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.     

A continuum damage mechanics model for crack initiation in mixed mode ductile fracture

This paper presents the development of a fracture criterion based on the postulation that the threshold condition of crack initiation in mixed mode ductile fracture is satisfied when the overall damage w in an element at the prospective direction of crack path reaches its critical value w_c. The validity of the proposed criterion is checked by predicting the fracture loads of thin aluminium plates containing an isolated crack inclined at the angle of =30, 45, 60 and 75 degrees and the predicted loads are compared satisfactorily with those determined experimentally. The analysis is performed based on the anisotropic model of continuum damage mechanics theory proposed earlier by the authors, thus providing additional proof of the consistency, applicability and versatility of the model. When the fracture loads of the mixed mode plates calculated using conventional fracture mechanics are compared with those determined using the proposed damage model, a maximum close to 30 percent over-estimation of the loads from the conventional approach is observed as opposed to within 7 percent discrepancy between the computed and measured fracture loads using the damage approach. The observation reveals the importance of including damage consideration in any ductile fracture analysis.The effect of varying damage coefficients on the fracture loads is examined and it is found that the crack initiation load decreases with the increase of anisotropic damage coefficient.     

Incipient fracture angle,fracture loci and critical stress for mixed mode loading

A prediction of the direction of incipient crack growth in brittle-like materials and the associated fracture loci under mixed mode loading is proposed. It is postulated that the direction of unstable crack propagation is determined by the “weakest” near-tip element defined as the one which would relax maximum potential energy upon prospective crack extension. Starting from the energy rate principle of crack extension (Eshelby energy-momentum tensor and Rice $\text{J-}\text{internal}$ vector) it is deduced that a crack will extent in the direction along which the following stress criterion is satisfied, $\text{(δ}{}_{\mathrm{\theta \theta }}{}^2\text{? δ}{}_{\mathrm{rr}}{}^2\text{) →}\text{maximum (for}\text{δ}{}_{\mathrm{\theta \theta \; >\; 0)}}$ The fracture angle in pure Mode II (70.4° away from the original straight path) is shown to be unstable in the sense that any slight tension along the crack (non-singular at the crack tip) affects considerably (up to 22%) the directionality of crack extension. It appears to be sensitive to the extent of the near-tip zone ($\text{r}{}_0$) in which linear elasticity does not hold and the non-singular stress term (squared).The fracture loci in mixed mode loading (generated by projecting the $\text{J-}\text{integral}$ vector along the prospective fracture path and letting this scalar function attain a critical value) is quadratic in $\text{K}{}_1$ and $\text{K}{}_2$ with an interactive cross product term $\text{K}{}_1\text{× K}{}_2$.The suggested criterion with its implication in predicting critical fracture load, exhibits behavior which is consistent with experimental observations collected from several sources. The common and uncommon features with respect to other known criteria are compared and discussed.     

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 mixed mode fracture specimen for mode II dominant deformation

Several theories have been proposed for the failure of metals, as well as for the angle of crack propagation in mixed mode loading. In order to demonstrate the validity of these theories, the majority of tests have been carried out with an oblique crack placed in a uniaxial stress field. Better testing conditions may be achieved by placing a crack in a uniform bidimensional stress field. A specimen which was recently developed for $\text{K}{}_{\mathrm{IIC}}$ measurement may be readily adapted to achieve a bidimensional stress field and be used for mixed mode testing for the case in which shear deformation is dominant. The main aims of this study are to examine both the cracked and uncracked specimen by means of photoelasticity and finite elements in order to analyze the capabilities and limitations of this specimen for mixed mode testing. It will be demonstrated that there exists a nearly uniform biaxial field in the uncracked specimen. Moreover, calibration formulas will be presented for $\text{K}{}_{\mathrm{I}}$ and $\text{K}{}_{\mathrm{II}}$.     

Quantification of constraint effects in fracture mechanism transition for cracked structures under mixed mode loading

The objective of the study is to evaluate the effects of plastic constraint on transition between tensile-type and shear-type fracture. The T -stress is employed as the quantifying parameter for constraint and is incorporated into the existing theoretical criteria for modelling this transition. It is found that different constraint levels can dramatically alter the transition point. To verify this finding, two sets of mixed mode tests with different constraint levels are carried out. Alongside the theoretical and experimental study, finite element simulation is performed to verify and support these findings. Substantially improved agreement is observed with experimental data if the effect of plastic constraint on transition is included.     

Two parameter approach for elastic-plastic fracture of short cracked specimens under mixed mode loading

For mode-I loading, in order to describe the near-tip stress field in a specimen under large scaled yielding, two parameter approaches such as J-T, J-Q and J-A₂ theories have been developed and proved well for their validity and limit. In this work elastic-plastic finite element analysis were performed to investigate the effects of mode mixity and T-stress upon near-tip stress distribution for a small-scale-yield model with the modified boundary layer and CTS (Compact Tension-Shear) configuration under large-scale-yield state. As the results, some peculiar characteristics were found as follows; As the mode mixity increases, normal stresses _rr and near the crack tip in the small-scale-yield model get significantly affected by the positive T-stress as well as the negative T-stress, while the shear stress _r is little affected by T-stress. Also, the near-tip stress distribution of short cracked CTS specimens under the large-scale-yield state agree fairly well with that of the small-scale-yield model with an appropriate positive T-stress. The two parameters approach with J-integral and T-stress seems to be a good tool for describing the near-tip stress field under a mixed mode loading and large-scale-yield state.     

Two parameter fracture mechanics: Fatigue crack behavior under mixed mode conditions

The fatigue crack path has been studied on a tensile specimen with holes. The experimental crack path trajectories were compared with those calculated numerically. To incorporate the influence of constraint on the crack curving, we predicted the fatigue crack path by using the two-parameter modification of the maximum tensile stress (MTS) criterion. The values of the mixed-mode stress intensity factors K_I, and K_II as well as the corresponding constraint level characterized by T-stress were calculated for the obtained curvilinear and reference crack path trajectories. It is shown that in the studied configuration the effect of T-stress on the crack path is not significant. On the other hand the effect of constraint on the fatigue crack propagation rate is more pronounced.     

The fourth mode of fracture in fractal fracture mechanics

This paper offers a systematic approach for obtaining the order of stress singularity for different self-similar and self-affine fractal cracks. Mode II and Mode III fractal cracks are studied and are shown to introduce the same order of stress singularity as Mode I fractal cracks do. In addition to these three classical modes, a Mode IV is discovered, which is a consequence of the fractal fracture. It is shown that, for this mode, stress has a weaker singularity than it does in the classical modes of fracture when self-affine fractal cracks are considered, and stress has the same order of singularity when self-similar cracks are considered. Considering this new mode of fracture, some single-mode problems of classical fracture mechanics could be mixed-mode problems in fractal fracture mechanics. By imposing a continuous transition from fractal to classical stress and displacement fields, the complete forms of the stress and displacement fields around the tip of a fractal crack are found. Then a universal relationship between fractal and classical stress intensity factors is derived. It is demonstrated that for a Mode IV fractal crack, only one of the stress components is singular; the other stress components are identically zero. Finally, stress singularity for three-dimensional bodies with self-affine fractal cracks is studied. As in the two-dimensional case, the fourth mode of fracture introduces a weaker stress singularity for self-affine fractal cracks than classical modes of fracture do.     

J dominance in mixed mode loading

Elastic-plastic plane-strain crack problems subject to combined mode I and mode II loadings have been analysed with modified boundary layer formulations using the first two terms, K and T of the asymptotic elastic field. Corresponding full field calculations have been performed on geometries in which the mode I component arises largely from bending or tension and in which the T stress varies from tensile to compressive. The conditions for J dominance have been considered in terms of the effect of the T stress on the asymptotic field. As in related work on the pure mode I problem, positive T stresses are shown to favour J dominance, while compressive T stresses cause the stresses to fall from the HRR field.     

An investigation of the edge-sliding mode in fracture mechanics

A boundary collocation procedure has been applied to the Williams stress function to determine the elastic stress distribution for the crack tip region of a finite, edge-cracked plate subjected to mode II loading at the crack tips. The asymmetric specimen selected was particularly suitable for the determination of plane strain fracture toughness for mode II loading. Numerical solutions for stress intensity factors for the edge-sliding mode obtained by the boundary collocation method were in close agreement with values obtained from photoelastic experiments.Fracture tests of several compact shear specimens of 2024-T4 aluminum were conducted in order to experimentally investigate the behavior of the edge-sliding mode. In each case a brittle shear failure was observed and mode II fracture toughness values were obtained. The average value for K_IIc obtained from two tests was 39.5 ksi(in)^{$\text{1}\text{2}$}. No K_Ic. data for 2024-T4 were available for comparison purposes; however, K_Ic values for a similar alloy, 2024-T351, have been reported as 34ksi(in)^{$\text{1}\text{2}$} which is only about 15 per cent below the corresponding K_IIc value.     

A modified specimen for evaluating the mixed mode fracture toughness of adhesives

This article introduces a specimen geometry that allows the separation of fracture energy release rates G _I and G _II in adhesively joined beams made of disparate materials. The analysis is based upon a Green's functions formulation for shear deformable beams and circumvents the need to employ finite element computations. The current method results in a system of non-singular integral equations, that can be discretized and reduced to a system of algebraic equations which may be solved by common numerical techniques. The analysis accounts for the dimensions and properties of the adhesive and provides results for a wide range of G _I, G _II and their ratio. Those results agree with finite element computational values to within less than 4%.     

A non-proportional loading finite element analysis of continuum damage mechanics for ductile fracture

This paper presents the development of a finite element analysis based on an anisotropic model of continuum damage mechanics theory proposed recently by the authors for ductile fracture under non-proportional loading. The condition of non-proportional loading is formulated by introducing a dynamic co-ordinate system of principal damage allowing the principal direction of damage during the loading to rotate accordingly. The finite element analysis developed under non-proportional loading is applied to predict the crack initiation load of a centre-cracked plate under uniform loading. The predicted load agrees satisfactorily with those determined experimentally with centre-cracked thin plates made of aluminium alloy 2024-T3. The analysis also reveals under non-proportional loading the hysteresis effect of the principal directions of damage and stress. In addition, the influence of varying anisotropic damage coefficients on the crack initiation load and the crack tip displacement profile is also examined. The larger the degree of the anisotropy, the higher the crack initiation load. The magnitude of the crack tip displacement profile is found to be proportional to the degree of material anisotropy.