A model for the Mullins effect during multicyclic equibiaxial loading

In this paper, we derive a model to describe the cyclic stress softening of a carbon-filled rubber vulcanizate through multiple stress–strain cycles with increasing values of the maximum strain, specializing to equibiaxial loading. Since the carbon-filled rubber vulcanizate is initially isotropic, we can show that following initial equibiaxial loading the material becomes transversely isotropic with preferred direction orthogonal to the plane defined by the equibiaxial loading. This is an example of strain-induced anisotropy. Accordingly, we derive nonlinear transversely isotropic models for the elastic response, stress relaxation, residual strain and creep of residual strain in order to model accurately the inelastic features associated with cyclic stress softening. These ideas are then combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation for the cyclic stress softening of a carbon-filled rubber vulcanizate. The model developed includes the effects of hysteresis, stress relaxation, residual strain and creep of residual strain. The model is found to compare extremely well with experimental data.     

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.     

Wechselverformungskurven und Mikrostruktur bei spannungskontrollierter Zug-Druck-Wechselbeanspruchung von Ck 45 im Temperaturbereich 295 K ≤ T ≤ 873 K

Cyclic deformation curves and microstructure of SAE 1045 after stress-controlled push-pull loading in the temperature range 295 K ≤ T ≤ 873 K Stress-controlled cyclic deformation tests were performed with normalized SAE 1045 in the temperature range 295 K ≤ T ≤ 873 K. From the measured mechanical hysteresis loops cyclic deformation curves were determined, which are characterized by temperature dependent cyclic softening and hardening processes. Due to the cyclic deformation processes at any temperature distinct dislocation structures are formed depending on the stress amplitudes and the number of cycles chosen. In the whole temperature range and for all stress amplitudes the plastic strain amplitude was proved to be a suitable parameter to describe the actual fatigue state.     

Key issues in cyclic plastic deformation: Experimentation

Cyclic plastic deformation phenomena include the Bauschinger effect, cyclic hardening/softening, strain range effect, loading history memory, ratcheting, mean stress dependent hardening, mean stress relaxation and non-proportional hardening. In this work, different cyclic plastic deformation responses of piping materials (SA333 C-Mn steel and 304LN stainless steel) are experimentally explored. Cyclic hardening/softening is depends upon loading types (i.e. stress/strain controlled), previous loading history and strain/stress range. Pre-straining followed by LCF and mean stress relaxation shows similar kind of material response. Substantial amount of non proportional hardening is observed in SA333 C-Mn steel during 90° out of phase tension-torsion loading. During ratcheting, large amount of permanent strain is accumulated with progression of cycles. Permanent strain accumulation in a particular direction causes cross-sectional area reduction and which results uncontrollable alteration of true stress in engineering stress controlled ratcheting test. In this work, true stress control ratcheting on piping materials has been carried out in laboratory environment. Effects of stress amplitude and mean stress on the ratcheting behaviors are analyzed. A comparison has also been drawn in between the true and engineering stress controlled tests, and massive difference in ratcheting life and strain accumulation is found.     

Softening response of cast Al-Si-Cu samples tested under low-cycle-fatigue and fatigue-creep regime and elevated temperate: Experimental and analysis

In this paper, the strain-controlled low-cycle-fatigue (LCF) test and low-cycle-fatigue-creep (LCFC) test of a cast Al-Si-Cu alloy at 350°C were investigated. A significant softening behavior can be observed in the tests, the strain amplitude has an effect on softening speed, and the dwell time has an effect on both softening ratio and softening speed. Microscopic observation indicated that the compressive load of creep could result in a significant amount of plastic deformation and lead to the eutectic silicon particles rupture and the propagation of dimples, causing the softening behavior and the decrease of failure life of Al-Si-Cu alloy. Based on the experiment data, a modified visco-plasticity model combining Chaboche model and θ-projection model was proposed to capture the cyclic stress response. The modified model could simulate the softening behavior and hysteresis loop for LCF test and LCFC test of Al-Si-Cu alloy accurately.     

A UNIAXIAL CYCLIC PLASTICITY MODEL INCLUDING TRANSIENT MATERIAL BEHAVIOUR

Abstract— A phenomenological uniaxial material model, which is a one-dimensional equivalent to a two-surface multiaxial plasticity model, is presented. The model takes into account the transient effects cyclic hardening/softening and mean stress relaxation by means of exponential relationships that are functions of number of reversals. The parameters describing these relationships are obtained from the strain-life curve and the cyclic stress-strain curve. The hysteresis loops are assumed to be bounded by two parallel, straight lines in tension and compression, i.e. bounding lines. The stress-strain curve approaches these lines with increasing strain through a nonlinearly decreasing plastic modulus. The transient effects are simulated by increasing/decreasing the distance between the lines for cycling hardening/softening at each stress or strain reversal. A positive mean stress yields translation downwards of the bounding lines if the material is subjected to strain control, and translation in the opposite direction if the mean stress is negative. Dynamic creep is simulated when the material is subjected to stress control and mean stress. The test data was obtained from an experimental programme on St52–3N, which is a normalized structural steel. The specimens were loaded in strain or stress control. The comparison between tests and simulations shows that the model described simulates the cyclic transient material behaviour quite well. The main inaccuracy is due to using parallel bounding lines, as the tests show that these lines are not completely parallel for the present steel material.     

HIGH TEMPERATURE FATIGUE-CREEP BEHAVIOUR OF SINGLE CRYSTAL SRR90 NICKEL BASE SUPERALLOYS: PART 1—CYCLIC MECHANICAL RESPONSE

Abstract—High temperature low cycle fatigue tests, with and without strain dwells, were conducted at 750°C, 950°C and 1050°C, on single crystal SRR99 nickel base superalloy, with different crystal orientations. At 750°C, SSR99 exhibited cyclic stability regardless of cycle type. The presence of strain dwells caused cyclic softening at 950°C compared with continuous cycling tests. At 1050°C, cyclic softening was observed for all the tests. The introduction of strain dwells produced significant stress relaxation at 950°C and 1050°C, but not at 750°C for the strain ranges in this study. Significant mean stress was observed at the three temperatures for tests with tensile or compressive strain dwells. The crystal orientation was found to have a dominating influence on the cyclic stress strain relationship and stress relaxation response. A simple approach is developed to correlate the effect of orientation on the cyclic mechanical response.     

Pseudo-elastic description of polymeric foams at finite deformation with stress softening and residual strain effects

Polymeric foams are typical materials for energy absorber in such areas as aircraft, car industry and in the field of electronic packaging. Besides the typical hyperelastic behaviour, non-linear stress–strain behaviour in large elastic deformation, polymeric foams may also exhibit some inelastic effects, like stress softening and residual strain. In this paper we first describe some experiment results that illustrate the stress softening in compressible expanded polypropylene (EPP) foams together with associated residual strain effects. Then, based on Ogden and Dorfmann’s results, a pseudo-elastic model is introduced to capture the stress softening and residual strain effects by including of two variables in the energy function. Numerical simulations of uniaxial-compression tests of two types of EPP foam are used to determine the material parameters of Ogden’s model, stress softening and residual strain effects. The numerical simulations indicate that the pseudo-elastic model provides reasonably accurate predictions of the inelastic behaviour of polymeric foam.     

Experimental and theoretical investigation on mechanical behaviors of TB991 weld sealant under cyclic loading

The mechanical behaviors of TB991 weld sealant under cyclic loading conditions were experimentally investigated. The evolution of relaxation stress, cyclic softening, and dissipated energy was evaluated with the effect of strain amplitude and mean strain. The experimental results showed that the stress–strain response curves of the first loading-unloading and cyclic loading-unloading were significantly different. The phenomenon of stress relaxation and cyclic softening occurred under cyclic strain loading conditions. Furthermore, the relaxation stress and dissipated energy decreased rapidly during the initial cyclic loading and then steadily decreased with the increase of cycle number, while the cyclic softening increased rapidly at first and then steadily. Besides, a viscoelastic constitutive model was proposed which can describe the different shapes of stress–strain curve between the initial loading-unloading and the cyclic loading-unloading and also considers the cyclic stress relaxation and cyclic softening of the materials under cyclic loading condition. Comparisons between the numerical results and the experimental data demonstrated that the proposed model can better describe the mechanical behavior of TB991 weld sealant under cyclic loading conditions.     

循环软化材料单轴时相关循环变形行为的实验研究

在室温下,对循环软化材料(调质42CrMo钢)的单轴时相关应变循环特性和时相关棘轮行为进行了实验研究。揭示了材料在不同加载速率、不同峰/谷值保持时间以及不同加载波形(三角波和正弦波)下的循环软化特性和棘轮行为特性。结果表明,在室温下,材料的循环软化和棘轮变形行为均体现出明显的时相关效应:其循环变形行为不仅依赖于加载速率,而且还明显依赖于保持时间以及加载波形的形状。研究有助于后续建立循环软化材料时相关循环本构模型。     

聚碳酸酯的单轴棘轮行为研究

在常温常湿下,对聚碳酸酯(PC)材料进行了一系列单轴应变循环和非对称应力循环实验。讨论了PC材料在不同加载水平、加载历史、应力率和峰值保持时间下的循环变形特征。结果表明:PC材料在应变循环过程中体现出了一定程度的循环软化特性,其响应应力幅值在应变循环中随着循环周次的增加而下降,但不是很明显;PC材料在非对称应力循环加载过程中产生明显的棘轮行为,棘轮应变随着平均应力和应力幅值的增加而增加,并且平均应力的影响大于应力幅值的影响;加载历史对于棘轮变形行为的影响较为明显,但对应变循环特性影响不大;PC材料的棘轮变形随着应力率的减小和峰值保持时间的增加而增加,体现明显的时相关性。     

INTERACTIONS BETWEEN TIME AND CYCLIC DEFORMATION PROCESSES IN A NIMONIC ALLOY

Interactive creep–fatigue behaviour of a nickel-base superalloy (IN 597) has been examined at 850 °C under various strain-limited, cyclic torsional loading conditions. In one test, forward creep deformation was reversed by creep under equal magnitude stress levels and strain limits. In other tests, forward creep strain was reversed by fast monotonic plasticity with and without a subsequent period of relaxation. These cycles were repeated within each test until fracture. This paper examines empirically the influence of a number of test variables upon cyclic creep curves, and demonstrates the usefulness of predictions based upon continuous low cycle fatigue and simple creep data when used in conjunction with a mechanical equation of state. A cyclic equilibrium condition was not achieved from these tests. Instead, a progressive softening occurred giving reductions to the amount of creep strain, creep time interval and reversed peak stress with each new cycle. Such reductions are expressed from derived formulae that embrace the range of inelastic strain, cycle number, creep dwell stress, reversed peak stress, and times expended in creep and relaxation.
Observations made on accumulated creep strain reveal the contribution to a creep–fatigue fracture from cyclic creep. This has led to a modified form of the linear damage rule which can provide conservative life predictions for components operating in service under similar cyclic conditions.     

Low-cycle fatigue behavior and microstructural evolution in a low-carbon carbide-free bainitic steel

This paper deals with the cyclic deformation behavior and microstructural evolution in a low-carbon carbide-free bainitic steel with two different microstructures. Low-cycle fatigue tests were performed at room temperature at various strain amplitudes under total strain control. The variations of the amount of retained austenite and the substructures versus the number of fatigue cycles were evaluated by the X-ray diffraction technique and electron microscopy. Fatigue test results demonstrate that the two microstructures exhibit very similar cyclic stress responses, i.e. initial cyclic hardening followed by cyclic softening or by cyclic saturation and softening till failure, depending on the strain amplitude applied. Parametric studies of the microstructure–property relationship indicate that the major cause for the initial cyclic hardening is neither martensitic transformation nor increased dislocation density. Based on these results and considering the initial high density of dislocations, which are pre-existent and mobile in the starting microstructure and which are entangled, rearranged or annihilated with cycling, the mechanisms responsible for the initial cyclic hardening followed by softening are analyzed.     

Cyclic loading and fatigue in ice

Isotropic polycrystalline ice was subjected to cyclic loading in uniaxial compression at ?5°C, with stress limits 0–2 and 0–3 MPa, and frequencies in the range 0.043 to 0.5 Hz. Stress-strain records showed hysteresis loops progressing along the strain axis at non-uniform rates. The effective secant modulus, which was about half the true Young's modulus, decreased during the course of a test. The elastic strain amplitude and the energy dissipated during a loading cycle both increased with increase of time and plastic strain. Strain-time records gave mean curves which were identical in form to classical constant-stress creep curves, with a small cyclic alternation of recoverable strain about the mean curve. The inflection point of the “creep curve”, marking the transition from strain hardening to strain softening, occurred at a plastic strain of 1% (±0.1%), which is about the same as the “ductile failure strain” found in constant stress creep tests and in constant strain-rate tests on ice of the same type at the same temperature. The dissipation of strain energy up to this “failure point” was much higher for the cyclic tests than for corresponding quasi-static tests ? 100 to 600 kPa (or kN-m/m³) in comparison to about 30 kPa. The number of cycles taken to reach the “failure point” was of no direct significance, varying greatly with stress amplitude and with frequency. The results of the tests suggest that maximum resistance under compressive cyclic loading occurs at an axial plastic strain of about 1%, which is essentially the same as the failure strain for ductile yielding under constant stress and under constant strain-rate.     

The effect of temperature on low-cycle fatigue behavior of prior cold worked 316L stainless steel

Low-cycle fatigue tests on cold worked 316L stainless steel were carried out at various temperatures from room temperature to 650 °C and tensile tests were conducted on the cold worked and solution-treated materials. At all test temperatures, the cold worked material showed the tendency of higher strength and lower ductility. Following initial cyclic hardening for a few cycles, cyclic softening behavior was observed to dominate until failure occurred during low-cycle fatigue deformation. The softening behavior strongly depends on temperature and strain amplitude. Several life prediction models were examined and it was found that it is important to select a proper life prediction parameter since stress and strain depend strongly on temperature. A phenomenological fatigue life prediction model is proposed to account for the influence of temperature on life. The model is correlated with the experimental results.     

Experimental Study and Numerical Modelling of Creep and Stress Relaxation of Dielectric Elastomers

Dielectric elastomers (DEs) are gaining acceptance as potential actuator materials because of their exhibition of a large amount of deformation when stimulated by electrostatic forces. However, time‐dependent behaviour such as creep and stress relaxation still pose a great challenge for the design, modelling and control of the DE‐based actuators. In this work, attempts are made for experimental estimation and modelling of creep and relaxation properties of one of the most widely used dielectric acrylic elastomers, VHB 4910. Experimental investigation shows that the material possesses strong time‐dependent creep and stress relaxation. It has been shown that creep and stress relaxation characteristics vary with the holding stress and holding strain respectively. Creep and stress relaxation properties are also shown to depend on the number of cycles in the case of cyclic loading. Results also show that Findley's power law can successfully model the creep and stress relaxation behaviour of the VHB 4910 elastomer.     

Uniaxial ratchetting in steels with different cyclic softening/hardening behaviours

The cyclic deformation of three structural steels, SS316L stainless steel, 40Cr3MoV bainitic steel and 25CDV4.11 steel, were studied experimentally by uniaxial cyclic straining or stressing tests at room temperature. The cyclic softening/hardening behaviours of the steels were discussed first by cyclic straining tests; and then the effects of cyclic softening/hardening behaviours on the uniaxial ratchetting of the materials were investigated by asymmetrical cyclic stressing tests. It is concluded from the experimental results that the ratchetting greatly depends on the cyclic softening/hardening behaviours of the materials, as well as the loading history. Different ratchetting and failure behaviours are observed for the prescribed steels. It is also stated that the proposed unified visco‐plastic constitutive model can provide a fairly reasonable simulation of the uniaxial ratchetting of SS316L stainless steel and 25CDV4.11 steel; but cannot simulate the ratchetting of 40Cr3MoV bainitic steel since the dependence of cyclic softening behaviours on the applied inelastic strain amplitude cannot be reasonably described in the discussed constitutive model. Some significant conclusions are obtained, which are useful to construct constitutive model to describe the ratchetting of the materials with different cyclic softening/hardening behaviours.     

Mullins effect characterization of elastomers by multi-axial cyclic tests and optical experimental methods

Besides the typical hyperelastic behaviour, large elastic deformations with non-linear stress–strain behaviour, rubber-like materials may also exhibit some inelastic effects, like hysteresis and permanent set. One of them is a particular damage phenomenon called Mullins effect. This is visible when cyclic tension tests are performed with increasing values of deformation. Material is deformed up to a fixed strain value and then unloaded. When a second load is applied it is possible to observe a stress softening effect. In the present work uniaxial and equibiaxial tension tests have been carried out by a standard tensile machine and by an hydraulic bulge test experimental rig, respectively. In both tests optical methods have been used for strain measurement. Experimental data have been successively introduced in a numerical procedure that permitted to extract the best material parameters for two of the most known pseudo-elastic models [Ogden, R.W., Roxburgh, D.G., 1999. A pseudo-elastic model for the Mullins effect in filled rubber. Proceedings of the Royal Society London A 455, 2861–2877; Dorfmann, A., Ogden, R.W., 2004. A constitutive model for the Mullins effect with permanent set in particle-reinforced rubber. International Journal of Solids and Structures 41, 1855–1878] accounting for both stress-softening behaviour and residual strain.     

Low cycle fatigue performance of DP900 steel under cyclic shear paths

In this study, a sheet fatigue shear test device is designed and applied to the low-cycle fatigue testing of DP900 with varying strain amplitudes within the range of 0.5%–6.0%. The microstructure is analyzed by using electron backscatter diffraction, and fracture surfaces are examined via scanning electron microscopy. Results indicate that the material exhibits cyclic softening behavior after the first two cycles of hardening, with a stable softening rate and a high damage evolution rate related to loading amplitude. The variation of the hysteresis curve in the cyclic process is shown. Total plastic strain energy absorbed increases as loading amplitude decreases and it reaches the peak at approximately 1%. The life prediction model based on plastic strain energy density and strain amplitude is verified to be suitable for the cyclic shear path. The influence of microinhomogeneity on the distribution of stress and strain, especially the deformation of martensite, is closely related to the bi-linear region of fatigue life curve.     

Biaxial cyclic deformation of an epoxy resin: Experiments and constitutive modeling

Biaxial (proportional and non-proportional) cyclic tests were conducted on thin-walled tubular specimens to investigate deformation behavior of an epoxy resin, Epon 826/Epi-Cure Curing Agent 9551. The focus was placed on the biaxial stress-strain response and their dependency on the load control mode, stress or strain range and loading path. Experimental results indicated that under strain-controlled equi-biaxial (proportional) cyclic loading, mean stress relaxation occurred in both axial and hoop directions, whereas under stress-controlled equi-biaxial cyclic loading, ratcheting strains accumulated in both principal directions. Under strain- or stress-controlled non-proportional cyclic loading, anisotropy in stress-strain responses was induced in both axial and hoop directions, and the axial and hoop hysteresis loops rotated in opposite directions. This was particularly evident at high stress or strain levels. The experimental results were further used to evaluate the predictive capabilities of a nonlinear viscoelastic constitutive model. Qualitative and quantitative comparison with the test data indicated a good agreement in predicting the complex stress-strain response under biaxial cyclic loading with various loading paths, applied stress or strain ranges and loading control modes.