Study of the relationship between J-integral and COD parameters under mixed mode I + II loading in aluminum alloy Ly 12

In this paper, the relationship between the J-integral and COD under mixed mode I+II loading was proposed and investigated. The J-integral was calculated by the Finite Element Method, and COD was defined by Rice`s model and measured by a duplicating method in an aluminum alloy Ly12. The critical values of the J-integral and COD for a stable mixed crack initiation were also determined by a resistance curve. It shows that: (1) the mixed J-integral, J _M, and the mixed COD satisfy the relations of J _M=dn₀CTOD+ds₀CTSD and J _M=d_yieldCOD, where dn, ds and d are coefficients; CTOD and CTSD are the mode I and mode II components of COD, respectively; ₀ and ₀ are the tensile and shear stresses at the crack tip strip, respectively, and (2) the initiation values of the J-integral and COD of mixed stable crack growth increase with an increasing mode II component, the J _IIC value is 2 times greater than that of J _IC, and the COD_i for a pure mode II crack is 6 times greater than that of COD_i for a pure mode I crack.     

Experimental and numerical investigation of cracked chevron notched Brazilian disc specimen for fracture toughness testing of rock

The cracked chevron notched Brazilian disc (CCNBD) specimen has been suggested by the International Society for Rock Mechanics to quantify mode I fracture toughness (K_Ic) of rock, and it has also been applied to mode II fracture toughness (K_IIc) testing in some research on the basis of some assumptions about the crack growth process in the specimen. However, the K_Ic value measured using the CCNBD specimen is usually conservative, and the assumptions made in the mode II test are rarely assessed. In this study, both laboratory experiments and numerical modeling are performed to study the modes I and II CCNBD tests, and an acoustic emission technique is used to monitor the fracture processes of the specimens. A large fracture process zone and a length of subcritical crack growth are found to be key factors affecting the K_Ic measurement using the CCNBD specimen. For the mode II CCNBD test, the crack growth process is actually quite different from the assumptions often made for determining the fracture toughness. The experimental and numerical results call for more attention on the realistic crack growth processes in rock fracture toughness specimens.     

Critical values of stress intensity factor in mode II fracture of cementitious composites

The problem of mode II of brittle matrix composites is considered. After a short discussion of the present knowledge and a review of test results, the importance of fracture toughness in mode II is stressed. The test results presented concern both modes of fracture, and obtained values of stress intensity factorsK _Ic andK _IIc are discussed, taking into consideration the results of observations on SEM micrographs. It is suggested that the fracture toughness of concretelike materials should be expressed as a combination of mode I and II characteristics.     

J II fracture testing of a plastically deforming material

A compact model II fracture specimen was previously analyzed and employed to determine the mode II fracture toughness K _IIc , of perspex. In employing this specimen for a more ductile material such as aluminium, it was observed that the load vs. crack sliding displacement record becomes nonlinear for small loads. Thus, concepts of linear elastic fracture mechanics cannot be employed. To this end, the specimen was calibrated for J-integral testing, so that J _IIc mesurements can be performed.In this study, mode I and II tests are carried out on an aircraft aluminium alloy, AI 7075-T7351. First, standard K _Ic tests are performed leading to a value of 27.9 15-1 which would be equivalent to a J _Ic of 10.7 kN/m. Then standard J _Ic tests are carried out on this material with specimen thicknesses, of 5, 7.5 and 9.9 mm, leading to an average J _Ic value of 10.5 kN/m. Methods for J _II testing are proposed; a series of specimens of six thicknesses between 5 and 16 mm are employed for testing. An average J _IIc value was found to be 40.2 kN/m which yields a K _IIc value of 54.1 15-2. Thus, K _IIc is seen to be approximately twice that of K _Ic for this material.     

Comparison of mode I and mode II elastic-plastic fracture toughness for two low alloyed high strength steels

In the present work, mode I and mode II tests were carried out on two low alloyed high strength steels. An asymmetrical four point bend specimen and J _II-integral vs. crack growth resistance curve technique were used for determining the mode II elastic-plastic fracture toughness, J _IIc · J _II-integral expression of the specimen was calibrated by finite element method. The results indicate that the present procedure for determining the J _IIc values is easy to use. Moreover, the mode I fracture toughness J _Ic is very sensitive to the rolling direction of the test steels, but the mode II fracture toughness J _IIc is completely insensitive to the rolling direction of the steels, and the J _IIc /J _Ic ratio is not a constant for the two steels, including the same steel with different orientations. Finally, the difference of the fracture toughness between the mode I and mode II is discussed with consideration of the different fracture mechanisms.     

Fracture mechanics analysis of anisotropic plates by the boundary element method

This paper presents an analysis of mixed mode fracture mechanics problems arising in anisotropic composite laminates. The boundary element method (BEM) and the J _k integral are presented as accurate techniques to compute the stress intensity factors K _I and K _II of two dimensional anisotropic bodies. Using function of a complex variable a decoupling procedure is derived to obtain the stress intensity factors. The procedure is based on the computation of the J ₁-integral and of the ratio of relative displacements at the crack faces, near the crack tip. Applications are presented for unidirectional and symmetric laminates of glass, boron and graphite-epoxy materials. Numerical examples of problems of pure mode I and mixed mode deformations are given, in order to demonstrate the accuracy of the method.     

Cracked Brazilian disc specimen subjected to mode II deformation

The centrally cracked Brazilian disc specimen has been used by many researchers to study mode I and mode II brittle fracture in different materials. However, the experimental results obtained in the past from this specimen indicate that the fracture toughness ratio (K_IIc/K_Ic) is always significantly higher than the theoretical predictions. It is shown in this paper that the increase in the ratio K_IIc/K_Ic can be predicted if a modified maximum tangential stress (MTS) criterion is used. The modified criterion takes into account the effect of T-stress in addition to the conventional singular stresses. The fracture toughness ratio K_IIc/K_Ic is calculated for two brittle materials using the modified criterion and is compared with the relevant published experimental results obtained from fracture tests on the cracked Brazilian disc specimen. A very good agreement is shown to exist between the theoretical predictions and the experimental results.     

A fracture mechanics analysis on the fatigue behaviour of cruciform joints of duplex stainless steel

The paper presents the results of an experimental and numerical study on the fatigue behaviour of cruciform load carrying joints made from the duplex stainless steel and failing from the weld root through the weld metal. Fatigue crack growth (FCG) data, obtained in specimens of the weld metal, are presented, as well as threshold data, both obtained for R= 0 and 0.5. The influence of stress ratio is discussed, and the FCGR results are compared with data for low carbon structural steels. S–N data were obtained in the joints, both for R= 0.05 and 0.5, and the fatigue cracking mechanisms were analysed in detail with the SEM. It was found that the cracks propagated very early in the lifetime of the joints, under mixed mode conditions (I + II), but the mode I component was found to be predominant over mode II. The geometries of the cracks were defined in detail from measurements taken in the fracture surfaces. A 2D FE analysis was carried out for the mixed mode inclined cracks obtained at the weld root, and the J‐integral formulations were obtained as a function of crack length and crack propagation angle. The values of the crack propagation angle, θ_i, were obtained for the J_max conditions, and it was found that, in the fatigue tests, the cracks propagated in directions very close to the predicted directions of maximum J. K_I and K_II formulations were obtained, and the K_I data were compared with the formulations given in the PD6493 (BS7910) document, and some differences were found. A more general formulation for K under mixed mode conditions was derived. The derived K solutions were applied to predict the fatigue lives of the joints under crack propagation, and an extremely good agreement was found with the experimental results obtained in the fatigue tests.     

Study on the behavior of elastic-plastic fracture under mixed I+II mode loading in aluminum alloy Ly12-J-integral analysis

The elastic-plastic fracture behavior of aluminum alloy Ly12 under mixed I+II mode loading was studied by finite element method and fracture test. A mixed mode elastic-plastic fracture criterion of J-integral was proposed by using the J-resistance curve, and the maximum fracture effective plastic strain _p max of different mixed ratios at crack tip were also calculated. The results show that(1) the initiation J-integral values of different mixed ratios have the equation

The behaviour of cracks in elastic-plastic materials under plane normal and shear loadings

Cracks in structures are often subjected to complex loading conditions. The direction of the crack extension depends on the normal and the shear components of the load. This paper is based on the kinking behaviour of cracks taking elastic-plastic behaviour of materials into account. The J-integral and the mixed-mode components J _I and J _II were determined after having performed several finite element analyses for different loading conditions. The path independence of J, J _I and J II is investigated for both, the line integral proposed by Rice and the volume integral proposed by deLorenzi. For correctly determined crack deflection angles the J _II-component vanishes when a FE-model with a kinked crack is considered. Hence, cracks propagate perpendicularly to the local mode I load.     

Methods for calculating stress intensity factors in anisotropic materials: Part I—z = 0 is a symmetric plane

The problem of a crack in general anisotropic material under LEFM conditions is presented. In Part I, three methods are presented for calculating stress intensity factors for various anisotropic materials in which z = 0 is a plane of symmetry. All of the methods employ the displacement field obtained by means of the finite element method. The first one is known as displacement extrapolation and requires the values of the crack face displacements. The other two are conservative integrals based upon the J-integral. One employs symmetric and asymmetric fields to separate the mode I and II stress intensity factors. The second is the M-integral which also allows for calculation of K_I and K_II separately.All of these methods were originally presented for isotopic materials. Displacement extrapolation and the M-integral are extended for orthotropic and monoclinic materials, whereas the J_I- and J_II-integrals are only extended for orthotropic material in which the crack and material directions coincide. Results are obtained by these methods for several problems appearing in the literature. Good to excellent agreement is found in comparison to published values. New results are obtained for several problems.In Part II, the M-integral is extended for more general anisotropies. In these cases, three-dimensional problems must be solved, requiring a three-dimensional M-integral.     

Mixed mode fatigue and fracture in planar geometries: Observations on Keq and crack path modelling

The mixed mode I‐II fatigue and fracture is briefly reviewed, addressing experimental and numerical modelling aspects, and focusing on planar specimens. One major challenge concerns the determination of equivalent stress intensity factor (K_eq) in mixed mode situations. Several approaches were compared through the determination of K_eq/K_I over a wide range of values of K_I/K_II or K_II/K_I. Whereas all different approaches converge to the same value as K_I/K_II increases, the same does not happen for large K_II/K_I, where differences between values of K_eq persist. In the regions of 0 < K_I/K_II < 2 and 0 < K_II/K_I < 2, no stable trend of results can be defined. Experimental fatigue crack growth results are presented for Al alloy AA6082‐T6. Compact tension specimens, modified with holes, and four‐point bending specimens under asymmetrical loading promoting mixed mode situations, were subjected to fatigue crack growth tests, where crack path and crack growth rate were measured. The presentation of the fatigue crack growth data was made using a Paris law based upon K_eq. Differences in the Paris law constants were found for the different K_eq criteria. Recent developments in numerical techniques, as the implementation of the extended finite element method (XFEM) in finite element software packages allows to determine accurately crack paths in mixed mode fracture. This article highlights concepts for mixed‐mode fatigue and fracture and supporting data, identifying challenges still to be overcome.     

Fracture of knitted randomly oriented short-fiber composite

Fracture behavior of a chopped-mat E-glass fiber reinforced hybrid resin composite knitted with continuous poly(ethylene terephthalate) (PET) fibers is investigated by a combined experimental and analytical study. In both opening-mode (mode-I) and shear-mode (mode-II) fracture studies of the composite, the macroscopic critical stress intensities or toughnesses of the material are found to be distinct along the warp and fill directions of the knitting PET fibers. Values of K _IIc ( ^F ) and K _IIc ( ^F ) are lower than those of K _Ic ( ^W ), owing to different fracture mechanisms involved. In the mixed-mode fracture of the composite, a failure envelope in K _I and K _II is constructed. The different mechanisms involved in opening-mode, mode-II and mixed-mode fracture are studied with SEM observations.     

Mixed mode I and II fracture of carbon black filled natural rubber

The J-integral was used to characterize initiation and rapid fracture under mixed mode loading conditions in carbon black filled natural rubber. The total critical J was geometry dependent, but the J analysis partitions the energy into that needed locally in the crack tip region and in the bulk. The critical conditions for pure mode II loading could not be determined because of buckling of the specimen; however, values obtained by extrapolation show J _IIc to be about twice J _Ic. The relation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% aabiqaaiGacaGaamqadaabaeaafiaakeaadaWcaaqaaiaadQeadaWg% aaWcbaGaamysaaqabaaakeaacaWGkbWaaSbaaSqaaiaadMeacaWGJb% aabeaaaaGccaqGGaGaae4kaiaabccadaWcaaqaaiaadQeadaWgaaWc% baGaamysaiaadMeaaeqaaaGcbaGaamOsamaaBaaaleaacaWGjbGaam% ysaiaadogaaeqaaaaakiaabccacaqG9aGaaeiiaiaabgdaaaa!5355!\[\frac{{J_I }}{{J_{Ic} }}{\text{ + }}\frac{{J_{II} }}{{J_{IIc} }}{\text{ = 1}}\]

---

Résumé On utilise l'intégrale J pour caractériser l'amorcage et le déploiement brutal d'une rupture dans du caoutchouc noir naturel soumis à des conditions de sollicitation de mode mixte. Pour une sollicitation purement de Mode II, on n'a pu déterminer les conditions critiques en raison d'un flambage de l'échantillon. Néanmoins, les valeurs obtenues par extrapolation montrent que J _IIc vaut à peu prés le double de J _Ic. Les conditions de rupture sous sollicitation de mode mixte sont décrites par la relation: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% aabiqaaiGacaGaamqadaabaeaafiaakeaadaWcaaqaaiaadQeadaWg% aaWcbaGaamysaaqabaaakeaacaWGkbWaaSbaaSqaaiaadMeacaWGJb% aabeaaaaGccaqGGaGaae4kaiaabccadaWcaaqaaiaadQeadaWgaaWc% baGaamysaiaadMeaaeqaaaGcbaGaamOsamaaBaaaleaacaWGjbGaam% ysaiaadogaaeqaaaaakiaabccacaqG9aGaaeiiaiaabgdaaaa!5355!\[\frac{{J_I }}{{J_{Ic} }}{\text{ + }}\frac{{J_{II} }}{{J_{IIc} }}{\text{ = 1}}\].

     

Influence of thickness on critical crack opening displacement (COD) and J values

The effect of specimen thickness on the critical values of COD and J integral at fracture was studied in a carbon manganese steel plate 100 mm thick. Ductile brittle transition data from preferred geometry bend specimens showed that K _IJ values obtained from J _Ic in 10 mm thick specimens can overestimate K _Ic measured in 100 mm thick specimens. Relaxation of stress triaxiality and changes in micromode of crack initiation with decreasing specimen thickness are considered to be factors causing the discrepancy between K _IJ and K _Ic. The problems associated with using K _IJ values, obtained from thin specimens, to predict defect sizes in thick structures are discussed.

---

Résumé L'effet de l'épaisseur d'une éprouvette sur la valeur critique du COD et l'intégrale J lors de la rupture a été étudié dans une tôle d'acier au carbone manganèse épaisse de 100 mm. Les données de transition ductile fragile établies à partir d'éprouvettes de flexion à géométrie définie ont montré que les valeurs de K _IJ établies pour les valeurs de J _Ic correspondant à des éprouvettes de 10 mm d'épaisseur peuvent surestimer les valeurs de K _Ic mesurées dans des éprouvettes de 100 mm d'épaisseurs. La relaxation de la triaxialité des contraintes et les changements d'amorçage de fissure au point de vue de leur mode avec les épaisseurs décroissantes d'une éprouvette sont considérées comme des facteurs susceptibles de causer des différences entre K _IJ et K _Ic. Les problèmes associés avec l'utilisation des valeurs K _IJ, obtenues à partir d'éprouvettes minces, en vue de prédire les dimensions d'un défaut dans des structures épaisses sont mises en discussion.

     

Continuum shape sensitivity analysis of mixed-mode fracture using fractal finite element method

This paper presents a new fractal finite element based method for continuum-based shape sensitivity analysis for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations predicts the first-order sensitivity of J-integral or mode-I and mode-II stress-intensity factors, K_I and K_II, more efficiently and accurately than the finite-difference methods. Unlike the integral based methods such as J-integral or M-integral no special finite elements and post-processing are needed to determine the first-order sensitivity of J-integral or K_I and K_II. Also a parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. Four numerical examples which include both mode-I and mixed-mode problems, are presented to calculate the first-order derivative of the J-integral or stress-intensity factors. The results show that first-order sensitivities of J-integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.     

Interlaminar shear fracture of chopped strand mat glass fibre-reinforced polyester laminates

The interlaminar shear fracture of chopped strand mat glass fibre-reinforced polyester laminates has been studied both experimentally and analytically. Lap shear (double-grooved) specimens were used to measure the interlaminar shear strength and the cracking mechanism was studied using photomicrography. The finite element method was used to calculate the stress distribution along the shear surface and the mixed-mode stress intensity factors K_I and K_II. The length of the shear surface was found to have a significant effect on the results. Based on the experimental and analytical results, the validity of the British Standard for GRP pressure vessels (BS4994, 1973) was evaluated and the critical stress intensity factors K_Ic and K_IIc for this material were estimated.     

Fracture Toughness and Creep Crack Growth in Polypropylene Under Combined Modes I and II Loading

Fracture toughness and creep crack growth characteristics under combined mode I and II loadings were studied using the compact tension shear (CTS) specimens of polypropylene. The K _I - K _II envelope for crack initiation was obtained under various combined mode loadings. The creep crack growth rates da/dt under combined mode I and mode II loadings can be correlated with a single effective stress intensity factor K _Ieff based on the combined mode fracture envelope.     

Fatigue crack growth under non-proportional mixed-mode loading in ferritic-pearlitic steel

Mode II fatigue crack growth tests as well as tests in sequential mode I and then mode II were performed on ferritic‐pearlitic steel. For ΔK_II ranging from 7 to , bifurcation occurs after 12–450 μm of coplanar growth at a decreasing speed. By contrast, hundreds of micrometres of constant speed coplanar growth were obtained under sequential mode I and then mode II loading, for and ΔK_I ranging from 0.25 to 1.0 ΔK_II . The crack growth rate is a simple sum of the contributions of each mode for ΔK_I= 0.25 ΔK_II but above this value a synergetic effect is found. The mechanism of this fast‐propagation mode is discussed in the light of strain range maps ahead of the crack tip obtained by digital SEM image correlation and elastic–plastic finite element calculations. The stability of the crack path according to the maximum growth rate criterion is demonstrated.     

MIXED-MODE FRACTURE MECHANISMS NEAR THE FATIGUE THRESHOLD OF AISI 316 STAINLESS STEEL

Abstract— Near threshold, mixed mode (I and II), fatigue crack growth occurs mainly by two mechanisms, coplanar (or shear) mode and branch (or tensile) mode. For a constant ratio of ΔK_I/ΔK_II the shear mode growth shows a self-arrest character and it would only start again when ΔK_I and ΔK_II are increased. Both shear crack growth and the early stages of tensile crack growth, are of a crystallographic nature; the fatigue crack proceeds along slip planes or grain boundaries. The appearance of the fracture surfaces suggest that the mechanism of crack extension is by developing slip band microcracks which join up to form a macrocrack. This process is thought to be assisted by the nature of the plastic deformation within the reversed plastic zone where high back stresses are set up by dislocation pile-ups against grain boundaries. The interaction of the crack tip stress field with that of the dislocation pile-ups leads to the formation of slip band microcracks and subsequent crack extension. The change from shear mode to tensile mode growth probably occurs when the maximum tensile stress and the microcrack density in the maximum tensile plane direction attain critical values.