Neuber's Rule Research Articles

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

AbstractThe 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.

Local ratcheting behavior in notched 1045 steel plates

In this paper, local ratcheting behavior of 1045 steel plates with circular cutout was investigated. Experimental tests were carried out by a Zwick/Roell HB 100 servo hydraulic machine. In order to measure the local strain at notch root, a data acquisition system with strain gauge was used. Various notch diameters and distances of strain gauges mounted from the notch root were found influential in the magnitude of local ratcheting strain. It was found that the local maximum principal stress plays a crucial role in increasing the local plastic deformation. Numerical simulation was done by ABAQUS software using nonlinear isotropic/kinematic hardening model. Material parameters of hardening model were attained from several stabilized cycles of flat specimens subjected to symmetric strain cycles. The nonlinear kinematic hardening model along with the Neuber's rule was employed to assess local ratcheting at the notch root of steel plates. The results of the numerical simulations agreed closely with those measured values in this study. Both ratcheting progress and mean stress relaxation occurred simultaneously at the notch root.

Deviatoric Neuber method for stress and strain analysis at notches under multiaxial loadings

Fatigue life prediction of notched components requires accurate local stresses and strains at notch areas. Researchers have developed a number of approximation methods to estimate local elasto-plastic stress and strain responses on the basis of elastic stress/strain results. The most two common methods are known as the Neuber rule and the equivalent strain energy density (ESED) method. It has been presented that the Neuber rule has a tendency to overestimate notch elasto-plastic strains and stresses while the ESED method tends to underestimate notch elasto-plastic strains and stresses. These drawbacks are attributed to the fact that the original Neuber rule was developed for notched bodies under pure shear loading, but the Neuber rule is applied to other multiaxial loadings where the present stress state can significantly vary from the pure shear stress state. Therefore, analytical approximation model based on the deviatoric form of the Neuber rule has been proposed to estimate elasto-plastic stresses and strains at notches. Predicted notch root stresses and strains from the proposed model are compared with non-linear Finite Element Analysis (FEA) results for SAE 1045 and SAE 1070 steel notched-bars. The SAE 1045 and SAE 1070 steel notch-bars are subjected to monotonic and cyclic non-proportional loadings respectively. The model shows very good agreements with the FEA data for both steel materials.

High‐temperature LCF life estimation based on stress gradient effect of notched GH4169 alloy specimens

AbstractBased on the Neuber's rule and Walker equation, a new measurement method considering the influence of mean stress and stress gradient is proposed. A stress gradient factor is defined to correct the traditional Walker equation; thus, it is easy to estimate the life of the structural components only based on the purely elastic finite element analysis. The low‐cycle fatigue experiment was carried out by using the direct aging GH4169 notched specimen with different stress concentration factors under evaluated temperature. The fracture morphology of specimens was analyzed to determine the crack initiation position and the damage mechanism. The data points of the experiment life and the estimation life are correlated well within a three times scatter band. It indicates that the Walker equation corrected by introducing a new kind of measurement of stress gradient is reliable to estimate the notched specimen fatigue life.

A Generalization of Neuber's Rule for Numerical Applications

In this paper, a generalization of Neuber's rule for a quick and easy elastic/plastic notch analysis is established and discussed. The proposed generalization allows for a numerical and/or graphical solution for any notch geometry as well as its associated stress concentration factor, kt, and fatigue notch factor, kf. It is shown that the so called Neuber's “master” curve, involved in such analysis, is unique and is only material dependent. This is obtained by a simultaneous solution of the Ramberg-Osgood relationship and Neuber's rule. These solutions are plotted in terms of the product of nominal stress, S, times stress concentration factor, kt (or kf), versus the actual notch root strain, ε. The Neuber's “master” curve can be interactive and is applicable for both monotonic and cyclic loading situations. The present formulation is pertinent to conditions when applied nominal stresses, S, is below material's yield stress, σ0, i.e. |S| ≤ σ0. The proposed method is particularly suitable for rapid fatigue life predictions and material screening during pre-prototype phase of notched component design.

Fatigue analysis of notched specimens made of direct-quenched ultra-high-strength steel under constant amplitude loading

Notched specimens made of a direct-quenched ultra-high-strength steel were investigated by applying the different methods available for the stress-strain analysis of notched components. The methods included the linear rule, Neuber's rule and the strain energy density method. The latter two methods were used in modified forms to improve their estimation capabilities under plane strain conditions. Meanwhile, an elastic-plastic finite element model equipped with the stabilized cyclic stress-strain curve as the material response was used to capture the strain values at the notch root required for the life estimation. The most advanced method based on the theory of critical distances, abbreviated as TCD, was utilized as well. This method, using the results of the elastic-plastic finite element model, was able to define a material characteristic length, which is a material constant, regardless of the notch configurations or the number of cycles in a low-cycle regime. The results showed that the finite element analysis and TCD method are able to estimate the lives closest to the experimental values. The TCD method also predicted the lives with the lowest level of unnecessary conservatism, especially in the case of specimens with sharper notches.

The stress raisers effects on the materials mechanical characteristics, analyzed by local microhardness measurements

There have been few studies on the effects of stress raisers on the parts that are plastically deformed, at least into a certain extent of their volume. Such a situation may arrive near a stress raiser, when the peak stress value rises over the material yield stress limit. Some tensile tests are described in the paper, on flat aluminum specimens, with and without the presence of a stress raiser on their surface, namely a through frontal hole, at the center of their calibrated region. Some of the mechanical characteristics (yield limit, elongation at break, Young's modulus) were affected by the stress raiser presence, but its ductile behavior and tensile strength were not. The effective values of stress and strain concentration coefficients were calculated using the Neuber's rule, but the results may be considered as overestimated. The plastic strain enlargement in the specimen volume was also evaluated by measuring the Vickers microhardness values in the stress raiser vicinity. The tests results were shown that the plastic deformation is more pronounced for the measuring points that are closer to the hole's edge; that fact was confirmed by the specimens appearance, after the material failure. A hardness values ratio is finally proposed as an evaluation of the effective stress concentration coefficient.

Plastic flow equations for the local strain approach in the multiaxial case

This paper presents a system of plastic flow equations which uses and generalizes to the multiaxial case a number of concepts commonly employed in the so-called Local Strain Approach to low cycle fatigue. Everything is built upon the idea of distance between stress points. It is believed that this will ease the generalization to the multiaxial case of the intuitive methods used in low cycle fatigue calculations, based on hysteresis loops, Ramberg?Osgood equations, Neuber or ESED rule, etc. It is proposed that the stress space is endowed with a quadratic metric whose structure is embedded in the yield criterion. Considerations of initial isotropy of the material and of the null influence of the hydrostatic stress upon yielding leads to the realization of the simplest metric, which is associated with the von Mises yield criterion. The use of the strain?hardening hypothesis leads in natural way to a normal flow rule and this establishes a linear relationship between the plastic strain increment and the stress increment.

Estimation of the effective notch root stresses and strains for plate with opposite U-shaped notches by nonlinear analysis

In this paper evaluation of realistic stresses and strains at the notch root for plate with U-shaped opposite side notches with shallow and deep notched geometry using nonlinear finite element analysis (NLFEA) is presented. The paper also focuses on the prediction of local stresses and strains on the notched specimens by notch stress-strain conversion (NSSC) rules viz, Neuber's rule and equivalent strain energy density (ESED) method. The results of linear finite element analysis (LFEA) are used for the assessment of the inelastic stresses and strains by NSSC rules. The accuracy of the result is assessed by comparing results with NLFEA.

On-line monitoring and control of thermal stresses in steam turbine rotors

The requirement of high operational flexibility of utility power plants creates a need of using on-line systems for monitoring and control of damage of critical components, e.g. steam turbine rotors. Such systems make use of different measurements and mathematical models enabling calculation of thermal stresses and their continuous control. The paper presents key elements of the proposed system and discusses their use from the point of view of thermodynamics, heat transfer and solid mechanics. Thermodynamic relationships, well proven in design calculations, are applied to calculate on-line the steam temperature at critical locations using standard turbine measurements as input signals. The model predictions are compared with operational data from a real power plant during a warm start-up and show reasonably good accuracy. The effect of variable heat transfer coefficient and material properties on thermal stresses is investigated numerically by finite element method (FEM) on a cylinder model, and a concept of equivalent Green's function is introduced to account for this variability in thermal stress model based on Duhamel's integral. This approach was shown to produce accurate results for more complicated geometries by comparing thermal stresses at rotor blade groove computed using FEM and Duhamel's integral. Finally, the applicability of Neuber's and the Glinka–Molski rule with ideal elastic and bilinear material model to estimating elasto-plastic strains at the rotor groove was investigated by FEM. For the considered groove geometry, Neuber's rule with bilinear material model resulted in most accurate strain predictions.

Cyclic Constitutive Response and Effective S–N Diagram of M50 NiL Case-Hardened Bearing Steel Subjected to Rolling Contact Fatigue

A combined experimental and numerical method is developed to estimate the continuously evolving cyclic plastic strain amplitudes in plastically deformed subsurface regions of a case-hardened M50 NiL steel rod subjected to rolling contact fatigue (RCF) over several hundred million cycles. The subsurface hardness values measured over the entire plastically deformed regions and the elastoplastic von Mises stresses determined from the three-dimensional (3D) Hertzian contact finite element (FE) model have been used in conjunction with Neuber's rule to estimate the evolved cyclic plastic strain amplitudes at various points within the RCF-affected zone. The cyclic stress–strain plots developed as a function of case depth revealed that cyclic hardening exponent of the material is greater than the monotonic strain-hardening exponent. Effective S–N diagram for the RCF loading of the case-hardened steel has been presented and the effect of compressive mean stress on its fatigue strength has been explained using Haigh diagram. The compressive mean stress correction according to Haigh diagram predicts that the allowable fatigue strength of the steel increases by a factor of two compared to its fatigue limit before mean stress correction, thus potentially allowing the rolling element bearings to operate over several hundred billion cycles. The methodology presented here is generalized and can be adopted to obtain the constitutive response and S–N diagrams of both through- and case-hardened steels subjected to RCF.

Effects of mean stress and stress concentration on fatigue behavior of short fiber reinforced polymer composites

AbstractAn experimental study was conducted to evaluate the effect of mean stress on fatigue behavior of two short glass fiber reinforced thermoplastic composites and the effect of stress concentration on fatigue behavior of an unreinforced and a short glass fiber reinforced thermoplastic. Load‐controlled fatigue tests were conducted on unnotched (smooth) specimens at R ratios of −1, 0.1, and 0.3 in different mold flow directions or fiber orientations and at a range of temperatures between −40 and 125 °C. Effect of mean stress on fatigue life was found to be significant at all temperatures. Several mean stress parameters including modified Goodman, Walker, and Smith–Watson–Topper were evaluated for their ability to correlate mean stress data. A general fatigue life prediction model was also used to account for the effect of mean stress, temperature, and fiber orientation. Notched fatigue tests of an unreinforced polymer and a short glass fiber thermoplastic composite were also conducted using plate type specimens with a central circular hole and with or without the presence of mean stress. Effect of stress concentration was found to be considerable, with or without mean stress and in both the longitudinal and transverse directions. The commonly used Neuber's rule for metallic materials, nonlinear finite element analysis, as well as critical distance approaches were utilized for notch deformation and fatigue life analyses.

Effect of surface topography on stress concentration factor

Neuber rule and Arola-Ramulu model are widely used to predict the stress concentration factor of rough specimens. However, the height parameters and effective valley radius used in these two models depend strongly on the resolution of the roughness-measuring instruments and are easily introduce measuring errors. Besides, it is difficult to find a suitable parameter to characterize surface topography to quantitatively describe its effect on stress concentration factor. In order to overcome these disadvantages, profile moments are carried out to characterize surface topography, surface topography is simulated by superposing series of cosine components, the stress concentration factors of different micro cosine-shaped surface topographies are investigated by finite element analysis. In terms of micro cosine-shaped surface topography, an equation using the second profile moment to estimate the stress concentration factor is proposed, predictions for the stress concentration factor using the proposed expression are within 10% error compared with the results of finite element analysis, which are more accurate than other models. Moreover, the proposed equation is applied to the real surface topography machined by turning. Predictions for the stress concentration factor using the proposed expression are within 10% of the maximum stress concentration factors and about 5% of the effective stress concentration factors estimated from the finite element analysis for three levels of turning surface topographies under different simulated scales. The proposed model is feasible in predicting the stress concentration factors of real machined surface topographies.

Tensile and fatigue behaviors of polymers for automotive applications

An experimental study was conducted on tensile and fatigue behaviors of two unreinforced polymers and two short glass fiber reinforced polymer composites. A number of effects including anisotropy, strain rate and temperature on tensile behavior was investigated. Fatigue behavior was evaluated with respect to effects of mold flow direction, temperature, mean stress, and stress concentration. Effect of mold flow direction on both tensile and fatigue behaviors of unreinforced materials was found to be negligible, while it was significant for reinforced materials. Tsai‐Hill criterion commonly used for continuous fiber composites was used to predict the off‐axis fatigue strengths of short fiber reinforced materials. Effect of strain rate on tensile properties was found to be significant for all materials. Test temperature also influenced tensile and fatigue behaviors of both unreinforced and reinforced materials. Mathematical equations were developed to represent variations of tensile properties with both temperature and strain rate. Walker equation with a mean stress correction parameter is proposed to correct for the effect of mean stress. Effect of stress concentration combined with mold flow direction and mean stress was also investigated and Neuber's rule resulted in accurate estimation of fatigue life.

Comparison of Spectral Methods for Fatigue Analysis in Non-gaussian Random Processes – Application to Elastic-plastic Behaviour

In this paper, we present techniques for fatigue damage evaluation using spectral methods and a model taking into account elastic-plastic behaviour. The model is associated with a nonlinear fatigue law, covering the whole endurance domain (low cycle and high cycle fatigue). It uses Neuber's rule and is valid for limited plasticity. To validate this modelling, we perform a correlation between classic spectral methods, modified spectral methods and Monte Carlo simulations in time domain. Results obtained in the case of a numerical design plan are presented.

An Interactive Web-based Tool for Fatigue Analysis and Life Prediction

This paper presents an interactive web-based tool for fatigue analysis and life prediction methodology of smooth and notched components. Users around the globe may access it via Internet by means of multiple platforms such as desktop and laptop computers, tablets and/or smart-phones. In particular, the users with a limited fatigue background would benefit from “on-the-fly” fatigue learning experience. It makes a self-explanatory and frustration free web-based education software, which allows users to learn the fatigue fundamentals while expanding their knowledge on modern fatigue analysis methods. The tool is aimed to assist a designer in fatigue analysis and file predictions of parts subjected to constant amplitude, block loading, and variable loading histories. For variable amplitude loading a dedicated spectrum software package is provided, which is essential for a potential clean-up and/or desired modifications of a raw spectrum data. Subsequently, a rainflow method is utilized and the corresponding hysteresis loops at the notch-root or critical location are determined and plotted. In analyses, three convenient and innovative “master curves” techniques are used, namely for Neuber's rule and also for stress- and strain-based fatigue life approaches.

A structural strain method for low-cycle fatigue evaluation of welded components

In this paper, a new structural strain method is presented to extend the early structural stress based master S–N curve method to low cycle fatigue regime in which plastic deformation can be significant while an elastic core is still present. The method is formulated by taking advantage of elastically calculated mesh-insensitive structural stresses based on nodal forces available from finite element solutions. The structural strain definition is consistent with classical plate and shell theory in which a linear through-thickness deformation field is assumed a priori in both elastic or elastic–plastic regimes. With considerations of both yield and equilibrium conditions, the resulting structural strains are analytically solved if assuming elastic and perfectly plastic material behavior. The formulation can be readily extended to strain-hardening materials for which structural strains can be numerically calculated with ease. The method is shown effective in correlating low-cycle fatigue test data of various sources documented in the literature into a single narrow scatter band which is remarkable consistent with the scatter band of the existing master S–N curve adopted ASME B&PV Code since 2007.With this new method, some of the inconsistencies of the pseudo-elastic structural stress procedure in 2007 ASME Div 2 Code can now be eliminated, such as its use of Neuber's rule in approximating structural strain beyond yield. More importantly, both low cycle and high cycle fatigue behaviors can now be treated in a unified manner. The earlier mesh-insensitive structural stress based master S–N curve method can now be viewed as an application of the structural strain method in high cycle regime, in which structural strains are linearly related to traction-based structural stresses according to Hooke's law. In low-cycle regime, the structural strain method characterizes fatigue damage directly in terms of structural strains that satisfy linear through-thickness deformation gradient assumption, material nonlinear behavior, and equilibrium conditions. The use of a pseudo-elastic structural stress definition is not fundamental, but merely a means to put low-cycle and high-cycle fatigue test data in a conventional stress-based S–N data representation which is typically preferred in engineering practice, than a strain-based representation.

A Modified Method for Calculating Notch-Root Stresses and Strains under Multiaxial Loading

Based on the analysis of notch-root stresses and strains in bodies subjected to multiaxial loading, a quantitative relationship between Neuber rule and the equivalent strain energy density method is found. In the case of elastic range, both Neuber rule and the equivalent strain energy density method get the same estimation of the local stresses and strains. Whereas in the case of elastic-plastic range, Neuber rule generally overestimates the notch-root stresses and strains and the equivalent strain energy density method tends to underestimate the notch-root stresses and strains. A modified method is presented considering the material constants of elastic-plastic Poisson's ratio, elastic modulus, shear elastic modulus, and yield stress. The essence of the modified model is to add a modified coefficient to Neuber rule, which makes the calculated results tend to be more precise and reveals its energy meaning. This approach considers the elastic-plastic properties of the material itself and avoids the blindness of selecting coefficient values. Finally the calculation results using the modified model are validated with the experimental data.

Life Estimation of a Steam Turbine Blade Using Low Cycle Fatigue Analysis

This work is primarily focused in utilising a technique to assess low cycle fatigue life of steam turbine blade. Two approaches are employed here, first is based on the linear elastic finite element analysis. Fictive elastic results are recalculated using Neuber's rule. Second approach is based on elastic-plastic Finite Element analysis (FEA). Strain amplitude approach is followed through Universal slope method and Coffin-Manson equation to determine the number of start-up and shut down cyclesProper blade design with conservative stress levels is important for reliability in rotor blade design. Finite Element Analysis based fatigue tools enable reliable fatigue life calculations to be done at the design stage of a development process and a proper design methodology is important to predict the catastrophic failure of turbine blade due to fatigue.

저압터빈 최종단 블레이드 손상해석

A steam turbine blade is one of the core parts in a power plant. It transforms steam energy into mechanical energy. It is installed on the rim of a rotor disk. Many failure cases have been reported at the final stage blades of a low-pressure (LP) turbine that is cyclically loaded by centrifugal force because of the repeated startups of the turbine. Therefore, to ensure the safety of an LP steam turbine blade, it is necessary to investigate the fatigue strength and life. In this study, the low cycle fatigue life of an LP steam turbine blade is evaluated based on actual damage analysis. To determine the crack initiation life of the final stage of a steam turbine, Neuber's rule is applied to elastic stresses by the finite element method to calculate the true strain amplitude. It is observed that the expected life and actual number of starts/stops of the blade were well matched.