Non-proportional Cyclic Hardening Research Articles

Micromechanics-Based Low Cycle Fatigue Life Prediction Model of ECAPed Aluminum Alloy

Ultrafine-grained aluminum alloys (UFG AA) show great potential in the design of fatigue-resistant lightweight alloys, and the methodology to assess low-cycle fatigue (LCF) life remains to be studied. In this work, a micromechanics-based LCF life prediction model is presented by conducting crystal plasticity finite element simulation (CPFEM). The fatigue indicator parameter (FIP) of maximum accumulated equivalent plastic strain energy, modulated by triaxiality, is developed to assess the material damage in the microstructure. Particularly, a new multiaxial strain parameter is proposed by considering the combined influence of the mean strain and non-proportional cyclic additional hardening effect, and then directly embedding into the cyclic J-integral. Finally, the reformulated Manson-Coffin relationship is theoretically constructed by correlating the crack tip opening displacement to the crack propagation equation. The results show the scatter fatigue life of UFG AA6061 is not only related to the inhomogeneous evolution of plastic deformation but also to the local stress state. Since the proposed approach considers both the deformation mechanisms at the micro-scale and the corresponding macroscopic responses, it can predict the LCF life of UFG AA with reasonable accuracy.

An Approach for Predicting the Low-Cycle-Fatigue Crack Initiation Life of Ultrafine-Grained Aluminum Alloy Considering Inhomogeneous Deformation and Microscale Multiaxial Strain.

In this paper, a low-cycle-fatigue (LCF) crack initiation life prediction approach that explicitly distinguishes nucleation and small crack propagation regimes is presented for ultrafine-grained (UFG) aluminum alloy by introducing two fatigue indicator parameters (FIPs) at the grain level. These two characterization parameters, the deformation inhomogeneity measured by the standard deviation of the dot product of normal stress and longitudinal strain and the microscale multiaxial strain considering the non-proportional cyclic additional hardening and mean strain effect, were proposed and respectively regarded as the driving forces for fatigue nucleation and small crack propagation. Then, the nucleation and small crack propagation lives were predicted by correlating these FIPs with statistical variables and cyclic J-integrals, respectively. By constructing a microstructure-based 3D polycrystalline finite element model with a free surface, a crystal plasticity finite element-based numerical simulation was carried out to quantify FIPs and clarify the role of crystallographic anisotropy in fatigue crack initiation. The numerical results reveal the following: (1) Nucleation is prone to occur on the surface of a material as a result of it having a higher inhomogeneous deformation than the interior of the material. (2) Compared with the experimental data, the LCF initiation life of UFG 6061 aluminum alloy could be predicted using the new parameters as FIPs. (3) The predicted results confirm the importance of considering the fatigue behavior of nucleation and small crack propagation with different deformation mechanisms for improving the fatigue crack initiation life prediction accuracy.

Determination of the Non-Proportional Cyclic Hardening Coefficient Sensitive to the Loading Amplitude

The research on determination of sensitivity of metal materials to non-proportional cyclic loading and the accompanying effects is analyzed. The indirect methods for determining the coefficient of non-proportional cyclic hardening of materials based on simple experiments are considered. A comparative analysis of methods for predicting non-proportional cyclic hardening for out-of-phase with a 90°displacement of biaxial cyclic loading with strain control is performed. It is shown that methods that use only the parameters of the stress–strain curve of materials are less effective, since they do not take into account the effect of cyclic loading amplitude on non-proportional hardening, which is quite significant for metals and alloys with face-centered cubic microstructure. Analysis of experimental data obtained at cyclic non-proportional deformation of stainless and carbon steels, aluminum and titanium alloys showed that the coefficient of non-proportional cyclic hardening varies depending on the loading amplitude. For most of the studied materials with face-centered cubic structure, there is a tendency to increase this hardening coefficient with increasing amplitude of cyclic loading and to its stability or slight decrease for metals with volume-centered cubic structure. Based on the behavior of the vast majority of materials, a logarithmic functional dependence is chosen to take into account the effect of the strain amplitude on the non-proportional cyclic hardening. A modification of the key parameter for determining mentioned hardening is proposed, taking into account both the characteristics of the stress–strain curve and the dimensionless value of the cyclic strain amplitude. It is shown that its use allows one to obtain a better correlation of experimental data with the approximation dependence proposed earlier by the author. The results of determining the coefficients of non-proportional cyclic hardening due to the improved approach meet the requirements of engineering use.

An improved Armstrong–Frederick-Type Plasticity Model for Stable Cyclic Stress–Strain Responses Considering Nonproportional Hardening

This paper modified an Armstrong–Frederick-type plasticity model for investigating the stable cyclic deformation behavior of metallic materials with different sensitivity to nonproportional loadings. In the modified model, the nonproportionality factor and nonproportional cyclic hardening coefficient coupled with the Jiang–Sehitoglu incremental plasticity model were used to estimate the stable stress–strain responses of the two materials (1045HR steel and 304 stainless steel) under various tension–torsion strain paths. A new equation was proposed to calculate the nonproportionality factor on the basis of the minimum normal strain range. Procedures to determine the minimum normal strain range were presented for general multiaxial loadings. Then, the modified model requires only the cyclic strain hardening exponent and cyclic strength coefficient to determine the material constants. It is convenient for predicting the stable stress–strain responses of materials in engineering application. Comparisons showed that the modified model can reflect the effect of nonproportional cyclic hardening well.

Prediction of non-proportional cyclic hardening and multiaxial fatigue life for FCC and BCC metals under constant amplitude of strain cycling

The fatigue lives are reduced accompanying an additional cyclic hardening under strain controlled non-proportional cyclic loading in which principal directions of stress and strain are altered within a cycle. This study predicts non-proportional cyclic hardening and multiaxial fatigue life for several BCC and FCC metals under constant amplitude strain cycling. A novel procedure to determine non-proportional cyclic hardening form uniaxial tensile properties has discussed in this study. Standard plastic strain energy density based fatigue criteria with considering the non-proportional cyclic hardening effect successfully predicts multiaxial fatigue lives. The predictions of non-proportional cyclic hardening and multiaxial fatigue life through models are validated by experimental results of various BCC and FCC metals which are collected from literatures.

Fatigue life prediction for some metallic materials under constant amplitude multiaxial loading

Based on the critical plane approach, the Sun-Shang-Bao (SSB) model is analyzed and verified. It is discovered that SSB model cannot take the non-proportional cyclic hardening into account and gives non-conservative fatigue life predictions under the non-proportional loading. To solve this problem, a stress-correlated factor is introduced to describe the degree of the non-proportional cyclic hardening as well as the effect of the non-zero mean stress. The accuracy of the proposed method is systematically checked against the experimental data found in literature for 16 different materials under constant amplitude multiaxial loading paths.

The non-proportional behaviour of a nickel-based superalloy at room temperature, and characterisation of the additional hardening response by a modified cyclic hardening curve

Multi-axial deformation strain-controlled testing was completed at Swansea University to characterise the non-proportional hardening response of a nickel-based superalloy. The data was used to assess the validity of a non-proportional cyclic hardening curve correction to represent the hardening response of this alloy. A square strain path, equivalent to a fully out-of-phase loading, was found to exhibit additional hardening at moderate peak applied strains compared to the proportional in-phase case. This additional hardening was not observed at low strain levels or at a test condition applying a lower level of non-proportionality in the applied strain path, and was dependent on strain range.

A Three-Parameter Model for Predicting Fatigue Life of Ductile Metals Under Constant Amplitude Multiaxial Loading

In this article, a fatigue damage parameter is proposed to assess the multiaxial fatigue lives of ductile metals based on the critical plane concept: Fatigue crack initiation is controlled by the maximum shear strain, and the other important effect in the fatigue damage process is the normal strain and stress. This fatigue damage parameter introduces a stress-correlated factor, which describes the degree of the non-proportional cyclic hardening. Besides, a three-parameter multiaxial fatigue criterion is used to correlate the fatigue lifetime of metallic materials with the proposed damage parameter. Under the uniaxial loading, this three-parameter model reduces to the recently developed Zhang’s model for predicting the uniaxial fatigue crack initiation life. The accuracy and reliability of this three-parameter model are checked against the experimental data found in literature through testing six different ductile metals under various strain paths with zero/non-zero mean stress.

An Instantaneous Power of Strain Approach to the Calculation of Elastoplastic Strain Energy Density in SUS 304 Steel

Presenting a method of identifying and calculating the elastoplastic strain energy density with the instantaneous power of strain, authors of this paper propose using it as a fatigue life/parameter for materials prone to nonproportional cyclic hardening effect.

Multiaxial fatigue: An overview and some approximation models for life estimation

A brief overview of some important issues in multiaxial fatigue and life estimation is presented. These include damage mechanisms and damage quantification parameters, material constitutive response and non-proportional hardening, cycle counting and damage accumulation in variable amplitude loading, and mixed-mode crack growth. It is shown that capturing the correct damage mechanism is essential to develop a proper damage quantification parameter for robust multiaxial fatigue life estimation. Additional cyclic hardening of some materials under non-proportional multiaxial loading and its dependence on the load path as well as material microstructure is also discussed. It is argued that critical plane damage models with both stress and strain terms are most appropriate since they can reflect the material constitutive response under non-proportional loading. Importance of a proper cycle counting method to identify cycles in a variable amplitude load history, and capability of the linear cumulative damage rule to sum damage from the counted cycles are also discussed. As mixed-mode crack growth can constitute a significant portion of the total fatigue life, analysis of crack growth rates and correlations under combined stresses is presented. Several models as well as some simple approximations in capturing the aforementioned effects in multiaxial fatigue life estimations are also described. The approximation models include an estimation model for obtaining material non-proportional cyclic hardening coefficient, and a simple multiaxial fatigue life estimation model for steels based on hardness as the only required material property.

Multiaxial fatigue evaluation using discriminating strain paths

Fatigue life and available cycle counting methodologies based on the critical plane approach are examined under discriminating axial-torsion strain paths with random and incremental changes in straining direction. Fatigue lives for quenched and tempered 1050 steel with no non-proportional hardening were found to be more sensitive to non-proportionality of loadings as compared to 304L stainless steel with significant non-proportional hardening. Proportional or in-phase axial-torsion cycles with different axial to shear strain ratios within an equivalent strain circle when applied in a random sequence resulted in significant additional hardening for 304L stainless steel, similar to the non-proportional cyclic hardening observed in 90° out-of-phase loading. In contrast, when such cycles are applied with a gradual increment of the axial to shear strain ratio, the stress response of 304L stainless steel is closer to that observed for in-phase loading. However, the sequence of loading did not significantly affect fatigue life for either material. Experimentally observed failure planes for all strain paths were in very good agreements with predicted failure planes based on the Fatemi–Socie critical plane parameter. Finally, fatigue lives for both materials under various strain paths were predicted satisfactorily employing Fatemi–Socie parameter, Palmgren–Miner linear damage rule, and either Bannantine–Socie or Wang–Brown cycle counting method.

Effect of microstructure and hardness on non-proportional cyclic hardening coefficient and predictions

Abstract This paper investigates the effects of microstructure and hardness on non-proportional cyclic hardening of metallic materials. Constant amplitude in-phase and 90° out-of-phase strain-controlled axial-torsion cyclic tests were conducted to evaluate the hardening. Tubular specimens made from 1050 steel in normalized, quenched and tempered, and induction hardened conditions as well as 304L stainless steel were used to study the effect of microstructure on multiaxial cyclic deformation. Reductions in the non-proportional cyclic hardening were observed as the microstructure of 1050 steel changed form pearlitic–ferritic with lower hardness to tempered martensite with higher hardness. Significant non-proportional cyclic hardening was also observed for 304L stainless steel with austenitic microstructure. Multiaxial data generated in this study as well as multiaxial deformation data of several materials from literature suggest non-proportional cyclic hardening can be related to uniaxial cyclic hardening. Non-proportional hardening coefficients predicted from a proposed equation based on this observation were found to be in very good agreement with the experimental values in this study and from the literature.

Thermodynamic based model for the evolution equation of the backstress in cyclic plasticity

A nonlinear kinematic hardening rule is developed here within the framework of thermodynamic principles. The derived kinematic hardening evolution equation has three distinct terms: two strain hardening terms and a dynamic recovery term that operates at all times. The proposed hardening rule, which is referred in this paper as the FAPC (Fredrick and Armstrong–Phillips–Chaboche) kinematic hardening rule, shows a combined form of the Frederick and Armstrong backstress evolution equation, Phillips evolution equation, and Chaboche series rule. A new term is incorporated into the Frederick and Armstrong evolution equation that appears to have agreement with the experimental observations that show the motion of the center of the yield surface in the stress space is directed between the gradient to the surface at the stress point and the stress rate direction at that point. The model is further modified in order to simulate nonproportional cyclic hardening by proposing a measure representing the degree of nonproportionality of loading. This measure represents the topology of the incremental stress path. Numerically, it represents the angle between the current stress increment and the previous stress increment, which is interpreted through the material constants of the kinematic hardening evolution equation. This new kinematic hardening rule is incorporated in a material constitutive model based on the von Mises plasticity type and the Chaboche isotropic hardening type. Numerical integration of the incremental elasto-plastic constitutive equations is based on a simple semi-implicit return-mapping algorithm and the full Newton–Raphson iterative method is used to solve the resulting nonlinear equations. Experimental simulations are conducted for proportional and non-proportional cyclic loadings. The model shows good correlation with the experimental results.

Nonproportional Low Cycle Fatigue Criterion for Type 304 Stainless Steel

This paper describes a multiaxial low cycle fatigue parameter for correlating Hues under nonproportional loadings. Constant amplitude low cycle fatigue tests were carried out under 14 proportional and complex nonproportional cyclic strain paths using type 304 stainless steel hollow cylinder specimens at room temperature. In nonproportional loading tests, fatigue lives are decreased by as much as a factor of 10 in comparison with those in proportional loading tests with the same strain range. Reduction in fatigue life due to nonproportional loading is closely related to additional nonproportional cyclic hardening. The product of the maximum principal stress and strain ranges correlated the nonproportional fatigue data. A nonproportional cyclic hardening parameter computed from the strain path is also proposed that allows life estimates to be obtained directly from the strain history without the need for a cyclic plasticity model.

CONSTITUTIVE MODELING OF PROPORTIONAL/NONPROPORTIONAL CYCLIC PLASTICITY FOR TYPE 316 STAINLESS STEEL APPLICABLE TO A WIDE TEMPERATURE RANGE

A constitutive model of cyclic plasticity is formulated to describe proportional and nonproportional hardening of type 316 stainless steel in the range from room temperature to 973K. For this purpose, we first examine the evolution of the size of isotropic and kinematic hardening range based on the experimental results performed previously by the present authors. It is elucidated that the complicated behavior of cyclic hardening under proportional cycles is induced by the evolution of kinematic hardening variable. Then, new evolution equations of kinematic hardening variables are established by incorporating this information into the nonlinear kinematic hardening rule. In particular, the dependence of cyclic hardening on the history of strain amplitude is taken into account. The final model is established by incorporating these evolution equations into the nonproportional viscoplastic model proposed by one of the present authors. It is elucidated that the proposed model can describe the behavior of proportional and nonproportional cyclic hardening under various loading conditions including strain cycles with amplitude variation in the range from room temperature to 973K.

Representation of cyclic hardening and softening properties using continuous variables

Abstract Our aim is the modeling of cyclic hardening, cyclic softening, cyclic mean stress relaxation, and additional nonproportional cyclic hardening. We do so by means of hardening functionals for back stress and yield stress without employing additional memory surfaces. Rather, we suppose all quantities to evolve simultaneously during elastic-plastic loading in a continuous manner. The basic idea is to formulate evolution equations for the hardening variables, which are of the “hardening/dynamic recovery” format with respect to a transformed arc length. The corresponding transformation is influenced by continuously evolving parameters, measuring strain amplitude and nonproportionality during the recent process history. Although the resulting, model has a very simple structure, it is capable of describing the basic phenomena under quite general loading conditions.

Effects of cyclic plasticity on subsequent creep for type 316 stainless steel at elevated temperature.

Multiaxialhistory effects of cyclic plasticity on creep were studied experimentally for type 316 stainless steel at an elevated temperature. Firstly, after a cyclic stabilization of prior cyclic tension-compression of constant total strain amplitude, constant stress creep tests were conducted under simple tension, simple torsion, and combined tension-torsion (√(3)τ/σ = 0, oo, 1). Three levels of creep stress were chosen which corresponded to the saturated cyclic stress amplitude and to certain values larger or smaller than that. Secondly, in order to elucidate the path shape effect of the prior strain cycle, a non-proportional strain cycle along a circular path was followed by a constant stress creep in tension where the strain amplitude was specified so as to result in the same saturated stress amplitude as in the tension-compression cycle compared. It was observed that the prior tension-compression cycle induced an anisotropic hardening for creep ; creep resistance was enhanced in the torsional direction, while somewhat decreased in the tensile direction. This is similar to the cross hardening effect reported in experiments on non-proportional cyclic strain hardening. The circular cycle, as expected, showed a much more significant hardening effect on creep.

Temperature-dependence of multiaxial non-proportional cyclic hardening of type 316 stainless steel.

The temperature dependence of the multiaxial cyclic behaviour of type 316 stainless steel was elucidated experimentally. Cyclic tests under constant total-strain amplitudes were performed for uniaxial tension-compression and circular strain paths at several temperatures ; room temperature, 200°C, 400°C, 500°C, 600°C and 700°C. The strain amplitudes of the cycles were specified to be 0.2, 0.3 and 0.4% under constant strain rate of 0.2% /min. It was observed that though strain hardening for circular cycles was larger than that of the tension-compression cycles at every temperature, the effect of nonproportionality in cyclic hardening decreased with the increase in temperature. The cyclic hardening was significant particularly in the temperature range between 400°C and 500°C.

Inelastic Behavior of Type 316 Stainless Steel Under Multiaxial Nonproportional Cyclic Stressings at Elevated Temperature

The stress-range and path-shape dependencies of multiaxial nonproportional cyclic hardening were studied for annealed type 316 stainless steel at 600°C by means of stress controlled tests. Cyclic experiments along circular stress paths with constant effective stresses in the axial-torsional stress plane were first performed. The significant cyclic hardening and its stress-range dependency observed for the circular stress cyclings were quantitatively shown in reference to the cyclic stress-strain curves resulted from uniaxial stress cyclings. Then, to discuss the effect of path-shape, the cyclic tests along square stress paths inscribed by the above circular paths, as well as the tests where uniaxial cyclings and torsional ones were alternated, were also carried out. As a result of these tests, the cyclic hardenings for square paths were found to be almost equivalent to those for their circumscribed circular paths. The other type of stress cyclings caused almost the same amount of cyclic hardenings as those for the circular cyclings of the identical stress-ranges.