Induced out‐of‐plane mode at the tip of blunt lateral notches and holes under in‐plane shear loading

As it is well known the Poisson's effect in a cracked plate subjected to anti‐symmetric plane loading leads to the generation of a coupled out‐of‐plane singular mode. Recent theoretical and numerical analyses have shown that this effect is present also in plates weakened by sharp V‐notches and might play a role in failure initiation phenomena of notched plates subjected to Mode II loading, especially in the presence of a large notch opening angle. Dealing with blunt notches with a large notch radius, and not just with sharp notches, the presence or not of an out‐of‐plane mode does not appear to have been systematically investigated in the past. The main aim of this work is to confirm the existence of the stress field associated with the out‐of‐plane mode (Mode O) and to describe its main features in the presence of a notch radius significantly different from zero. The analyses include U‐notches, as well as circular and elliptic holes. The strain energy density in a 3D control volume is utilized to identify the most critical zone (with respect to failure initiation) through the plate thickness at the notch tip.     

Fatigue limit evaluation of notches,small cracks and defects: an engineering approach

In this paper, the average stress method for the fatigue limit evaluation of stress raising geometrical features is revised and extended. In particular, an analytical close‐form approach was used and the linear elastic stress equations were modified by taking into account the effect of nominal stress on the local stress distribution. Hence, the average tangential stress was correctly evaluated over a distance of 2a₀, where a₀ was El Haddad's short crack constant, for long and small notches as well as for crack‐like notches. When this model is applied to a wide range of geometrical features subjected to mode I fatigue loading, the classical shape of the curves of the Kitagawa–Takahashi diagram was obtained for changes in crack‐like notch size. Similarly, notch sensitivity was estimated by reducing the notch tip radius. The accuracy of the proposed method in predicting fatigue limits was then checked by using experimental data taken from the literature and generated on testing specimens weakened by rounded and sharp notches as well as by small artificial defects.     

A numerical method for evaluation of J‐integral in plates made of functionally graded materials with sharp and blunt V‐notches

The main purpose of the paper is to propose a numerical method for evaluation of J‐integral in plates made of functionally graded materials (FGM) with sharp and blunt V‐notches under Mode I loading. The material properties have been assumed to be varied exponentially along the specimen width (notch direction). Using the proposed method, the effect of material gradient on the J‐integral for two cases of sharp and blunt V‐notches has been studied. The results have shown that in FGMs with sharp V‐notches, the J‐integral is not proportional to . So, the parameter J_L is path dependent. It has been observed that the material gradient has larger effect on the J‐integral in sharp V‐notch compared with that in blunt V‐notch.     

The role of first non‐singular stress terms in mixed mode brittle fracture of V‐notched components: an experimental study

This paper investigates the effects of the first non‐singular stress terms on the fracture assessment of sharp V‐notches under mixed mode loading. First, numerical studies have been performed on a fracture test configuration called single V‐notched ring (SVR) specimen. Then, the notch stress intensity factors as well as the coefficients of the first non‐singular stress terms, which are vital parameters in brittle fracture of V‐notched components, were calculated via a finite element over‐deterministic algorithm for a wide range of loading and geometry conditions. The obtained results demonstrate that the SVR specimen is able to provide a complete range of mode mixities from pure mode I to pure mode II loading conditions. The numerical results, next, have been converted to dimensionless parameters and are illustrated in several graphs. Indeed, these graphs can be easily employed by the engineers for rapid calculation of the corresponding notch stress intensity factors and the coefficients of the first non‐singular stress terms in the SVR specimen. The obtained fracture parameters are then submitted to the maximum tangential stress criterion to assess the effects of the first non‐singular terms on fracture behaviour of the specimen. Finally, an experimental study has been performed on the SVR specimen made of Nayriz Marble rock for two notch angles with a complete range of mode mixities. The obtained experimental data confirm the significant role of the first non‐singular stress terms. In fact, these results show that considering only the singular stress terms may induce an average error of 38% in the predicted fracture loads, which can be decreased to about 12% just by adding the contribution of the first non‐singular terms to the maximum tangential stress criterion.     

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.     

Fracture analysis of V‐notched rubbers: An experimental and theoretical study

In this study, the rupture load in rubbers weakened by sharp V‐notch is investigated under mode I loading. To this end, first, mode I fracture tests are performed on V‐notched samples made of styrene‐butadiene rubbers and the corresponding rupture loads are obtained. Then, the effective stretch (ES) criterion, which was recently developed by the present authors for rupture assessment of cracked rubber parts, is extended and used for the V‐notched rubbers. It is shown that similar to cracked rubbers, the state of stress near the notch tip is also nearly uniaxial. By employing the ES criterion, the critical displacements corresponding to the rupture in the tested samples are calculated. Finally, the predictions of the criterion are compared with the corresponding experimental values, and good consistency is shown to exist.     

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.     

Computation of the stress intensity factors of sharp notched plates by the fractal‐like finite element method

The fractal‐like finite element method (FFEM) is an accurate and efficient method to compute the stress intensity factors (SIFs) of different crack configurations. In the FFEM, the cracked/notched body is divided into singular and regular regions; both regions are modelled using conventional finite elements. A self‐similar fractal mesh of an ‘infinite’ number of conventional finite elements is used to model the singular region. The corresponding large number of local variables in the singular region around the crack tip is transformed to a small set of global co‐ordinates after performing a global transformation by using global interpolation functions. In this paper, we extend this method to analyse the singularity problems of sharp notched plates. The exact stress and displacement fields of a plate with a notch of general angle are derived for plane‐stress/strain conditions. These exact analytical solutions which are eigenfunction expansion series are used to perform the global transformation and to determine the SIFs. The use of the global interpolation functions reduces the computational cost significantly and neither post‐processing technique to extract SIFs nor special singular elements to model the singular region are needed. The numerical examples demonstrate the accuracy and efficiency of the FFEM for sharp notched problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Fictitious notch rounding concept applied to sharp V-notches: Evaluation of the microstructural support factor for different failure hypotheses: Part II: Microstructural support analysis

On the basis of the comprehensive and accurate stress field equations for sharp rounded V-notches derived in Part I of this contribution, the microstructural support factor of these notches is determined which quantifies the fictitious notch radius in Neuber’s elastic microstructural support concept. By means of Filippi’s equations and considering different failure criteria (Rankine, von Mises and Beltrami) the fictitious notch radius is evaluated for different notch opening angles as a function of the actual notch radius and the microstructural support length. Plane stress and, alternatively, plane strain conditions are introduced. Once the fictitious radius has been found, the support factor s is derived from the expression: fictitious notch radius minus actual notch radius divided by microstructural support length. The support factor s is found to be very sensitive to the notch opening angle, but constant ‘plateau values’ are determined for an actual radius greater than the microstructural support length. The dependence of s on the failure criterion and the multiaxiality conditions (plane stress or plane strain) is also investigated. Various numerical analyses using the FE method have been carried out to compare the theoretical stress concentration factor to the effective stress concentration factor, the former obtained by considering fictitiously rounded notches under tension loading using the plateau values of s, the latter obtained by integrating the relevant stress over the microstructural support length along the bisector of the pointed V-notch. Finally, dealing with out-of-plane shear loading, Neuber’s corresponding solution valid for sharp rounded notches is re-evaluated and the numerical analysis described above is extended to this loading case. All the comparisons above are preceded by elementary solutions for pointed notches in general. It is shown that the plateau values of s are well suited for engineering usage in structural strength assessments.     

A three‐dimensional stress field solution for pointed and sharply radiused V‐notches in plates of finite thickness

By making use of the generalized plane strain hypothesis, an approximate stress field theory has been developed according to which the three‐dimensional governing equations lead to a system where a bi‐harmonic equation and a harmonic equation should be simultaneously satisfied. The former provides the solution of the corresponding plane notch problem, and the latter provides the solution of the corresponding out‐of‐plane shear notch problem. The system can be applied not only to pointed three‐dimensional V‐notches but also to sharply radiused V‐notches characterized by a notch tip radius small enough. Limits and degree of accuracy of the analytical frame are discussed comparing theoretical results and numerical data from FE models.     

Brittle fracture in rounded-tip V-shaped notches

Two failure criteria are proposed in this paper for brittle fracture in rounded-tip V-shaped notches under pure mode I loading. One of these criteria is developed based on the mean stress criterion and the other based on the point stress criterion which both are well known failure criteria for investigating brittle fracture in elements containing a sharp crack or a sharp V-notch. To verify the validity of the proposed criteria, first the experimental data reported by other authors from three-point bend (TPB) and four-point bend (FPB) tests on PMMA at −60 °C and Alumina–7% Zirconia ceramic are used. Additionally, some new fracture tests are also carried out on the rounded-tip V-notched semi-circular bend (RV-SCB) specimens made of PMMA for various notch opening angles and different notch tip radii. A very good agreement is shown to exist between the results of the mean stress criterion and the experimental data.     

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.     

Generalised stress intensity factors for rounded notches in plates under in-plane shear loading

The Notch Stress Intensity Factors (NSIFs) quantify the intensities of the asymptotic linear elastic stress distributions of sharp (zero radius) V-shaped notches. When the notch tip radius is different from zero, the singular sharp-notch field diverges from the rounded-notch solution in the close neighborhood of the notch tip. Nevertheless the NSIFs might continue to be parameters governing fracture if the notch root radius is small enough. Otherwise they can be seen simply as stress field parameters useful in quantifying the stress distributions ahead of the specific notch. Taking advantage of some analytical formulations which are able to describe stress distributions ahead of parabolic, hyperbolic and V-shaped notches with end holes, the paper discusses the form and the significance of the NSIFs with reference to in-plane shear loading, considering explicitly the role played by the notch opening angle and the notch tip radius. These parameters quantify the stress redistribution due to the root radius with respect to the sharp notch case to which they should naturally tend for decreasing values of the notch radius.     

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.     

On Scale Effect in Plates Weakened by Rounded V-Notches and Subjected to In-Plane Shear Loading

It is now well-known that in plate problems with through-the-thickness cracks in-plane shear and anti-plane loadings generate coupled three-dimensional fracture modes. The dominance domain and intensity of the singular states associated with these 3D fracture modes are functions of the intensity of the primary loading (K_II and K_III) and Poisson’s ratio. A similar situation takes place for V-shaped notches. However, for geometrically similar notch geometries subjected the same nominal stress the intensity of the coupled modes is also a function of the plate thickness. Despite this almost all 3D effects are currently ignored in industrial standards and fracture assessment codes. Recent theoretical and numerical studies have demonstrated that in many practical situations the intensities of the coupled fracture modes for cracks and sharp notches are not negligible and can influence fracture conditions. The current paper extends this conclusion to rounded notches. By using the finite element modelling it is demonstrated that the intensity of the stress states associated with the coupled fracture modes in a sufficiently thick plate weakened by a rounded notch can exceed the magnitude of stresses due to the primary loading. This means that the coupled modes can dominate the stress state in the vicinity of the notch root and be primary responsible for fracture initiation.     

Determination of high‐order fields for multianisotropic material antiplane V‐notches and inclusions by the asymptotic expansion technique and an overdeterministic method

In this paper, the singular behavior for anisotropic multimaterial V‐notched plates is investigated under antiplane shear loading condition. Firstly, the elastic governing equations are transformed into eigen ordinary differential equations through introducing the asymptotic expansions of displacements near the notch tip. The stress singularity exponents, including the higher‐order terms, and the corresponding eigen angular functions are then obtained by solving the established equations by using the interpolating matrix method. Thus, using the combination of the results from finite element analyses and the derived asymptotic expansion, an overdeterministic method is employed to calculate the amplitudes of the coefficients in the asymptotic expansions. Finally, the stress and displacement fields in the vicinity of the notch tip, consisting of both singular terms and higher‐order terms, are determined. The effects of material properties and geometry characteristic on the singular behaviour of the notch tip are discussed in detail.     

A Finite Fracture Mechanics approach to the asymptotic behaviour of U‐notched structures

A Finite Fracture Mechanics (FFM) criterion is formalized to predict the critical failure loads of brittle U‐notched specimens, subjected to mode I loading. The criterion, recently applied to V‐notched structures, requires the contemporaneous fulfilment of stress requirements and energy conditions for fracture to propagate: the stress field ahead of the notch tip and the stress intensity factor related to a crack stemming from the root are involved. Both the apparent fracture toughness and the critical crack advancement result to be structural parameters. For sufficiently slender notches, the root radius becomes the only relevant geometric dimension. The consistency of the approach is proved by the comparison with experimental data available in the Literature.     

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.     

Singularity analysis near the vertex of the V‐notch in Reissner's plate by coupling the asymptotic expansion technique with the interpolating matrix method

A new numerical method for calculating the singularity orders of V‐notches in Reissner's plate is proposed in this paper. By introducing the asymptotic expansion of the generalised displacement field at the notch tip into the equilibrium equations of a plate, a set of characteristic ordinary differential equations with respect to the singularity order are established. In addition, by adopting the variable substitution technique, the obtained non‐linear characteristic equations are transformed into linear ones, which are solved by the interpolating matrix method. The singularity orders of moments and shear forces can be obtained simultaneously and can be distinguished from the corresponding characteristic angular functions conveniently. Four types of boundary conditions are proposed to investigate the influence of boundary conditions on the singularity order values. The effect of the Poisson's ratio on the singularity orders of the V‐notch in Reissner's plate is discussed. The present method is versatile for the singularity analysis of single material V‐notches and bi‐material V‐notches, and can be easily extended to multi‐material V‐notches.     

An element‐free method for in‐plane notch problems with anisotropic materials

This study developed an element‐free Galerkin method (EFGM) to simulate notched anisotropic plates containing stress singularities at the notch tip. Two‐dimensional theoretical complex displacement functions are first deduced into the moving least‐squares interpolation. The interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the Galerkin method. In the numerical validation, an interface layer of the EFGM is used to combine the mesh between the traditional finite elements and the proposed singular notch EFGM. The H‐integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method. The comparisons indicate that the proposed method obtains more accurate results for the displacement, stress, and energy fields than those determined from the standard finite element method. Copyright © 2013 John Wiley & Sons, Ltd.