Crack paths and the problem of global directional stability

Crack paths of original Griffith crack and edge crack under biaxial remote mode-I loading after different local disturbances are calculated by using integro-differential equations of first-order perturbation and numerical simulation with FEM respectively. Considering the asymptotic behaviour for large crack lengths the problem of global directional stability is reinvestigated extending the work by Melin. For the Griffith crack, correct power functions for the asymptotic path with one or both crack tips growing have been determined. The well- known critical stress biaxiality ratio R _c = 1 for the global directional stability has been obtained independently whether the crack is disturbed by local imperfections in geometry or in loading. For an edge crack the calculated critical stress biaxiality ratio for the global directional stability R _c  = 0.616, also irrespective of the local disturbances, corresponds to a positive T-stress and is considerably smaller than the value R  >  0.95 estimated by Melin (2002). In general, cracks need not propagate asymptotically in the direction perpendicular to the largest principal stress (without crack). This is found to be due to the effect of the boundaries. Considering the initial crack growth exclusively, it is shown that the solution for crack path prediction in series expansion form as derived by Cotterell and Rice (1980) for traction-free crack faces (after correction of a misprint) is exact in the two first terms in all cases. Thus, for small crack growth the Cotterell and Rice solution is universal with respect to all loading and geometrical situations.     

Dynamic fracture mechanics analysis of failure mode transitions along weakened interfaces in elastic solids

Dynamic fracture mechanics theory was employed to analyze the crack deflection behavior of dynamic mode-I cracks propagating towards inclined weak planes/interfaces in otherwise homogenous elastic solids. When the incident mode-I crack reached the weak interface, it kinked out of its original plane and continued to propagate along the weak interface. The dynamic stress intensity factors and the non-singular T-stresses of the incident cracks were fitted, and then dynamic fracture mechanics concepts were used to obtain the stress intensity factors of the kinked cracks as functions of kinking angles and crack tip speeds. The T-stress of the incident crack has a small positive value but the crack path was quite stable. In order to validate fracture mechanics predictions, the theoretical photoelasticity fringe patterns of the kinked cracks were compared with the recorded experimental fringes. Moreover, the mode mixity of the kinked crack was found to depend on the kinking angle and the crack tip speed. A weak interface will lead to a high mode-II component and a fast crack tip speed of the kinked mixed-mode crack.     

An experimental investigation of mixed-mode creep crack growth in Jethete M152 at 550°C

Creep crack growth tests have been made on Jethete M152 at 550°C under initially mixed-mode (i.e. K_I/K_II ≈ 1.6) and mode-II crack tip conditions using compact mixed-mode (CMM) specimens. The results of these tests have been compared with mode-I data obtained from compact tension (CT) tests, using a C* approach. The correlation between the mode-I, mode-II and mixed-mode data is reasonably good. However, the scatter band is greater than that obtained from the mode-I results only. The results indicate that the C* approach, which has been used successfully in mode-I situations, may also be useful for predicting creep crack growth in more general situations.     

Fatigue cracks emanating from sharp notches in high-strength aluminium alloys: The effect of loading direction, kinking, notch geometry and microstructure

The effect of notch geometry on the propagation of fatigue cracks emanating from sharp V-shaped notches was investigated. To this purpose, an experimental campaign has been conducted on Al-7075–T651 specimens carrying notches with aperture angles of 45°, 90°, and 135°. In order to investigate the role of microstructure texture, specimens were extracted from the plates with the main axis either in the longitudinal rolling direction (L-samples) or in the transversal direction (T-samples), or 45° inclined with respect to both directions (LT-samples). The effect of stress amplitude was investigated by performing tests at two load levels. Three loading directions θ = 0°, 45° and 90° were considered. Some specimens experienced pure Mode I loading condition, whereas the remaining ones were subjected to combined Mode I and Mode II loading condition. The crack deflection induced by the variation in loading direction was determined by measuring the kinking angle. A linear elastic fracture mechanics approach was adopted for the analysis of experimental results. Stress intensity factors (SIF) of straight cracks were calculated using an appropriate weight function set up for studying inclined edge cracks emanating from sharp V-notches. On the contrary, a finite element model has been built up to derive the SIFs at the tip of the kinked cracks. The influence of K_II on the crack propagation was discussed on the basis of theoretical and semi-empirical models. It has been found that (i) the crack initiation at the notch root occurred in mixed mode conditions, (ii) a decreasing Mode II component with growing crack length was observed under initial loading direction θ₀ = 45° and θ₀ = 90°, (iii) a crack deflection was observed after 45° rotation of the initial loading direction; a good prediction of the kinking angle was obtained using the maximum tangential stress criterion, and (iv) a fairly good rationalization of all the collected crack growth rate data is obtained if the driving force for crack propagation is expressed in terms of K_I.     

Effect of geometry on the fracture behavior of lead-free solder joints

Copper bars were soldered along their length with a thin layer of lead-free Sn3.0Ag05.Cu alloy under standard surface mount processing conditions to prepare double cantilever beam (DCB) specimens. The geometry of the DCBs was varied by changing the thickness of the solder layer and the copper bars. These specimens were then fractured under mode-I and two mixed-mode loading conditions. The initiation strain energy release rate, G_ci, increased with the relative fraction of mode-II, but was unaffected by the changes in either the substrate stiffness or the solder layer thickness. However, the steady-state strain energy release rate, realized after several millimeters of crack growth, was found to increase with the solder layer thickness at the various mode ratios. The crack path was found to be influenced by mode ratio of loading and followed a path that maximizes the von Mises strain rather than maximum principal stress. Finally, some preliminary results indicated that the loading rate significantly affects the G_ci.     

Mixed-mode fracture load prediction in lead-free solder joints

Double cantilever beam (DCB) fracture specimens were made by joining copper bars with both continuous and discrete SAC305 solder layers of different lengths under standard surface mount (SMT) processing conditions. The specimens were then fractured under mode-I and various mixed-mode loading conditions. The loads corresponding to crack initiation in the continuous joints were used to calculate the critical strain energy release rate, J_ci, at the various mode ratios using elastic–plastic finite element analysis (FEA). It was found that the J_ci from the continuous joint DCBs provided a lower bound strength prediction for discrete 2 mm and 5 mm long joints at the various mode ratios. Additionally, these J_ci values calculated from FEA using the measured fracture loads agreed reasonably with J_ci estimated from measured crack opening displacements at crack initiation in both the continuous and discrete joints. Therefore, the critical strain energy release rate as a function of the mode ratio of loading is a promising fracture criterion that can be used to predict the strength of solder joints of arbitrary geometry subject to combined tensile and shear loads.     

Fracture criterion for kinking cracks in a tri-material adhesively bonded joint under mixed mode loading

The fracture behavior of a composite/adhesive/steel bonded joint was investigated by using double cantilever beam specimens. A starter crack is embedded at the steel/adhesive interface by inserting Teflon tape. The composite adherend is a random carbon fiber reinforced vinyl ester resin composite while the other adherend is cold rolled steel. The adhesive is a one-part epoxy that is heat cured. The Fernlund-Spelt mixed mode loading fixture was employed to generate five different mode mixities. Due to the dissimilar adherends, crack turning into the adhesive (or crack kinking) associated with joint failure, was observed. The bulk fracture toughness of the adhesive was measured separately by using standard compact tension specimens. The strain energy release rates for kinking cracks at the critical loads were calculated by a commercial finite element analysis software ABAQUS in conjunction with the virtual crack closure technique. Two fracture criteria related to strain energy release rates were examined. These are (1) maximum energy release rate criterion (G_max) and, (2) mode I facture criterion (G_II = 0). They are shown to be equivalent in this study. That is, crack kinking takes place at the angle close to maximum G or G_I (also minimum G_II, with a value that is approximately zero). The average value of G_IC obtained from bulk adhesive tests using compact tension specimens is shown to be an accurate indicator of the mode I fracture toughness of the kinking cracks within the adhesive layer. It is concluded that the crack in tri-material adhesively bonded joint tends to initiate into the adhesive along a path that promotes failure in pure mode I, locally.     

Effects of T-stresses on fracture initiation for a closed crack in compression with frictional crack faces

This paper studies crack extension resulting from a closed crack in compression. The crack-tip field of such a crack contains a singular field relative to K _II and non-singular T-stresses T _x and T _y parallel and perpendicular to the crack plane, respectively. Using a modified maximum tensile stress criterion with the singular and non-singular terms, the kinking angle at the onset of crack growth is determined by a two parameter field involving the mode-II stress intensity factors and T-stresses, and at fracture initiation a wing crack may be created at an arbitrary angle from 0° to 90°. A compressive T _y increases the kinking angle and reinforces apparent mode-II fracture toughness, while a compressive T _x decreases the kinking angle and enhances apparent mode-II fracture toughness. The direction and resistance of fracture onset is strongly affected by T-stresses as well as frictional stress. The von Mises effective stress is determined for small-scale yielding near the crack tip. The effective stress contour shape exhibits a marked asymmetrical behavior unless 2T _x  = T _y  ≤ 0 for plane stress state. Coulomb friction between two crack faces generally increases the kinking angle, shrinks the size enclosed by the effective stress contour and enhances apparent fracture toughness. Field evidence and experimental observations of many phenomena involving the growth of closed cracks in compression agree well with theoretical predictions of the present model.     

Constraint effects on plastic crack-tip fields for plane strain mode-I, II and III cracks in non-hardening materials

To explore constraint effects on fully plastic crakc-tip fields, analytical solutions are examined for mode-I, II and III loading in non-hardening materials under plane strain conditions. The results reveal that under mode-II and III loading the crack-tip stress fields are unique, and thus can be characterized by a `single parameter'. Under mode-I loading, however, the crack-tip stress field is non-unique but can be characterized by two sets of solutions or `two parameters'. One set of the solutions is the well-known Prandtl field and the other is a plastic T-stress field. This conclusion corroborates the observation of McClintock (1971) that the slip-line field is non-unique for plane strain tensile cracks. A two-term plastic solution which combines the Prandtl field and the plastic T-stress field with two parameters B ₁ and B ₂ can then characterize the crack-tip stress field of plane strain mode-I crack over the plastic region and quantify the magnitude of crack-tip constraints. These characters are similar to those for hardening materials. Analyses and examples show that the two-term plastic solution can match well with the slip-line field or finite element results over plastic region. Thus the parameters B ₁ and B ₂ can be used to characterize the constraint level for mode-I finite-sized crack specimens in non-hardening materials under plane strain conditions.     

Effect of particles on interlaminar crack growth in cross-plied carbon-fibre/epoxy laminates

This paper discusses the effect of particulate additions on the mode-I and mode-II interlaminar fracture toughness of a cross-plied, carbon-fibre-reinforced, epoxy-resin laminate. Particles of graphite, silicon carbide and polyethylene were mixed with the epoxy resin prior to laminating with woven carbon-fibre cloth. Tests have been performed on double cantilever beam (DCB) and end-notched-flexure (ENF) specimens to obtain the critical-strain energy-release rates,G _IC andG _IIC, for the laminates with and without particulate additions. The dependences of the values ofG _IC andG _IIC on the crack length are also considered. The results indicate that the interlaminar-fracture-toughness (mode-I and mode-II) values of the CFRP laminate increase with increases in the particle content up to about 3%, and thereafter they decrease with further increases in the particle content. This was thought to be due to an increase in multiple-crack formation in the middle region of the cracked-plate samples. Furthermore, mode-I tests indicate that the propagation values ofG _IC are dependent on the crack length.     

Fractal effects of crack propagation on dynamic stress intensity factors and crack velocities

Crack extension paths are often irregular, producing rough fracture surfaces which have a fractal geometry. In this paper, crack tip motion along a fractal crack trace is analysed. A fractal kinking model of the crack extension path is established to describe irregular crack growth. A formula is derived to describe the effects of fractal crack propagation on the dynamic stress intensity factor and on crack velocity. The ratio of the dynamic stress intensity factor to the applied stress intensity factor K(L(D, t), V)/K(L(t), 0), is a function of apparent crack velocity V_o, microstructure parameter d/a (grain size/crack increment step length), fractal dimension D, and fractal kinking angle of crack extension path . For fractal crack propagation, the apparent (or measured) crack velocity V_o, cannot approach the Rayleigh wave speed Cr. Why V_o is significantly lower than Cr in dynamic fracture experiments can be explained by the effects of fractal crack propagation. The dynamic stress intensity factor and apparent crack velocity are strongly affected by the microstructure parameter (d/a), fractal dimension D, and fractal kinking angle of crack extension path . This is in good agreement with experimental findings.     

Mode I fatigue crack growth under biaxial loading

This paper is centred on the role of the T-stress during mode I fatigue crack growth. The effect of a T-stress is studied through its effect on plastic blunting at crack tip. As a matter of fact, fatigue crack growth is characterized by the presence of striations on the fracture surface, which implies that the crack grows by a mechanism of plastic blunting and re-sharpening (Laird C. The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In: Fatigue crack propagation, STP 415. Philadelphia: ASTM; 1967. p. 131–68 [8]). In the present study, plastic blunting at crack tip is a global variable ρ, which is calculated using the finite element method. ρ is defined as the average value of the permanent displacement of the crack faces over the whole K-dominance area. The presence of a T-stress modifies significantly the evolution of plastic deformation within the crack tip plastic zone as a consequence of plastic blunting at crack tip. A yield stress intensity factor K_Y is defined for the cracked structure, as the stress intensity factor for which plastic blunting at crack tip exceeds a given value. The variation of the yield stress intensity factor was studied as a function of the T-stress. It is found that the T-stress modifies significantly the yield point of the cracked structure and that the yield surface in a (T, K_I) plane is independent of the crack length. Finally, a yield criterion is proposed for the cracked structure. This criterion is an extent of the Von-Mises yield criterion to the problem of the cracked structure. The proposed criterion matches almost perfectly the results obtained from the FEM. The evolution of the yield surface of the cracked structure in a (T, K_I) plane was also studied for a few loading schemes. These results should develop a plasticity model for the cracked structure taking into account the effect of the T-stress.     

Calculation of KII in crack face contacts using X-FEM. Application to fretting fatigue

In this work, the modeling of LEFM problems that imply crack face closure and contact using the extended finite element method (X-FEM) is presented aiming at its application to fretting fatigue problems. An assessment of the accuracy in the calculation of K_II is performed for two different techniques to model crack face contacts in X-FEM: one is based on the use of additional elements to establish the contact and the other on a segment-to-segment (or mortar) approach. It is concluded that only the segment-to-segment approach can lead to optimal convergence rates of the error in K_II. The crack face contact modeling has also been applied to a fretting fatigue problem, where the estimation of K_II under crack closure conditions plays an important role in the stage I of fatigue crack propagation. The effect of the crack face friction coefficient has been studied and its influence on the range of K_II has been ascertained during loading and unloading cycles.     

Elastic–plastic J and COD estimation schemes for 90° elbow with throughwall circumferential crack at intrados under in-plane opening moment

Leak-before-break (LBB) assessment of primary heat transport piping of nuclear reactors involves detailed fracture assessment of pipes and elbows with postulated throughwall cracks. Fracture assessment requires the calculation of elastic–plastic J-integral and crack opening displacement (COD)¹ for these piping components. Analytical estimation schemes to evaluate elastic–plastic J-integral and COD simplify the calculations. These types of estimation schemes are available for pipes with various crack configurations subjected to different types of loading. However, such schemes for elbow (or pipe bend), which is one of the important components for LBB analyses, is very meager. Recently, elastic–plastic J and COD estimation scheme has been developed for throughwall circumferentially cracked elbow subjected to closing bending moment. However, it is well known that the elbow deformation characteristics are distinctly different for closing and opening bending modes because the ovalisation patterns of elbow cross section are different under these two modes. Development of elastic–plastic J and COD estimation scheme for an elbow with throughwall circumferential crack at intrados subjected to opening bending moment forms the objective of the present paper. Experimental validation of proposed J-estimation scheme has been provided by comparing the crack initiation, unstable ductile tearing loads and crack extension at instability with the test data. The COD estimation scheme has been validated by comparing the COD of test data with the predictions of the proposed scheme.     

Intersonic shear crack growth along weak planes

Classical dynamic fracture theories predict the Rayleigh surface wave speed (c _R ) to be the limiting speed of propagation for mode-I cracks in constitutively homogeneous, isotropic, linear elastic materials subjected to remote loading. For mode-II cracks, propagating along prescribed straight line paths, the same theories, while excluding the possibility of crack growth in the speed regime between c _R and the shear wave speed, c _s , do not exclude intersonic (c _s <υ<c _l ) crack tip speeds. In the present study, we provide the first experimental evidence of intersonic crack growth in such constitutively homogeneous and isotropic solids, ever recorded in a laboratory setting. Intersonic shear dominated crack growth, featuring shear shock waves, was observed along weak planes in a brittle polyester resin under far-field asymmetric loading. The shear cracks initially propagate at speeds just above c _s and subsequently accelerate rapidly to the longitudinal wave speed (c _l ) of the solid. At longer times, when steady state conditions are attained, they propagate at speeds slightly higher than √2–c _s . The experimental results compare well with existing asymptotic theories of intersonic crack growth, and the significance of the preferred speed of √2–c _s is discussed. Received: 13 September 1999 / Reviewed and occerted: 19 November 1999     

Analysis of vacuum fatigue crack growth results and its implications

Conventionally, the reduction of ΔK_th with load ratio R has been interpreted in terms of crack closure arising from plasticity, oxide or crack surface roughness. Since, plane-strain conditions exist near-threshold, plasticity-induced closure is absent. Therefore, to account for R-ratio effects near-threshold, the oxide and roughness closure mechanisms have been proposed. Further analysis has shown that these other two closure effects also are small, when the results taken in vacuum were included. The present analysis shows that there is a unique relation of the oxide thickness with a threshold K_max, rather than with a threshold ΔK. This threshold K_max (K*_max,th) depends on environment. When the applied K_max<K*_max,th, the crack is stationary in the presence of the oxide formation and grows only when the applied K_max>K*_max,th. Thus, the oxide thickness—K_max relation seems to have a bearing on the criterion for crack growth in the presence of the environment. Oxide formation passivates the crack surfaces and retards the environmental damage ahead of the crack-tip. Examples from CrMo and NiCrMo steels support this viewpoint and suggest that there is a pressing need for quantifying the crack-tip environmental effects and fatigue thresholds.     

Mixed‐Mode Stress Intensity Factors for Partially Opened Cracks

Failure considerations under mixed-mode loading require knowledge about the influence of friction between partially closed crack faces in the case of a negative mode-I stress intensity factor. A simple relation is derived, which allowes to compute friction contributions to the mode-II stress intensity factor K_II for the case of negative mode-I stress intensity factors K_I. The relation is exact for the limit case of an edge crack in a half-space. It can be shown that small cracks in finite bodies can also be described with sufficient accuracy.     

Crack branching and arrest in environmental cracking of polyethylene

At intermediateK values a region where the crack speed is constant, that is, independent ofK, is observed in detergent cracking of low density polyethylene. This region is terminated at highK by one of two processes: in thin specimens the crack arrests, and in thick specimens it branches. The mechanism of crack branching involves the crack front twisting and then the initiation of a new crack at the centre of the specimen. A model has been proposed to explain this mechanism and also the fact that crack arrest or branching occur at approximately the sameK value. After branching the cracks continue to propagate at constant speed along paths whereK _| 0 and if the specimen is wide enough, can branch again.     

The kinked interface crack

The integral equation of the kinked interface crack is solved numerically. The values of K _I, K _II and G for an interface crack with an infinitesimal kink are used to predict the kinking angle for two different material combinations under uniaxial tension.