Measurement of mode I and mode II rock dynamic fracture toughness with cracked straight through flattened Brazilian disc impacted by split Hopkinson pressure bar

In order to find an effective and convincing method to measure rock dynamic fracture toughness for mode I and mode II, cracked straight through flattened Brazilian disc specimens of marble, which were geometrically similar for three size, were diametrically impacted by split Hopkinson pressure bar on the flat end of the specimen with three load angle respectively. History of stress intensity factors (K_I(t) for opening mode I, and K_II(t) for sliding mode II), mode mixture ratio (K_I(t)/K_II(t)), as well as mode I and mode II dynamic fracture toughness at crack initiation (K_Id and K_IId) were determined with the experimental–numerical method. It is found that there is a unique size effect for dynamic fracture test with the specimens, the mode mixture ratio is not solely determined by load angle (the angle between load direction and crack line) as in the static loading; the pure mode II load angle is 19° for the ?50 mm specimen, however it is 10° for the ?130 mm and ?200 mm specimens; the mode II load angle decreases with increment of specimen size. Realization of pure mode II is justified by the mode mixture ratio approaching zero, it can be realized under certain load angle and loading rate for the specimen of specified size. K_IId is generally greater than K_Id. Both K_Id and K_IId increase with increment of specimen size, and this trend for K_IId is more remarkable than that for K_Id.     

A new method to estimate the plane strain fracture toughness of materials

Although the testing method for fracture toughness K_IC has been implemented for decades, the strict specimen size requirements make it difficult to get the accurate K_IC for the high‐toughness materials. In this study, different specimen sizes of high‐strength steels were adopted in fracture toughness testing. Through the observations on the fracture surfaces of the K_IC specimen, it is shown that the fracture energy can be divided into 2 distinct parts: (1) the energy for flat fracture and (2) the energy for shear fracture. According to the energy criterion, the K_IC values can be acquired by small‐size specimens through derivation. The results reveal that the estimated toughness value is consistent with the experimental data. The new method would be widely applied to predict the fracture toughness of metallic materials with small‐size specimens.     

Dynamic fatigue behaviour of a steatite ceramic

Dynamic fatigue of a low dielectric loss steatite was investigated. To this end, the values of n and B, the so‐called subcritical crack growth (SCG) parameters were experimentally determined. The steatite exhibited the expected dynamic fatigue behaviour, so that the stress corrosion susceptibility parameter, n, of 24 was obtained. In addition, the material/environment parameter B, which is a constant for a given test environment, was also attained. These parameters are instrumental in predicting the lifetime of components under stress. When the applied load is such that the resulting strength equals half of the inert strength (σ_i), defined as the strength of a sample tested in an inert environment or at a fast stress rate, i.e. where no subcritical crack growth occurs, the time to failure (t_f) of the material was found to be ～140 h. Measurement of the fracture toughness of steatite is also of upmost importance and so it was determined using three test methods. A value of K_Ic = 1.91 ± 0.29 MPa m^1/2 was attained by the indentation fracture method through measurement of the cracks emanating from the Vickers indentation. This value is in good agreement with those determined using the K_Isc (surface crack in flexure) test method (2.21 ± 0.07 MPa m^1/2) and fractography analysis test method (2.00 ± 0.44 MPa m^1/2). Differences in test procedure and analysis causing the values from each test method to be different are discussed.     

Fracture initiation under impact

In the first part of this paper the influence of temperature T and loading rate K_I upon the fracture toughness K_IC of structural steels is considered. A review of experimental results is presented over a wide range of loading rate and temperature in the form of the cross-sections of the constitutive surface K_INC = f(K_I,T). The hypothesis is proposed that both yield stress σ_y in uniaxial tension and fracture toughness K_IC are controlled by the same process of thermally activated movements of dislocations. Consequently, an introduction of the characteristic time t_c leads to the master plot K_IC (σ_y) in double logarithmic coordinates which is temperature and rate-independent. Such an approach provides a simple method for estimating the value of K_IC under a given set of imposed conditions (T,K?_I)₁ provided it is known for another set of imposed conditions (T,K?_I)₂.In the next part of this paper an attempt is presented to model the effect of T and K?_I on fracture toughness K_IC [15]. A model is discussed which combines correlations between critical cleavage stress σ_F, yield stress σ_y and the concept of thermally activated plastic flow from side and the local fracture criteria from the other [15]. It has been demonstrated that this approach can be useful in the proper predictions of changes of K_IC as a function of loading rate and temperature. For some steels, however, a minimum of fracture toughness is observed and typically occurs for $\text{K}{}_{\mathrm{I}}\text{? 1×10}{}^4$ MPa/pv/m/s at room temperature. The last part of this study deals with this important phenomenon [34]. It is concluded that the behavior of the constitutive surface K_IC = f(K_I,T) is highly nonlinear for steels.     

Estimation of high‐temperature fracture parameters for small punch specimen with a surface crack

An outline of a newly proposed methodology for evaluating creep crack growth (CCG) parameters using cracked small‐punch (SP) specimens is explained. Three‐dimensional finite element analyses were performed to calculate the stress intensity factor along the crack front for a surface crack formed at the centre of a SP specimen. Effects of crack ratio, (a/t); crack aspect ratio, (a/c); and thickness of the specimen, (t), on the fracture parameters were studied. It was observed that the minimum variation of K‐value along the crack front can be achieved when a/c was 0.50 except the location very near the intersection of the crack and free surface. This condition is similar to the case of constant K‐values along the crack front of the conventional compact tension specimen. Thus, it can be argued that the SP specimen with a surface crack is a suitable specimen geometry for CCG testing. The proposed CCG test method was found to be practically applicable for the crack geometry of 0.10 to 0.30 of a/t with constant aspect ratio of 0.50. An estimation of the K and C_t‐parameter under the small scale creep condition was derived. Future work for further development of the suggested CCG testing is discussed.     

Application of artificial neural network for predicting plain strain fracture toughness using tensile test results

A back‐propagation neural network was applied to predicting the K_IC values using tensile material data and investigating the effects of crack plane orientation and temperature. The 595 K_IC data of structural steels were used for training and testing the neural network model. In the trained neural network model, yield stress has relatively the most effect on K_IC value among tensile material properties and K_IC value was more sensitive to K_IC test temperature than to crack plane orientation valid in the range of material data covered in this study. The performance of the trained artificial neural network (ANN) was evaluated by comparing output of the ANN with results of a conventional least squares fit to an assumed shape. The conventional linear or nonlinear least squares fitting methods gave very poor fitting results but the results predicted by the trained neural network were considerably satisfactory. This study shows that the neural network can be a good tool to predict K_IC values according to the variation of the temperature and the crack plane orientation using tensile test results.     

Effect of annealing on the fracture behavior of La0.58Sr0.4Co0.8Fe0.2O3‐δ, a potential energy material for oxygen separation

The perovskite material La_0.58Sr_0.4Co_0.8Fe_0.2O_3‐δ, offers high oxygen permeability at elevated temperature and is considered as a potential material for oxygen separation membranes. It can enhance the efficiency of oxy‐fuel combustion at high temperatures (> 800 °C) and hence due to the high reliability demands, required by the long term operation at elevated temperatures, it requires a thorough investigation from the view point of structural stability. Aiming towards long term stability, the present work is a detailed and systematic study on the effect of annealing on the mechanical behavior of dense La_0.58Sr_0.4Co_0.8Fe_0.2O_3‐δ. The study reveals that the indentation fracture toughness of the material increases with increase in annealing temperature. In most of the indentation loads, the subsurface crack profile was Palmqvist in nature with low value of the ratio of crack length versus indentation size (c/a). A consistent pattern of variation of c/a and indentation fracture toughness (K_IC) at all indentation loads was observed. Systematic drop in c/a and subsequent increase in fracture toughness in the as prepared test pieces has been attributed to residual stress accumulation during preparation.     

Evaluation of the KIC and JIC fracture parameters in a sand cast AZ91 magnesium alloy

The critical fracture parameters K_IC and J_IC were determined on compact tension (CT) specimens designed according to ASTM standards. The minimum specimen thickness B required for a valid K_IC test is suggested to be directly proportional to the σ_ys/E ratio, which is very high for magnesium alloys. As consequence, the value of B is more than 128 mm. Valid K_IC values were obtained using specimens with B=110 mm. Research on the effect of specimen thickness on the K_Q value was performed. The specimen thickness proposed by ASTM E 399 is overestimated. The experimental results show that not so large specimens are needed for a valid K_IC test. K_IC experimental value was 22 MPa m^1/2, that is two times the value reported in the literature for this alloy. The determination of both K_IC and J_IC has offered also the possibility of evaluating the real efficacy of formulas relating the two parameters: starting from the J_IC experimental data, formulas give some underestimation of the K_IC values. SEM fractographic examinations revealed the influence of grain size on the fracture resistance of AZ91.     

A Padé‐based factorization‐free algorithm for identifying the eigenvalues missed by a generalized symmetric eigensolver

When computing the solution of a generalized symmetric eigenvalue problem of the form Ku =λ Mu , the Sturm sequence check, also known as the inertia check, is the most popular method for reporting the number of missed eigenvalues within a range [σ_L,σ_R]. This method requires the factorization of the matrices K ?σ_L M and K ?σ_R M . When the size of the problem is reasonable and the matrices K and M are assembled, these factorizations are possible. When the eigensolver is equipped with an iterative solver, which is nowadays the preferred choice for large‐scale problems, the factorization of K ?σ M is not desired or feasible and therefore the inertia check cannot be performed. To this effect, the purpose of this paper is to present a factorization‐free algorithm for detecting and identifying the eigenvalues that were missed by an eigensolver equipped with an iterative linear equation solver within an interval of interest [σ_L,σ_R]. This algorithm constructs a scalar, rational, transfer function whose poles are exactly the eigenvalues of the symmetric pencil ( K , M ), approximates it by a Padé expansion, and computes the poles of this approximation to detect and identify the missed eigenvalues. The proposed algorithm is illustrated with an academic numerical example. Its potential for real engineering applications is also demonstrated with a large‐scale structural vibrations problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Transferability of the two‐parameter fracture criterion for 2219 aluminium alloy cracked configurations

Two‐dimensional elastic–plastic finite‐element fracture simulations with the critical crack‐tip‐opening‐angle fracture criterion were used to evaluate the two‐parameter fracture criterion (TPFC). Three different crack configurations under tension and bending loads made of thin‐sheet 2219‐T87 aluminium alloy were analysed. A very wide range of widths (w = 76 to 2440 mm) and initial crack‐length‐to‐width ratios (c_i/w = 0.05 to 0.95) were considered. A relation from the original TPFC was shown to fit the simulated fracture behaviour fairly well for the three different specimen types for net‐section stresses less than the yield stress (σ_y) of the material. Comparisons were also made on measured and simulated fracture tests on middle‐crack‐tension specimens. A relation between the elastic stress‐intensity factor, K_Ie, and net‐section stress, S_n, at failure was found to be linear for S_n < σ_y. The results demonstrated the transferability of the TPFC for different crack configurations for S_n < σ_y, but further study is needed for S_n > σ_y.     

Notched fatigue testing of Inconel 718 prepared by selective laser melting

Selective laser melting (SLM) was used to prepare notched high‐cycle fatigue test specimens made from nickel‐based superalloy Inconel 718. Samples were designed to have 1 of 3 different notch geometries, including V notches with K_t of 2.2 or 3.1, a U notch with K_t of 2.0, and were printed in either vertical or horizontal orientations. Samples were tested with as‐printed dimensions and surfaces after heat treatment, but a separate set of SLM samples were printed as plates and machined to final dimensions comporting to the V‐notch specimen with K_t = 3.1. High‐cycle fatigue testing showed that machined SLM specimens behaved similar to wrought Inconel 718 plate specimens, but testing with as‐produced surfaces led to a decrease in fatigue life. The explanation for this difference is based on approximations of linear elastic fracture mechanics solutions for short cracks emanating from notch roots, with intrinsic surface features of SLM materials serving as the cracks. Analysis of the actual notch geometries after SLM fabrication indicates that stress intensity in the presence of these features plays a prominent role in determining number of cycles before fatigue crack initiation and propagation occurs.     

Energy density zone model and fatigue life prediction considering microscopic effects

An energy density zone (EDZ) model is developed for the prediction of fatigue life. The microscopic effects can be involved in the EDZ model. Three scale transitional functions in the model are utilized to describe the trans‐scale behaviours of fatigue failure from micro‐scale to macro‐scale. Fatigue failure behaviours of a low‐alloy and ultra‐high‐strength steel material (i.e. 40CrNi2Si2MoVA steel) is investigated. Two fatigue parameters in the model are determined from the experimental S–N curves for the smooth cylinder specimens (the stress concentration factor, SCF, K_t = 1). Then, fatigue lives of notched specimens with SCFs K_t = 2 and K_t = 3 are predicted respectively by the proposed model. The predicted S–N curves are satisfactory in comparison with the experimental results. Scatter of the fatigue test data can be depicted when the microscopic effects are considered. Influences of microscopic effects on the fatigue behaviours are explored by means of numerical simulations.     

Critical evaluation of indentation fracture toughness measurements with Vickers indenter on yttria‐stabilized zirconia dental ceramics

The purpose of this study was to investigate and analyze fracture toughness (K_Ic) of yttria stabilized tetragonal zirconia (Y‐TZP) dental ceramics by the Vickers indentation fracture test. In order to determine fracture toughness, the Vickers indenter was used under the load of 294.20 N (HV30). The cracks, which occur from the corners of a Vickers indentation, were measured and used for fracture toughness determination, through five mathematical models according to (I) Anstis, (II) Evans and Charles, (III) Tanaka, (IV) Niihara, Morena and Hasselman and (V) Lankford. Morphology of indentation cracking was determined by scanning electron microscope. The most adequate model for determination of fracture toughness (K_Ic) of yttria stabilized tetragonal zirconia dental ceramics by the Vickers indentation fracture test is Lankford model.     

Mean stress and plasticity effect prediction on notch fatigue and crack growth threshold,combining the theory of critical distances and multiaxial fatigue criteria

In the present work, we propose a robust calibration of some bi‐parametric multiaxial fatigue criteria applied in conjunction with the theory of critical distances (TCD). This is based on least‐square fitting fatigue data generated using plain and sharp‐notched specimens tested at two different load ratios and allows for the estimation of the critical distance according to the point and line method formulation of TCD. It is shown that this combination permits to incorporate the mean stress effect into the fatigue strength calculation, which is not accounted for in the classical formulation of TCD based on the range of the maximum principal stress. It is also shown that for those materials exhibiting a low fatigue‐strength‐to‐yield‐stress ratio σ_fl,R = ?1/σ_YS, such as 7075‐T6 (σ_fl,R = ?1/σ_YS = 0.30), satisfactorily accurate predictions are obtained assuming a linear‐elastic stress distribution, even at the tip of sharp notches and cracks. Conversely, for any materials characterized by higher values of this ratio, as quenched and tempered 42CrMo4 (σ_fl,R = ?1/σ_YS = 0.54), it is recommended to consider the stabilized elastic‐plastic stress/strain distribution, also for plain and blunt‐notched samples and even in the high cycle fatigue regime still with the application of the TCD.     

Stress concentration formulae useful for any shape of notch in a round test specimen under tension and under bending

In this work stress concentration factors, K_t , for a round bar with a circular-arc or V-shaped notch are considered on the basis of exact solutions for special cases and accurate numerical results. Then, a set of K_t formulae useful for any shape of notch is proposed. The conclusions can be summarized as follows. (i) For the limiting cases of deep (d) and shallow (s) notches, the body force method is used to calculate the K_t values. Then, the formulae are obtained as K_td and K_ts . (ii) On the one hand, upon comparison of K_t and K_td it is found that K_t is nearly equal to K_td if the notch is deep or blunt. (iii) On the other hand, if the notch is sharp or shallow, K_t is mainly controlled by K_ts and the notch depth. (iv) The notch shape is classified into several groups according to the notch radius and notch depth. Then, the least-squares method is applied for the calculation of K_t /K_td and K_t /K_ts . (v) Finally, a set of convenient formulae is proposed that are useful for any shape of notch in a round test specimen. The formulae give SCFs with <1% error for any shape of notch.     

THE EFFECTS OF MEAN STRESS AND STRESS CONCENTRATION ON FATIGUE UNDER COMBINED BENDING AND TWISTING

The Gough test data on fatigue under combined bending and twisting with superimposed mean bending and torsion stresses with and without stress raisers has been re-investigated in terms of the stresses acting on the plane of maximum range of shear stress. It has been shown that the allowable amplitude of shear stress on this plane can be predicted from an equation of the form τ_a= [t - c₁ (K_t×σ_a)^1.5?c₂σ²_m]/K_t where σ_a and σ_m are the normal stress amplitude and mean normal stress respectively on the plane of maximum range of shear stress, c₁ and c₂ are defined material constants and K_t is the theoretical stress concentration factor.     

Stress concentration formula useful for any dimensions of shoulder fillet in a round bar under tension and bending

In this work stress concentration factors (SCFs), K_t for a round bar with a fillet are considered on the basis of exact solutions, now available for special cases, and accurate numerical results. Then, a convenient K_t formula useful for any dimensions of the fillet is proposed. The conclusions can be summarised as follows: (i) For the limiting cases of deep (d) and shallow (s) fillet, the body force method is used to calculate the K_t values. Then, the formula are obtained as K_td and K_ts. (ii) On the one hand, upon comparison of K_t and K_td, it is found that K_t is nearly equal to K_td if the fillet is deep or blunt. (iii) On the other hand, if the fillet is sharp or shallow, K_t is mainly controlled by K_ts and the fillet depth. (iv) The fillet shape is classified into several groups according to the fillet radius and fillet depth. Then the least squares method is applied for calculation of K_t/K_td and K_t/K_ts. (v) Finally, a convenient formula is proposed that is useful for any dimensions of fillet in a round bar. The formula give SCFs with less than 1% error in most cases for any dimensions of fillet under tension and bending.     

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.     

Experimental Evaluation of Mixed Mode Stress Intensity Factor for Prediction of Crack Growth by Phoelastic Method

Determination of stress intensity factors K _I, K _II, and constant stress term, σ _ox is investigated. A theory of determining the stress intensity factors using photo-elastic method is formulated taking three stress terms. Three-parameter method of fracture analysis for determining the mixed mode stress intensity factors under biaxial loading conditions from photo-elastic isochromatic fringe data is used. A special biaxial test rig is designed and fabricated for loading the specimen biaxially. A simplified and accurate method is proposed to collect the data from isochromatic fringes. Taking specimen geometry and boundary conditions into account, regression models are developed for estimation of fracture parameters.     

Evaluation by indentation of fracture toughness of ceramic materials

A transition fracture mode from Palmqvist to median has been observed in a number of ceramic materials. A new expression to determine the fracture toughness (K _IC) by indentation is presented. The K _IC values calculated by this formula are independent of the crack profile (median or Palmqvist) and of the applied load. This formula has been obtained by modifying the universal curve of Evans and Charles to incorporate Palmqvist and median cracks over a wide range of loads in the case of brittle materials with different mechanical properties (elastic properties: E, v, K _IC).