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.     

Fracture of V-notched specimens under mixed mode (I + II) loading in brittle materials

The purpose of this research is threefold. First, to provide experimental results of fracture loads for V-notched beams loaded under mixed mode. Second, to check the suitability of fracture criteria based on the cohesive zone model and strain energy density when applied to those samples. And, third, to suggest a very simple fracture criterion, based on the dominance of the local mode I, for notched samples (with different V-notch angles and notch root radii) loaded under mixed (I + II) mode. This proposal unifies predictions for the experimental results obtained under mode I and mixed mode loading. To this end, 36 fracture tests on V-notched beams were performed and reported: three V-notched angles were investigated (90°, 60°, 30°, four different loadings (mixed modes I and II) were selected and three samples were tested for each configuration.     

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.     

Investigation of mixed mode brittle fracture in rounded-tip V-notched components

A criterion is proposed for brittle fracture analysis in rounded-tip V-notched components. This criterion, called RV-MTS, is developed based on the maximum tangential stress (MTS) criterion proposed earlier for investigating mixed mode brittle fracture in sharp cracks. Using the RV-MTS criterion, a set of fracture curves is presented based on the notch stress intensity factors (NSIFs) for predicting mixed mode and also pure mode II fracture toughness of rounded-tip V-notches. The criterion is also able to predict fracture initiation angles under mixed mode loading. The validity of the criterion is evaluated by several fracture tests performed on the rounded-tip V-notched Brazilian disc (RV-BD) specimens made of PMMA. A good agreement is shown to exist between the theoretical predictions and the experimental results for various notch opening angles and different notch radii.     

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.     

CRACK CLOSURE EFFECT ON CRACK GROWTH RATE AT 650°C IN DOUBLE NOTCHED SPECIMENS OF A NICKEL BASE SUPERALLOY

Abstract— Creep-fatigue tests were performed at 650°C in air on a N18 nickel base superalloy, using double notched and smooth specimens. The deformation mechanisms observed by TEM at the notch root are shown to be compatible with the constitutive set of equations used in the finite element analysis which is presented. For a given K _max at the notch root, the crack growth rate is much higher in a notched specimen than in a smooth one. This effect can be explained by a variation of the crack closure stress level with the local R ratio and the local stress. A strong accelerating effect of the R ratio, especially for negative values, is found in smooth specimens. Introducing a K _op correction in the experimental results leads to a good agreement between the measured crack growth rate plotted versus K _eff in notched and smooth samples.     

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.     

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.     

Fracture initiation at sharp notches under mode I, mode II, and mild mixed mode loading

In the context of linear elasticity, a stress singularity of the type Kⁿr^δ(δ<0) may exist at sharp re-entrant corners, with an intensity Kⁿ. In general the order of the stress singularity δ and the stress intensity differ for symmetric (mode I) and antisymmetric (mode II) loading. Under general mixed-mode loadings, the magnitudes of the mode I and II intensities fully characterize the stress state in the region of the corner. A failure criterion based on critical values of these intensities may be appropriate in situations where the region around the corner dominated by the singular fields is large compared to intrinsic flaw sizes, inelastic zones, and fracture process zone sizes. We determined the mode I and II stress intensities for notched mode I tensile specimens and notched mode II flexure specimens using a combination of the Williams (1952) asymptotic method, dimensional considerations, and detailed finite element analysis. We carried out a companion experimental study to extract critical values of the mode I and II stress intensities for a series of notched polymethyl methacrylate (PMMA) tensile and flexure specimens with notch angles of 90-. The data show that excellent failure correlation is obtained, in both mode I and II loading, through the use of a single parameter, the critical stress intensity. We then analyzed and tested a series of T-shaped structures containing 90- corners. The applied tensile loading results in mixed-mode loading of the 90- corners. Failure of the specimens is brittle and can be well-correlated with a critical mode I stress intensity criterion using the results of the notched mode I tensile tests. This is attributed to large difference in the strength of the stress singularities in modes I and II: δ= -0.4555 and -0.0915 for modes I and II for a 90- notch. As a result, the mode I loading dominates the failure process for the 90- corner in the T-structure. This revised version was published online in August 2006 with corrections to the Cover Date.     

THE SIGNIFICANCE OF MEAN STRESS ON THE FATIGUE CRACK GROWTH THRESHOLD FOR MIXED MODE I + II LOADING

Abstract— The effect of mean stress on near threshold fatigue crack growth behaviour under mixed mode I+II loading has been studied in a structural steel BS4360 50D in laboratory air at room temperature. It was found that the branch crack threshold decreased significantly as R ratio ( P _min/ P _max) increased from 0.1 to 0.7. A simple model is proposed to predict the branch crack threshold behaviour for R -ratio sensitive materials. Further investigation is required to model the mode II dominant situations where the branch crack thresholds tend to converge on a high value.     

An investigation into the location of crack initiation sites in alumina, polycarbonate and mild steel

The effects of notch root radius on fracture toughness and crack initiation sites have been investigated in this paper using three different classes of materials. Data on alumina which represent ceramics, mild steel from the metals ffeily and polycarbonate representing plastics were obtained and analysed. The locations of crack initiation sites have been pinpointed by scanning electron microscopy. These identified sites more or less are located within the critical process zone or the theoretical plastic zone. The critical process zone size ( D _c ) or the theoretical plastic zone size ( R _YF ) are independent of the notch root radius unlike the plain-strain fracture toughness of notched specimens [ K _{IC (ρ)].} The authors emphasize why the parameters D _c and R _YF are useful for a quantitative evaluation of the reliability of structural materials.     

Effects of notch geometry on the local cleavage fracture stress σf

An elastic–plastic finite element method (FEM) is used to analyse the stress and strain distributions ahead of notches with various depths and flank angles in four-point bending (4PB) specimens of a C–Mn steel. By accurately measuring the distances of the cleavage initiation sites from the notch roots, the local cleavage fracture stress σ _f is measured. By increasing the notch depth and notch flank angle from 2.25 to 8.25 mm and 10 to 90°, respectively, the distributions of high stress and strain at the moment of fracture show considerable variations. However, the value of σ _f stays relatively constant. The critical fracture event is thus shown to be identical, i.e. the propagation of a ferrite grain-sized crack into the neighbouring matrix. It is concluded that σ _f is mainly determined by the length of the critical microcrack, while the notch geometry and its associated stress volume have little effect on the value of σ _f . The cleavage site ahead of a notch is determined by the stress distributions and the positions of the weakest grains.     

ANALYSIS OF CRACK TIP HYDROGEN DISTRIBUTION UNDER I/II MIXED MODE LOADS

Abstract— The distribution of hydrogen in the vicinity of a crack tip was studied using SIMS (Secondary Ion Mass Spectrometry) under different ratios of I/II mixed mode loads. Modified WOL specimens with kinked slits were employed in the course of the experimental work. Spectrographic measurements show that under I/II mixed mode loading, both in the HIC and in the r ^max_p directions, there are two hydrogen accumulation peaks ahead of the crack tip, corresponding to the location of the maximum hydrostatic stress and maximum equivalent plastic strain, respectively. Based on results obtained over a range of loading conditions from mode I to a high K_II/ K_I, ratio, it is shown that the mode II component has a clear influence on both peaks. The conditions for hydrogen redistribution are discussed in terms of crack tip stress-strain fields.     

CTODi MEASURED BY MEANS OF A MODIFIED CHARPY-V-SPECIMEN

Abstract— Measurements of CTOD_i on Charpy-V-specitnens of mild steel St 37 and pressure-vessel steel 22NiMoCr37 have been carried out. Slotted and precracked specimens have been used besides the original V-notched ones. A definition of CTOD = 2(R – R_o) has been proposed which corresponds to δ₄₅, defining the CTOD of fatigue cracks. The symbols R_o and R represent the original and the actual crack tip radii respectively. Additionally, this definition presents the opportunity to measure CTOD and CTOD_i by a direct metallographic method. It is demonstrated that COD testing, based on the hinge model, can also be applied to slotted bars, delivering CTOD and CTOD_i values which are equal to those evaluated by direct metallographic measurements.
The results obtained on four different tip radii, R_o, show a linear increase of CTOD_i as a function of R_o, which is steeper for the softer material St 37. The extrapolation to the tip radius R_o=0 gives a CTOD_i, which is equal to those determined from precracked specimens.     

Failure criteria for brittle elastic materials

The validity of several known failure initiation criteria at reentrant corners in brittle elastic materials is examined and a simple one is proposed. Their predictions, under mode I stress field, are compared to experimental observations carried out on PMMA (polymer) and Alumina-7%Zirconia (ceramic) V-notched specimens. Because all realistic V-notched reentrant corners are blunt, a detailed experimental procedure has been followed, focusing on specimens with different notch tip radii. It is demonstrated that by assuming a sharp V-notch, some failure criteria predict reasonably well the experimental findings, and that corrections are needed in order for these to take into consideration the realistic radius at the notch tip.     

Transformation zones, crack shielding, and crack-growth resistance of Ce-TZP/alumina composite in mode II and combined mode II and mode I loading

Crack-tip transformation zones, crack shielding and crack-growth-resistance (R-curve) behaviors of a transformation-toughened ceria-partially stabilized zirconia–alumina (Ce-TZP/alumina) composite were studied in mode II and combined mode I and mode II loading using compact-tension-shear (CTS) specimens. The mode II and mode I stress intensities for both the initial straight cracks and the subsequent kinked cracks were assessed by the method of caustics using geometrically equivalent specimens of polymethyl methacrylate (PMMA). The angle of formation of the transformation zones as well as of extension of the cracks increased systematically with increasing ratio of the mode II and the mode I stress intensities and approached a value of θ^*=−72° in pure mode II loading. This angle was close to the angle for maximum hoop tension in the stress field of a mode II crack (θ^*=−70.5°). A crack-initiation toughness envelope was constructed on a K_I–K_II diagram using the critical loads for incremental crack extension. The crack-initiation toughness in pure mode II loading was less than the corresponding toughness in mode I loading. This result was consistent with calculations that indicated no shielding from the asymmetric and elongated zones developed in mode II loading. The fracture toughness measured for the kinked cracks at long kink lengths approached the maximum fracture toughness measured for a mode I crack.     

MODE I STRESS INTENSITY FACTOR EQUATIONS FOR CRACKS AT NOTCHES AND CAVITIES

Abstract— In this paper, the notch-crack problem is treated in two different ways: if the non-dimensional crack length l /ρ ( l = crack length; ρ= notch root radius) is smaller than the transition crack length l _T/ρ, it is treated as an edge crack lying within the local stress field around the notch tip; if l/ ρ is larger than l _T/ρ, the notch-crack is considered as a simple flat crack problem subjected to remote loading, the flat crack size being the sum of notch depth and the real crack length. Based on currently available numerical data, expressions for the transition crack length, l _T, and for the geometric factor F = K _I/(1.1215K_tσ√π l ) are developed for various notch problems for the crack length range l ≦ l _T. It is found that the stress (σ_yy) normalized by the peak stress (σ_peak), σ_yy/σ_peak, for the pre-cracked component is very similar to the geometric factor for short cracks.     

FRACTURE BEHAVIOUR OF UNTREATED AND SILANE-TREATED TALC-FILLED POLYPROPYLENE COMPOSITES

Abstract— Fracture behaviour of injection-moulded polypropylene filled with silane-treated talc was studied as a function of filler volume fraction (0–20%) and compared to that of polypropylene filled with untreated talc. High-rate tests (0.57 m/s) on SENB specimens were carried out using an instrumented Charpy impact pendulum, and linear elastic fracture mechanics (LEFM) was applied to calculate the fracture parameters, K _C and G _C. It was found that moderate fractions of talc which were added to the polypropylene matrix increased the fracture toughness of the composite independent of the talc surface treatment. This general improvement seems to be due to the peculiar orientation of the talc platelets in the injection-moulded specimens. The fracture behaviour of the composites was also studied at low strain rate (1 mm/min) by tests on J -integral type specimens with the same SENB geometry. In this case, the composites with silane-treated talc presented poor J -integral values compared to those of the samples with untreated talc. This was attributed to a reduction of the plastic zone at the crack tip, since the improved coupling between the talc platelets and matrix increased the yield strength of the composite. All the results are explained on a basis of morphological and microstructural details.     

Assessment of the period to fatigue macrocrack initiation near stress concentrators by means of strain parameters

A new approach to the experimental assessment of the local strain at a stress concentrator has been presented. It is based on a procedure of notch opening displacement measurements at certain points in the vicinity of a notch related to the effective notch radius ρ _eff = ρ + d * , where ρ is the notch radius and d * is a material constant. Different stress concentrators in structural elements were modelled for a wide variation of notch radii ( ρ = 0.1–6.5 mm) and different geometries of specimens. Hence, a basic relationship, which directly relates the local strain range Δ ε* to the period of fatigue macrocrack initiation N _i has been established. Thus, by applying the value of Δ ε* , assessed from a direct measurement at the notch root, it is possible to determine the period N _i to initiate a fatigue macrocrack of length a _i = d * for some structural components of complicated geometry.     

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.