Stress intensity factors for corner cracks in single‐edge notch bend specimen by a three‐dimensional weight function method

A three‐dimensional (3D) weight function method is employed to calculate stress intensity factors of quarter‐elliptical corner cracks at a semi‐circular notch in the newly developed single‐edge notch bend specimen. Corner cracks covering a wide range of geometrical parameters under pin‐loading and remote tension conditions are analysed. Stress intensity factors from the 3D weight function analysis agree well with ABAQUS‐Franc3D finite element results. An engineering similitude approach previously developed for the half‐elliptical surface crack in single‐edge notch bend specimen is also applied to the present corner crack configuration. The results compare well with those from the present weight function analysis.     

Stress intensity factors for surface cracks in single‐edge notch bend specimen by a three‐dimensional weight function method

Stress intensity factors for half‐elliptical surface cracks at a semi‐circular notch in a recently developed single‐edge notch bend specimen are determined for a wide range of geometrical parameters using a three‐dimensional weight function method. Two load cases of pin loading and uniform remote tension are considered. The results are in good agreement with abaqus/franc3d finite element analysis. It is found that the Ziegler–Newman engineering similitude approach (programmed into the Fatigue Crack Growth Structural Analysis life‐prediction code) produces good results for a wide range in a/c ratios. Expressions by multi‐variable curve fitting to the weight function results are presented for easy engineering applications.     

A coupled FE–EFG approach for modelling crack growth in ductile materials

In this work, a coupled finite element–element free Galerkin approach has been used to model crack growth in ductile materials under monotonic and cyclic loads. In this approach, a small discontinuous domain near crack is modelled by EFG method, whereas the rest of the domain is modelled by FEM to exploit the advantages of both the methods. A ramp function has been used in the transition region to maintain the continuity between FE and EFG domains. Two plasticity models (GTN and von‐Mises) and three hardening rules (isotropic, kinematic and mixed) have been used to model the nonlinear material behaviour. Four different problems, i.e. single edge notched tension specimen, double edge notched tension specimen, compact tension specimen and three‐point bend specimen, are solved under plane strain condition using J–R curve approach. Finally, a CT specimen problem is also solved by coupled approach using three hardening rules and two plasticity models under cyclic loading.     

Numerical simulations of fracture initiation in ductile solids under mode I, dynamic loading

In this paper, an overview of some recent computational studies by the authors on ductile crack initiation under mode I, dynamic loading is presented. In these studies, a large deformation finite element procedure is employed along with the viscoplastic version of the Gurson constitutive model that accounts for the micro-mechanical processes of void nucleation, growth and coalescence. A three-point bend fracture specimen subjected to impact, and a single edge notched specimen loaded by a tensile stress pulse are analysed. Several loading rates are simulated by varying the impact speed or the rise time and magnitude of the stress pulse. A simple model involving a semi-circular notch with a pre-nucleated circular hole situated ahead of it is considered. The growth of the hole and its interaction with the notch tip, which leads to plastic strain and porosity localization in the ligament connecting them, is simulated. The role of strain-rate dependence on ductile crack initiation at high loading rates, and the specimen geometry effect on the variation of dynamic fracture toughness with loading rate are investigated.     

A weight function method for mixed modes hole‐edge cracks

A weight function approach is proposed to calculate the stress intensity factor and crack opening displacement for cracks emanating from a circular hole in an infinite sheet subjected to mixed modes load. The weight function for a pure mode II hole‐edge crack is given in this paper. The stress intensity factors for a mixed modes hole‐edge crack are obtained by using the present mode II weight function and existing mode I Green (weight) function for a hole‐edge crack. Without complex derivation, the weight functions for a single hole‐edge crack and a centre crack in infinite sheets are used to study 2 unequal‐length hole‐edge cracks. The stress intensity factor and crack opening displacement obtained from the present weight function method are compared well with available results from literature and finite element analysis. Compared with the alternative methods, the present weight function approach is simple, accurate, efficient, and versatile in calculating the stress intensity factor and crack opening displacement.     

RATE-DEPENDENT DEFORMATION AND FRACTURE OF α-TITANIUM

Aspects of combined rate-dependent deformation and crack growth in α-titanium at room temperature are examined. Results are presented for tests carried out on pre-cracked three point loaded single edge notch bend and compact tension specimens subjected to constant crack opening displacement rates and constant load. Curves of the ratio of the reference stress to the yield stress as a function of the ratio of the plastic displacement to specimen width are found to be different for different rates. The stress difference between continuously loaded curves and curves obtained from load relaxation tests (“relaxed” curves) is found to be similar to uniaxial results. Earlier uniaxial tests show that the “relaxed” curve represents a boundary below which no further creep takes place. The pre-cracked specimen constant load curves cross the “relaxed” curve, even though the contribution from crack growth to the overall deformation is found to be small. Sustained load crack growth is observed to take place under contained yielding conditions and the sustained load resistance curves are found to be different for different reference stresses.     

A finite element method for contact problems related to fracture mechanics

A finite element method for contact problems in crack mechanics is developed on the basis of the penalty function method. The method is successfully applied to three important problems in fracture mechanics: a crack propagated from a pin hole, a two-point supported specimen with an edge crack loaded by a stamp, and a thick plate with a through-wall crack under bending force.     

Evaluation of elastic fracture mechanics parameters for bend specimens

Existing solutions for stress intensity factor and mouth opening displacement of three-point bend specimens are shown to overestimate these quantities for shallow cracks by up to ±4.5 percent, because they do not account for the disturbance of the bending stress distribution by the concentrated force at the loading point. This error is far larger than the accuracy claimed by these solutions (0.2 to 0.5 percent).New expressions are therefore developed for stress intensity, crack mouth opening displacement and crack mouth open angle of single edge notched bend specimens loaded in three-point bending. As a reference, and to show the accuracy of the solutions, also the pure bending situation is treated. Rigorously derived asymptotic solutions are used for the shallow and deep crack limits, in order to prescribe both the proper limit values and gradients to the interpolation functions, of which the intermediate values are derived from refined finite element analyses.The crack mouth opening angle solutions are primarily intended to facilitate crack mouth opening measurement at other locations then the specimen surface, i.e. at an offset from the specimen surface as is the case when removable knife edges are applied. No solutions of the crack mouth opening angle of three-point bend specimens were available until now. For use with unloading compliance crack length measurement, also an inverse crack mouth opening relation is developed. This also includes crack mouth opening measurement at an offset from the specimen surface, which is lacking in presently available expressions of this kind.     

On the use of an anti‐symmetric four‐point bend specimen for mode II fracture experiments

The edge‐cracked beam specimen subjected to anti‐symmetric four‐point bend (ASFPB) loading has been conventionally used in the past for investigating the pure mode II fracture experiments in many engineering materials. However, it is shown through finite element analysis that the ASFPB specimen sometimes fails to produce pure mode II conditions. For anti‐symmetric loads applied close to the crack line, there are considerable effects from K_I and T‐stress in the ASFPB specimen. Pure mode II is provided only when the applied loads are sufficiently far from the crack plane.     

Crack growth predictions using a strip‐yield model for variable‐amplitude and spectrum loading

Fatigue‐crack‐growth tests were conducted on compact, C(T), specimens made of D16Cz aluminum alloy. Constant‐amplitude tests were conducted over a range of stress ratios (R = P_min/P_max = 0.1 to 0.75). Comparisons were made between test data from middle‐crack tension, M(T), specimens from the literature and C(T) specimens. A crack‐closure analysis was used to collapse the rate data from both specimen types into a fairly narrow band over many orders of magnitude in rates using proper constraint factors. Constraint factors were established from single‐spike overload and constant‐amplitude tests. The life‐prediction code, FASTRAN, which is based on the strip‐yield‐model concept, was used to calculate the crack‐length‐against‐cycles under constant‐amplitude (CA) loading and the single‐spike overload (OL) tests; and to predict crack growth under variable‐amplitude (VA) loading on M(T) specimens and simulated aircraft loading spectrum tests on both specimen types. The calculated crack‐growth lives under CA and the OL tests were generally within ±20 % of the test results, the predicted crack‐growth lives for the VA and Mini‐Falstaff tests on the M(T) specimens were short by 30 to 45 %, while the Mini‐Falstaff+ results on the C(T) specimens were within 10 %. Issues on the crack‐starter notch effects under spectrum loading are discussed, and recommendations are suggested on avoiding these notch effects.     

Wide‐range and accurate mixed‐mode weight functions and stress intensity factors for cracked discs

The Wu‐Carlsson displacement‐based weight function method is extended to obtain the mode I and mode II weight functions for the edge‐ and centre‐cracked discs. Compared with Fett's direct adjustment weight functions for the edge‐cracked discs, the present weight functions are more accurate and are applicable for a wider range of crack lengths. Using the present weight functions, extensive and highly accurate mixed‐mode stress intensity factors are obtained for the cracked discs subjected to diametrically compressive forces. Assuming perfect contact and using Coulomb friction law and the present weight functions, the mode II stress intensity factors for the cracked discs with consideration of friction are obtained and widely compared with the corresponding results from finite element analyses.     

Effect of loadline shift on applied crack tip stress intensity in WOL specimens

A change in applied stress intensity due to shifting of load line from the pin hole to a wedge located on the outside edge of the notch has been investigated by: (1) finite element analysis, (2) measurements of front face crack opening displacement and (3) strain relaxation near the crack tip.

Results show that this wedge loading procedure will result in a significant drop, up to a factor of two, in applied stress intensity. The drop in stress intensity is inversely related to the crack length (expressed by a/W). This drop in stress intensity is due to overall specimen distortion because of load line shift and local deformation of the wedge and notch surfaces. Implications of this drop on Stress Corrosion Cracking results are discussed. For reliable stress corrosion testing, modifications in specimen geometries and loading procedures are suggested.     

A New Procedure for Mode I Fracture Characterization of Cement‐Based Materials

Fracture characterization under mode I loading of a cement‐based material using the single‐edge‐notched beam loaded in tree‐point‐bending was performed. A new method based on beam theory and crack equivalent concept is proposed to evaluate the Resistance‐curve, which is essential to determine fracture toughness with accuracy. The method considers the existence of a stress relief region in the vicinity of the crack, dispensing crack length monitoring during experiments. A numerical validation was performed by finite element analysis considering a bilinear cohesive damage model. Experimental tests were performed in order to validate the numerical procedure. Digital image correlation technique was used to measure the specimen displacement with accuracy and without interference. Excellent agreement between numerical and experimental load–displacement curves was obtained, which validates the procedure.     

Automatic 3-D crack propagation calculations: a pure hexahedral element approach versus a combined element approach

This article presents an evaluation of two different crack prediction approaches based on a comparison of the stress intensity factor distribution for three example problems. A single edge notch specimen and a quarter circular corner crack specimen subjected to shear displacements and a three point bend specimen with a crack inclined to the mid-plane are examined. The stress intensity factors are determined from the singular stress field close to the crack front. Two different fracture criteria are adopted for the calculation of an equivalent stress intensity factor and crack deflection angle. The stress intensity factor distributions for both numerical methods agree well to available reference solutions. Deviations are recorded at crack front locations near the free surface probably due to global contraction effects and the twisting behaviour of the crack front. Crack propagation calculations for the three point bending specimen give results that satisfy intuitive expectations. The outcome of the study encourages further pursuit of a crack propagation tool based on a combination of elements.     

Rectangular Tensile Sheet with Double Edge Notch Cracks

This paper deals with the rectangular tensile sheet with symmetric double edge notch cracks. Such a crack problem is called an edge notch crack problem for short. By using a hybrid displacement discontinuity method (a boundary element method), two edge notch models are analyzed in detail. By changing the geometrical forms and parameters of the edge notch, and by comparing the stress intensity factors (SIFs) of the edge notch crack problem with those of the double edge cracked plate tension specimen (DECT), which is a model frequently used in fracture mechanics, the effect of the geometrical forms and parameters of the edge notch on the SIFs of the DECT specimen, is revealed. Some geometric characterestic parameters are introduced here, which are used to formulate the notch length and the branch crack length, which are to be determined in mechanical machining of the DECT specimen So we can say that the geometric characterestic parameters and the formulae used to determine the notch length and the branch crack length presented in this paper perhaps have some guidance role for mechanical machining of the DECT specimen.     

Experimentally determined key curves for fracture specimens

Crack extension during fracture toughness tests of ferritic structural steels cannot be determined from measurements of unloading compliance or electric potential change when the specimen is dynamically tested. Measurements of crack extension in fracture toughness tests are also very difficult when the test temperature is high or the test environment is aggressive. To circumvent this limitation, researchers for years have been developing key curve and normalization function methods to estimate crack extension in standard elastic-plastic fracture toughness test geometries. In the key curve method (Ernst et al., 1979; Joyce et al., 1980) a load-displacement curve is measured for a so-called `source' specimen that is sub size or has a blunt notch so that the crack will not initiate during elastic-plastic loading. The load and displacement are then converted to normalized stress-strain units to obtain a key curve that can be used to predict crack extension in geometrically similar `target' specimens of same material loaded at similar loading rates and tested under similar environmental conditions. More recently Landes and coworkers (Herrera and Landes, 1990; Landes et al., 1991) proposed the normalization data reduction technique – Annex A15 of ASTM 1820 specification – that presents an alternative to the standard E1820 unloading compliance procedure. Although the normalization method works well in many cases, it has serious drawbacks: the load, displacement and crack length at the end of the test must be measured; the prescribed functional form that is fitted to the initial and final data may not be accurate for all materials; and the iterative method of inferring crack length from the combination of the data and the normalization function is complex. The compliance ratio (CR) method developed in this paper determines key curves for predicting crack extension as follows. First, a statically loaded source specimen with the unloading compliance procedure specified in ASTM 1820. Second, the so-called CR load-displacement curve is calculated for the source specimen, which is the load-displacement record that would have been obtained if the crack had not extended. Third, non-dimensionalizing the CR load by the maximum load and the displacement by the elastic displacement at the maximum load, P ^* _i/P _max and v _i/v _{el max} from the source specimen yields the adjusted key curve. Analysis of extensive data shows that the key curve is independent of notch type, initial crack length and temperature. But it is dependent on specimen size and steel type. Assuming that the key curves of the source and target specimens are one and the same, the compliance of the target specimens are calculated with a reverse application of the compliance ratio method, and the crack length is obtained using the equations in ASTM E1820. The CR Method is found to be much simpler than the normalization method described in the Annex A15 of ASTM 1820. With the compliance ratio method, Joyce et al. (2001) successfully predicted crack extension in dynamically loaded specimens using a key curve of a statically loaded specimen.     

Book Reviews

‘Early’ creep-fatigue crack growth rates have been measured in complex-cycle large single edge notched bend feature-specimen tests on a 1¼ CrMoV turbine casting steel at 550°C. Crack propagation rates initially accelerate with increasing distance below the stress concentration to a peak value. The depth at which this maximum occurs depends on the notch geometry and the magnitude of any superimposed primary loading. ‘Early’ creep-fatigue crack growth rates are dependent on crack size, notch root strain range and any creep damage accumulated due to primary and secondary loading.     

The determination of the elastic T-term using higher order weight functions

It has been shown in a recent work [1] that the elastic T-term at the tip of a mixed mode crack can be determined by the so-called second order weight functions through a work-conjugate integral that is akin to that of the Bueckner-Rice weight function method for evaluating stress intensity factors. In this paper, the development of the second order weight functions is reviewed. These second order weight functions are determined using a unified finite element method introduced in [2]. The finite element procedure handles both traction and displacement boundaries and it permits the Bueckner-Rice weight functions and the second order weight functions for the elastic T-term to be determined in one single finite element run. The accuracy of the computed weight functions is assessed by comparing the computed results with special closed form solutions. The numerical values of the elastic T-term for single edge notch specimens under tension, pure bending and three-point bend are given. The corresponding second order weight functions are tabulated.     

Experimental investigation on low cycle fatigue and fracture behaviour of a notched Ni‐based superalloy at elevated temperature

In order to investigate the effects of stress concentration on low cycle fatigue properties and fracture behaviour of a nickel‐based powder metallurgy superalloy, FGH97, at elevated temperature, the low cycle fatigue tests have been conducted with semi‐circular and semi‐elliptical single‐edge notched plate specimens at 550 and 700 °C. The results show that the fatigue life of the notched specimen decreases with the increase of stress concentration factor and the fatigue crack initiation life evidently decreases because of the defect located in the stress concentration zone. Moreover, the plastic deformation induced by notch stress concentration affects the initial crack occurrence zone. The angle α of the crack occurrence zone is within ±10° of notch bisector for semi‐circular notched specimens and ±20° for semi‐elliptical notched specimens. The crack propagation rate decreases to a minimum at a certain length, D, and then increases with the growth of the crack. The crack propagation rate of the semi‐elliptical notched specimen decelerates at a faster rate than that of the semi‐circular notched specimen because of the increase of the notch plasticity gradient. The crack length, D, is affected by both the applied load and the notch plasticity gradient. In addition, the fracture mechanism is shown to transition from transgranular to intergranular as temperature increases from 550 to 700 °C, which would accelerate crack propagation and reduce the fatigue life.     

In‐plane and out‐of‐plane constraint parameters along a three‐dimensional crack‐front stress field under creep loading

Full‐field three‐dimensional (3D) numerical analyses was performed to determine in‐plane and out‐of‐plane constraint effect on crack‐front stress fields under creep conditions of finite thickness boundary layer models and different specimen geometries. Several parameters are used to characterize constraint effects including the non‐singular T‐stresses, the local triaxiality parameter, the T_z ‐factor of the stress‐state in a 3D cracked body and the second‐order‐term amplitude factor. The constraint parameters are determined for centre‐cracked plate, three‐point bend specimen and compact tension specimen. Discrepancies in constraint parameter distribution on the line of crack extension and along crack front depending on the thickness of the specimens have been observed under different loading conditions of creeping power law hardening material for various configurations of specimens.