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Approaches based on calculating Residual Notch Stress Intensity Factors (R-NSIFs) assume the weld toe to be a sharp V-notch that gives rise to a residual singular stress distribution close to the weld toe. Once R-NSIFs are determined, they might be included in local fatigue criteria for the structural strength assessment of welded joints based on NSIFs due to external cyclic loading. However, the numerical calculation of R-NSIFs through finite element (FE) simulations of the welding process requires extremely refined meshes to properly capture the residual stress singularity. In this context, the Peak Stress Method (PSM) has recently been adopted to estimate R-NSIFs due to residual stresses by means of coarse meshes of 2D 4-node plane or 3D 8-node brick elements. The aim of this work is to investigate the applicability of the PSM to estimate R-NSIFs in a butt-welded joint using coarse meshes of 3D 10-node tetra elements. The R-NSIF distribution at the weld toe line is estimated by applying the PSM to coarse meshes of 3D 10-node tetra elements, and the results are in agreement with those obtained using 3D 8-node brick elements. Accordingly, the PSM based on tetra elements further enhances the rapid estimation of R-NSIFs using coarse meshes and could be effective in analyzing complex 3D joint geometries.

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 RC of the averaging zone is carefully identified with reference to conventional arc welding processes. RC for welded joints made of steel and aluminium considered here is 0.28 mm and 0.12 mm, respectively. Different values of RC 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.

The experimentally obtained tensile load-bearing capacity of fifteen U-notched polycrystalline graphite plates reported in literature was theoretically estimated by means of two well-known brittle fracture models, namely the mean stress (MS) and the point stress (PS) criteria. The results showed that while the mean discrepancies between the experimental and the theoretical results for both the models are very good and approximately equal, the discrepancies are significantly different for various notch tip radii. Meanwhile, the results of MS and PS criteria were compared with the results of the strain energy density (SED) criterion reported in literature. Relatively similar value of mean discrepancy was also obtained for the SED model. It was demonstrated in this research that for small values of the notch tip radius, the MS model is the most appropriate failure criterion while the PS and SED criteria are much better models for medium radii. Moreover, for large notch tip radii, the MS and PS criteria are better choices for tensile fracture assessment of U-notched graphite plates than the SED criterion.

In order to simplify the calculation of notch stress field and N-SIF for V-notched structures, the concept of equivalent intensity factor was introduced and the simplified semi-analytical formula was deduced and fitted for calculating the notch stress and N-SIF of V-shaped notch. By establishing a large number of multi-parameter models, the equivalent intensity factor and simple formula of notch stress field under complex tension-bending load were quantitatively described and fitted. Furthermore, equivalent intensity factor as was used to solve dimensionless N-SIF. Numerical results show that the semi-analytical method proposed in this paper has high prediction accuracy. For the convenience of engineering application, the relationship between welded geometry and V-notch model parameters is established on the premise that the equivalent singular intensity factors are equal. The simple formula is extended to the prediction of notch stress field and N-SIF of typically welded joints, and the convenience and reliability of the simple formula are further verified by fatigue strength evaluation.

The fatigue strength of structural components is strongly affected by notches and imperfections. Both can be treated similarly, as local notch fatigue strength methods can also be applied to interior defects. Even though Murakami’s √area approach is commonly used in the threshold-based fatigue design of single imperfections, advanced concepts such as the Theory of Critical Distances (TCD), Notch Stress Intensity Factors (N-SIF), or Elastic Strain Energy Density (ESED) methods provide additional insight into the local fatigue strength distribution of irregularly shaped defects under varying uniaxial load vectors. The latter methods are based on the evaluation of the elastic stress field in the vicinity of the notch for each single load vector. Thus, this work provides numerically efficient methods to assess the local fatigue strength by means of TCD, N-SIF, and ESED, targeting the minimization of the required load case count, optimization of stress field evaluation data points, and utilization of multi-processing. Furthermore, the Peak Stress Method (PSM) is adapted for large opening angles, as in the case of globular defects. In detail, two numerical strategies are devised and comprehensively evaluated, either using a sub-case-based stress evaluation of the defect vicinity with an unchanged mesh pattern and varying load vector on the exterior model region with optimized load angle stepping or by the invocation of stress and strain tensor transformation equations to derive load angle-dependent result superposition while leaving the initial mesh unaltered. Both methods provide numerically efficient fatigue post-processing, as the mesh in the evaluated defect region is retained for varying load vectors. The key functions of the fatigue strength assessment, such as the evaluation of appropriate planar notch radius and determination of notch opening angle for the discretized imperfections, are presented. Although the presented numerical methods apply to planar simulation studies, the basic methodology can be easily expanded toward spatial fatigue assessment.

Brittle failure of components weakened by cracks or sharp and blunt V-notches is a topic of active and continuous research. It is attractive for all researchers who face the problem of fracture of materials under different loading conditions and deals with a large number of applications in different engineering fields, not only with the mechanical one. This topic is significant in all the cases where intrinsic defects of the material or geometrical discontinuities give rise to localized stress concentration which, in brittle materials, may generate a crack leading to catastrophic failure or to a shortening of the assessed structural life. Whereas cracks are viewed as unpleasant entities in most engineering materials, U- and V-notches of different acuities are sometimes deliberately introduced in design and manufacturing of structural components. Dealing with brittle failure of notched components and summarising some recent experimental results reported in the literature, the main aim of the present contribution is to present a review of the research work developed by Professor Paolo Lazzarin. The approach based on the volume strain energy density (SED), which has been recently applied to assess the brittle failure of a large number of materials. The main features of the SED approach are outlined in the paper and its peculiarities and advantages accurately underlined. Some examples of applications are reported, as well. The present contribution is based on the author’s experience over about 15 years and the contents of his personal library. This work is in honor and memory of Prof. Paolo Lazzarin who suddenly passed away in September 2014.

The averaged value of the strain energy density over a well-defined volume is used to predict the static strength of U-notched specimens under mixed-mode conditions due to combined bending and shear loads. The volume is centered in relation to the maximum principal stress present on the notch edge, by rigidly rotating the crescent-shaped volume already used in the literature to analyse U- and V-shaped notches subject to mode I loading. The volume size depends on the ultimate tensile strength σu and the fracture toughness KIC of the material. In parallel, an experimental programme was performed. All specimens are made of polymethyl-metacrylate (PMMA), a material which exhibits quasi-brittle behaviour at -60°C. Good agreement is found between experimental data for the critical loads to failure and theoretical predictions based on the constancy of the mean strain energy density over the control volume.

In this paper a comprehensive investigation is carried out with regard to the state of the stress and strain in the neighbourhood of notches in bodies subjected to an anti-plane state of shear stress, within the context of a strain limiting theory of elasticity. Taking advantage of a unified analytical framework, the strain-limiting theory of elasticity is used to determine the full stress and strain field close to a pointed or radiused notch with any notch opening angle. An extensive discussion is provided that highlights the main features of stress and strain distributions, and the implications of the new theory for fracture assessments. In particular, it is proved that the obtained stress and strain solution predicts finite strains at the notch tip and allows the intensity of the stress field to be written as a function of the elastic Notch Stress Intensity Factor K 3 , as in the case of conventional linearized elasticity theory. This makes the strain limiting elasticity an excellent vehicle for justifying theoretically a K based-approach to the fracture of brittle elastic solids, within the context of a self consistent theory, unlike the classical linearized theory that predicts singularities for the strain at crack tips.

The concept of a stress singularity is a cornerstone of modern fracture mechanics. A new mode of stress singularity, the out-of-plane singular mode or K O mode, was recently identified for V-shaped notches subjected to in-plane loading. This new mode is coupled with the shear mode and related to Poisson’s effect. The previous studies based on the first order plate theory focused on the derivation of a characteristic equation describing the strength of this singularity and provide no information regarding the intensity, extent and relevance of this mode to practical problems. This paper aims to fill this gap. The approach utilises the 3D finite element method and a standard regression technique for characterisation of the notch singular modes. A comprehensive study of the influence of Poisson’s ratio, plate thickness and notch opening angle on the value of the notch stress intensity factor associated with this mode is conducted for infinite plates. Additionally, to demonstrate the relevance of this new singular mode to practical problems an investigation of the out-of-plane mode and associated three-dimensional effects are presented for the case of a welded lap joint. Main areas of further research are also identified.

Fracture behaviour of V-notched specimens is assessed using two energy based criteria namely the averaged strain energy density (SED) and Finite Fracture Mechanics (FFM). Two different formulations of FFM criterion are considered for fracture analysis. A new formulation for calculation of the control radius Rc under pure Mode II loading is presented and used for prediction of fracture behaviour. The critical Notch Stress Intensity Factor (NSIF) at failure under Mode II loading condition can be expressed as a function of notch opening angle. Different formulations of NSIFs are derived using the three criteria and the results are compared in the case of sharp V-notched brittle components under in-plane shear loading, in order to investigate the ability of each method for the fracture assessment. For this purpose, a bulk of experimental data taken from the literature is employed for the comparison among the mentioned criteria.