Neuber's Rule Research Articles

Local strain computation of crack initiation life of an unstiffened steel tubular T-joint

This paper presents the results of the fatigue crack initiation life estimation for offshore tubular welded joints. The results are obtained by using the fatigue strength exponent, b, and fatigue ductility exponent, c, computed by a backcalculation procedure from an extensive experimental crack initiation estimation of both small scale and medium scale welded steel tubular joints. A general purpose finite element package, ABAQUS, was used for the stress analysis. This allows for the accurate determination of the critical location where the crack/cracks is/are likely to initiate. From the stress analysis, the stress and strain concentration factors (SCF and SNCF) along the intersection of the tube were obtained. The peak SCF and SNCF were used in modified Neuber's rule and the result introduced into a Manson-Coffin model for crack initiation life computation. The results obtained were compared with the experimental results previously obtained at the Canadian Cooperative Fatigue Studies Program.

Calculation of elastic-plastic strains and stresses in notches under torsion load

An application of the equivalent strain energy density method for calculation of elastic-plastic notch-tip strains in circumferential notches under torsion is presented. The mathematical formulation of the method is presented together with its numerical interpretations. Finally, the calculated notch-tip strains are compared with the finite element data obtained for several materials and notches. Included are also results obtained from Neuber's rule. Good notch-tip strain predictions based on the equivalent strain energy density concept were achieved for sharp and blunt notches.

Advanced creep-fatigue evaluation rule for fast breeder reactor components: generalization of elastic follow-up model

High temperature operative conditions of fast breeder reactors (FBRs) cause plastic and creep deformations of structures, which lead to the magnification of the creep-fatigue damage of the materials. To take these effects into account, a structural design code for the Japanese prototype FBR “Monju” has adopted an elastic follow-up model. The elastic follow-up model shows a potential for expressing both the plastic and creep behaviors of structures. However, this model has weaknesses which mean that it cannot be used to treat material properties quantitatively and also cannot be adaptive to the existence of stress redistributions. Consequently, we have generalized the elastic follow-up model, by introducing the following concepts: (1) an elastic follow-up model using a power law, for the purpose of the quantitative evaluation of material properties; (2) combined elastic follow-up models, for consideration of stress redistributions; (3) an equivalent elastic follow-up model of Neuber's rule. By applying the generalized elastic follow-up model to structural discontinuity portions, we proposed an advanced creep-fatigue evaluation rule based on elastic analysis. The proposed method is verified through structural strength tests and inelastic finite element calculations. As a result, the design stresses are raised significantly compared with those of the structural design code for “Monju”.

AN ANALYTICAL PROCEDURE FOR PREDICTING OPENING LOADS OF CRACKS AT NOTCHES

Abstract— An analytical procedure for the prediction of opening load levels for cracks growing from notches is presented, and estimates are made using experimental crack opening load data. The procedure is based on McClung's original procedure for estimating the local elastic‐plastic stress—strain situation at the point of a hypothetical crack tip in a notched but uncracked body. A crack opening stress value for a crack in an urmotched body under the same stress—strain condition is then predicted. The corresponding opening load level for the crack stress‐strain configuration in the notched but uncracked body is then predicted with a simple backward calculation. For the evaluation of the elastic‐plastic stress distribution, simple analytical relationships, such as Neuber's rule, are used. Estimation of the crack opening stress level is performed using various crack opening stress formulae reported in the literature for cracks in unnotched specimens under a uniform stress distribution. The accuracy of all the investigated crack opening stress formulae within the scope of the presented procedure is checked, and discussed in detail using experimental data sets for opening load levels of comer cracks and through‐thickness cracks in notched specimens of FeE460 and A15086. The predicted results using Newman's crack opening stress formulae, simply modified in respect to the definition of the flow stress, have been identified to be qualitatively and quantitatively consistent with the trends observed in experimental crack opening data.

FATIGUE CRACK INITIATION LIFE PREDICTION IN HIGH STRENGTH STRUCTURAL STEEL WELDED JOINTS

AbstractThe local approach method is used to calculate the fatigue crack initiation/early crack growth lives (Ni) in high strength structural steel weldments. Weld‐toe geometries, welding residual stresses and HAZ (heat affected zone) cyclic mechanical properties are taken into account in the Ni estimation procedure. Fatigue crack initiation lives are calculated from either a Basquin type or a Manson‐Collin type equation. The local (HAZ) stress and strain amplitudes and the local mean stress are determined from an analysis based on the Neuber rule and the Molski‐Glinka energy approach. The accuracy of the different methods is evaluated and discussed. Finally the previous methods are used with HAZ cyclic mechanical properties estimated from hardness measurements.

A Constitutive Model for Estimating Multiaxial Notch Strains

True stress and strain components at a notch are the essential parameters for fatigue life predictions. Nonlinear finite element analysis (FEA) could be the perfect solution to the notch stress and strain calculation, but its usage may be very limited due to intensive CPU time consumption. Estimation techniques for the multiaxial notch stresses and strains from linear FEA results are important and needed. The existing approach—the Hoffmann and Seeger theory—is limited for monotonic loading cases. It appears difficult to extend its application to nonproportional and variable amplitude loading cases. A generalized method for estimating multiaxial notch stresses and strains on the basis of elastic stress solutions is presented here. This method utilizes a two-surface model with the Mroz hardening equation and the associated flow rule to simulate the local notch stress and strain responses for any geometrical constraints of specimens under monotonic, in-phase and out-of-phase loading. The uniaxial material properties associated with the two-surface model are determined by: the Neuber rule, the Glinka rule and FEA results. Comparisons are made with the notch strains calculated by nonlinear FEA and those obtained from strain gages. Reasonable correlations between the measured and predicted notch strains are observed for SAE 1045 material.

A VARIABLE Kf‐NEUBERS RULE FOR PREDICTING FATIGUE CRACK INITIATION LIFE

AbstractAs is well known, the fatigue strength reduction factor due to a notch (Kf) in the modified Neuber's rule should not be considered a constant since it varies with different stress levels. In this paper, the definition and the variation of Kf are discussed on the basis of theoretical and experimental analyses, leading to the presentation of a variable Kf‐Neuber's rule. Comparing the results of experimental stress analyses and the fatigue life tests of specimens, the variable Kf‐Neuber's rule gives good predictions and is applied to the life prediction of an aircraft component.

Crack initiation criterion at notches in Zr-2.5Nb alloys

This work investigates the experimental conditions necessary for initiation of delayed hydride cracking in Zr-2.5Nb alloys. The experiments were performed on 240 notched cantiliver beam specimens loaded in pure bending to a wide range of stresses. The test temperature was 250°C with weekly cycles between 60 and 295°C. The results are interpreted in the light of a recently proposed fracture criterion for hydrides at cracks and notch tips. This criterion is based on the premise that crack initiation occurs if the local tensile stress at the hydride exceeds that for hydride fracture. The local stress at the hydride is expressed as the sum of the peak stress at the notch and the stress in the hydride arising from the hydride formation process. The fracture stress of the hydride is obtained experimentally. An approximate analytical method, based on a modified Neuber's rule, was used to calculate the plastic zone size needed to estimate the peak stress at the notch. These approximate calculations show that the threshold notch tip stress for cracking is between 675 and 750 MPa.

Evaluation of the fatigue strength of adhesive joints based on interfacial fracture mechanics: Yuuki, R., Liu, J.-Q., Xu, J.-O., Ohira, T. and Ono, T. J. Soc. Mater. Sci. Jpn (Aug. 1992) 41 (467) 1299–1304 (in Japanese)

Fatigue tests that simulate an autofrettaged thick-walled pressure vessel with an external groove under pulsating internal pressure loading conditions were performed using specimens taken from an autofrettaged thick-walled pressure vessel. Load-controlled simulation fatigue tests using rectangular, elliptical, and shot-peened elliptical grooved specimens were performed for three different autofrettage levels of 50, 75, and 100% overstrains. In order to estimate the fatigue life of the pressure vessel subjected to pulsating internal pressure, a local strain approach was considered to access the crack formation life. A cyclic stress-strain relation and fatigue damage models determined from strain-controlled low-cycle fatigue tests were employed to estimate the fatigue life of the pressure vessel. Larger local stresses and strains obtained from Neuber's rule compared with the linear rule led to the coservative fatigue life estimations. Estimated fatigue lives were obtained within factors of 2–4, compared with the experimental fatigue lives determined from the simulation fatigue tests.

The effect of proof loading on the fatigue behaviour of stud link chain

Although stud link chain is routinely proof loaded during its manufacture, the effect of residual stresses imposed by this operation on the fatigue strength of the chain has not been quantitatively investigated. This paper discusses the results of constant amplitude fatigue tests on stud link chain segments which have received proof loading at various levels. The chain was heat treated before fatigue testing to relieve manufacturing residual stresses and then proof loaded at various levels from zero to 86% of its break strength. Tests were performed at two mean load levels and three load amplitudes. Failure site trends are reported and correlated with results of a finite element stress analysis. Residual stress estimates are used with standard fatigue damage parameters to estimate the fatigue life of the chain and predictions are compared to the experimental data. Low cycle fatigue parameters were derived from a previous test programme for open link chain. Proof loading was shown to increase substantially the fatigue life of the stud link chain. Residual stresses can explain the increase in fatigue life. Neuber's rule demonstrated the ability to model the data trends.

Fatigue crack growth in the nozzle corner of a pressure vessel: Gao, Z., Xu, L. and Zhang, K.Int. J. Pressure Vessels Piping 1990 42, (1), 1–13

In this paper, two analytical models for estimating surface notch-root elastic–plastic, strain–stress histories in particulate metal matrix composites (PMMCs) when subjected to multiaxial cyclic loads are presented and compared. The models are based on the concept of incremental average stress, the endochronic theory of plasticity, and modified forms of the incremental equivalent strain energy density method (ESED) and Neuber's rule. Each model provides a complete set of equations capable of predicting the elastic–plastic strain–stress histories at the notch root given a corresponding elastic solution. The preliminary analytical results obtained by applying the formulations to a circumferentially notched round bar show that each model can independently be used to predict the nonlinear heterogeneous behavior at the notch root in PMMC components.

NOTCHED FATIGUE BEHAVIOUR OF TWO COLD ROLLED STEELS

Abstract— A SAE1010 plain carbon steel and a SAE945X HSLA steel were cold rolled to various thickness reductions. Centre notched specimens were tested under stress control at a stress ratio of—1. The effect of loading direction on the fatigue strength was examined. The notched specimen fatigue strength was only slightly increased by cold rolling, since two opposing factors: the smooth specimen fatigue strength and the notch sensitivity, were increased by cold rolling. The notched specimen fatigue strength in the transverse direction was approximately the same as that in the longitudinal direction. An empirical equation and equations derived from fracture mechanics and Neuber's rule were applied to predict the fatigue notch factor for the sharp and blunt notch geometries examined. A reasonable agreement between the predictions and the experimental results was observed for the sharp notches. For the blunt notches, the predicted fatigue notch factors were conservative.

Isothermal and nonisothermal fatigue behavior of a metal matrix composite: Gabb, T.P., Gayda, J. and MacKay, R.A. J. Compos. Mater. June 1990 24, (6), 667–686

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.

Definition of hot spot Stress in welded structure for fatigue assessment (2nd Report)

In design of a large structure, the assessment of fatigue strength is one of major design considerations. In evaluation of fatigue strength of welded toes, the development of FE analysis and detailed model tests have enhanced the use of “Hot Spot Stress” or “Peak Strain” concepts.In the first report, the hot spot stress definition for plate type structures was discussed, using the experimental data of the model tests by SR202.In this paper, fatigue strength assessment in the use of “Peak Strain” concept is presented. Fatigue strength assessment methods with the peak stress are shown in ASME Boiler and Pressure Code Sec. III and NK rule Part P.For the short of TMCP steel low cycle fatigue data and the short of application for model tests data in the peak strain concept application, in this research, low cycle fatigue tests of round bars with notch were done and analysis based on the peak strain concept for the model tests by SR202 have been made.In the analysis, the NK method with some choice of detailed assessment process have been investigated.Finally, detailed calculation method for the peak strain with the object of assessing low to high cycle fatigue strength of welded structures is proposed, that is the use of Neuber rule for plastic strain concentration and Manson method for stable stress strain relation.

Use of Neuber's rule to estimate the fatigue life of notched specimens of ASME SA 106-B steel piping in 288 °C air: Terrell, J.B. Int. J. Pressure Vessels Piping 1989 40, (1), 17–40

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.

Effect of notch root radius on fatigue crack initiation

The initiation of a fatigue crack terminates the useful life of small components. The initiation process is intimately related to cyclic plastic deformation and thus occurs mostly at notches where stresses are elevated appreciably. The root radius of such notches dictates the severity of stress and strain concentration. Using notched specimens of varying root radii the initiation lives were determined experimentally. Both the fictitious elastic notch root stress and the local strain at the notch root have been used to correlate with the initiation life. The local strain approach, together with Neuber's rule, is observed to give less scatter. The design-oriented nominal mean stress method of Juvinall yields results comparable with the local strain approach.

LOW CYCLE FATIGUE LIVES OF NOTCHED COMPONENTS

Abstract— In low cycle fatigue situations, the plastic behaviour of the material at the root of stress concentrators is of prime importance in determining the cyclic life. However, simple procedures such as Neuber's rule do not adequately describe the development of plastic behaviour at a notch root, while the expense of a finite element analysis is not justified in many instances. This paper describes a simple, approximate numerical method of calculating plastic notch stresses and strains that would be of use in such situations. The usefulness of the technique is demonstrated by comparing low cycle fatigue lives predicted from notch plastic strains with those determined by fatigue testing of smooth specimens subjected to similar plastic strain ranges.

Use of elastic calculations in analysis of fatigue (factor [formula omitted

Evaluation of fatigue damage necessitates determination of the strain range. Elastic calculations have certain advantages but the raw results need to be processed. It is demonstrated that this processing must be carried out in two stages. The first stage covers the component as a whole and is based on evaluation of its spring effect. The second is local and relates to stress raisers, using an extension of the Neuber rule. The two stages are described in detail and illustrated by examples.

Low-cycle fatigue life prediction of circumferentially notched component based on elastic-plastic stress-strain analyses.

The circumferentially notched medium carbon steel specimens of root radii of 2, 3 and 5 mm were low-cycle fatigued. Meanwhile the stress-strain response was calculated at the notch root by means of the F. E. M. and the Neuber's rule. The crack initiation lives of the notched specimens were controlled well by the equivalent stress, the equivalent plastic strain and the axial plastic strain produced at the notch roots. The axial plastic strain versus crack initiation life curve estimated through Neuber's rule predicts the shorter and the longer lives in the higher and the lower plastic strain regions, respectively. The applicability of Neuber's rule to the estimation of the crack initiation life extends much more in the notched specimen of the smaller root radius. The above three quantities are also the dominant factors for the failure life of the notched specimen, nevertheless the stress and strain gradients develop in the cross section of the notched specimen. The necessity of the fracture mechanics approach to the surface crack produced at the notch root was suggested.

Use of Neuber's rule to estimate the fatigue life of notched specimens of ASME SA 106-B steel piping in 288°C air

Fatigue strain-life tests were conducted on notched specimens of ASME SA 106-B piping steel at PWR operating temperatures (288°C (550°F)), under completely reversed loading. Fatigue limits at 10 7 cycles were estimated for smooth specimens to be 185 MPa (26·8 ksi) at 24°C and 232 MPa (33·7 ksi) at 288°C. The higher fatigue strength observed at the PWR temperature is postulated to be caused by dynamic strain aging processes. However, a reduction in fatigue strength in the low cycle fatigue regime was observed in 288°C air environment tests, which may indicate that the current ASME Section III design curve for carbon steels is nonconservative in its positioning. Notch strain histories were estimated for the notched specimen tests using various interpretations of Neuber's rule. It was concluded that the use of the fatigue notch concentration factor (K f) in the Neuber relation in conjunction with the uniaxial cyclic stress-strain curve provided the best correlation of notched specimen fatigue data with results obtained from smooth sp̀ecimen tests. The notched specimen strain-life results derived from the application of Neuber's rule alone proved to be conservative when compared with smooth specimen test results to such an extent that Neuber-generated notch stresses and strain amplitudes cannot accurately be compared with the mean data curves derived from the ASME Section III fatigue curves for carbon steels which are based on net section stress measurements.