new expression of consistent tangent stiffness. Error maps for implicitly integrating stress under non‐proportional as well as proportional loading are also given to show that the multilinear case of the general form provides high accuracy even if strain increment is very large. Copyright © 2002 John Wiley & Sons, Ltd.     

热等静压对圆形加载路径下A319铝合金多轴疲劳特性的影响

采用MTS809拉扭复合疲劳试验机、扫描电镜(SEM)研究了热等静压(Hot isostatic pressing,HIP)处理前后,圆形加载路径下,A319铝合金多轴疲劳特性。结果表明,HIP处理后,材料中的孔洞缺陷数量减少,疲劳失效过程中产生的微裂纹的数量减少且尺寸减小。相同等效应变幅值下产生的轴向应力幅值、切向应力幅值、等效应力幅值均显著增加。材料轴向上表现为先产生循环硬化而后循环软化,切向则表现为先产生循环硬化后趋于循环稳定,HIP处理前后循环软化硬化趋势大致相同。轴向应力应变滞后回线、切向扭矩扭角滞后回线面积有所降低,附加强化效果增强。     

Low-cycle fatigue model of damage accumulation - The strain approach

The paper presents a model of damage accumulation designed to analyse fatigue life of structural elements exploited in multiaxial, non-proportional, low-cycle loading conditions. The discussed approach consists of two calculation blocs. In the first bloc the components of stress and strain tensor are determined. This module, in which Mroz’s multisurface model was used, contains constitutive relations and the law of kinematic hardening. The second bloc contains the dependencies which determine the growth of anisotropic measure of damage accumulation (associated with the physical plane) and crack initiation criterion. The growth of the damage accumulation measure was associated with the loading damage accumulation function and the increment of non-dilatational plastic strain on the physical plane. It was assumed that crack initiation occurs when stress or a measure of damage accumulation on any physical plane reaches critical values.     

A complete stress update algorithm for the non-associated Drucker-Prager model including treatment of the apex

Numerical techniques based on convex analysis are applied to the non-associated Drucker-Prager model (without hardening) for which the plastic behavior is completely described by a unique function, called bi-potential. Among advantages of the present approach, motivated by mechanical considerations, a variational stress update algorithm along with coupled extremum principles can be derived. The time-integration algorithm is considered in detail and it is shown how the method can conveniently treat the singular point present in the Drucker-Prager model (apex). The existence of weak extremum principles allows using Mathematical Programming techniques and thereby obtains a robust algorithm even in the presence of large time increment and strong non-associativity. Numerical examples of incremental limit analysis for both the associated and the non-associated cases are presented.     

Non-local line method for notched elements with use of effective length calculated in an elasto-plastic condition

The paper presents a non-local line method used to the fatigue life calculation of notched elements. The presented method is based on the concept of an effective length which determines the size of the equivalent fatigue zone. Effective values of normal stress calculated in the critical plane with a weight function were applied when determining the equivalent fatigue zone. Simulation studies were performed for two types of steel and two types of loading. Five different series of tests and simulations were used. Experimental studies were carried out for 40 HM-T and EA4T steels. These materials are used in railway industry, including the manufacturing of coupling bars. The notched test specimens contained notches with a tip radius of 0.2, 0.5, 0.8 and 1 mm. Stress calculations were performed using the finite element method by adopting cyclic material properties described by the model of a multi-linear hardening. Non-local calculations were performed in a defined critical plane for normal stress distribution and a weight function. As a result, the function of variation of the effective length depending on the loading level and geometry of the notch has been determined.     

Effect of alternating stress on the creep of frozen soils

Cyclic creep behaviour of a frozen saturated sand and frozen clay was studied by superimposing alternating stresses over a mean static compressive stress on the samples. Cyclic loading increased the creep rate of these materials considerably, which could be due to the increase in unfrozen water content generated by the transient thermal energy produced by mechanical cycling. The strain increment per cycle was found to increase with increasing strain in the frozen sand; whereas it decreased in the frozen clay, showing a tendency for hardening. The difference could be due to the fact that the mean stress on the sample was larger than a threshold stress in the former and smaller in the latter material.     

Effect of elastic–plastic matrix on thermo-mechanical response of continuous anisotropic fiber-reinforced composites

Based on a concentric cylinder model, the analytical elastic–plastic solution of deformations and stresses for the composites reinforced with transversely isotropic, circumferentially orthotropic and radially orthotropic fibers subjected to axisymmetric thermo-mechanical loading is developed. How the plasticity, volume fraction, physical and mechanical properties of the matrix affect the elastic–plastic thermo-mechanical response of the composites is investigated. The plasticity of the matrix decreases greatly the axial compressive stress in the matrix, but more noticeably increases the axial compressive stress in the fiber. For the composites reinforced with transversely isotropic, circumferentially orthotropic and radially orthotropic fibers, decreasing the volume fraction, thermal expansion coefficient and Young's modulus, and increasing the yield stress and hardening parameter of the matrix can lower the maximum equivalent stress of the fiber. However, increasing the yield stress and hardening parameter of the matrix raises the maximum equivalent stress of the matrix.