Ratcheting progress at notch root of 1045 steel samples over asymmetric loading cycles: Experiments and analyses

The present study examines ratcheting response of steel samples with various notch diameters through conducting several cyclic tests. Ratcheting strain values were measured through strain gauges mounted at different distances from the notch root. Local ratcheting at the notch region was highly influenced by notch diameter, notch shape, distance from the notch root, and magnitude of the nominal mean/amplitude of loading cycles. Nominal force‐controlled cycles were kept below the yield point and the Neuber's rule accommodated for the maximum/minimum local stress components along those local strains measured through the strain gauges at the notch region. Plastic strains at the vicinity of notch root over loading cycles were further accumulated by means of the Chaboche hardening model. The local ratcheting strain while progressed at the notch root plastic zone over loading cycles resulted in mean stress relaxation controlled by the model.     

Ratcheting assessment of materials based on the modified Armstrong–Frederick hardening rule at various uniaxial stress levels

The present study evaluates ratcheting response of materials by means of the Armstrong–Frederick (A–F) hardening rule, the modified A–F rule (Bower's model), and further modifications of the hardening rule based on new introduced coefficients. The implemented modifications on the A–F‐based hardening rule aims to address stages of ratcheting over stress cycles. The modified hardening rule predicts the ratcheting strain rate decay over stage I and the constant rate of strain accumulation during stage II. The modified hardening rule consisted of the coefficients of the hardening rule controlling stress–strain hysteresis loops generated over stress cycles during ratcheting process (Bower's modification on A–F rule) plus the coefficients controlling rates over stages of materials ratcheting deformation. Stress–strain‐dependent coefficients in the modified rule are responsible to compromise overprediction of ratcheting of A–F during stage I and the premature plastic shakedown beyond stage I induced by Bower's model. Ratcheting strain rate coefficients improved the hardening rule capability to calibrate and control the rate of ratcheting in stages I and II and enabled the modified hardening rule to predict ratcheting strain over a prolonged domain of stress cycles. The modified hardening rule was employed to assess ratcheting response of 304, 42CrMo, 316L steel and copper samples under uniaxial loading conditions. The predicted ratcheting values based on the modified hardening rule and the experimental ratcheting strains were found in good agreements.     

A notch strain calculation of a notched specimen under axial-torsion loadings

In this paper, a notch analysis model is presented for the numerical prediction of multiaxial strains of a notched 1070 steel specimen under combined axial and torsion loadings. The proposed model is based on the notion of a structural yield surface and uses a small-strain cyclic plasticity model to describe stress–strain relations. A notch load–strain curve is calculated with Neuber’s rule and incremental nonlinear finite element analysis. The presented model is applied to simulate the notch root deformations of a circumferentially notched specimen under cyclic tension–compression–torsion loading histories. The model predictions are evaluated with strain measurements at the notch root of the specimen in a comprehensive set of cyclic tests. The computed strain loops were in accord with experimental data and matched qualitatively with measured shear–axial strain histories irrespective of loading path of the test. In proportional balanced torsion-axial loading, the nonlinear shear strain–axial strain loops were calculated properly. The modeling errors were determined to be a function of the loading path shape, and compared to shear strains, axial strain predictions were more accurate.     

Notched fatigue behavior including load sequence effects under axial and torsional loadings

Notch effects on axial and torsion fatigue behaviors of low carbon steel were investigated. Fully-reversed tests were conducted on thin-walled tubular specimens with or without a transverse circular hole. A shear failure mechanism was observed for both smooth and notched specimens and under both axial and torsion loadings. The notch effect was more pronounced under axial loading, in spite of higher stress concentration factor in torsion. The commonly used nominal S–N approach with fatigue notch factor in conjunction with von Mises effective stress resulted in overly conservative life predictions of both smooth and notched torsion fatigue lives. Neuber’s rule yielded notch root stress and strain amplitudes close to the FEA results for both axial and torsion loadings. The local strain approach based on effective strain obtained from Neuber’s rule or FEA resulted in poor correlation of the fatigue life data of smooth and notched specimens. The Fatemi–Socie critical plane parameter represented the observed failure mechanism and resulted in very good correlations of smooth and notched specimens fatigue data under both axial and torsion loadings. In block loading tests with equal number of alternating axial and torsion cycles at the same stress level, beneficial effect of axial loading was observed. Possible potential reasons for this unexpected behavior are discussed.     

Isotropic‐kinematic hardening framework to assess ratcheting response of steel samples undergoing asymmetric loading cycles

The present study intends to characterize ratcheting response of several steel alloys subject to asymmetric loading cycles through coupling the Ahmadzadeh‐Varvani kinematic hardening rule with isotropic hardening rules of Lee and Zavrel, Chaboche, and Kang. The Ahmadzadeh‐Varvani kinematic hardening rule was developed to address ratcheting progress over asymmetric stress cycles with relatively a simple framework and less number of coefficients. Inclusion of isotropic hardening rules to the framework improved ratcheting response of materials mainly over the first stage of ratcheting. Lee and Zavrel model (ISO‐I) developed an exponential function to account for accumulated plastic strain as yield surface is expanded over stage I and early stage II of ratcheting. Isotropic models by Chaboche (ISO‐II) and Kang (ISO‐III) encountered yield surface evolution in the framework by introducing an internal variable that takes into account the prior maximum plastic strain range. The choice of isotropic hardening model coupled to the kinematic hardening model is highly influenced by material softening/hardening response.     

Ratcheting response of nylon fiber reinforced natural rubber/styrene butadiene rubber composites under uniaxial stress cycles: Experimental studies

The present study intends to study the ratcheting response of nylon fiber reinforced natural rubber (NR)‐styrene butadiene rubber (SBR) composite samples under asymmetric stress cycles. Uniaxial tests conducted on composite samples have shown how influential the weight fraction of short nylon fibers on the stress‐strain curves/loops under monotonic and cyclic loads is. NR/SBR composite samples with various fiber contents of 0, 10, 20, 30, and 40 per hundred rubber (phr) were tested under asymmetric stress cycles. In these tests, stress‐strain hysteresis loops were progressively shifted over stress cycles resulting in progressive plastic strain accumulation. Over stress cycles, ratcheting strain progressed within the first few cycles with a relatively high rate, and as the number of cycles increased, this rate decayed resulting in a plateau in strain accumulation (shakedown). The ratcheting strain rate and magnitude resulting in shakedown were highly affected by the nylon fiber content. The experimental observations showed that this plateau (shakedown) occurred after a number of cycles in NR/SBR composite samples where the widths of hysteresis loops stayed unchanged. Samples with no fiber and that with 10 phr fiber content possessed high ratcheting rates leading samples to failure after a few stress cycles. Fracture surfaces in these samples were further analyzed through SEM investigation.     

State‐of‐the‐art review on extended stress intensity factor concepts

The stress intensity factor concept for describing the stress field at pointed crack or slit tips is well known from fracture mechanics. It has been substantially extended since Williams' basic contribution (1952) on stress fields at angular corners. One extension refers to pointed V‐notches with stress intensities depending on the notch opening angle. The loading‐mode‐related simple notch stress intensity factors K₁, K₂ and K₃ are introduced. Another extension refers to rounded notches with crack shape or V‐notch shape in two variants: parabolic, elliptic or hyperbolic notches (‘blunt notches’) on the one hand and root hole notches (‘keyholes’ when considering crack shapes) on the other hand. Here, the loading‐mode‐related generalised notch stress intensity factors K_1ρ, K_2ρ and K_3ρ are defined. The concepts of elastic stress intensity factor, notch stress intensity factor and generalised notch stress intensity factor are extended into the range of elastic–plastic (work‐hardening) or perfectly plastic notch tip or notch root behaviour. Here, the plastic notch stress intensity factors K_1p, K_2p and K_3p are of relevance. The elastic notch stress intensity factors are used to describe the fatigue strength of fillet‐welded attachment joints. The fracture toughness of brittle materials may also be evaluated on this basis. The plastic notch stress intensity factors characterise the stress and strain field at pointed V‐notch tips. A new version of the Neuber rule accounting for the influence of the notch opening angle is presented.     

Development of new cyclic plasticity model for 304LN stainless steel through simulation and experimental investigation

Cyclic plastic deformation characteristics of 304LN stainless steel material have been studied with two proposed cyclic plasticity models. Model MM-I has been proposed to improve the simulation of ratcheting phenomenon and model MM-II has the capability to simulate both cyclic hardening and softening characteristics of the material at various strain ranges. In the present paper, strain controlled simulations are performed with constant, increasing and decreasing strain amplitudes to verify the influences of loading schemes on cyclic plasticity behaviors through simulations and experiments. It is observed that the material 304LN exhibits non Masing characteristics under cyclic plastic deformation. The measured deviation from Masing is well established from the simulation as well as from experiment. Simulation result shows that the assumption of only isotropic hardening is unable to explain the hardening or softening characteristics of the material in low cycle fatigue test. The introduction of memory stress based cyclic hardening coefficient and an exponentially varying ratcheting parameter in the recall term of kinematic hardening rule, have resulted in exceptional improvement in the ratcheting simulation with the proposed model, MM-II. Plastic energy, shape and size of the hysteresis loops are additionally used to verify the nature of cyclic plasticity deformations. Ratcheting test and simulation have been performed to estimate the accumulated plastic strain with different mean and amplitude stresses. In the proposed model MM-I, a new proposition is incorporated for yield stress variation based on the memory stress of loading history along with the evolution of ratcheting parameter with an exponential function of plastic strain. These formulations lead to better realization of ratcheting rate in the transient cycles for all loading schemes. Effect of mean stress on the plastic energy is examined by the simulation model, MM-I. Finally, the micro structural investigation from transmission electronic microscopy is used to correlate the macroscopic and microscopic non Masing behavior of the material.     

Ratcheting assessment of steel alloys under uniaxial loading: a parametric model versus hardening rule of Bower

This study intends to compare ratcheting response of 42CrMo, 1020, SA333 and SS304 steel alloys over uniaxial stress cycles evaluated by a parametric ratcheting model and Bower's hardening rule. The parametric ratcheting equation was formulated to describe triphasic stages of ratcheting deformation over stress cycles. Mechanistic parameters of mean stress, stress amplitude, material properties and cyclic softening/hardening response of materials were employed to calibrate parametric equation. Based on the framework of cyclic plasticity theory, the modified Armstrong–Frederick nonlinear hardening rule of Bower was employed to assess ratcheting response of steel alloys under uniaxial stress cycles. Bower's model was chosen mainly due to simplicity of the model and its lower number of constants required to predict ratcheting strain over stress cycles as compared with other hardening rules. Ratcheting strain values predicted by Bower's model showed good agreements over stage I of stress cycles as compared with experimental values of ratcheting strain. Beyond of stage I stress cycles, Bower ratcheting strain rate stayed constant resulting in an arrest in ratcheting process. The predicted ratcheting strains based on the parametric equation were found in good agreements over three stages of ratcheting as compared with those of experimentally obtained.     

Estimation of elastic‐plastic strain response at two‐dimensional notches

It is widely recognized that the accuracy of notch fatigue calculations can be improved significantly when those calculations are based on the elastic‐plastic response strain at the notch root, as opposed to the remotely applied loads or stresses. Two of the most widely used approximations for this response are Neuber's rule and Glinka's equivalent strain energy density method. In the present work, a survey of some of the many published evaluations of these methods was first conducted, and then, additional detailed comparisons with elastic‐plastic finite element analyses for a series of semicircular and V‐shaped notch configurations were performed. Based on the observed limitations of both the Neuber and Glinka approaches, and with the guidance of the elastic‐plastic finite element results, a new (and more robust) approach for the estimation of notch response strains is proposed. This approach calls for the definition of a generalized notch response curve (GNRC), which is dependent on both the material stress–strain curve and the notch geometry. Once defined, the GNRC allows the determination of the response strain for any applied stress.     

State‐of‐the‐art review on the local strain energy density concept and its relation to the J‐integral and peak stress method

The local average strain energy density (SED) approach has been proposed and elaborated by Lazzarin for strength assessments in respect of brittle fracture and high‐cycle fatigue. Pointed and rounded (blunt) V‐notches subjected to tensile loading (mode 1) are primarily considered. The method is systematically extended to multiaxial conditions (mode 3, mixed modes 1 and 2). The application to brittle fracture is documented for PMMA flat bar specimens with pointed or rounded V‐notches inclusive of U‐notches. Results for other brittle materials (ceramics, PVC, duraluminum and graphite) are also recorded. The application to high‐cycle fatigue comprises fillet‐welded joints, weld‐like shaped and V‐notched base material specimens as well as round bar specimens with a V‐notch. The relation of the local SED concept to comparable other concepts is investigated, among them the Kitagawa, Taylor and Atzori–Lazzarin diagrams, the Neuber concept of fictitious notch rounding applied to welded joints and also the J‐integral approach. Alternative details of the local SED concept such as a semicircular control volume, microrounded notches and slit‐parallel loading are also mentioned. Coarse FE meshes at pointed or rounded notch tips are proven to be acceptable for accurate local SED evaluations. The peak stress method proposed by Meneghetti, which is based on a notch stress intensity factor consideration combined with a globally even coarse FE mesh and is used for the assessment of the fatigue strength of welded joints, is also presented.     

Ratcheting of 304 Stainless Steel Alloys subjected to Stress‐Controlled and mixed Stress‐ and Strain‐Controlled Conditions evaluated by Kinematic Hardening Rules

The present study predicts ratcheting response of SS304 tubular stainless steel samples using kinematic hardening rules of Ohno–Wang (O–W), Chen‐Jiao‐Kim (C–J–K) and a newly modified hardening rule under various stress‐controlled, and combined stress‐ and strain‐controlled histories. The O–W hardening rule was developed based on the critical state of dynamic recovery of backstress. The C–J–K hardening rule further developed the O–W rule to include the effect of non‐proportionality in ratcheting assessment of materials. The modified rule involved terms , and in the dynamic recovery of the Ahmadzadeh–Varvani (A–V) model to respectively track different directions under multiaxial loading, account for non‐proportionality and prevent plastic shakedown of ratcheting data over multiaxial stress cycles. The O–W model persistently overestimated ratcheting strain over the multiaxial loading paths. The C–J–K model further lowered this overprediction and improved the predicted ratcheting curves. The predicted ratcheting curves based on the modified model closely agreed with experimental data under various loading paths.     

Effect of stress parameters on ratcheting deformation stages of polycrystalline OFHC copper

Engineering stress‐controlled ratcheting tests under different sets of stress amplitudes and mean stresses show that ratcheting deformation in polycrystalline OFHC copper occurs in three different stages. A plateau region with almost no accumulation of inelastic strain follows general ratcheting deformation during initial loading cycles. With breakdown of the plateau region inelastic ratcheting deformation occurs at an increasingly rapid rate. The effect of the stress amplitude on the ratcheting process is found to be more than mean stress effect. Reconstruction of the ratcheting curves clearly separates the conditions for stress‐controlled low cycle fatigue with zero mean stress and ratcheting with tensile mean stress.     

Concurrent ratcheting–fatigue damage analysis of uniaxially loaded A‐516 Gr.70 and 42CrMo Steels

This study intends to investigate the concurrent interaction of fatigue damage and ratcheting strain in two commonly used steel alloys of (American Society for Testing and Materials) ASTM A‐516 Gr.70 and 42CrMo, respectively for pressure vessels and high grade machinery parts over uniaxial stress cycles. Ratcheting extension and fatigue damage progress were both characterized cycle‐by‐cycle over life cycles of tested materials. The interaction of ratcheting and fatigue damage was defined based on mechanistic parameters involving the effects of mean stress, stress amplitude and cyclic softening/hardening response of materials. The extent of ratcheting effect was defined by product of average ratcheting strain per cycle, and maximum stress value during a cycle, while fatigue damage was analysed based on earlier developed energy‐based models of Xia–Ellyin, and Smith–Watson–Topper. Overall damage due to ratcheting and fatigue was calibrated through a weighting factor at various mean/ cyclic amplitude stresses. An algorithm was developed to evaluate overall damage due to ratcheting and fatigue stress cycles of materials subjected to various mean and amplitude stresses. The estimated lives at different mean stresses and stress amplitudes for ASTM A‐516 Gr.70 and 42CrMo samples showed good agreements as compared with those of reported experimental data.     

The Equivalent Strain Energy Density approach re-formulated and applied to sharp V-shaped notches under localized and generalized plasticity

In the case of a rounded notch, the stress and strain at the notch tip can be determined by the traditional Neuber rule or by the Equivalent Strain Energy Density (ESED) approach, as formulated by Glinka and Molski. In the latter case the elastoplastic strain energy density at the notch tip is thought of as coincident with that determined under purely elastic conditions. For sharply V‐shaped notches this approach is not directly applicable, since the strain energy density at the notch tip tends toward infinity both for a material obeying an elastic law and a material obeying a power hardening law. By using the notch stress intensity factors, the present paper suggests a re‐formulation of the ESED approach which is applied no longer at the notch tip but to a finite size circular sector surrounding the notch tip. In particular we have adopted the hypothesis that, under plane strain conditions, the value of the energy concentration due to the notch is constant and independent of the two constitutive laws. When small scale yielding conditions are present, such a hypothesis immediately results in the constancy of the strain energy averaged over the process volume. As a consequence, plastic notch stress intensity factors valid for sharp V‐shaped notches can be predicted on the basis of the linear elastic stress distributions alone.     

A Coupled Armstrong-Frederick Type Plasticity Correction Methodology for Calculating Multiaxial Notch Stresses and Strains

Based on the pseudo-strain method, a computational modeling technique coupling with Armstrong-Frederick type nonlinear kinematic hardening rule (Jiang-Sehitoglu model) is developed to calculate the multiaxial stress-strain responses of notched components. The pseudo-strain-true notch stress curve is determined using Neuber’s rule. The material constants in Jiang-Sehitoglu model are calculated using the Ramberg-Osgood curve. The presented method is applied to simulate the notch-tip deformations of circumferentially notched 1070 steel and S460N steel shafts subjected to various loadings, including box, circle, V-shape, zigzag-shape, and butterfly-shape loading paths. The calculated strain loops are in accord with experimental data and show reasonable accuracy.     

Ratcheting assessment of steel alloys under step-loading conditions

The present study examines the capability of a recently modified hardening rule to characterize ratcheting response of materials subjected to multi-step uniaxial stress cycles. The modified hardening rule was developed based on Armstrong–Frederick (A–F) hardening rule through implementing new ratcheting rate dependent coefficients γ₂ and δ. These coefficients were estimated by means of calibrated curves for any given stress levels defined from the uniaxial single-step ratcheting response at various cyclic stress levels. At a constant mean stress, ratcheting strain progressively increased as stress amplitude over steps of loading history increased. Similar response was also evident for step-loading with constant stress amplitude while the values of mean stress increased. For high–low histories, the trend of predicted ratcheting strain from higher to lower magnitudes found agreeable with that of experimental data. The discrepancy of the predicted and experimentally ratcheting strain values in the high–low step loading however was due to constancy in the shape and size of translating yield surface in the modified kinematic hardening rule. The modified hardening rule was employed to assess ratcheting response of SS316L, SA333, SS316L(N) and 1070 steel alloys under various step-loading conditions. Predicted ratcheting data at various stress level were found in good agreements as compared with the experimental ratcheting strains.     

概率局部应力应变法

实际工程中的结构件往往具有多个不确定因素,包括材料、几何、载荷等。这些不确定因素导致构件的局部应力应变响应和疲劳寿命响应具有随机性。因此对低周疲劳分析中的局部应力应变法进行了概率分析。通过基本随机变量将诺伯法中的循环应力应变曲线(迟滞回线)和诺伯公式表示为概率曲线,基本随机变量反映了构件的不确定因素。通过建立近似拟合多项式的方法,求得局部应力应变的随机响应。将应变寿命曲线视为概率曲线,采用随机累积损伤理论,通过同样方法得到疲劳寿命的随机响应。算例表明结果与蒙特卡罗模拟的结果十分接近。该方法是一个简单有效的疲劳寿命概率分析方法。     

Theoretical fatigue–effective notch stresses at spot welds

Notch stress formulae are derived for the application of a notch stress approach to the fatigue assessment of spot welds. A keyhole notch is assumed to describe the edge of the weld spot between the overlapping plates. The stress fields at the keyhole notch under 'singular' and 'non-singular' in-plane loading modes inclusive of the stress concentration factors K _t are derived from the relevant Airy stress functions. The formulae are applied to typical loading cases of spot welds and compared with finite element solutions. Fatigue-effective notch stresses inclusive of fatigue notch factors K _f are calculated by applying the microstructural support hypothesis of Neuber. The notch stresses at the keyhole are also derived for out-of-plane shear loading based on the relevant harmonic stress functions. The multiaxial notch stresses at the weld spot edge are thus completely described.