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.     

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.     

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.     

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.     

Control volumes and strain energy density under small and large scale yielding due to tension and torsion loading

In the recent literature some researchers proposed the use of the mean value of the Strain Energy Density (SED) over a well-defined control volume for static and fatigue strength assessment of components weakened by sharp V-shaped notches. In those papers the SED was expressed in terms of Notch Stress Intensity Factors (NSIFs), whose accurate evaluation needs a very fine mesh when based on local stress determined along the notch bisector. This contribution shows that when the material behaviour is ideally linear elastic or obeys a power hardening law, the mean value of the SED over the control volume can also be precisely determined from a coarse mesh. This result is of interest in the practical application of the SED approach to real components. Eventually, NSIFs can be evaluated a posteriori, just on the basis of the local SED. While discussing some results from elastic-plastic analyses carried out on a V-notched plate under tension loading and on a V-notched round bar under torsion, the different roles played by local and large scale yielding are highlighted. The result is used here to provide a justification for the different slopes, 3.0 and 5.0, reported by Eurocode 3 and other Standards in force for welded details subjected to tensile or shear stresses, respectively.     

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.     

The Equivalent Strain Energy Density approach re-formulated and applied to sharp V-shaped notches under localized and generalized plasticity

In the case of a rounded notch, the stress and strain at the notch tip can be determined by the traditional Neuber rule or by the Equivalent Strain Energy Density (ESED) approach, as formulated by Glinka and Molski. In the latter case the elastoplastic strain energy density at the notch tip is thought of as coincident with that determined under purely elastic conditions. For sharply V‐shaped notches this approach is not directly applicable, since the strain energy density at the notch tip tends toward infinity both for a material obeying an elastic law and a material obeying a power hardening law. By using the notch stress intensity factors, the present paper suggests a re‐formulation of the ESED approach which is applied no longer at the notch tip but to a finite size circular sector surrounding the notch tip. In particular we have adopted the hypothesis that, under plane strain conditions, the value of the energy concentration due to the notch is constant and independent of the two constitutive laws. When small scale yielding conditions are present, such a hypothesis immediately results in the constancy of the strain energy averaged over the process volume. As a consequence, plastic notch stress intensity factors valid for sharp V‐shaped notches can be predicted on the basis of the linear elastic stress distributions alone.     

Numerical and experimental applications of the least-squares method to determine mode-II stress intensity factors for double fillet welded lap joints

Due to its simplicity, the least-squares method provides an efficient means to evaluate the stress intensity factors (SIFs) of cracks in complicated structures. This paper demonstrates numerical and experimental applications of the least-squares method to study mode-II SIFs of double fillet welded lap joints. In the numerical application, double fillet welded lap joints with different geometric parameters, including overlap length, weld leg size, plate thickness and plate length, were systematically analysed by the finite-element method combined with the least-squares method. The computed SIF results were then employed to develop the general formulae of the shearing fracture mode (mode-II) stress intensity factors. To validate the numerical results, three double fillet welded lap joint specimens were tested by a non-contact optical experiment using a common digital camera and a proposed image processing scheme. The measured crack shearing displacements near the crack tip were substituted into the least-squares procedure to obtain the SIFs of the specimens. The numerical and experimental results were in good agreement with the existing numerical results for double fillet welded lap joints provided in the handbook (Murakami, 1987). The non-contact optical experiment makes the field measurement of SIFs possible, which is very useful for fracture analysis or fatigue evaluation of structures like steel bridges, naval structures and offshore structures.     

Stress distribution and fatigue crack propagation analyses in welded joints

An approach is presented, based on the weight function method to calculate the stress intensity factors of semielliptical surface cracks originating from the notch root of welded joints. The stress distribution along the potential crack plane required in the weight function method is constructed on the basis of the notch stress intensity factor approach in the highly stressed zone and of the equivalent linearized stress distribution and is compared with those determined by the finite element method and existing predictions. The stress intensity factors determined by the proposed approach are compared with available solutions. These comparisons show that the results determined by the proposed approach generally agree well with the existing solutions. For the cases where the agreement is poor, the reasons are identified. One important feature of the proposed approach is that the stress singularity at sharp notch tip can be considered, which cannot be appropriately simulated by the finite element method. Finally, to demonstrate the applicability of the proposed approach, the fatigue life and the fatigue crack shape evolution of welded joints are predicted and they are compared with experimental results.     

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.     

Local fatigue strength parameters for welded joints based on strain energy density with inclusion of small-size notches

A fatigue strength parameter for (seam-)welded joints is presented which is based on the averaged elastic strain energy density (SED) criterion applied to full circle and semicircular ‘control volumes’, the latter centred by the expected crack path. The parameter is applicable both at weld toes and weld roots, at least in the medium-cycle and high-cycle fatigue range where elastic conditions are prevailing. Based on a rectangular slit-plate model representing the weld root and analysed by the finite element method, the effect of the following influencing conditions is investigated: tension loading (mode 1) and shear loading (mode 2), slit-parallel tension loading acting on a rounded slit tip, pointed slit tip versus small-size key-hole at the slit tip, semicircle and narrow sector versus full circle or full sector SED evaluations, distortional SED versus total SED under plane strain conditions. The following conclusions are drawn from the numerical results. The SED approach should be based on the full circle or full sector evaluation of the total SED, with R₀ = 0.28 mm for steels. In cases of a markedly unilateral angular SED distribution, the semicircle evaluation centred by the expected crack path is more appropriate. The use of small-size reference notches instead of pointed notches provides no advantage. The endurable remote stresses for fatigue-loaded welded joints according to the SED approach are well in correspondence with those according to the fictitious notch rounding approach. High accuracy of the results can already be achieved with a rough meshing at the pointed notches.     

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.     

Approximate stress intensity factors for cracked V-notched specimens based on asymptotic solutions with application to T-joints

This paper presents an approximate method based on asymptotic solutions for estimating the stress intensity factor K for semi-elliptic surface cracks at stress concentrations. The proposed equations make use of a reference solution to interpolate over the entire range from shallow to deep cracks. The reference solution is obtained by considering the current crack emanating from the associated specimen with a sharp notch. It is shown that the proposed formulae satisfy the shallow and deep crack asymptotes. The asymptotic solutions are applied to a T-joint with a fillet-weld-shaped transition. The accuracy of the predictions is assessed using numerical calculations.     

Thermal stress intensity factors of an infinite orthotropic layer with a crack

Stress intensity factors are determined for a crack in an infinite orthotropic layer. The crack is situated parallel to the plane surfaces of the layer. Stresses are solved for two kinds of the boundary conditions with respect to temperature field. In the first problem, the upper surface of the layer is heated to maintain a constant temperature T ₀, while the lower surface is cooled to maintain a constant temperature –T ₀. In the other problem, uniform heat flows perpendicular to the crack. The surfaces of the crack are assumed to be insulated. The boundary conditions are reduced to dual integral equations using the Fourier transform technique. To satisfy the boundary conditions outside the crack, the difference in temperature at the crack surfaces and differences in displacements are expanded in a series of functions that vanish outside the crack. The unknown coefficients in each series are evaluated using the Schmidt method. Stress intensity factors are then calculated numerically for a steel layer that behaves as an isotropic material and for a tyrannohex layer that behaves as an orthotropic material.     

Plastic notch stress intensity factors for pointed V-notches under antiplane shear loading

The paper deals with a work-hardening, elastic–plastic, stress analysis of pointed V-notches under antiplane shear deformation loading both under small and large scale yielding. Stress and strain field intensities are expressed in terms of plastic Notch Stress Intensity Factors, which are analytically linked to the corresponding linear elastic ones under small scale yielding. The near tip stress and strain fields are then used to give closed-form expressions for the Strain Energy Density over a circular sector surrounding the notch tip, and for the J-integral parameter, both as a function of the relevant plastic NSIFs, these expressions being valid both under small and large scale yielding.     

Elastodynamic stress intensity factors for a semi-infinite crack due to three-dimensional concentrated shear loadings on the crack faces

The dynamic stress intensity factors for a semi-infinite crack in an otherwise unbounded elastic body is investigated. The crack is subjected to a pair of suddenly-applied shear point loads on its faces at a distance l away from the crack tip. This problem is treated as the superposition of two problems. The first problem considers the disturbance by a concentrated shear force acting on the surface of an elastic half space, while the second problem discusses a half space with its surface subjected to the negative of the tangential surface displacements induced by the first problem in the front of the crack edge. A fundamental problem is proposed and solved by means of integral transforms together with the application of the Wiener–Hopf technique and Cagniard–de Hoop method. Exact expressions are then derived for the mode II and III dynamic stress intensity factors by taking integration over the fundamental solution. Some features of the solutions are discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

From a local stress approach to fracture mechanics: a comprehensive evaluation of the fatigue strength of welded joints

This paper investigates the possibility of unifying different criteria concerned with the fatigue strength of welded joints. In particular, it compares estimates based on local stress fields due to geometry (evaluated without any crack-like defect) and residual life predictions in the presence of a crack, according to LEFM. Fatigue strength results already reported in the literature for transverse non-load-carrying fillet welds are used as an experimental database. Nominal stress ranges were largely scattered, due to large variations of joint geometrical parameters. The scatter band greatly reduces as soon as a 0.3-mm virtual crack is introduced at the weld toe, and the behaviour of the joints is given in terms of Δ K _I versus total life fatigue. Such calculations, not different from residual life predictions, are easily performed by using the local stress distributions determined near the weld toes in the absence of crack-like defects. More precisely, the analytical expressions for K _I are based on a simple combination of the notch stress intensity factors K ₁^N and K ₂^N for opening and sliding modes. Then, fatigue strength predictions, as accurate as those based on fracture mechanics, are performed by the local stress analysis in a simpler way.     

A definition and evaluation procedure of generalized stress intensity factors at cracks and multi-material wedges

This paper proposes a definition of generalized stress intensity factors that includes classical definitions for crack problems as special cases. Based on the semi-analytical solution obtained from the scaled boundary finite-element method, the singular stress field is expressed as a matrix power function with its dimension equal to the number of singular terms. Not only real and complex power singularities but also power-logarithmic singularities are represented in a unified expression without explicitly determining the type of singularity. The generalized stress intensity factors are evaluated directly from the scaled boundary finite-element solution for the singular stress field by following standard stress recovery procedures in the finite element method. The definition and evaluation procedure are valid to multi-material wedges composed of any number of isotropic and anisotropic materials. Numerical examples, including a cracked homogeneous plate, a bimaterial plate with an interfacial crack, a V-notched bimaterial plate and a crack terminating at a material interface, are analyzed. Features of this unified definition are discussed.     

Dynamic stress intensity factor due to concentrated loads on a propagating semi-infinite crack in orthotropic materials

The elastodynamic response of an infinite orthotropic material with a semi-infinite crack propagating at constant speed under the action of concentrated loads on the crack faces is examined. Solution for the stress intensity factor history around the crack tip is found for the loading modes I and II. Laplace and Fourier transforms along with the Wiener-Hopf technique are employed to solve the equations of motion. The asymptotic expression for the stress near the crack tip is analyzed which lead to a closed-form solution of the dynamic stress intensity factor. It is found that the stress intensity factor for the propagating crack is proportional to the stress intensity factor for a stationary crack by a factor similar to the universal function k(v) from the isotropic case. Results are presented for orthotropic materials as well as for the isotropic case.     

Effect of a graded interlayer on the stress intensity factor of cracks in a joint under thermal loading

In a bimaterial joint with and without a graded interlayer, the stress intensity factor of cracks perpendicular to the interface was calculated for a thermal loading by a homogeneous change in the temperature. In joints without an interlayer, the stress intensity factor increases to infinity as the crack approaches the interface for the case of the Young’s moduli E₁/E₂>1 (crack in material 1). Introducing a graded interlayer with a continuous transition in the material properties between the two joined materials leads to a continuous change in the stress intensity factor if the crack propagates from material 1 into material 2. Results are presented for different transition functions of the material properties and for different thickness ratios of the layers. The possible beneficial effect of a graded interlayer is discussed.