Multi-axial fatigue behaviour of a severely notched carbon steel

The paper deals with multi-axial fatigue strength of notched specimens made of C40 carbon steel (normalised state), subjected to combined tension and torsion loading, both in-phase and out-of-phase (Φ=0 and 90°). V-notched specimens have been tested under two nominal load ratios, R=−1 and 0, while keeping constant and equal to the unity the biaxiality ratio, λ=σ_a/τ_a. All specimens have the same geometry, with notch tip radius and depth equal to 0.5 and 4 mm, respectively, while the V-notch angle is equal to 90°. The results determined are discussed together with those deduced under pure tension or torsion loading on plain and notched specimens as well as on small shafts with shoulders. The application of an energy-based approach allows all the fatigue data obtained from the notched specimens to be summarised in a single scatter band, in terms of the total strain energy density evaluated at the notch tip against cycles to failure.     

Low‐cycle fatigue/high‐cycle fatigue interactions in notched Ti‐6Al‐4V*

Combined low‐cycle fatigue/high‐cycle fatigue (LCF/HCF) loadings were investigated for smooth and circumferentially V‐notched cylindrical Ti–6Al–4V fatigue specimens. Smooth specimens were first cycled under LCF loading conditions for a fraction of the previously established fatigue life. The HCF 10⁷ cycle fatigue limit stress after LCF cycling was established using a step loading technique. Specimens with two notch sizes, both having elastic stress concentration factors of K_t = 2.7, were cycled under LCF loading conditions at a nominal stress ratio of R = 0.1. The subsequent 10⁶ cycle HCF fatigue limit stress at both R = 0.1 and 0.8 was determined. The combined loading LCF/HCF fatigue limit stresses for all specimens were compared to the baseline HCF fatigue limit stresses. After LCF cycling and prior to HCF cycling, the notched specimens were heat tinted, and final fracture surfaces examined for cracks formed during the initial LCF loading. Fatigue test results indicate that the LCF loading, applied for 75% of total LCF life for the smooth specimens and 25% for the notched specimens, resulted in only small reductions in the subsequent HCF fatigue limit stress. Under certain loading conditions, plasticity‐induced stress redistribution at the notch root during LCF cycling appears responsible for an observed increase in HCF fatigue limit stress, in terms of net section stress.     

A notch stress intensity approach to assess the multiaxial fatigue strength of welded tube-to-flange joints subjected to combined loadings

Full penetration T butt weld joints between a tube and its flange are considered, subjected to pure bending, pure torsion and a combination of these loading modes. The model treats the weld toe like a sharp V‐notch, in which mode I and mode III stress distributions are combined to give an equivalent notch stress intensity factor (N‐SIF) and assess the high cycle fatigue strength of the welded joints. The N‐SIF‐based approach is then extended to low/medium cycle fatigue, considering fatigue curves for pure bending and pure torsion having the same slope or, alternatively, different slopes. The expression for the equivalent N‐SIF is justified on the basis of the variation of the deviatoric strain energy in a small volume of material surrounding the weld toe. The energy is averaged in a critical volume of radius R_C and given in closed form as a function of the mode I and mode III N‐SIFs. The value of R_C is explicitly referred to high cycle fatigue conditions, the material being modelled as isotropic and linear elastic. R_C is thought of as a material property, independent in principle of the nominal load ratio. To validate the proposal, several experimental data taken from the literature are re‐analysed. Such data were obtained by testing under pure bending, pure torsion and combined bending and torsion, welded joints made of fine‐grained Fe E 460 steel and of age‐hardened AlSi1MgMn aluminium alloy. Under high cycle fatigue conditions the critical radius R_C was found to be close to 0.40 mm for welded joints made of Fe E 460 steel and close to 0.10 mm for those made of AlSi1MgMn alloy. Under low/medium cycle fatigue, the expression for energy has been modified by using directly the experimental slopes of the pure bending and pure torsion fatigue curves.     

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

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

Fatigue strength of notched specimens made of 40CrMoV13.9 under multiaxial loading

The work deals with multiaxial fatigue strength of notched round bars made of 40CrMoV13.9 steel and tested under combined tension and torsion loading, both in-phase and out-of-phase. The axis-symmetric V-notches present a constant notch root radius, 1 mm, and a notch opening angle of 90°; the notch root radius is equal to 4 mm in the semi-circular notches where the strength in the high cycle fatigue regime is usually controlled by the theoretical stress concentration factor, being the notch root radius large enough to result in a notch sensitivity index equals to unity. In both geometries the diameter of the net transverse area is 12 mm.The results from multi-axial tests are discussed together with those obtained under pure tension and pure torsion loading from notched specimens with the same geometry. Altogether more than 120 new fatigue data are summarised in the present work, corresponding to a one-year of testing programme.All fatigue data are presented first in terms of nominal stress amplitudes referred to the net area and then re-analysed in terms of the mean value of the strain energy density evaluated over a given, crescent shape volume embracing the stress concentration region. For the specific steel, the radius of the control volume is found to be independent of the loading mode.     

Experimental results and fatigue life evaluation of magnesium laserbeam‐welded joints under proportional and non‐proportional multiaxial fatigue loading with variable amplitudes

Fatigue life of magnesium laserbeam‐welds (AZ31 and AZ61 alloys) was assessed experimentally under variable amplitude loadings. The specimens were subjected to load‐controlled cyclic loadings. The tests were carried out using a Gauss‐distributed amplitude sequence of length L_S = 5 · 10⁴ cycles and loading ratio R = –1 under pure axial, pure torsion as well as in‐phase and out‐of‐phase combined loadings. The notch stresses were obtained from a linear‐elastic FE‐model using the reference radius approach with r_ref  = 0.05 mm. The stress‐based hypotheses were applied: Effective equivalent stress hypothesis (EESH), shear stress intensity hypothesis (SIH), Findley, and modified Gough‐Pollard. A non‐proportionality factor is introduced and steps required for computing are presented in order to improve fatigue life assessment under non‐proportional loadings.     

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

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

Shear mode threshold for a small fatigue crack in a bearing steel

The fatigue behaviour of small, semi‐elliptical surface cracks in a bearing steel was investigated under cyclic shear‐mode loading in ambient air. Fully reversed torsion was combined with a static axial compressive stress to obtain a stable shear‐mode crack growth in the longitudinal direction of cylindrical specimens. Non‐propagating cracks less than 1 mm in size were obtained (i) by decreasing the stress amplitude in tests using notched specimens and (ii) by using smooth specimens in constant stress amplitude tests. The threshold stress intensity factor ranges, ΔK_IIth and ΔK_IIIth, were estimated from the shape and dimensions of non‐propagating cracks. Wear on the crack faces was inferred by debris and also by changes in microstructure in the wake of crack tip. These effects resulted in a significant increase in the threshold value. The threshold value decreased with a decrease in crack size. No significant difference was observed between the values of ΔK_IIth and ΔK_IIIth.     

Mode I Fracture Toughness of CNT‐Reinforced PMMA

The influence of carbon nanotube (CNT) concentration on the fracture toughness of poly(methyl methacrylate) (PMMA) was examined on single‐edge V‐notched‐beam (SEVNB) specimens. Six groups of SEVNB specimens containing 0.5, 1, 2, 4, 8.5 wt% of CNTs and neat PMMA as a reference were tested. First, a notch was introduced into the specimens by a specially made disk whose edge is V‐shaped with a 30^° angle and a 30 μm tip width. As suggested by an American Society for Testing and Materials Standard for polymers, induction of a natural crack was attempted, without success. Therefore, fracture toughness values were determined with the ‘sharp’ machined notch by means of a calibration formula. These were compared to values obtained using a stress concentration factor and found to differ by less than 3%. The latter calculation takes into account the geometry of the notch. Results showed a decrease in the fracture toughness values with an increase in the CNT concentration. For specimens in which a natural crack was attempted, referred to as a razor‐cut notch, a significant increase in the apparent fracture toughness was observed, as a result of the induced damage.     

Stress intensity factors for surface cracks in single‐edge notch bend specimen by a three‐dimensional weight function method

Stress intensity factors for half‐elliptical surface cracks at a semi‐circular notch in a recently developed single‐edge notch bend specimen are determined for a wide range of geometrical parameters using a three‐dimensional weight function method. Two load cases of pin loading and uniform remote tension are considered. The results are in good agreement with abaqus/franc3d finite element analysis. It is found that the Ziegler–Newman engineering similitude approach (programmed into the Fatigue Crack Growth Structural Analysis life‐prediction code) produces good results for a wide range in a/c ratios. Expressions by multi‐variable curve fitting to the weight function results are presented for easy engineering applications.     

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

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

The effect of notch geometry on critical distance high cycle fatigue predictions

A critical distance method for predicting the fatigue limit stresses of notched specimens was implemented for notched specimens with a wide range of notch dimensions. Circumferentially notched cylindrical specimens (k_t=1.97–4.07) taken from Ti–6Al–4V forged plate were cycled to failure (R=0.1 and 0.5) using a step loading method for estimating the 10⁶ cycle fatigue limit stresses. These experimental data were used in combination with finite element solutions for all specimen geometries to determine a ‘critical distance’, a quantity or parameter determined from the stress distribution surrounding the notch in combination with fatigue limit stress data from unnotched specimens. A unique parameter was not found for all of the specimen geometries. However, predictions for the fatigue limit stresses of the larger notch geometries may be made with some amount of accuracy using a single value of the critical distance parameter, while reasonable predictions for the specimens with the smallest notch dimensions may be made upon the recognition of an apparent size effect.     

Fatigue assessment of laserbeam‐welded aluminium joints under multiaxial loading

This paper presents the results and evaluation of the multiaxial fatigue behaviour of laserbeam‐welded overlapped tubular joints made from the artificially hardened aluminium alloy AlSi1MgMn T6 (EN AW 6082 T6) under multiaxial loadings with constant and variable amplitudes. Several fatigue test series under pure axial and pure torsional loadings as well as combined axial and torsional proportional and non‐proportional loadings have been carried out in the range of 2·10⁴ to 2·10⁷ cycles. The assessment of the investigated thin‐walled joints is based on a local notch stress concept. In this concept the fatigue critical area of the weld root is substituted by a fictitious notch radius r_ref = 0.05 mm. The equivalent stresses in the notch, considering especially the fatigue life reducing influence of non‐proportional loading in comparison to proportional loading, were calculated by a recently developed hypothesis, which is called the Stress Space Curve Hypothesis (SSCH). This hypothesis is based on the time evolution of the stress state during one load cycle. In addition, the fatigue strength evaluation of multiaxial spectrum loading was carried out using a modified Gough‐Pollard algorithm.     

Fracture investigation of U‐notch made of tungsten–copper functionally graded materials by means of strain energy density

The averaged strain energy density over a well‐defined control volume was employed to assess the fracture of U‐notched specimens made of tungsten–copper functionally graded materials under prevalent mode II loading. The boundary of control volume was evaluated by using a numerical method. Power law function was employed to describe the mechanical properties (elasticity modulus, Poisson's ratio, fracture toughness and ultimate tensile stress) through the specimen width. The effect of notch tip radius and notch depth on notch stress intensity factors and mode mixity parameter χ were assessed. In addition, a comparison based on fracture load between functionally graded and homogeneous W–Cu was made. Furthermore, in this research, it was shown that the mean value of the strain energy density over the control volume can be accurately determined using coarse meshes for functionally graded materials.     

Numerical prediction of notch bluntness effect on fracture resistance of SM490A carbon steel

The present work investigates the notch radius effect on fracture resistance using the finite element (FE) damage analysis based on the multiaxial fracture strain model. The damage model was determined from experimental data of notched bar tensile and fracture toughness test data using a sharp‐cracked compact tension specimen. Then, the FE damage analysis was applied to simulate fracture resistance tests of SM490A carbon steel specimens with different notch radii. Comparison of simulated results with experimental data showed good agreement. Further simulation was then performed to see effects of the specimen size, thickness, and side groove on J‐R curves for different notch radii. It was found that effects of the specimen size and thickness became more pronounced for the larger notch radius. Furthermore, it was found that without side groove, tearing modulus for notched specimens was similar to that for cracked specimens, regardless of the notch radius.     

Tensile fracture of centre-notched angle ply (0/±45/0)S and (0/90)2S graphite — epoxy composites

The fracture behaviour of centre-notched (0/± 45/0)_S and (0/90)_2S laminates with increasing notch length has been studied. Two test series have been investigated: specimens of constant width (W=20 mm) and small notch length (2a 12 mm), and specimens with various notch lengths (5 2a 35 mm) and a constant relative notch length (2a/W=0.5). An X-ray technique showed that the damage at the notch tip, which is formed at increasing load, consists mainly of subcracks parallel to the fibres of the constituent layers. The damage zone causes the crack opening displacement (COD) to deviate from the original linearity. TheK _R curve concept has been applied assuming that the COD deviation from linearity is completely the result of original crack extension. This approach fails to describe the notch length effect, because a tangent point between theK _R andK curves was not found and because of a strong dependency of the maximum fracture resistanceK _Rmax on notch length. The fracture behaviour of 20 mm wide specimens could be explained with the point and average stress criteria, based on characteristic lengths which are independent of notch length. At various notch lengths at a constant 2a/W=0.5, however, the characteristic lengths increased with increasing notch length.     

The Modified Wöhler Curve Method applied along with the Theory of Critical Distances to estimate finite life of notched components subjected to complex multiaxial loading paths

This paper is concerned with the use of the Modified Wöhler Curve Method (MWCM) applied in conjunction with the Theory of Critical Distances (TCD) to estimate fatigue lifetime of mechanical components subjected to multiaxial cyclic loading and experiencing stress concentration phenomena. In more detail, our engineering approach takes as its starting point the idea that accurate estimates can be obtained by simply assuming that the value of the critical length, L_M, to be used to evaluate fatigue damage in the medium–cycle multiaxial fatigue regime is a function of the number of cycles to failure, N_f. In other words, the MWCM, which is a bi‐parametrical critical plane approach, is suggested here to be applied by directly post‐processing the linear‐elastic stress state damaging a material point whose distance from the notch tip increases as N_f decreases. According to the main feature of the TCD, the above L_M versus N_f relationship is assumed to be a material property to be determined experimentally: such an hypothesis results in a great simplification of the fatigue assessment problem because, for a given material, the same critical length can be used to estimate fatigue damage independent of the considered geometrical feature. The accuracy of the devised approach was checked by analysing about 150 experimental results we generated by testing V‐notched cylindrical samples made of a commercial cold‐rolled low‐carbon steel. The above specimens were tested under in‐phase and out‐of‐phase combined tension and torsion, considering the damaging effect of superimposed static stresses as well. Moreover, in order to better check its accuracy in assessing notched components subjected to complex loading paths, our method was also applied to several data sets taken from the literature. This extensive validation exercise allowed us to prove that the MWCM applied along with the TCD is successful in estimating medium‐cycle multiaxial fatigue damage (N_f values in the range 10⁴–10⁶), resulting in predictions falling within the widest scatter band between the two used to calibrate the method itself. Such a high accuracy level is very promising, especially in light of the fact that the proposed approach predicts multiaxial fatigue lifetime by post‐processing the linear elastic stress fields in the fatigue process zone: this makes our method suitable for being used to assess real components by performing the stress analysis through simple linear‐elastic FE models.     

The fracture behaviour of notched specimens of polymethylmethacrylate

The fracture behaviour of notched specimens of polymethylmethacrylate has been examined for a wide range of geometries in Charpy impact tests, and in tensile and slow bend fracture tests. It was found that the failure of the very sharply notched specimens was consistent with linear elastic fracture mechanics and defined a constant fracture toughness K _IC for a constant notch tip radius, whereas the blunt notched specimens failed at a constant critical stress at the root of the notch.     

Constraint‐corrected J–R curve based on three‐dimensional finite element analyses

Three‐dimensional (3D) finite element analyses are carried out on single‐edge bend [SE(B)] specimens for which the J‐integral resistance curves (J–R curves) have been experimentally determined to develop the constraint‐corrected J–R curves for the X80 grade pipe steel. The constraint parameters considered in this study include Q_HRR, Q_SSY, Q_{SSY_m}, Q_LM, Q_BM1, Q_BM2, A₂, h and T_z. The constraint‐corrected J–R curves were developed on the basis of the constraint parameters obtained from finite element analysis and experimentally determined J–R curves associated with deeply cracked and medium‐cracked SE(B) specimens and validated against shallow‐cracked SE(B) specimens. The analysis results indicate that all the constraint parameters considered in this study except Q_HRR, Q_SSY, Q_{SSY_m} and Q_LM lead to reasonably accurate constraint‐corrected J–R curves if the crack extensions are relatively small (≤0.7 mm). For larger crack extensions (≤1.5 mm), the Q_BM1‐based constraint‐corrected J–R curve leads to the most accurate predictions of J among all the constraint parameters considered.