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.     

Fatigue strength assessment of partial and full‐penetration steel and aluminium butt‐welded joints according to the peak stress method

In fatigue design of welded joints, the local approach based on the notch stress intensity factors (NSIFs) assumes that the weld toe profile is a sharp V‐notch having a tip radius equal to zero, while the root side is a pre‐crack in the structure. The peak stress method (PSM) is an engineering, FE‐oriented application of the NSIF approach to fatigue design of welded joints, which takes advantage of the elastic peak stresses from FE analyses carried out by using a given mesh pattern, where the element type is kept constant and the average element size can be chosen arbitrarily within a given range. The meshes required for the PSM application are rather coarse if compared with those necessary to evaluate the NSIFs from the local stress distributions. In this paper, the PSM is extended for the first time to butt‐welded joints in steel as well as in aluminium alloys, by comparing a number of experimental data taken from the literature with the design scatter bands previously calibrated on results relevant only to fillet‐welded joints. A major problem in the case of butt‐welded joints is to define the weld bead geometry with reasonable accuracy. Only in few cases such geometrical data were available, and this fact made the application of the local approaches more difficult. Provided the local geometry is defined, the PSM can be easily applied: a properly defined design stress, that is, the equivalent peak stress, is shown (i) to single out the crack initiation point in cases where competition between root and toe failure exists and (ii) to correlate with good approximation all analysed experimental data.     

An approach to calculate the J‐integral by digital image correlation displacement field measurement

This paper presents a combined experimental‐numerical technique for the calculation of the J‐integral as an area integral in cracked specimens. The proposed technique is based on full‐field measurement using digital image correlation (DIC) and the finite element method. The J‐integral is probably the most generalised and widely used parameter to quantify the fracture behaviour of both elastic and elastoplastic materials. The proposed technique has the advantage that it does not require crack length measurements nor is it limited to elastic fracture mechanics, provided that only small scale yielding is present. Evaluated are three test geometries; compact tension, three‐point bend and the double torsion beam. Possible errors and their magnitude and the limitations of the method are considered.     

3D J‐integral evaluation using the computation of line and surface integrals

In the analysis of fracture mechanics of structures using three‐dimensional (3D) J‐integral, an integral evaluation of line and surface is required. However, because surface integral evaluation requires the calculation of the second derivative of displacement field and commercial finite element codes cannot calculate it, then this portion of the integral is neglected in some research. In this paper, a method for computing 3D J‐integral is presented using finite element analysis. In the analysis, the second derivative evaluation of displacement field is employed. The method is implemented in calculating the J‐integral of some 3D cracks and results are compared to well‐known reference values. The results show that the method is reliable and is suitable for applications in engineering. The portion of 3D J‐integral, namely the surface integral value is investigated and it is shown that neglecting this portion can introduce considerable error in the final results.     

Fatigue crack growth law of API X80 pipeline steel under various stress ratios based on J‐integral

Load‐controlled three‐point bending fatigue tests were conducted on API X80 pipeline steel to investigate the effects of stress ratio and specimen orientation on the fatigue crack growth behaviour. Because of the high strength and toughness of X80 steel, crack growth rate was measured and plotted versus ΔJ with stress ratio. The fatigue crack length is longer in the transverse direction, whereas the fatigue crack growth rates are nearly the same in different orientations. Finally, a new fatigue crack growth model was proposed. The effective J‐integral range was modified by ΔJ_p in order to correlate crack closure effect due to large‐scale yield of crack tip. The model was proved to fit well for fatigue crack growth rate of API X80 at various stress ratios of R > 0.     

Fatigue failure transition analysis in load‐carrying cruciform welded joints based on strain energy density approach

This paper details a study of the application of notch stress intensity theory to the fatigue failure mode analysis of the transition in load‐carrying cruciform welded joints. The weldment fatigue crack initiation point is difficult to predict precisely because it usually occurs in the vicinity of the weld toe or weld root. To investigate the relationship between fatigue failure location and the geometry of the weldments, we analysed the weld toe and root asymptotic notch stress fields were analysed using the notch stress intensity factors on the basis of the Williams' solution in Linear Elastic Fracture Mechanics (LEFM). Numerous configurations of cruciform joints of various plate thicknesses, transverse plate thickness, weld sizes and incomplete penetration size were used to investigate the location of the fatigue failure. The strain energy density (SED) surrounding the notch tip was introduced to unify the scalar quantity and preclude the inconsistency of the dimensionality of the notch stress intensity factors for various notch opening angles. The results of the investigation showed that the SED approach can be used to determine the transition zone for a variety of joint geometries. The validity of the SED criteria was verified by comparing the experimental results of this study with the complied results for load‐carrying cruciform welded joints reported in literature.     

Fatigue strength assessment of laser stake‐welded T‐joints subjected to reversed bending

The paper investigates the fatigue strength of laser stake‐welded T‐joints subjected to reversed bending. The fatigue tests are carried out with the load ratio, R ≈ ?0.8. The experimental data is firstly analysed using the nominal stress approach and then by the J‐integral as the local fatigue strength parameter in the finite element (FE) assessment. The nominal stress approach demonstrated that the fatigue strength of the investigated T‐joints is lower than encountered for any other steel joint under reversed tensile loading. The results also showed that the fatigue strength of this joint under the load ratio R ≈ ?0.8 increases with respect to R = 0 bending by 22.6% in the case of the nominal stress approach and 13% in the case of the J‐integral approach. However, the slopes of the fatigue resistance curves for different load ratios appear very similar, suggesting that the load ratio has an insignificant influence to the slope. In contrast to the similar slopes, the scatter indexes were different. The nominal stress approach shows that the scatter index is 3.4 times larger for R ≈ ?0.8 than R = 0 bending. The J‐integral approach showed that the scatter index for R ≈ ?0.8 is only 67% larger than in the R = 0 case because the weld geometry is modelled in the FE analysis.     

An efficient fatigue and creep‐fatigue life prediction method by using the hysteresis energy density rate concept

Fatigue damage, time‐dependent creep damage and their interaction are considered as the main failure mechanisms for many high temperature structural components. A generalized methodology for predicting both the high temperature low cycle fatigue (HTLCF) and creep‐fatigue lives by using the hysteresis energy density rate (HEDR) and fatigue damage stress concepts was proposed. Experimental data for HTLCF and creep‐fatigue in Alloy 617, Haynes 230 and P92 steel were respectively collected to validate the method. A better prediction capacity and most of the data points that fall within a 1.5 scatter band were obtained compared with the traditional energy‐based method, time fraction rule and ductility exhaustion model. Moreover, a creep‐fatigue damage diagram was also constructed by using the proposed approach.     

The local strain energy density approach applied to pre‐stressed components subjected to cyclic load

Ahead of sharp V‐notches, residual stresses, arising from the solidification of a fusion zone, have the same asymptotic nature of the stress field induced by mechanical loads. This stress field significantly affects the engineering properties of structural components, notably fatigue life and corrosion resistance of welded joints. Tensile residual stresses can reduce the fatigue strength of welded joints particularly in the high‐cycle regime, where no stress redistribution due to local plasticity phenomena is expected to be present. The aim of this work is to analyse, by means of the numerical simulation, the residual stress redistribution near a V‐notch tip induced by cyclic loads and to propose a method, based on the local strain energy approach, for the fatigue resistance estimation of pre‐stressed components. The numerical solutions of the problem were carried out under the hypothesis of generalized plane strain conditions by means of SYSWELD and SYSTUS codes.     

An extrapolation method to determine the effective notch stress in circular hollow section X‐joints

This paper presents an extrapolation method to determine the effective notch stress (ENS) on the weld toes of welded circular hollow section (CHS) X‐joints in‐line with the extrapolation method to determine the structural hot‐spot stress in existing design guidelines. This investigation verifies and extends the recently proposed ENS extrapolation scheme to CHS X‐joints, which elevates significantly the difficulty in generating a weld toe radius along the brace‐to‐chord intersection as required in existing guidelines. An extensive numerical study then investigates the ENS extrapolation for CHS X‐joints under three loading conditions, namely, the brace axial load, the brace in‐plane bending load, and the brace out‐of‐plane bending load. The numerical study derives a set of recommended extrapolation parameters that ensures accurate ENS estimation for each loading condition. This paper demonstrates that the proposed extrapolation method for the CHS X‐joints yields good agreement with the standard ENS calculation procedure described in the International Institute of Welding (IIW) guideline.     

A High‐order extended finite element method for extraction of mixed‐mode strain energy release rates in arbitrary crack settings based on Irwin's integral

An analytical formulation based on Irwin's integral and combined with the extended finite element method is proposed to extract mixed‐mode components of strain energy release rates in linear elastic fracture mechanics. The proposed formulation extends our previous work to cracks in arbitrary orientations and is therefore suited for crack propagation problems. In essence, the approach employs high‐order enrichment functions and evaluates Irwin's integral in closed form, once the linear system is solved and the algebraic degrees of freedom are determined. Several benchmark examples are investigated including off‐center cracks, inclined cracks, and two crack growth problems. On all these problems, the method is shown to work well, giving accurate results. Moreover, because of its analytical nature, no special post‐processing is required. Thus, we conclude that this method may provide a good and simple alternative to the popular J‐integral method. In addition, it may circumvent some of the limitations of the J‐integral in 3D modeling and in problems involving branching and coalescence of cracks. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiaxial fatigue of V‐notched steel specimens: a non‐conventional application of the local energy method

The paper deals with the multi‐axial fatigue strength of notched specimens made of 39NiCrMo3 hardened and tempered steel. Circumferentially V‐notched specimens were subjected to combined tension and torsion loading, both in‐phase and out‐of‐phase, under two nominal load ratios, R=?1 and R= 0, also taking into account the influence of the biaxiality ratio, λ=τ_a/σ_a. The notch geometry of all axi‐symmetric specimens was a notch tip radius of 0.1 mm, a notch depth of 4 mm, an included V‐notch angle of 90° and a net section diameter of 12 mm. The results from multi‐axial tests are discussed together with those obtained under pure tension and pure torsion loading on plain and notched specimens. Furthermore the fracture surfaces are examined and the size of non‐propagating cracks measured from some run‐out specimens at 5 million cycles. Finally, all results are presented in terms of the local strain energy density averaged in a given control volume close to the V‐notch tip. The control volume is found to be dependent on the loading mode.     

Evaluation of the strain energy density control volume for a nanoscale singular stress field

Fracture mechanics at micro‐ and nano‐scale has become a very attractive topic in the last years. However, the results are still few, mostly because of the lack of effective analytical tools and of the difficult to conduct experimental tests at those scales. In this study, the authors report preliminary analysis on the application of the Strain Energy Density (SED) method at nano‐scale. In detail, starting from mechanical properties experimentally evaluated on small single crystal silicon cracked specimens, a first evaluation of the control volume due to a nano‐size singular stress field is carried out. If the extension of the SED approach at micro‐ nano‐scale is given in near future, an easy and fast tool to design against fatigue will be provided for micro‐ nano‐devices such as MEMS and NEMS, resulting in a significant technological impact and providing an easy and fast tool to conduct static and fatigue assessment at micro‐ and nano‐scale.     

A direct analytical method to extract mixed‐mode components of strain energy release rates from Irwin's integral using extended finite element method

A new analytical approach, within the extended finite element framework, is proposed to compute mixed‐mode components of strain energy release rates directly from Irwin's integral. Crack tip enrichment functions in extended FEM allow for evaluation of integral quantities in closed form (for some crack configurations studied) and therefore resulting in a simple and accurate method. Several benchmark examples on pure and mixed‐mode problems are studied. In particular, we analyze the effects of high‐order enrichments, mesh refinement, and the integration limits of Irwin's integral. The results indicate that high‐order enrichment functions have significant effect on the convergence, in particular when the integral limits are finite. When the integral limits tend to zero, simpler strain energy release rate expressions are obtained, and high‐order terms vanish. Nonetheless, these terms contribute indirectly via coefficients of first‐order terms. The numerical results show that high accuracy can be achieved with high‐order enrichment terms and mesh refinement. However, the effect of the integral limits remains an open question, with finite integration intervals chosen as h ∕ 2 tending to give more accurate results. Copyright © 2013 John Wiley & Sons, Ltd.     

Brittle failures from U- and V-notches in mode I and mixed, I + II, mode: a synthesis based on the strain energy density averaged on finite-size volumes

A large bulk of static test results carried out on notched specimens are presented in a unified way by using the mean value of the strain energy density (SED) over a given finite-size volume surrounding the highly stressed regions. In plane problems, when cracks or pointed V-notches are considered, the volume becomes a circle or a circular sector, respectively, with R _C being the radius. R _C depends on the fracture toughness of the material, the ultimate tensile strength and the Poisson's ratio. When the notch is blunt, the control area assumes a crescent shape and R _C is its width as measured along the notch bisector.
About 900 experimental data, taken from recent literature, are involved in the local SED-based synthesis. They have been obtained from (a) U- and V-notched specimens made of different materials tested under mode I loading; (b) U- and V-notched specimens made of polymethyl-metacrylate (PMMA) and an acrylic resin, respectively, tested in mixed, I + II, mode; (c) U-notched specimens made of ceramics materials tested under mode I.
The local SED values are normalized to the critical SED values (as determined from unnotched specimens) and plotted as a function of the R / R _C ratio. A scatter band is obtained whose mean value does not depend on R / R _C, whereas the ratio between the upper and the lower limits are found to be about equal to 1.6. The strong variability of the non-dimensional radius R / R _C (ranging here from about zero to around 1000) makes stringent the check of the approach based on the mean value of the local SED on a material-dependent control volume.     

An extended maximum tangential strain energy density criterion considering T‐stress for combined mode I–III brittle fracture

The maximum tangential strain energy density (MTSED) criterion was modified by taking the influences of stress intensity factors and T‐stress into consideration for combined mode I–III brittle fracture. Furthermore, the Poisson's ratio and T‐stress influencing the fracture characteristics of cracked components were discussed by using the extended MTSED criterion. Moreover, the predicted values of this extended MTSED criterion and some testing results were comparatively analysed. The results indicate that the Poisson's ratio and T‐stress have no impact on the out‐of‐plane initiation angle; however, their effects on fracture resistance ratios are significant especially for pure mode III. A positive T‐stress increases the fracture resistance ratio, and it is opposite for a negative T‐stress. The predicted values calculated by the extended MTSED criterion agree very well with the testing data obtained with edge‐notched disc bend samples especially for pure mode III case.     

Fatigue strength of steel and aluminium welded joints based on generalised stress intensity factors and local strain energy values

Weld bead geometry cannot, by its nature, be precisely defined. Parameters such as bead shape and toe radius vary from joint to joint even in well-controlled manufacturing operations. In the present paper the weld toe region is modelled as a sharp, zero radius, V-shaped notch and the intensity of asymptotic stress distributions obeying Williams’ solution are quantified by means of the Notch Stress Intensity Factors (NSIFs). When the constancy of the angle included between weld flanks and main plates is assured and the angle is large enough to make mode II contribution non-singular, mode I NSIF can be directly used to summarise the fatigue strength of welded joints having very different geometry. By using a large amount of experimental data taken from the literature and related to a V-notch angle of 135°, two NSIF-based bands are reported for steel and aluminium welded joints under a nominal load ratio about equal to zero. A third band is reported for steel welded joints with failures originated from the weld roots, where the lack of penetration zone is treated as a crack-like notch and units for NSIFs are the same as conventional SIF used in LEFM. Afterwards, in order to overcome the problem related to the variability of the V-notch opening angle, the synthesis is made by simply using a scalar quantity, i.e. the mean value of the strain energy averaged in the structural volume surrounding the notch tips. This energy is given in closed form on the basis of the relevant NSIFs for modes I and II and the radius R_C of the averaging zone is carefully identified with reference to conventional arc welding processes. R_C for welded joints made of steel and aluminium considered here is 0.28 mm and 0.12 mm, respectively. Different values of R_C might characterise welded joints obtained from high-power processes, in particular from automated laser beam welding. The local-energy based criterion is applied to steel welded joints under prevailing mode I (with failures both at the weld root and toe) and to aluminium welded joints under mode I and mixed load modes (with mode II contribution prevailing on that ascribable to mode I). Surprising, the mean value of ΔW related to the two groups of welded materials was found practically coincident at 2 million cycles. More than 750 fatigue data have been considered in the analyses reported herein.     

On combination of the equivalent material concept and J‐integral criterion for ductile failure prediction of U‐notches subjected to tension

The aim of the present research is to check the capability of the equivalent material concept (EMC) combined with the J‐integral failure criterion, called EMC‐J criterion, in predicting the load‐carrying capacity (LCC) of U‐notched ductile aluminium plates subjected to tension by considering the 2 moderate and large‐scale yielding regimes. For this purpose, first, a set of experimental results on LCC of 2 groups of thin U‐notched rectangular plates made of Al 7075‐T6 and Al 6061‐T6 are gathered from the recent literature. Then, because the Al 7075‐T6 and Al 6061‐T6 plates have ductile behaviour, EMC is employed to avoid performing elastic‐plastic failure analysis for LCC predictions. Up to now, different failure models in the context of the linear‐elastic notch fracture mechanics have been successfully utilized in combination with EMC for ductile failure prediction of notched members. However, this is the first time in this research that J‐integral, as a well‐known brittle failure criterion, is linked to EMC for predicting LCC of the U‐notched rectangular aluminium plates. Finally, it is shown that EMC‐J criterion can predict well the experimental results of tensile LCC.     

Fatigue design of welded double‐sided T‐joints and double‐sided cruciform joints in steel marine structures: A total stress concept

Fatigue is a governing design limit state for marine structures. Welded joints are important in that respect. The weld notch stress (intensity) distributions contain essential information and formulations have been established to obtain a total stress fatigue damage criterion and corresponding fatigue resistance curve; a total stress concept. However, the involved weld load carrying stress model does not provide the required estimates and trends for varying geometry dimensions and loading & response combinations. A new one has been developed and performance evaluation for T‐joints and cruciform joints in steel marine structures shows that in comparison with the nominal stress, hot spot structural stress and effective notch stress concept based results up to 50% more accurate fatigue design life time estimates can be obtained. Taking advantage of the weld notch stress formulations, the effective notch stress concept performance has improved adopting a stress‐averaged criterion rather than a fictitious notch radius‐based one.     

Residual stress/strain evolution due to low‐cycle fatigue by removing local material volume and optical interferometric data

Three experimental methods, based on optical interferometric measurements of deformation response to local material removing, have been implemented for residual stresses determination. Two first techniques are employed to characterize initial residual stress values and their evolution near welded joints of aluminium plates under low‐cycle fatigue. The hole‐drilling method gives high‐accurate dependencies between residual stress components and number of cycles. The second approach comprises cracks modelling by narrow notches to describe residual stress distributions in more wide spatial range near the weld. The results demonstrate residual stress evolution is of complex character and cannot be uniquely qualified as a gradual relaxation. Besides, the secondary hole drilling method is developed and used as a fast and reliable tool to quantify the redistribution of residual strains near cold‐expanded holes due to low‐cycle fatigue. Dependencies of circumferential residual strains along the secondary hole edge versus number of cycles are constructed.