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.     

Investigation of Mixed Mode I/II Brittle Fracture Using ASCB Specimen

Using the asymmetric semi-circular bend specimen (ASCB) with vertical crack, a series of mixed mode I/II fracture tests were performed on two different rigid polyurethanes materials. The experimental results obtained for brittle fracture are compared with the theoretical predictions based on MTS and SED criteria. The results demonstrate that the SED criterion gives better results for mode II loading conditions, while for mode I and predominantly mode I both criteria are accurate.     

Fracture assessment of U-notches under three point bending by means of local energy density

The main purpose of the paper is twofold. First, to provide a new set of experimental results on fracture of U-notched samples, made of two different materials; second, to apply a fracture criterion based on the strain energy density (SED) averaged over a control volume to assess the fracture load of blunt-notched components under three point bending. Two different materials are considered in the tests: a composite material (Al–15%SiC) tested at room temperature and a steel with a ferritic–pearlitic structure tested at −40 °C. All samples are weakened by U-notches characterized by different values of notch root radius and notch depth. The theoretical loads to failure as determined according to the SED criterion are compared with the experimental data from more than 40 static tests and with a SED-based scatter band recently reported in the literature for a number of materials exhibiting a brittle behaviour under static loads.     

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.     

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.     

Fracture of U-notched specimens under mixed mode: Experimental results and numerical predictions

The first part of the paper gives an account of 153 fracture tests on blunted notched specimens (with notches of root radius ranging from 0.3 to 4.0 mm), loaded under mixed mode (ranging from almost pure mode I to mode II, and beyond). Maximum loads and initial crack angles were measured as a function of notch root radius and loading mixity. Such results can help in evaluating numerical models of the fracture of notched components. The second part of the paper deals with the suitability of the cohesive crack concept for predicting fracture loads under mixed mode. Use of local mode I was considered for numerical computations. Comparison of experimental results with numerical predictions was significantly accurate. Diagrams of fracture loci for notched components loaded under mixed mode are discussed.     

Sudden Fracture from U-Notches in Fine-Grained Isostatic Graphite Under Mixed Mode I/II Loading

The U-notched maximum tangential stress (UMTS) criterion, proposed originally and utilized previously by the author and his co-researcher for predicting mixed mode I/II fracture in plexi-glass (PMMA) and also pure mode II fracture in PMMA and soda-lime glass, was employed to estimate the experimental results reported in literature dealing with brittle fracture of many U-notched fine-grained isostatic graphite plates under combined tensile/shear loading conditions. By using the fracture curves of the UMTS criterion, which can predict the onset of brittle fracture in terms of the notch stress intensity factors (NSIFs) in the entire domain from pure mode I to pure mode II, the mixed mode fracture toughness (i.e. the load-bearing capacity) of U-notched graphite plates was successfully estimated.     

A new fixture for fracture tests under mixed mode I/II/III loading

In this paper, a new loading device for general mixed mode I/II/III fracture tests is designed and recommended. Finite element analyses are conducted on the proposed apparatus to evaluate the fracture parameters of the tested samples under various mixed mode loading conditions. The numerical results revealed that the designed loading fixture can generate wide varieties of mode mixities from pure tensile mode to pure in‐plane and out‐of‐plane shear modes. The accuracy of the proposed fixture is evaluated by conducting a wide range of fracture tests on compact tension shear (CTS) specimens made of polymethyl methacrylate (PMMA). The experimental results are then compared with the theoretical predictions obtained by the Richard criterion. A good consistency is observed between the experimental results and theoretical predictions.     

A generalised notch stress intensity factor for U-notched components loaded under mixed mode

A novel notch stress intensity factor (NSIF) for U-notched specimens loaded under mixed mode is examined in this article. The concept is based on the averaged strain energy density criterion, or alternatively on the cohesive zone model, as well as the equivalent local mode approach. To a certain extent, it is a generalisation of Glinka’s NSIF for mode I, where σ_tip is replaced by σ_max.The applicability of a fracture criterion based on this new NSIF is checked against 171 fracture tests with PMMA (at −60 °C) performed on U-notched specimens, with different notch root radii and loaded under mixed mode. The asymptotic behaviour of the new NSIF as the notch becomes a crack (when the notch root radius tends to zero) or when the notch disappears (when the notch root radius tends to infinity) is also discussed.     

Failure criteria for mixed mode delamination in glass fibre epoxy composites

Critical strain energy release rate of glass/epoxy laminates using the virtual crack closure technique for mode I, mode II, mixed-mode I + II and mode III were determined. Mode I, mode II, mode III and mixed-mode I + II fracture toughness were obtained using the double cantilever beam test, the end notch flexure test, the edge crack torsion test and the mixed-mode bending test respectively. Results were analysed through the most widely used criteria to predict delamination propagation under mixed-mode loading: the Power Law and the Benzeggagh and Kenane criteria. Mixed-mode fracture toughness results seem to represent the data with reasonable accuracy.     

Influence of the matrix constituent on mode I and mode II delamination toughness in fiber-reinforced polymer composites under cyclic fatigue

Mode I and mode II fracture behaviour under static and dynamic loading was analyzed in two composites made up of the same reinforcement though embedded in two different matrices. Specifically, the delamination energy under static and dynamic loading was obtained for both materials and both fracture modes, i.e. the number of cycles necessary for the onset of fatigue delamination. Subsequently, the crack growth rate (delamination rate) was obtained for different percentages of the critical energy rate. The main goal of the study was to ascertain the influence of the matrix on the behaviour of the laminate under fatigue loading.From the experimental results for the onset of delamination, similar fatigue behaviour was observed at a low number of cycles for both matrices and both fracture modes, while in fatigue at a high number of cycles, a higher fatigue limit was obtained in the composite with the modified resin (higher toughness) for both fracture modes. From the point of view of crack growth rate, both materials behaved similarly for different levels of stress under fatigue and the two fracture modes for small crack lengths (initial growth zone < 5 mm), although the growth rate increased for large crack lengths. This behaviour was the same in both loading modes.     

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.     

Investigation of blunting line and evaluation of fracture toughness under mixed mode I/III loading in commercially pure titanium

Abstract

The blunting line and fracture toughness in commercially pure titanium under mode I and mixed mode I/III loading was studied. A modified compact tension geometry was used for determining the blunting line as well as mixed mode I/III fracture toughness. The results showed that the constraint factor m in the blunting line equation under mode I loading was 1.84. Also, there was no effect of notch root radius on the slope of the blunting line. The blunting line slope under mixed mode I/III loading was found to be lower than that under mode I loading and agreed with empirical correlations. The fracture toughness under mode I loading was found to be higher for specimens with larger notch root radius. However, notch root independent fracture toughness could be obtained from blunt notch specimen tests using stretch zone width measurements. The fracture toughness was found to decrease with increasing mode III loading.     

A critical shear displacement criterion for ductile–brittle transition under mixed mode I and II loading

Abstract

Micromechanisms producing ductile and brittle damage operate in parallel at a crack tip. The dominant mode of failure depends upon which of the two (ductile or brittle) damage parameters first reaches its critical value. This has been shown by a study of ductile–brittle transition behaviour in HY100 steel under mixed mode I and II loading. The transition from ductile to brittle behaviour in HY100 steel was found to be affected by mixed mode I and II ratio (ratio of imposed tensile and shear loading) in a manner such that with increasing shear the transition temperature decreased. In the present paper, a criterion is proposed based on the shear strain ahead of a notch tip, to predict the fracture behaviour at any given temperature and mixed mode ratio.     

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.     

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.     

Constraint effects on ductile fracture processes near a notch tip under mixed-mode loading

In this work, the effect of constraint on ductile fracture process of microvoid growth and coalescence near a notch tip in a ductile material under mode I and mixed mode loading (involving modes I and II) is investigated. To this end, two sets of finite element simulations are carried out under two-dimensional plane strain conditions. In the first set, a modified boundary layer formulation is employed in which the mixed mode elastic K–T field is prescribed as remote boundary conditions. Several analyses are carried out corresponding to different values of T-stress and remote elastic mode-mixity. Next, ductile four point bend specimens subjected to mode I and mixed mode loading are considered. In both sets of simulations, the interaction between a notch tip and a pre-nucleated hole ahead of it is modelled. The background material is represented by the Gurson constitutive model and micro-void nucleation at uniformly distributed small scale particles is also taken into account. The accumulation of matrix plastic strain and porosity in the ligament between the notch tip and the hole as well as the growth of the hole are studied. Finally, the effect of crack tip constraint on the relationship between the fracture toughness and mode mixity is examined.     

Mixed mode brittle fracture in epoxy/multi-walled carbon nanotube nanocomposites

The effects of multi-walled carbon nanotubes (MWNTs) on fracture behavior of epoxy under mixed mode I/II loading have been studied. A number of test specimens based on different contents of MWNTs were prepared and the fracture tests were carried out. The increase in fracture resistance of the nanocomposite depended on the mode mixity. To find the reason, the fracture mechanisms in different modes of fracture were studied by examination of the fracture surfaces. It was found that in addition to the mechanisms, which contribute in mode I fracture, some extra mechanisms participate in mixed mode and mode II loading.     

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.     

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.