Calculation of stress intensity factors for functionally graded materials by using the weight functions derived by the virtual crack extension technique

The weight function method provides a powerful approach for calculating the stress intensity factors for a homogeneous cracked body subjected to mechanical loadings. In this paper, the basic equations of weight function method for mode I and mixed mode crack problems in a two-dimensional functionally graded crack system are derived based on the Betti’s reciprocal theorem. The weight functions derived by the virtual crack extension technique are further used to calculate the stress intensity factors of functionally graded materials (FGMs). The practicability and accuracy of this proposed method has been confirmed by the comparison with theoretical or numerical solutions available in the literatures. On account that the repeated extractions of the distributions of normal stress and shear stress in the uncracked component along the prospective crack line under different loadings can be avoided using the method presented in this paper, this method can be potentially used for optimal design for FGMs under multiple-load cases.     

Rib loading effects on weld root fatigue failure modes at rib‐to‐deck welded joint

The service life of orthotropic steel decks is dependent on the fatigue resistance of rib‐to‐deck welded joints, which is often tested using two kinds of experimental models in terms of the rib loading condition. Different weld root fatigue failure modes have been observed in the different models, but the role of rib loading remains unclear. This paper aims to clarify the effect of rib loadings on the weld root fatigue failure modes at rib‐to‐deck welded joints. The loadings are decomposed into the deck loadings and rib loadings according to the principle of superposition. Formulae of the weld root notch stress intensity factors and T‐stress under rib loadings are developed by multiparameter regression analysis and subsequently used for the local stress analysis. The fatigue failure modes determined from the local stress field agree well with the experimental results. The results reveal that the weld root failure modes depend on the rib loadings but are independent of the weld geometries. The averaged strain energy density (SED) that can capture both weld geometry and loading condition effects is used to correlate the fatigue test data of different weld root failure modes. The SED is capable of evaluating the fatigue strength of the rib‐to‐deck welded joint failed by different weld root failure modes with a narrow scatter band.     

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.     

Wide‐range and accurate mixed‐mode weight functions and stress intensity factors for cracked discs

The Wu‐Carlsson displacement‐based weight function method is extended to obtain the mode I and mode II weight functions for the edge‐ and centre‐cracked discs. Compared with Fett's direct adjustment weight functions for the edge‐cracked discs, the present weight functions are more accurate and are applicable for a wider range of crack lengths. Using the present weight functions, extensive and highly accurate mixed‐mode stress intensity factors are obtained for the cracked discs subjected to diametrically compressive forces. Assuming perfect contact and using Coulomb friction law and the present weight functions, the mode II stress intensity factors for the cracked discs with consideration of friction are obtained and widely compared with the corresponding results from finite element analyses.     

Evaluation of the strain energy density control volume for a nanoscale singular stress field

Fracture mechanics at micro‐ and nano‐scale has become a very attractive topic in the last years. However, the results are still few, mostly because of the lack of effective analytical tools and of the difficult to conduct experimental tests at those scales. In this study, the authors report preliminary analysis on the application of the Strain Energy Density (SED) method at nano‐scale. In detail, starting from mechanical properties experimentally evaluated on small single crystal silicon cracked specimens, a first evaluation of the control volume due to a nano‐size singular stress field is carried out. If the extension of the SED approach at micro‐ nano‐scale is given in near future, an easy and fast tool to design against fatigue will be provided for micro‐ nano‐devices such as MEMS and NEMS, resulting in a significant technological impact and providing an easy and fast tool to conduct static and fatigue assessment at micro‐ and nano‐scale.     

A semi-infinite-crack model for determining mode I stress intensity factors using crack surface displacements

A semi-infinite-crack model is used to supplement the conic section simulation method for determining stress intensity factors of finite cracked bodies under mode I loadings. The actual displaced crack surface profile is found by finite element analysis. For each crack surface segment between two neighbouring nodes, a set of model parameters is found by using the displacements of these two nodes. A stress intensity factor estimate is then calculated from the closed-form formula associated with the model. It is found that near-tip crack surface displacements produce model parameters that are sufficient for quantifying the stress intensity factor. The semi-infinite-crack model can be used either as a stand alone model or in conjunction with the ellipse simulation procedure to form a systematic approach. It is shown that this model can be applied to different geometries and loadings with excellent accuracy.     

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.     

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.     

Averaged strain energy density and J-integral for U- and blunt V-shaped notches under torsion

Analytical solutions for the J-integral and the strain energy density (SED) averaged over a control volume are presented for U- and blunt V-shaped notches under torsion. The influence of the notch tip radius and the notch opening angle are fully included in the new proposed relationships. These expressions take advantage of some recent solutions for the stress fields ahead of blunt notches under Mode III loading and can be seen as a synthesis of the efforts carried out during the last years by the present authors on this topic. Afterwards, the expressions are applied to a recent set of experimental data from cracked, notched and plain specimens tested under torsion at low temperature \((-60\,^{\circ }\hbox {C})\) . The large scatter shown by notched specimens in terms of maximum elastic stresses strongly reduces when the J-integral or the local SED are used in combination with the specific control volumes.     

Complex fracture parameters for an interface crack between twohardening materials: a photoelastic study

In this paper, complex stress intensity factors (SIFs) at an interface crack are determined for a range of applied loads, crack lengths and remote mode mixes using automated photoelasticity. The specimen geometries comprise epoxy resin and aluminium alloy halves bonded together, and are loaded in either compact tension in mixed‐mode conditions or in three‐point bend under mode I conditions. In the experiments, full‐field isochromatic data were obtained from the epoxy half using an established phase‐stepping technique. A reworked approach to the determination of the SIFs was developed by combining a least‐squares over‐deterministic method for fitting crack‐tip stress equations to the data and a weighting factor that ensures that only data in the singularity zone are used. For comparison, some of the specimens were tested using a linear‐elastic finite element (FE) analysis and/or by experiment using homogeneous test specimens. Excellent agreement between the experimental and numerical SIF moduli was achieved for remote mode I loadings. However, for good agreement to be made between the phase angle results requires an additional phase term to be added to the FE solution at each load to account for the development of a crack‐tip plastic zone. Further, results for the SIFs from remote mixed‐mode loadings of the compact tension specimen only have a meaningful interpretation in light of small‐scale yielding conditions. It is shown, qualitatively, that the experiments verify some of the predictions made in the literature of asymptotic behaviour at interface crack tips from results of elasto‐plastic FE analyses.     

Fracture assessment of Brazilian disc specimens weakened by blunt V-notches under mixed mode loading by means of local energy

The main purpose of this research is to re-analyse experimental results of fracture loads from blunt V-notched samples under mixed mode (I + II) loading considering different combinations of mode mixity ranging from pure modes I to II. The specimens are made of polymethyl-metacrylate (PMMA) and tested at room temperature. The suitability of fracture criterion based on the strain energy density (SED) when applied to these data is checked in the paper. Dealing with notched samples, characterized by different notch angles and notch root radii, the SED criterion used in combination with the concept of local mode I, valid in the proximity of the zone of crack nucleation, permits to provide a simple approximate but accurate equation for the SED in the control volume. This proposal unifies predictions for the experimental results obtained under modes I, II and mixed mode loading.     

MODE I AND MIXED MODE I/II FATIGUE CRACKING IN Ni3Al(CrB) SINGLE CRYSTALS

Abstract— Nominal mode I and mixed mode I/II fatigue tests were carried out using the intermetallic compound Ni₃Al(CrB) in the form of single crystal specimens. The effects of crystal orientation and load mode on fatigue crack initiation and growth were studied. The fracture surfaces of the single crystals were characterized by a cleavage-like appearance and cracking occurred either on a single {111} plane or on multiple {111} planes irrespective of whether mode I or mixed mode I/II loadings were applied. It was found that the crack initiation and growth behaviour are dependent on both crystal orientation and applied loading mode. The cracking behaviour predicted by three mixed mode fracture criteria (MTS, SED and G criteria) in polycrystalline materials under mixed mode loading can be understood from the present results on single crystals.     

Calculation of mixed mode stress intensity factors for an elliptical subsurface crack under arbitrary normal loading

Normal loading causes mixed fracture modes in an elliptical subsurface crack because of the nonsymmetrical geometry with respect to the crack face. In this paper, mixed mode weight functions (MMWFs) for elliptical subsurface cracks in an elastic semi‐infinite space under normal loading are derived. Reference mixed mode stress intensity factors (MMSIFs), calculated by finite element analysis, under uniform normal loading are used to derive MMWFs. The cracks have aspect ratios and crack depth to crack length ratios of 0.2–1.0 and 0.05 to infinity, respectively. MMWFs are used to calculate MMSIFs for any point of the crack front under linear and nonlinear two‐dimensional (2D) loadings. So, in order to evaluate the fatigue crack growth phenomenon under complicated 2D stress distributions, MMWFs can be easily used. The comparison between the MMSIFs obtained from the MMWFs and finite element analysis indicates high accuracy.     

An investigation of T‐stresses in a plate with an inclined crack under biaxial loadings

For plates with an inclined crack of wide‐range aspect ratios under biaxial loadings, T‐stress values are calculated with three‐dimensional finite element method. The results show that the normalized T‐stress is crack length and orientation dependent. A linear equation for the relationship between normalized T‐stresses and biaxility factors is proposed to describe the normalized T‐stresses for different crack lengths and crack angles under different biaxial loadings, which is more convenient and involves wider biaxility ratios compared with the existing solutions. The plate thickness effect and the trend of normalized T‐stresses along the crack front thickness are also studied for mode I and I–II mixed‐mode cracks. Based on the analyses and comparisons, it is necessary to take the thickness effect into consideration when the crack length is long enough (a/W = 7/10). When the component of mode II is significant (β > 45°), and the biaxility ratios are negative, T‐stresses near the free surface are lower than those at other positions, which are the opposite of mode I crack and most of I–II mixed‐mode crack.     

Brittle failures from U- and V-notches in mode I and mixed, I + II, mode: a synthesis based on the strain energy density averaged on finite-size volumes

A large bulk of static test results carried out on notched specimens are presented in a unified way by using the mean value of the strain energy density (SED) over a given finite-size volume surrounding the highly stressed regions. In plane problems, when cracks or pointed V-notches are considered, the volume becomes a circle or a circular sector, respectively, with R _C being the radius. R _C depends on the fracture toughness of the material, the ultimate tensile strength and the Poisson's ratio. When the notch is blunt, the control area assumes a crescent shape and R _C is its width as measured along the notch bisector.
About 900 experimental data, taken from recent literature, are involved in the local SED-based synthesis. They have been obtained from (a) U- and V-notched specimens made of different materials tested under mode I loading; (b) U- and V-notched specimens made of polymethyl-metacrylate (PMMA) and an acrylic resin, respectively, tested in mixed, I + II, mode; (c) U-notched specimens made of ceramics materials tested under mode I.
The local SED values are normalized to the critical SED values (as determined from unnotched specimens) and plotted as a function of the R / R _C ratio. A scatter band is obtained whose mean value does not depend on R / R _C, whereas the ratio between the upper and the lower limits are found to be about equal to 1.6. The strong variability of the non-dimensional radius R / R _C (ranging here from about zero to around 1000) makes stringent the check of the approach based on the mean value of the local SED on a material-dependent control volume.     

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.     

Review of the fatigue strength of welded joints based on the notch stress intensity factor and SED approaches

Several approaches exist for the fatigue strength assessment of welded joints. In addition to the traditional nominal stress approach, various approaches were developed using a local stress as fatigue parameter. In recent times, the N-SIF based approaches using the notch stress intensity at the weld toe or root have been developed. Based on this, the more practical strain energy density (SED) and the Peak Stress approaches were proposed. This paper reviews the proposed design S–N curves of the N-SIF and SED approaches questioning in particular the consideration of misalignment effects, which should be included on the load side of local approaches in order to consider them individually in different types of welded joints. A re-analysis of fatigue tests evaluated for the effective notch stress approach leads to slight changes of the design S–N curves and of the radius of the control volume used for averaging the SED at the notches. Further, on purpose fatigue tests of artificially notched specimens show that the fatigue assessment using a single-point fatigue parameter might be problematic because the crack propagation phase, being part of the fatigue life, is strongly affected by the stress distribution along the crack path that may vary considerably between different geometries and loading cases.     

A mixed mode crack growth model taking account of fracture surface contact and friction

Fracture surface interactions, of whatever origin, can significantly affect the stress intensity factor, and consequently, can also be relevant to fatigue crack propagation. In the occurrence of interaction between fracture surfaces, the effective loading cycle experienced by material near the crack tip may be very different from that evaluated on the basis of the external loadings only. The purpose of the work described in this paper is to obtain the effective mode II stress intensity factor, k _IIeff, in a surface cracked elasto-plastic plate with a factory roof fracture surface subjected to an in-plane shear (mode II) loading. A new model estimating the magnitude of the frictional mode II stress intensity factor, k _f, arising from the mismatch of the fracture surface roughness during in-plane shear, is developed. Furthermore, the results of this study are employed in modeling the fatigue response of the surface cracked plates subjected to mixed mode loading.     

A lamellar inhomogeneity near a multiphase reinforcement

In composites, the stress intensity factors (SIFs) of a lamellar inhomogeneity near a multiphase reinforcement are of interest. Based on extension of Eshelby’s equivalent inclusion method, a unified approach is presented to study the effect of a multiphase inhomogeneity on the SIF at the tip points of two- and three-dimensional lamellar inhomogeneities under nonuniform far-field loadings. Alteration of the SIF due to the presence of a coating layer around the inhomogeneity is addressed. Furthermore, the effect of geometry and stiffness of each phase of a multiphase reinforcement on the mixed mode SIFs of a lamellar inhomogeneity is investigated. In contrast to cracks whose SIFs are the same for uniaxial and multiaxial far-field loadings, all axial far-field applied stresses, which are parallel or perpendicular to the anticrack plane, result in the square root stress singularity at the anticrack tip points. However, only those components of the far-field shear stress whose couple vector is perpendicular to the anticrack plane would generate nonvanishing mixed mode SIFs, whereas for the shear components with couple vectors parallel to the anticrack plane, the SIF vanishes.     

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.