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.     

Analyse the role of the non‐singular stress in brittle fracture by BEM coupled with eigen‐analysis

A coupled model resulting from the boundary element method and eigen‐analysis is proposed in this paper to analyse the stress field at crack tip. This new combine method can yield several terms of the non‐singular stress in the Williams asymptotic expansion. Then the maximum circumferential stress (MCS) criterion taken the non‐singular stress into account is introduced to predict the brittle fracture of cracked structures. Two earlier experiments are re‐examined by the present numerical method and the role of the non‐singular stress in the brittle fracture is investigated. Results show that if more terms of non‐singular stress are taken into account, the predicted crack propagation direction and the critical loading by MCS criterion are much closer to the existing experimental results, especially for dominating mode II loading conditions. Moreover, numerical results manifest that Williams series expansion can describe the stress field further from the crack tip if more non‐singular stress terms are adopted.     

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.     

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.     

On the I–II mixed mode fracture of granite using four‐point bend specimen

Four‐point bend experiments on black granite are conducted. The fracture behaviours of granite under pure mode I, pure mode II and I–II mixed mode are investigated, and the corresponding stress intensity factors K_I , K_II and the non‐singular term T‐stress are obtained through numerical–experimental method. The results are compared with the theoretical predictions of generalized maximum tangential stress criterion and other conventional criteria. It shows that generalized maximum tangential stress criterion fits the experimental results better for considering the effect of T‐stress. Contrasting with other loading configurations, the values of T‐stress for asymmetric four‐point bend specimens are much smaller, especially for pure mode II specimens, which provide an asymmetric deformation field where the T‐stress is approaching zero.     

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.     

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.     

Out‐of‐plane singular stress fields in V‐notched plates and welded lap joints induced by in‐plane shear load conditions

When a crack or sharp notch is subjected to antisymmetric plane loading the Poisson's effect leads to the generation of a coupled out‐of‐plane singular mode. The latter was known to exist for problems with cracks for a long period of time; meanwhile this mode was largely ignored in theoretical studies of V‐shaped notches subjected to in‐plane loading as well as in practical fracture problems associated with such geometries. Only recently a characteristic equation describing the strength of the singularity of this mode was derived within the first order plate theory. Preliminary numerical investigations confirmed that a highly localized out‐of‐plane singular state linked to the transverse shear stress components does exist in the close vicinity of the notch tip with the singular behaviour as theoretically predicted. However, until now it is unclear how significant this mode is and whether it has to be taken into consideration in the stress analysis of engineering structures. This paper is aimed to discuss important features of this recently identified singular mode, out‐of‐plane singular mode, conduct a comprehensive three‐dimensional numerical study of a typical problem of a welded lap joint to investigate the contribution of this mode into the overall stress state in the close vicinity of the notch tip and discuss the implementation of these new results to the failure and integrity assessment of plate structures with sharp notches.     

Fracture analysis of functionally graded materials with U‐ and V‐notches under mode I loading using the averaged strain‐energy density criterion

One of the most powerful criteria to predict the critical fracture load in plates with notches is the strain‐energy density averaged over a well‐defined control volume ahead of the notch tip. Although the averaged strain‐energy density (ASED) criterion has been proposed for homogeneous materials, it has been shown in this paper that this criterion can also be applied for non‐homogeneous materials, especially for functionally graded materials (FGMs). A numerical method has been used to evaluate the control volume boundary, the averaged strain‐energy density over the control volume, and also the critical fracture load in FGMs under mode I loading. A new set of experimental results on fracture of blunt V‐notched samples made of austenitic–martensitic functionally graded steel under mode I loading have been provided.     

Low‐cycle fatigue/high‐cycle fatigue interactions in notched Ti‐6Al‐4V*

Combined low‐cycle fatigue/high‐cycle fatigue (LCF/HCF) loadings were investigated for smooth and circumferentially V‐notched cylindrical Ti–6Al–4V fatigue specimens. Smooth specimens were first cycled under LCF loading conditions for a fraction of the previously established fatigue life. The HCF 10⁷ cycle fatigue limit stress after LCF cycling was established using a step loading technique. Specimens with two notch sizes, both having elastic stress concentration factors of K_t = 2.7, were cycled under LCF loading conditions at a nominal stress ratio of R = 0.1. The subsequent 10⁶ cycle HCF fatigue limit stress at both R = 0.1 and 0.8 was determined. The combined loading LCF/HCF fatigue limit stresses for all specimens were compared to the baseline HCF fatigue limit stresses. After LCF cycling and prior to HCF cycling, the notched specimens were heat tinted, and final fracture surfaces examined for cracks formed during the initial LCF loading. Fatigue test results indicate that the LCF loading, applied for 75% of total LCF life for the smooth specimens and 25% for the notched specimens, resulted in only small reductions in the subsequent HCF fatigue limit stress. Under certain loading conditions, plasticity‐induced stress redistribution at the notch root during LCF cycling appears responsible for an observed increase in HCF fatigue limit stress, in terms of net section stress.     

An improved semi-circular bend specimen for investigating mixed mode brittle fracture

An edge cracked semi-circular specimen subjected to asymmetric three-point bend loading was suggested for investigating mixed mode fracture in brittle materials. Using finite element analysis, the crack parameters were obtained for various crack lengths and different locations of loading points. It was shown that by selecting appropriate positions for the loading points, full mode mixities from pure mode I to pure mode II could be achieved. Then, a series of fracture tests were conducted on PMMA using the proposed specimen. Very good agreement was found between the experimental results and those predicted from the generalized maximum tangential stress criterion.     

State‐of‐the‐art review on the local strain energy density concept and its relation to the J‐integral and peak stress method

The local average strain energy density (SED) approach has been proposed and elaborated by Lazzarin for strength assessments in respect of brittle fracture and high‐cycle fatigue. Pointed and rounded (blunt) V‐notches subjected to tensile loading (mode 1) are primarily considered. The method is systematically extended to multiaxial conditions (mode 3, mixed modes 1 and 2). The application to brittle fracture is documented for PMMA flat bar specimens with pointed or rounded V‐notches inclusive of U‐notches. Results for other brittle materials (ceramics, PVC, duraluminum and graphite) are also recorded. The application to high‐cycle fatigue comprises fillet‐welded joints, weld‐like shaped and V‐notched base material specimens as well as round bar specimens with a V‐notch. The relation of the local SED concept to comparable other concepts is investigated, among them the Kitagawa, Taylor and Atzori–Lazzarin diagrams, the Neuber concept of fictitious notch rounding applied to welded joints and also the J‐integral approach. Alternative details of the local SED concept such as a semicircular control volume, microrounded notches and slit‐parallel loading are also mentioned. Coarse FE meshes at pointed or rounded notch tips are proven to be acceptable for accurate local SED evaluations. The peak stress method proposed by Meneghetti, which is based on a notch stress intensity factor consideration combined with a globally even coarse FE mesh and is used for the assessment of the fatigue strength of welded joints, is also presented.     

Sharp V‐notches in viscoplastic solids: Strain energy rate density rule and fracture toughness

Different from Neuber's rule or Glinka's energy method which are always adopted to characterize the notch tip field under elastoplastic condition, in this paper, the strain energy rate density (SERD) rule is used for viscoplastic materials. In particular, based on the definition of generalized notch stress intensity factor (G‐NSIF) for sharp V‐notch in viscoplastic solids, the concept of SERD for sharp V‐notch in viscoplastic solids is presented. Subsequently, by taking as a starting point the SERD, the averaged strain energy density (SED) for sharp V‐notch in viscoplastic solids is derived with integration of time. The fracture toughness relation between sharp V‐notch specimens and crack specimen in viscoplastic materials is given based on the transformation of SERD. A numerical approach is presented to compute the SERD and SED based on finite element method. Some crucial comments on the G‐NSIF have been discussed. Some typical solutions for SERD and SED for sharp V‐notched specimens are investigated.     

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.     

Fracture investigation of U‐notch made of tungsten–copper functionally graded materials by means of strain energy density

The averaged strain energy density over a well‐defined control volume was employed to assess the fracture of U‐notched specimens made of tungsten–copper functionally graded materials under prevalent mode II loading. The boundary of control volume was evaluated by using a numerical method. Power law function was employed to describe the mechanical properties (elasticity modulus, Poisson's ratio, fracture toughness and ultimate tensile stress) through the specimen width. The effect of notch tip radius and notch depth on notch stress intensity factors and mode mixity parameter χ were assessed. In addition, a comparison based on fracture load between functionally graded and homogeneous W–Cu was made. Furthermore, in this research, it was shown that the mean value of the strain energy density over the control volume can be accurately determined using coarse meshes for functionally graded materials.     

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.     

A criterion for brittle fracture in U-notched components under mixed mode loading

A failure criterion is proposed for brittle fracture in U-notched components under mixed-mode static loading. The criterion, called UMTS, is developed based on the maximum tangential stress criterion and also a criterion proposed in the past for mode I failure of rounded V-shaped notches [Gomez FJ, Elices M. A fracture criterion for blunted V-notched samples. Int J Fracture 2004;127:239-64]. Using the UMTS criterion, a set of fracture curves are derived in terms of the notch stress intensity factors. These curves can be used to predict the mixed mode fracture toughness and the crack initiation angle at the notch tip. An expression is also obtained from this criterion for predicting fracture toughness of U-notched components in pure mode II loading. It is shown that there is a good agreement between the results of UMTS criterion and the experimental data obtained by other authors from three-point bend specimens.     

Brittle Fracture Analysis Using a Ring-Shape Specimen Containing Two Angled Cracks

A ring shaped specimen is used for studying mixed mode fracture in brittle materials. The ring specimen is subjected to a compressive diametral load and contains two angled central cracks. A series of fracture tests are conducted under various mode mixities using the ring specimen. It is shown that the obtained experimental results are in a very good agreement with theoretical predictions of the modified maximum tangential stress criterion.