Stress redistribution in notched specimens during fatigue cycling

Most materials exhibit a change in stress-strain relationship when subjected to fatigue stresses. In this work, the effect of this change on the stress distribution across the throat of notched-plate specimens of mild steel is examined. Using a set of strain gages, the strain distribution across the specimens was determined under dynamic conditions for various numbers of cycles. Tests of unnotched specimens were used to obtain the cyclic stress-strain properties for corresponding numbers of cycles, and from these data the stress distribution in the notched specimens was determined. Tests in which the strain amplitude at the notch root was held constant revealed a decreasing maximum stress with fatigue cycles. In another series of tests, in which the load amplitude was constant, the maximum stress amplitude was observed to decrease with number of fatigue cycles, despite an increasing strain amplitude. In both types of tests, the stress-concentration factor was observed to decrease with increasing number of fatigue cycles.     

Verification of a neuber-based notch analysis by the companion-specimen method

Measurements of actual notch-root stress-strain behavior by means of the companion-specimen method are compared with predictions of the notch analysis using both a constant (long-life) fatigue-notch factor and a variable fatigue-notch factor determined from fatigue-life data. The results, using cyclicly stabilized specimens, show thatK _f is dependent upon the stress level of the degree of local straining. The amount of error incurred in the use of either the variable or constantK _f was about 7 percent in the prediction of the local strains. In addition, test results using specimens that had not been cyclicly hardened, and specimens subjected to a random-load history, indicated the effects of material-deformation phenomena such as cyclic hardening, stress-strain hysteresis-loop shape, and material memory, on local stress-strain predictions.     

Q235钢缺口试样拉压低循环疲劳的实验研究

以Q235钢制U型缺口板试样为研究对象,用有限元方法计算其缺口根部等效应变幅对应的试样标距段位移,以此控制试验机进行拉压循环疲劳试验。然后用局部应力应变法对试验测得的寿命结果进行分析。结果表明：无论用有限元还是修正Neuber公式计算缺口根部的应力应变,局部应力应变法的疲劳寿命评估只适用于缺口半径较大的试样;对缺口半径较小试样的估计寿命明显低于实测值,且有限元法比修正Neuber法更保守。进而又对试样缺口区域应变梯度的影响进行了探讨：参照有限元计算的应变梯度,利用Taylor模型估算了缺口根部的屈服应力和流动应力;在此基础上重新计算应变分布并估计试样的疲劳寿命,结果证实考虑应变梯度影响可改善缺口试样的疲劳寿命估计。     

Modeling of fatigue crack growth from a notch

When a fatigue crack is nucleated and propagates into the vicinity of the notch, the crack growth rate is generally higher than that can be expected by using the stress intensity factor concept. The current study attempted to describe the crack growth at notches quantitatively with a detailed consideration of the cyclic plasticity of the material. An elastic–plastic finite element analysis was conducted to obtain the stress and strain histories of the notched component. A single multiaxial fatigue criterion was used to determine the crack initiation from the notch and the subsequent crack growth. Round compact specimens made of 1070 steel were subjected to Mode I cyclic loading with different R-ratios at room temperature. The approach developed was able to quantitatively capture the crack growth behavior near the notch. When the R-ratio was positive, the crack growth near a notch was mainly influenced by the plasticity created by the notch and the resulted fatigue damage during crack initiation. When the R-ratio was negative, the contact of the cracked surfaces during a part of a loading cycle reduced the cyclic plasticity of the material near the crack tip. The combined effect of notch plasticity and possible contact of cracked surface were responsible for the observed crack growth phenomenon near a notch.     

Method of calculating residual stresses in semicircular notches in a surface hardened hollow cylindrical specimen

A method is proposed to study the distribution of residual stresses in a semicircular notch in a hollow cylindrical specimen after advanced surface plastic deformation. The initial information used in the method is one or two experimentally determined components of the residual stress tensor in the hardened layer of the smooth specimen. The problem is solved using a finite element technique taking into account initial plastic strains, which are set in correspondence to the residual stresses according to the laws of elasticity. The effect of the hardening technology and notch depth on the distribution of residual stresses is studied. Experimental verification of the method showed that the calculated and experimental data on the stress distribution over the depth of the layer are in good agreement.     

Modelling of rotational factor in notched bend specimen under general and local yield situation

The location of the plastic hinge axis in a three point SEN bend specimen is a highly controversial issue. An unambiguous and reliable estimation of rotational factor (r_p) is very essential for the accurate determination of CTOD data. In contrast to the numerous studies reported on the r_p determination in a cracked situation, limited information is available for a blunt notch situation, although many engineering structures do contain notchlike defects with finite root radius. An attempt is made to determine r_p for two situations, namely well below the general yield and around the general yield. The work is based on a theoretical estimation of the plastic zone size using the stress concentration factor and the elastic as well as the elastic-plastic stress distribution. A theoretical estimation of r_p in both the pseudo-elastic and the elastic-plastic situation is estimated through analytical modelling involving factors like plastic zone size, bend angle and notch opening displacement. The values of the rotational factor are found to increase from a small value to around 0.29 in a well below general yield situation to 0.53 to 0.54 in a general yield situation with continued loading. A wide discrepancy in the P/P_GY ratio separating the two situations, i.e. well below general yield and around general yield, is observed. Consideration of the elastic and the elasto-plastic stress distribution indicates a much smaller value of P/P_GY as compared to the ratio obtained from experimental load-displacement plots.     

Yielding and intrinsic plasticity of Ti–Zr–Ni–Cu–Be bulk metallic glass

Bulk metallic glass with composition Ti₄₀Zr₂₅Ni₈Cu₉Be₁₈ exhibits considerably high compressive yield stress, significant plasticity (with a concomitant vein-like fracture morphology) and relatively low density. Yielding and intrinsic plasticity of this alloy are discussed in terms of its thermal and elastic properties. An influence of normal stresses acting on the shear plane is evidenced by: (i) the fracture angle (<45°) and (ii) finite-element simulations of nanoindentation curves, which require the use of a specific yield criterion, sensitive to local normal stresses acting on the shear plane, to properly match the experimental data. The ratio between hardness and compressive yield strength (constraint factor) is analyzed in terms of several models and is best adjusted using a modified expanding cavity model incorporating a pressure-sensitivity index defined by the Drucker–Prager yield criterion. Furthermore, comparative results from compression tests and nanoindentation reveal that deformation also causes strain softening, a phenomenon which is accompanied with the occurrence of serrated plastic flow and results in a so-called indentation size effect (ISE). A new approach to model the ISE of this metallic glass using the free volume concept is presented.     

On the measurement of elastoplastic stresses

In principle, one should be able to measure elastoplastic stresses in the same manner as one does elastic stresses; i.e., measure the strains and compute the stresses from the constitutive law. In practice, this is rarely done because of the more complicated material response and the anisotropy of the plastic behavior. Further, elastoplastic stresses should be computed incrementally in the general case. This paper presents procedures for computing stresses from elastoplastic strains measured incrementally in a test under microcomputer control. The approach is evaluated for four different materials—two obeying the assumptions of classical plasticity and two showing anisotropic behavior—by computing the stresses in a smooth specimen from measured principal strains. A useful application is presented by computing the stresses at a notch root from biaxial strains measured with laser-based interferometry. The general conclusion is that even in situations where the material is clearly anisotropic, this approach can give a reasonableestimate of the largest local principal stress. Paper was presented at the 1991 SEM Spring Conference on Experimental Mechanics held in Milwaukee, WI on June 9–13.     

Cyclic-inelastic deformation and fatigue resistance of notched-thin aluminum plates

The results of experiments designed to develop data to assess the accuracy and utility of the critical location concept in applications of fatigue-crack nucleation analysis at notch roots are presented and discussed. Fully reversed and nonzero mean-stress data are presented over a range of lives which encompass both elastic and inelastic deformations for thin-notched specimens and smooth specimens made of 2024 T351 aluminum-alloy sheet. Notch-root strains were measured via an extensometer, whereas the formation of small cracks was detected via an eddy-current transducer. Data reported indicated the validity of the assumption that smooth and notched specimens form cracks at the same cycle number when identical deformation histories are imposed at their respective critical locations. They also serve to demonstrate the accuracy and utility of the critical-location approach in analysis to predict the formation of small cracks at notches in coupons and components.     

A unified expression of elastic–plastic notch stress–strain calculation in bodies subjected to multiaxial cyclic loading

In this paper, a simplified thermodynamics analysis of cyclic plastic deformation is performed in order to establish an energy transition relation for describing the elastic–plastic stress and strain behavior of the notch-tip material element in bodies subjected to multiaxial cyclic loads. Based on the thermodynamics analysis, it is deduced that in the case of elastic–plastic deformation, Neuber’s rule inevitably overestimates the actual stress and strain at the notch tip, while the equivalent strain energy density (ESED) method tends to underestimate the actual notch-tip stress and strain. According to the actual energy conversion occurring in the notch-tip material element during cyclic plastic deformation, a unified expression for estimating the elastic–plastic notch stress–strain responses in bodies subjected to multiaxial cyclic loads is developed, of which Neuber’s rule and the ESED method become two particular cases, i.e. upper and lower bound limits of the notch stress and strain estimations. This expression is verified experimentally under both proportional and non-proportional multiaxial cyclic loads and a good agreement between the calculated and the measured notch strains has been achieved. It is also shown that in the case of multiaxial cyclic loading, the unified expression distinctly improves the accuracy of the notch-tip stress–strain estimations in comparison with Neuber’s rule and the ESED method. The unified expression of the notch stress–strain calculation developed in this paper can thus provide a more logical approximate approach for estimating the elastic–plastic notch-tip stress and strain responses of components subjected to lengthy multiaxial cyclic loading histories for local strain approach-based fatigue-crack-initiation life prediction.     

Fatigue crack opening stress based on the strip-yield model

The modified strip-yield model based on the Dugdale model and two-dimensional approximate weight function method were utilized to evaluate the effect of in-plane constraint, transverse stress, on the fatigue crack closure. The plastic zone sizes and the crack opening stresses considering transverse stress were calculated for four specimens: single edge-notched tension (SENT) specimen, single edge-notched bend (SENB) specimen, center-cracked tension (CCT) specimen, double edge-notched tension (DENT) specimen under uniaxial loading. And the crack opening behavior of the center-cracked specimen under biaxial loading was also evaluated. Normalized crack opening stresses σ_op/σ_max for four specimens were successfully described by the normalized plastic zone parameter Δω^′_rev/ω^′ considering transverse stress, where Δω^′_rev and ω^′ are the size of the reversed plastic zone at the moment of first crack tip closure and the size of the forward plastic zone for maximum stress, respectively. The normalized plastic zone parameter with transverse stress also was satisfactorily correlated with the behavior of crack closure for CCT specimen under biaxial loading.     

An energetic approach to predict fatigue crack initiation life of LY12CZ aluminum and 16 Mn steel

The local strain range is considered to be the main factor controlling the fatigue damage process. An energetic approach is applied to correlate the elastic-plastic stress and strain near a notch with the remotely applied stress. Fatigue crack initiation lives of LYI2CZ aluminum and 16 Mn steel are predicted from a knowledge of the uniaxial data involving parameters such as the elastic modulus, strain hardening strength and strain hardening exponent. These quantities are contained and identified with the fatigue strength coefficient C^* which together with the equivalent stress range provide an estimate for the fatigue life of metals. The results agree well with the test data available in the open literature.     

The effect of notches on the fracture behavior in NiTi shape memory alloys

Tensile fracture experiments have been performed on double-notch plate form specimens with different notch types and sizes. Specimen without notch is also studied. The macro-mechanical responses as well as detail examination of the fracture surface have been carried out. The stress, plastic strain and phase transformation fields are analyzed by finite element (FE) simulations using a pseudoelastic constitutive model which considers the permanent plastic deformation. Experimental results show that different type of notches can influence not only the macro-mechanic pseudoelastic but also plastic behaviors of the specimens. Both notch type and notch size affect the mechanism of crack initiation. Notch size influences the specimen behavior in different way for different type of notches. Most of the experimental observations are interpreted properly by the FE results.     

A new criterion for mixed mode fracture initiation based on the crack tip plastic core region

In this work, we propose a new criterion for mixed mode I-II crack initiation angles based on the characteristics of the plastic core region surrounding the crack tip. The shape and size of the plastic core region are thoroughly analyzed under different loading conditions and a new formulation for the non-dimensional variable radius of the core region is presented for mixed mode (K_I−K_II) fracture. The proposed criterion states that the crack extends in the direction of the local or global minimum of the plastic core region boundary depending on the resultant stress state at the crack tip. The results show a well-defined correlation between the plastic core region characteristics and crack extension angles predicted by other criteria. The proposed criterion is formulated for various loading conditions and is compared with other available criteria against the limited available experimental data. It is shown that the proposed criterion provides a better agreement with the experimental data.     

Crack initiation at the rectangular step notch surface under combined loading

Summary  Under external forces acting on the face of a notch, cracks originate at corners, and the system is liable to fail. An analysis is presented of the stress field in the neighborhood of the notch tips, based on the integral representation of the biharmonic solution and on numerical methods. Computations were performed for constant loading or constant displacement distributed along one face of the notch. The coefficients in the principal terms of the asymptotic formulae for the circumferential and shear stresses depend on the angle and height of the notch face and on the boundary conditions. The maximal values of these coefficients determine the stress intensity factors for the opening and shear modes. The angles corresponding to the maximal values of the intensity factors indicate the directions of initiation of opening and sliding cracks. Received 30 May 2000; accepted for publication 3 April 2001     

Elastic-viscoplastic notch correction methods

Neuber’s type methods are dedicated to obtain fast estimation of elastic–plastic state at stress concentrations from elastic results. To deal with complex loadings, empirical rules are necessary and do not always give satisfying results. In this context, we propose a new approach based on homogenization techniques. The plastic zone is viewed as an inclusion in an infinite elastic matrix which results in relationships between the elastic solution of the problem and estimated stress–strain state at the notch tip. Three versions of the notch correction method are successively introduced, a linear one which directly uses Eshelby’s solution to compute stresses and strains at the notch, a non-linear method that takes into account plastic accommodation through a β$\beta$-rule correction and, finally, the extended method that is based on the transformation field analysis methods. All the notch correction methods need calibration of localization tensors. The corresponding procedures are proposed and analyzed. The methods are compared on different simulation cases of notched specimens and the predictive capabilities of the extended method in situations where plasticity is not confined at the notch are demonstrated. Finally, the case of a complex multiperforated specimen is addressed.     

Plastic stress-strain history at notch roots in tensile strips under monotonic loading

Various authors have proposed analytic relations for predicting the plastic behavior at the root of a notch. Among these are relations by Neuber and by Hardrath and Ohman, the latter having generalized on earlier work by Stowell. These methods have had limited experimental confirmation and it was the objective of the investigation reported herein to assess the predictive value of each of the two theories by comparison with test data on externally notched tensile specimens, monotonically loaded to large strains. Since maximum notch-root strain and net-section stress are the only parameters which can be directly measured in a test, theoretical predictions of these same parameters were developed. This was done using a piece-wise analytical representation of the stress-strain curve. Computer programs were developed for analyzing the stress-strain data and for computing the theoretical results. A comparison of theory with tests on flat, notched specimens of AISI 4340 steel, heat treated, with initial elastic concentration factors of 1.5 to 2.0 showed a systematic discrepancy which is attributed in part to notch strengthening due to triaxial stress. The discrepancy is of the order of 5 percent for the Neuber theory and larger for the Hardrath-Ohman theory for notch strains less than of 0.015 in./in. and becomes progressively larger for both theories at notch strains in excess of 0.015 in./in. For the mild notches studied here, the Neuber theory has better predictive value.     

Computer-aided photoelastic analysis of orthogonal 3D textile composites: Part 2. Combining least squares and finite-element methods for stress analysis

An approach combining least squares methods and finite element methods (FEM) is presented for subsequent photoelastic stress analysis of orthogonal 3D textile composites withR and α obtained in Part 1. Through this approach, these photoelastic stresses are obtained over a region of interest as if the composites were homogeneous materials. The least squares method is used for requiring the solution strain fields to best correlate with the distribution of the two photoelastic strain data of ɛ _x − ɛ _y and γ _xy calculated directly from the measuredR and α. The FEM uses the homogenized composite properties to construct the nodal force equilibrium equations as constraints in the least squares formulation. As a result of combining this least squares method and FEM with lagrange multipliers, a linear system of equations is formulated with the unknown nodal displacements. Once these nodal displacements are solved, the strains and stresses can be calculated through FEM formulations. This approach is tested with the two experimental results completed in Part 1 for the aluminum and composite plates. The stresses obtained for the aluminum plate show close agreement with those obtained with the plain FEM computation. In the case of the orthogonal 3D composite plate, the local variations as observed inR and α are already necessarily eliminated from these solved photoelastic stresses. Furthermore, these stresses also match well with those computed with the plain FEM from the homogenized composite properties.     

Elastic–plastic FE analysis of a notched shaft under multiaxial nonproportional synchronous cyclic loading

An elastic–plastic finite element analysis is presented for a notched shaft subjected to multiaxial nonproportional synchronous cyclic tension/torsion loading. The elastic–plastic material property is described by the von Mises yield criterion and the kinematic hardening rule of Prager/Ziegler. The finite element program system ABAQUS is used to solve the boundary value problem. Special emphasis is given to explore the effects of the stress amplitude, the mean-stress, and the mutual interactions on the local stress–strain responses at the notch root.