Weibull stress model for cleavage fracture under high-rate loading

This paper examines the effects of loading rate on the Weibull stress model for prediction of cleavage fracture in a low-strength, A515-70 pressure vessel steel. Interest focuses on low-to-moderate loading rates ( K˙ _I < 2500  MPa √m  s⁻¹ ). Shallow cracked SE(B) specimens were tested at four different loading rates for comparison with previous quasi-static tests on shallow notch SE(B)s and standard C(T)s. To utilize these dynamic experimental data, we assume that the Weibull modulus ( m ) previously calibrated using quasi-static data remains invariant over the loading rates of interest. The effects of dynamic loading on the Weibull stress model enter through the rate-sensitive material flow properties, the scale parameter ( σ _u ) and the threshold Weibull stress ( σ _w-min ). Rate-sensitive flow properties are modelled using a viscoplastic constitutive model with uniaxial, tension stress–plastic strain curves specified at varying plastic strain rates. The analyses examine dependencies of σ _w-min and σ _u on K˙ _I . Present results indicate that σ _w-min and σ _u are weak functions of loading rate K˙ _I for this pressure vessel steel. However, the predicted cumulative probability for cleavage exhibits a strong sensitivity to σ _u and, consequently, the dependency of σ _u on K˙ _I is sufficient to preclude use of the static σ _u value for high loading rates.     

Effects of loading rate on the local cleavage fracture stress σf in notched specimens

Four point bending (4PB) notched specimens with different notch sizes are tested at various loading rates at a temperature of −110 °C for a C-Mn steel. An elastic-plastic finite element method (FEM) is used to determine the stress and strain distributions ahead of notches. By accurately measuring the distances of the cleavage initiation sites from the notch roots, the local cleavage fracture stress σ_f is measured. The results show that the local cleavage fracture stress σ_f does not essentially change with loading rate V and notch size. The reason for this is that the cleavage micromechanism does not change in the different specimens at various loading rates. The cleavage micromechanism involves competition of two critical events of crack propagation and crack nucleation in the high stress and strain volume ahead of notch root. The large scatter of σ_f and notch toughness are mainly caused by the different critical events in different specimens.     

Effects of loading rate, notch geometry and loading mode on the local cleavage fracture stress of a C–Mn steel

In this work, notched specimens with two notch geometries were tested in two loading modes (four-point bending (4PB) and three-point bending (3PB)) at various loading rates at a temperature of − 110°C for a C–Mn steel. An elastic–plastic finite-element method (FEM) is used to determine the stress distributions ahead of notches. By accurately measuring the distances of the cleavage initiation sites from the notch roots, the local cleavage fracture stress σ _f is measured. The results obtained and combining with previous studies by the authors show that the local cleavage fracture stress σ _f is closely related to the cleavage fracture mechanism (critical events) in steels. The σ _f values do not change with loading rate, notch geometry and loading mode, as long as the critical event of cleavage fracture does not change at various testing conditions. The σ _f is mainly determined by the steel microstructure, and its scatter is mainly caused by the size distribution of the weakest constituent in steels (ferrite grain or pearlite colony with large sizes and large second phase particles) and the change of the critical events in cleavage process. The σ _f can characterize the intrinsic toughness of steels and may be used in a “local approach” model for assessing integrity of flawed structures. The σ _f values could be measured by both 4PB and 3PB tests.     

PROBABILISTIC ASPECTS OF CLEAVAGE CRACK INITIATION SITES AND FRACTURE TOUGHNESS

Abstract— Fracture toughness tests were performed in the ductile-brittle transition temperature range using 110 specimens of the three-point bend and CT types. Probabilistic characteristics of fracture toughness and cleavage crack initiation sites were analysed in detail, together with the fibrous crack shape, from which the plane strain region in the specimen was deduced. The criterion for obtaining plane strain at the mid plane of the specimen was established as: B ≤ 0.004{ K _c( J )/σ _y }²+ 0.01. The thickness effect of cleavage fracture toughness for the specimen satisfying this equation is mainly caused by the statistical distribution of the weakest points ahead of the crack front (the Weibull volume effect).     

On the mechanism of fatigue failure in the superlong life regime (N>107 cycles). Part II: influence of hydrogen trapped by inclusions

High cycle fatigue fracture surfaces of specimens in which failure was initiated at a subsurface inclusion were investigated by atomic force microscopy and by scanning electron microscopy. The surface roughness R _a increased with radial distance from the fracture origin (inclusion) under constant amplitude tension–compression fatigue, and the approximate relationship: R _a ≅ C Δ K ²_I holds. At the border of a fish-eye there is a stretched zone. Dimple patterns and intergranular fracture morphologies are present outside the border of the fish-eye. The height of the stretch zone is approximately a constant value around the periphery of the fish-eye. If we assume that a fatigue crack grows cycle-by-cycle from the edge of the optically dark area (ODA) outside the inclusion at the fracture origin to the border of the fish-eye, we can correlate the crack growth rate d a/ d N , stress intensity factor range Δ K _I and R _a for SCM435 steel by the equation
   
and by d a/ d N proportional to the parameter R _a .
Integrating the crack growth rate equation, the crack propagation period N _p2 consumed from the edge of the ODA to the border of the fish-eye can be estimated for the specimens which failed at N _f > 10⁷. Values of N _p2 were estimated to be ∼1.0 × 10⁶ for the specimens which failed at N _f ≅ 5 × 10⁸. It follows that the fatigue life in the regime of N _f >10⁷ is mostly spent in crack initiation and discrete crack growth inside the ODA.     

MODE I STRESS INTENSITY FACTOR EQUATIONS FOR CRACKS AT NOTCHES AND CAVITIES

Abstract— In this paper, the notch-crack problem is treated in two different ways: if the non-dimensional crack length l /ρ ( l = crack length; ρ= notch root radius) is smaller than the transition crack length l _T/ρ, it is treated as an edge crack lying within the local stress field around the notch tip; if l/ ρ is larger than l _T/ρ, the notch-crack is considered as a simple flat crack problem subjected to remote loading, the flat crack size being the sum of notch depth and the real crack length. Based on currently available numerical data, expressions for the transition crack length, l _T, and for the geometric factor F = K _I/(1.1215K_tσ√π l ) are developed for various notch problems for the crack length range l ≦ l _T. It is found that the stress (σ_yy) normalized by the peak stress (σ_peak), σ_yy/σ_peak, for the pre-cracked component is very similar to the geometric factor for short cracks.     

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.     

A Weibull stress model to predict cleavage fracture in plates containing surface cracks

This study applies recent advances in probabilistic modelling of cleavage fracture to predict the measured fracture behaviour of surface crack plates fabricated from an A515-70 pressure vessel steel. Modifications of the conventional, two-parameter Weibull stress model introduce a non-zero, threshold parameter (σ_w-min ). The introduction of σ_w-min brings numerical predictions of scatter in toughness data into better agreement with experimental measurements, and calibration of this new parameter requires no additional experimental data. The Weibull modulus ( m ) and scaling parameter (σu ) are calibrated using a new strategy based on the toughness transferability model, which eliminates the non-uniqueness that arises in calibrations using only small-scale yielding toughness data. Here, the Weibull stress model is calibrated using toughness data from deep-notch C(T) and shallow-notch SE(B) specimens, and is then applied to predict the measured response of surface crack plates loaded in different combinations of tension and bending. The model predictions accurately capture the measured distributions of fracture toughness values.     

DETERMINATION OF THE ENERGETIC PARAMETERS CONTROLLING CLEAVAGE FRACTURE INITIATION IN STEELS

Abstract— The process of brittle fracture in steels can be divided into three distinct steps: (1) initiation of a microcrack in a brittle particle, (2) propagation of the microcrack into the surrounding matrix and, finally, (3) crack progression through the matrix. Depending on microstructure, temperature and loading rate, the critical step which controls cleavage fracture is subject to change. In this work the behaviour of different microalloyed steels is considered and the energies γ_pm and γ_mm, which define the stress necessary for the microcrack to surmount steps 2 and 3 have been experimentally determined. While the γ_pm value remains constant around 7 J/m², it has been observed that γ_mm is dependent on temperature. At −196°C the value is lower than 50 J/m² and at room temperature it is higher than 200 J/m². This increase in the matrix-matrix energy with temperature increases the probability of microcracks, generated in particles, arresting at grain boundaries. This is the reason why refinement of grain size has an important effect in improving the fracture toughness at room temperature.     

ELASTIC-PLASTIC FRACTURE MECHANICS FOR INITIATION AND PROPAGATION OF NOTCH FATIGUE CRACKS

Abstract Initiation and propagation are considered to be controlled by the extent of total plastic shear deformation φ. Crack initiation and crack propagation occur when φ, exceeds a critical threshold value which can be equated to threshold conditions determined from linear elastic fracture mechanics analyses. When a crack is in a plastically deformed zone φ_t=φ _p +φ _e . where φ _p is the component of φ _t due to notch bulk plasticity and φ _e , is the component of φ _t due to a linear elastic fracture mechanics (LEFM) analysis of the crack tip plasticity field.
When cracks initiate at notch roots φ _t > φ_th. As the crack propagates in the notch plastic zone the rate of decrease of v _p will be different from the rate of increase of φ _e and it is possible for φ _t to decrease to a level below φ_th thereby creating a non-propagating crack.     

Very high cycle fatigue behaviour of 2000-MPa ultra-high-strength spring steel with bainite–martensite duplex microstructure

The very high cycle fatigue and fatigue crack growth (FCG) behaviours of 2000-MPa ultra-high-strength spring steel with different bainite–martensite duplex microstructures (designated as B-M1 and B-M2) obtained through isothermal quenching and fully martensite (designated as M) for comparison were studied in this paper by using ultrasonic fatigue testing and compact-tension specimens. It was found that for the B-M1 sample with well-controlled thin and uniformly distributed bainite, the fatigue crack threshold Δ K _th is higher and FCG rate da / dN at an early stage is lower than those of the M sample. Therefore, the former has rather longer fatigue life at high stress amplitude, though both have almost identical fatigue strength. However, the fatigue properties of bainite–martensite duplex microstructure are significantly deteriorated with the formation of large bainite. Furthermore, like that of the M sample, the S–N curves of the B-M1 and B-M2 samples also display continuous declining type and fish-eye marks were always observed on the fracture surface in the case of internal fractures, which were mainly induced by inclusion. A granular bright facet (GBF) was observed in the vicinity around the inclusion. For each of the three samples, the stress intensity factor range at the boundary of inclusion (Δ K_inc ) decreases with increasing the number of cycles to failure ( N _f), while the stress intensity factor range at the front of GBF(Δ K _GBF) is almost constant with N _f and equals to its Δ K _th. This indicates that Δ K _GBF might be the threshold value governing the beginning of stable crack propagation.     

Effects of geometry of notched specimens on the local cleavage fracture stress σf of C-Mn steel

An elastic-plastic three-dimensional finite element method analysis is used to determine the stress and strain distributions ahead of notches of four-point bending (4PB) specimens with various sizes (W, B and a) and widths (B). By measuring the location of the cleavage initiation sites for a C-Mn steel, the local cleavage fracture stress σ_f is accurately determined. With increasing specimen sizes and widths the fracture load P_f increases considerably, but σ_f remains nearly constant. The reason that the σ_f of the specimen with minimum size is slightly larger than that of the other specimens is analyzed by an active zone model of cleavage fracture for notched specimens. The critical event for cleavage fracture is the propagation of a ferrite grain-sized crack into the neighboring matrix, and is independent of specimen sizes and widths. σ_f is mainly determined by the length of the critical microcrack, and the specimen sizes and widths have little effect on it.     

Cleavag Fracture Criterion of Low Alloy Steal and Weld Metal in Notched Specimens

The cleavage fracture criterion of low alloy steel and weld metal in notched specimens is investigated in detail based on a great number of experimental data. It has been found that the most cleavage fractures initiate at a distance shorter (left side) than that of the peak stress location below a notch root, and the cleavage fracture in notched specimens must satisfy a dual criterion, i.e., a critical plastic strain (ε_p ≥ ε_pc) for initiating a crack nucleus, and a critical tensile stress (σ_yy ≥ σ_f) for its propagation. According to the dual criterion model, the great number of experimental data of 4PB (four-point bending) tests for the low alloy steel and weld metal and their statistical distribution are explained. The effects of temperature , the local fracture stress σ_f and the critical plastic strain ε_pc on the locations of cleavage initiation sites and the controlling steps of cleavage fracture process are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

INVERSION OF THE STRAIN-LIFE AND STRAIN-STRESS RELATIONSHIPS FOR USE IN METAL FATIGUE ANALYSIS

Abstract— Two methods are described for inverting the strainrange/life and strainrange/stressrange equations commonly used in fatigue analysis in order to obtain closed-form expressions for life and stressrange in terms of strainrange. In the Collocation approach the form used is N _f= A (Δε—Δε₀)γ or N _f= A (Δε)Ψ(Δε—Δε₀)γ. In the Spline-Function approach the curve is divided into two regions. At strainranges above where the elastic and plastic lines intersect the equation is N _f= N _T R ^1/c exp δ R α; at lower strainranges it is N _f= N _T R ^1/b exp δ R β, where N _T is transition life, R is strainrange normalised to transition strainrange, and b, c , α, β, δ are constants determinable from the constants of the equation to be inverted. Similar expressions are derived for the cyclic stress/strain curve in terms of the same constants. The methods are illustrated by an example, and found to have close conformity to the basic equations to be inverted.     

Assessment of the period to fatigue macrocrack initiation near stress concentrators by means of strain parameters

A new approach to the experimental assessment of the local strain at a stress concentrator has been presented. It is based on a procedure of notch opening displacement measurements at certain points in the vicinity of a notch related to the effective notch radius ρ _eff = ρ + d * , where ρ is the notch radius and d * is a material constant. Different stress concentrators in structural elements were modelled for a wide variation of notch radii ( ρ = 0.1–6.5 mm) and different geometries of specimens. Hence, a basic relationship, which directly relates the local strain range Δ ε* to the period of fatigue macrocrack initiation N _i has been established. Thus, by applying the value of Δ ε* , assessed from a direct measurement at the notch root, it is possible to determine the period N _i to initiate a fatigue macrocrack of length a _i = d * for some structural components of complicated geometry.     

Stability analysis and experimental verification for optimum notch configuration in chevron-notched round bar specimens

In some earlier communications [Ray and Poddar, FFEMS, 27 (2004), Sarkar and Ray, FFEMS, 31 (2008)], a methodology to estimate the minimum normalized stress intensity factor for chevron-notched round bar (CVNRB) specimens has been outlined. The primary aim of this report is to theoretically analyse stable crack propagation in CVNRB specimens in order to estimate conservative fracture toughness value associated with this specimen geometry. The theoretical analyses have been substantiated using fracture toughness tests on CVNRB specimens of steel having initial normalized notch lengths (α₀) between 0.2 and 0.5. The major inferences drawn from this investigation are: (1) the optimum notch geometry in CVNRB specimens corresponds to 0.2 < α₀ < 0.3 for the maximum stable crack extension, and (2) the estimated fracture toughness of the steel using CVNRB specimens indicates minimum K_ICV at α₀= 0.29 in good agreement with the theoretical analyses.     

Fracture behaviour and cleavage initiation in hypoeutectoid pearlitic steel

The paper reports on an investigation of the micromechanism of cleavage fracture in hypoeutectoid pearlitic R7T steel, commonly used for producing railway wheels. The steel possesses extensive Lüders deformation, which somewhat complicates finite element (FE) modelling and analyses of fracture behaviour. Standard Charpy V-notch specimens were used in order to analyse the fracture behaviour at quasistatic and impact loading. Finite element 3D calculations were performed and the elastic-plastic behaviour of notched bars up to the fracture was simulated. Detailed fractographic analysis was carried out on a number of Charpy V-notch specimens in order to investigate the origin site of cleavage fracture initiation and its distance from the notch root. The suitability of the three-criterion micromechanical model (Chen et al. Acta Materialia 51:1841–1855, 2003) for cleavage initiation was verified. The R7T steel under investigation exhibited a cleavage fracture stress of 1,837 MPa. Its independence on temperature evidenced the micromechanism of cleavage fracture to be microcrack propagation-controlled. For the investigated blunt-notched bend bars, an active volume exists ahead of the notch root in which pearlite colony-associated initiation sites are located. The cleavage fracture initiation of the steel is thus governed by the sites lying in the active volume. The active volume is determined by the values of three parameters. A plastic strain lying in interval from to (for the steel investigated from 0.033 to 0.108) is necessary to create a cleavage crack nucleus at any location within the active volume depending on the local pearlite properties. A stress triaxiality parameter ranging from h _min to h _max (from 0.93 to 1.39) is supposed to prevent the blunting process at the site of the cleavage nucleus. Once the main principal stress σ ₁ exceeds the local cleavage fracture stress σ _CFmin, an unstable global cleavage fracture occurs in a blunt-notched bar.     

ON THE TRANSITION FROM NEAR-THRESHOLD TO INTERMEDIATE GROWTH RATES IN FATIGUE

Abstract— Evidence is presented that the cyclic stress intensity threshold for fatigue crack growth in A1 2219-T851 is associated with a critical maximum value of stress intensity, K _c. This relationship is discovered by measuring the local value of stress intensity at the crack tip which is less than the applied stress intensity because of fatigue induced compressive residual stresses in the plastic zone. Crack growth rates and values of the crack tip residual stress are measured as functions of load ratio. For local stress intensities greater than K _c, the growth rate follows a power-law relationship, increasing monotonically with δ K . For local stress intensities below K _c, growth rates are also sensitive to the cyclic stress range, δσ. If the stress range is small, a threshold to growth, typical of long cracks, is seen. When the cracks are short and δσ exceeds a critical value, growth rates are a complex function of both δσ and δ K . This behavior is attributed to the effect of δσ on the propagation of the crack front past obstacles such as grain boundaries.     

Technical note A study on stress, reflection and double shot peening to inrease compressive residual stress

The technique of shot peening is commonly used to increase the fatigue limit of a steel. However, there are many practical difficulties in applying it to very high HV steel and complicated components. To overcome these problems, the authors proposed two new methods: stress double shot peening and stress reflection double shot peening. Both techniques were applied to quench and tempered steel (QT steel) and induction-heated steel (IH steel). The main results were as follows: (a) by double shot peening, the compressive residual stress near the sample surface was increased considerably; (b) by stress shot peening, the maximum compressive residual stress ( σ _max ) and the surface compressive residual stress ( σ _s ) were greatly increased; (c) by stress double shot peening, very high compressive residual stresses ( σ _max = −1710 MPa and σ _s = −1320 MPa) were successfully introduced into a hard steel (HV = 700); (d) in the new method (stress reflection double shot peening), very high compressive residual stresses ( σ _max = −1760 MPa and σ _s = −1460 MPa) were successfully introduced into a hard steel (HV = 700).     

A GENERAL CRITERION FOR HIGH CYCLE MULTIAXIAL FATIGUE FAILURE

Abstract— A new simple general criterion of failure for high cycle multiaxial fatigue, τ_a/ t _A.B+σ_{n. max}/2σ_T= 1 presented. The failure criterion is based on a critical plane approach where fatigue strength is a function of the shear stress amplitude and the maximum normal stress on the critical plane of maximum shear stress amplitude. The criterion takes account of whether case A cracks, growing along the surface, or case B cracks, growing into the surface, occur. It requires knowledge of the material properties, tensile strength, σ_T, and reversed shear fatigue strength for case A, t _A, or case B, t _B, cracking, whichever is relevant. t _A is the reversed torsion fatigue strength and t _B is found from a case B cracking test case. The criterion is applicable in the region, 0.5 t ≤ t _a≤ t , and 0 ≤σ_n.max≤σ_T.