T‐stress corrected notch stress intensity factors with application to welded lap joints

The definition, content and application of the notch stress intensity factors (NSIFs) characterizing the stress field at rounded slit tips (keyholes) is discussed. The same is done in respect of the T‐stress transferred from the corresponding pointed slit tips. A T‐stress based correction of the NSIF K_1,ρ is found to be necessary. The applicability of the T‐stress term supplemented by higher‐order terms in Williams’ solution to the slit tip stresses in tensile‐shear loaded lap joints is discussed in more detail. The role of the T‐stress in constituting the near‐field stresses of rounded slit tips is shown to cause a difference between internal and external slit tip notches. The notch stress equations for lap joints proposed by Radaj based on structural stress and by Lazzarin based on a finite element model of the rounded notch are reconsidered and amended based on the derivations above.     

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 superposition approach to the stress fields for various blunt V‐notches

This paper deals with the problems of blunt V‐notch with various notch shapes. The purpose is to develop a new method capable of obtaining more accurate solutions for the stress fields around a blunt V‐notch tip under opening and sliding modes. The key method is to use the principle of superposition for linear elastic materials. On the basis of the superposition method and the conventional stress fields for a sharp V‐notch, the stress fields useful for any shapes of blunt V‐notch is proposed. The notch stress intensity factors are estimated by the numerical analysis with finite element analysis, and then the effectiveness and validation of the proposed superposition approach are discussed by comparison with the results from the literature.     

Fatigue-relevant stress field parameters of welded lap joints: pointed slit tip compared with keyhole notch

The notch stress intensity factor (NSIF) based analytical frame is applied to the slit tips (or weld roots) of welded joints with inclusion of the T-stress component. This T-stress can be determined from FE models evaluating the ligament stresses close to the pointed slit tip. An alternative analytical frame is presented for the corresponding keyhole notches based on analytical solutions from the literature, which are applied to the ligament stresses.
In the slit tip models, the mean local strain energy density (SED) with inclusion of the T-stress effect is determined analytically and numerically in comparison, using two different fatigue-relevant control radii,  R ₀= 0.28 mm and  R ₀= 0.15 mm, the former value well proven for thick-sheet welded joints made of structural steel. The latter smaller value is tentatively proposed for thin-sheet welded joints, in the direction suggested in the recent literature where a reduction of the microstructural support length for laser beam welds and resistance spot welds is recommended. The FEM-based and analytical stress concentration factors (SCF) for the lap joint keyhole model and also the SED values for the corresponding pointed slit tips are found to be in good agreement. The  J -integral consisting of the first and second component (the latter containing the T-stress) is compared with the corresponding SED values.     

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.     

Calculation of mixed mode (I/II) stress intensities at sharp V‐notches using finite element notch opening and sliding displacements

In this paper, a simple, robust, and an efficient technique has been proposed for accurate estimation of mixed mode (I/II) notch stress intensity factors (NSIFs) of sharp V‐notched configurations using finite element notch opening and sliding displacements at the selected number of nodes along the notch flanks. Unlike the crack problems, displacement field is rarely employed in the notch problems due to complexities introduced by the presence of rigid body displacements. One of the main emphasis of the present work is to neatly bypass these rigid body displacements and develop a simple approach for accurate computation of the NSIFs so that it can be easily incorporated in the existing code. Several benchmark problems have been analyzed. The results obtained using the present method show excellent agreement with the solutions available in the literature. Some new results have also been reported in the present work.     

Asymptotic thermal and residual stress distributions due to transient thermal loads

The paper deals with the development of thermal and residual stress distributions arising from the solidification of a fusion zone near a V-notch tip. A set of numerical solutions of the problem was carried out under the hypothesis of generalized plane strain conditions by means of SYSWELD code. The intensity of the thermal and residual asymptotic stress fields at the sharp V-notch tip was studied for a given V-notch specimen geometry and a predefined fusion zone dimension after simulations on materials with different thermal, mechanical and phase transformation properties and after changing the clamping conditions at the specimen's boundary. The results were compared in terms of the elastic or elastic-plastic notch stress intensity factors giving a contribution to the interpretation of the experimental behaviour of welded joint.     

On the computation of stress fields on polygonal domains with V‐notches

The interior stress problem is solved numerically for a single‐edge notched specimen under uniaxial load. The algorithm is based on a modification of a Fredholm second‐kind integral equation with compact operators due to Muskhelishvili. Several singular basis functions for each of the seven corners in the geometry enable high uniform resolution of the stress field with a modest number of discretization points. As a consequence, notch stress intensity factors can be computed directly from the solution. This is an improvement over other procedures where the stress field is not resolved in the corners and where notch stress intensity factors are computed in a roundabout way via a path‐independent integral. Numerical examples illustrate the superior stability and economy of the new scheme. Copyright © 2001 John Wiley & Sons, Ltd.     

Generalized stress intensity factors of V-shaped notch in a round bar under torsion, tension, and bending

In this study, generalized stress intensity factors K_I,λ1, K_II,λ2, and K_III,λ4 are calculated for a V-shaped notched round bar under tension, bending, and torsion using the singular integral equation of the body force method. The body force method is used to formulate the problem as a system of singular integral equations, where the unknown functions are the densities of body forces distributed in an infinite body. In order to analyze the problem accurately, the unknown functions are expressed as piecewise smooth functions using three types of fundamental densities and power series, where the fundamental densities are chosen to represent the symmetric stress singularity and the skew-symmetric stress singularity. Generalized stress intensity factors at the notch tip are systematically calculated for various shapes of V-shaped notches. Normalized stress intensity factors are given by using limiting solutions; they are almost determined by notch depth alone, and almost independent of other geometrical parameters. The accuracy of Benthem-Koiter’s formula proposed for a circumferential crack is also examined through the comparison with the present analysis.     

Theoretical investigation of elastoplastic notch fields under triaxial stress constraint

In this paper, an exact elastic-plastic solution has been obtained based on the J ₂-deformation theory of plasticity for a plate having a circular hole under biaxial tension and triaxial stress constraint in linear elastic strain-hardening materials. The theoretical solution shows that a linear elastic solution of the equivalent strain can be used to linear elastic-power hardening plastic situation just by a simple variable replacement. Then a strain equivalent rule (SER) is proposed to predict the elastoplastic notch fields by use of the elastic solution. Validations against theoretical analyses and finite element calculation for various combinations of material properties, triaxial stress constraints, load levels show that the SER can be used to predict stress-strain distributions in the whole plastic zone effectively and conveniently.     

Concurrent ratcheting and stress relaxation at the notch root of steel samples undergoing asymmetric tensile loading cycles

The current study intends to develop a framework model to assess ratcheting and stress relaxation at the notch root of 1045 steel samples over asymmetric loading cycles. The framework involves the Ahmadzadeh‐Varvani (A‐V) kinematic hardening rule to control ratcheting progress and Neuber rule to accommodate for local stress and strain components at the vicinity of notch root. Plastic strain at notch root was first coupled with its counterpart in the A‐V model to establish a relation between local stress and backstress components. Calculated local stress and strain values at turning points enabled the A‐V model to assess ratcheting strain over each loading cycle. The stepwise drop in stresses at peak‐valley tips of hysteresis loops at the notch root was associated to coupled framework of the A‐V model and Neuber rule through constancy in local strain while ratcheting progressed over each cycle. This relaxed out the local stresses at tips of hysteresis loops to position on Neuber hyperbolic curve. Predicted ratcheting values at notch root of various diameters closely agreed with those of measured in steel samples over stress cycles.     

Magneto stress analysis of a strip with a semi elliptical notch under uniform magnetic field

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin strip with a semi-elliptical notch subjected to uniform magnetic field. The strip is a finite plate of a simply connected region. A linear constitutive equation is used for the stress analysis. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. Therefore, the magneto elastic stress is analyzed using Maxwell stress. In the present problem, as a result, the plane stress state does not arise, and the σ_z in the direction of the plate thickness and the shear deflection (anti-plane shear stress) arise for soft ferromagnetic material. The stress σ_z in the plate is strong compressive stress for a soft ferromagnetic material. A rational mapping function is used for the stress analysis, and the each solution is obtained as a closed form. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition is completely satisfied without any linear assumptions on the boundary. The anti-plane shear stress causes Mode III stress intensity factor when the notch is a crack. Stress concentration values are investigated for a notch problem, of which expression is given. Figures of the anti-plane shear stress distribution, Mode III stress intensity factor, and stress concentration values are shown.     

Rapid evaluation of notch stress intensity factors using the peak stress method: Comparison of commercial finite element codes for a range of mesh patterns

The peak stress method (PSM) is an engineering, finite element (FE)‐oriented method to rapidly estimate the notch stress intensity factors by using the singular linear elastic peak stresses calculated from coarse FE analyses. The average element size adopted to generate the mesh pattern can be chosen arbitrarily within a given range. Originally, the PSM has been calibrated under pure mode I and pure mode II loadings by means of Ansys FE software. In the present contribution, a round robin between 10 Italian universities has been carried out to calibrate the PSM with 7 different commercial FE codes. To this aim, several two‐dimensional mode I and mode II problems have been analysed independently by the participants. The obtained results have been used to calibrate the PSM for given stress analysis conditions in (i) FE software, (ii) element type and element formulation, (iii) mesh pattern, and (iv) criteria for stress extrapolation and principal stress analysis at FE nodes.     

Some remarks on the Neuber rule applied to a control volume surrounding sharp and blunt notch tips

The paper deals with the small scale yielding estimation of nonlinear stresses and strains at the root of sharp and blunt notches through the mechanical model of antiplane shear loadings. The frame stems from the relation existing between the elastic and plastic averaged strain energy densities evaluated over the control volume drawn by the energy contour lines ahead of the notch tip. The analysis proves that there exist different relationships in terms of point‐wise elastic and plastic stresses and strains at the notch tip depending whether the notch is sharp (small notch tip radius) or blunt. For sharp notches, the analysis confirms previous results obtained by the present authors, according to which . This equation accounts for the influence of the material law through the hardening exponent n. Differently, when the notch can be regarded as blunt, calculations over the control volume give , in agreement with the Neuber rule.     

Estimations of stress intensity factor for a short crack in an elastic–plastic notch

This note presents a simple method for estimating the stress intensity factor (SIF) for a short crack emanating from an elastic–plastic notch.     

A notch stress intensity approach applied to fatigue life predictions of welded joints with different local toe geometry

In the Notch Stress Intensity Factor (N‐SIF) approach the weld toe region is modelled as a sharp V‐shaped corner and local stress distributions in planar problems can be expressed in closed form on the basis of the relevant mode I and mode II N‐SIFs. Initially thought of as parameters suitable for quantifying only the crack initiation life, N‐SIFs were shown able to predict also the total fatigue life, at least when a large part of the life is spent as in the propagation of small cracks in the highly stressed region close to the notch tip. While the assumption of a welded toe radius equal to zero seems to be reasonable in many cases of practical interest, it is well known that some welding procedures are able to assure the presence of a mean value of the weld toe radius substantially different from zero. Under such conditions any N‐SIF‐based prediction is expected to underestimate the fatigue life. In order to investigate the degree of conservatism, a total of 128 fillet welded specimens are re‐analysed in the present work by using an energy‐based N‐SIF approach. The local weld toe geometry, characterised by its angle and radius, has been measured with accuracy for the actual test series. The aim of the work is to determine if the N‐SIF‐based model is capable of taking into account the large variability of the toe angle, and to quantify the inaccuracy in the predictions due to the simplification of setting the toe radius equal to zero.     

Evaluation of stress concentration of U‐shaped notch based on residual magnetic field measurements

In this research, the correlation between the stress concentration and the residual magnetic field (RMF) of 30Cr steel was investigated. Tensile tests were carried out to measure the RMF signals on the surface of U‐shaped defect specimens. It was found that the tangential RMF signal at the defect area is correlated to the applied load and the stress concentration factor. A new method based on magnetic field to evaluate the stress concentration degree is proposed. This research provides a potential possibility for quantitative inspection of the stress concentration in ferromagnetic steels using the RMF measurements.     

Effects of notch geometry on the local cleavage fracture stress σf

An elastic–plastic finite element method (FEM) is used to analyse the stress and strain distributions ahead of notches with various depths and flank angles in four-point bending (4PB) specimens of a C–Mn steel. By accurately measuring the distances of the cleavage initiation sites from the notch roots, the local cleavage fracture stress σ _f is measured. By increasing the notch depth and notch flank angle from 2.25 to 8.25 mm and 10 to 90°, respectively, the distributions of high stress and strain at the moment of fracture show considerable variations. However, the value of σ _f stays relatively constant. The critical fracture event is thus shown to be identical, i.e. the propagation of a ferrite grain-sized crack into the neighbouring matrix. It is concluded that σ _f is mainly determined by the length of the critical microcrack, while the notch geometry and its associated stress volume have little effect on the value of σ _f . The cleavage site ahead of a notch is determined by the stress distributions and the positions of the weakest grains.     

Elastic notch stress intensity factors for sharply V-notched rounded bars under torsion

The paper deals with the determination of analytical expressions for the mode III notch stress intensity factors for circumferentially-sharply-notched rounded bars under torsion loading, starting from the theoretical stress concentration factors of the corresponding notch problem.An exact, closed-form solution for the NSIFs is obtained for deep notches; subsequently the solution is extended also to finite notched components taking advantage of a shape function determined by a numerical best fitting procedure.     

Asymptotic deformation and stress fields at the tip of a sharp notch in an elastic-plastic material

The singular elastic-plastic stress, strain and the displacement fields at the tip of a sharp notch for both plane stress and plane strain conditions are investigated analytically. The material is assumed to be governed by the deformation theory of plasticity with linear strain hardening characteristic. Since the elastic strain is retained in the analysis, the final strain and displacement fields can be separated into the elastic and the plastic parts. In the case with zero notch angle, the results reduce to the classical crack problem. The relationship of the amplitude of the near crack tip elastic-plastic field to the elastic far field is obtained. Both mode I and mode II cases are investigated. The mixed mode case is also discussed.