Determination of double-Determination of double-K criterion for crack propagation in quasi-brittle fracture Part I: experimental investigation of crack propagation

The results of experimental investigations using laser speckle interferometry on small size three-point bending notched beams and using photoelastic coating and the strain gauges on very large size compact tension specimens of concrete are presented in detail. The investigations showed that there exists a stage of stable crack propagation before unstable fracture occurs. The results are in agreement with other researchers' investigations using moire interferometry, holographic interferometry, dye-impregnation method and microscope. Further detailed study shows that the three different states, i.e., crack initiation, stable crack propagation and unstable fracture can be distinguished in the fracture process in concrete structures. In order to predict the crack propagation during the fracture process in quasi-brittle materials a double-K criterion is proposed. The double-K criterion consists of two size-independent parameters. Both of them are expressed in terms of the stress intensity factors. One of them reflects the initial cracking toughness, denoted with Kⁱⁿⁱ, which can be directly evaluated by the initial cracking load, P_ini, and the precast crack length, a₀, using a formula of LEFM. The other one refers to the unstable fracture toughness, denoted with K^un, which can be obtained inserting the maximum load, P_max, and the effective crack length, a, into the same formula of LEFM. The values of the two parameters, K _Ic ⁱⁿⁱ and K _Ic ^un , obtained from the small size three-point bending notched beams and the large size compact tension specimens show that K _Ic ⁱⁿⁱ and K _Ic ^un are size-independent. Evaluating with the K-resistance curves obtained from the same test data, it is found that the proposed double-K criterion is equivalent to it in basic principle, but, the double-K criterion can be applied more easily than the K-resistance curve. Finally, as a practical example, the application of the double-K criterion to the prediction of the crack propagation in a concrete dam is discussed.     

Determination of double-K criterion for crack propagation in quasi-brittle fracture, Part II: Analytical evaluating and practical measuring methods for three-point bending notched beams

In this paper, an attempt is made to determine the double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un using three-point bending notched beams. First, based on the knowledge from extensive investigations which showed that the nonlinearity of P-CMOD curve is mainly associated with crack propagation, a linear asymptotic superposition assumption is proposed. Then, the critical effective crack length a is analytically evaluated by inserting the secant compliance c into the formula of LEFM. Furthermore, an analytical result of a fictitious crack with cohesive force in an infinite strip model was obtained. The double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un as well the critical crack tip opening displacement CTOD_c were analytically determined. The experimental evidence showed that the double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un are size-independent and can be considered as the fracture parameters to describe cracking initiation and unstable fracture in concrete structures. The testing method required to determine K _Ic ⁱⁿⁱ and K _Ic ^un is quite simple, without unloading and reloading procedures. So, for performing this test, a closed-loop testing system is not necessary.     

Determination of double-K criterion for crack propagation in quasi-brittle fracture, Part III: Compact tension specimens and wedge splitting specimens

This paper shows how the double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un can be determined for concrete using CT-specimens and wedge splitting specimens. The experimental results collected from the fracture tests the very large size CT-specimens and small size wedge splitting specimens carried out by many researchers are utilized to investigate the characters of the obtained double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un . It was found that the double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un determined from fracture tests on the large size CT-specimens are size-independent. And the values of K _Ic ⁱⁿⁱ and K _Ic ^un determined from small size wedge splitting specimens with same dimensions are independent of the relative preformed notch length a₀/D. However, when the dimensions of small size wedge splitting specimens change from 150×150×150 mm³ to 450×450×450 mm³, the values of K _Ic ⁱⁿⁱ and K _Ic ^un slightly depend on the heights of the specimens and do not depend on the thickness of the specimens.     

A simplified method for determining double-K fracture parameters for three-point bending tests

A simplified method for determining the double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un for three-point bending tests is proposed. Two empirical formulae are used to describe the crack mouth opening displacement CMOD and the stress intensity factor K _I ^c caused by the cohesive force (x) on the fictitious crack zone for three-point bending beams. It has been found that the two empirical formulae are accurate for a large practical region of a/D. Experiments carried out by many researchers showed that the new formula of CMOD for three-point bending beams can be directly used to predict the initial crack length for precracked beams, the notch depth and the critical effective crack length, as well as the crack length in the post-critical situation with a satisfactory accuracy. Further verification is demonstrated to determine the double-K parameters K _Ic ⁱⁿⁱ and K _Ic ^un. They are very close to those determined by the method proposed in our previous work. Using the simplified procedure, the experiments can be performed even without a closed-loop testing facility and the calculation can be carried out on a pocket calculator.     

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.     

Nature of scatter of fracture toughness in static loading

The previously proposed model of unstable fatigue crack growth is used to explain a large (in comparison with other mechanical characteristics) scatter of static fracture toughness for 15Kh2MFA and 15Kh2NMFA steels at temperatures below the tactile-brittle transition temperature. The results show that for the materials for which K_fc ¹ < K_Ic the critical stress intensity factor K_Ic depends on the specific energy of inelastic strain W at the tip of the initial fatigue crack in its formation stage. The value of W is a function of the number of load cycles (in the conditions with a constant range of the stress intensity factor K) as a result of irregular fatigue crack growth. Here K_fc ¹ is the minimum cyclic fracture toughness. A method is proposed of evaluating the minimum fracture toughness of the material in static loading based on inspection of the process of irregular fatigue crack growth in the stage of crack initiation.Translated from Problemy Prochnosti, No. 2, pp. 10–16, February, 1990.     

Thickness effect on the mode III fracture resistance and fracture path of rock using ENDB specimens

Study of the thickness effect in predicting the crack growth behavior and load bearing capacity of rock‐type structures is an important issue for obtaining a relation between the experimental fracture toughness of laboratory subsized samples and the real rock structures with large thickness. The fracture of rock masses or underground rock structures at deep strata may be dominantly governed by the tensile or tear crack growth mechanism. Therefore, in this research, a number of mode I and mode III fracture toughness experiments are conducted on edge notch disc bend (ENDB) specimen made of a kind of marble rock to investigate the effect of specimen thickness on the corresponding K_Ic and K_IIIc values. It is observed that the fracture toughness of both modes I and III are increased by increasing the height of the ENDB specimen. Also, the ratio of K_IIIc/K_Ic obtained from each thickness of the ENDB specimens is compared with those predicted by some fracture criteria, and it was shown that the minimum plastic radius (MPR) criterion is the main suitable criterion for investigating the fracture toughness ratio K_IIIc/K_Ic . Also, the effect of ENDB height on fracture trajectory of tested samples is assessed. It is shown that the crack grows curvilinearly in thicker ENDB samples and cannot extend along the crack front in small specimens.     

Strength Assessment for Butt Joints of Steel 50 by the Fracture Toughness Criteria

We propose an experimental-theoretical engineering procedure for assessing the strength of butt-welded joints using the force (K_Ic) and strain (_c) criteria of the brittle and quasibrittle fracture mechanics, respectively. For this purpose, beam specimens with square and bevel welds and an initial edge crack or notch in the weld metal are tested under three-point bending. The parameter K_Imax controlling fracture of a bent beam with an inclined (Mode I + Mode II) crack is assessed taking into account the values of the stress intensity factors K_I and K_II, and the crack inclination angle ^*. We also studied the plastic zone at the crack tip and the crack propagation kinetics depending on the weld geometry and the V-notch tip radius for butt-welded joints. The data obtained allow one to rate such joints by their strength according to the fracture toughness criteria K_Ic and _c.     

Effect of Strain Rate and Preloading on the Fracture Toughness of ZrO2-Based Ceramics

We study the variation of the fracture toughness K_Ic ofZrO₂ - Y₂ O₃ ceramics (density 98%) as a function of the testing machine crosshead speed (0.005–50 mm/min) and preloading at K_I < K_c. The fracture toughness is shown to be practically constant in the speed range from 0.05 to 5 mm/min. At a loading rate of 50 mm/min, the quantity K_Ic substantially decreases (by a factor of more than two), whereas at a rate of 0.005 mm/min it slightly increases. Preloading leads to a 1.5-fold increase in K_Ic. Variation of the fracture toughness is associated with structural transformations.     

Fracture parameters for sintered steels

Measurements of the plane strain fracture toughness K _Ic of sintered steels have frequently been invalid because the requirement that P _max/P _Q<1.1 (where P _max = maximum load and P _Q=load used to calculate K _Ic) has not been met. We show that the reason for the criterion not being met is that sintered steels have a considerable crack growth resistance K _R. Values obtained in the past for K _Ic probably have been over-estimates of the initiation value of the crack growth resistance K _i and under-estimates of the maximum crack growth resistance K . The important point is that the assessment of the toughness of sintered steels by a single parameter is not appropriate. Test methods to determine the crack growth resistance of sintered steels are discussed. Crack growth, which is difficult to detect by visual observation, can be determined by compliance techniques. Because of the porous nature of sintered steel, fatigue cracks are unnecessary at the tip of the notch and indeed are undesirable as they can easily cause errors in toughness measurements through inadvertent overloading. The thickness requirement for plane strain measurements can also be relaxed.     

Deformation and fracture of cork in tension

Various properties related to the deformation and fracture of cork in tension were experimentally determined, including the Young's modulus, the stress and strain at fracture and the fracture toughnessK _Ic. The transverse isotropy of cork implies that there are three independent systems of mode I crack propagation andK _Ic was measured for each. The mechanisms of deformation and fracture were identified by SEM microscope observation ofin situ deformation and of the fracture surfaces and crack paths. Two fundamental mechanisms of fracture occur: crack propagation along the lateral cell walls in non-radial tension, withK _Ic = 94±16 kPam^1/2 and crack propagation by breaking the cell walls in radial tension withK _Ic=125±14 kPam^1/2. In radial tension, local fractures that do not propagate due to crack stopping were observed which lead to serrations in the tensile curves for that direction. The strain to fracture in this direction is considerably larger than in the perpendicular (non-radial) directions.     

Fracture toughness of Si3N4 measured with short bar chevron-notched specimens

The short bar chevron-notched specimen was used to measure the plane strain fracture toughness of hot-pressed Si₃N₄. Specimen proportions and chevron-notch angle were varied, thereby varying the amount of crack extension to maximum load (upon which K_ic was based). The measured toughness (4.68 ± 0.19 MN m^3/2) was independent of these variations, inferring that the material has a flat crack growth resistance curve.Nomenclature a crack length - a _A crack length at arrest of unstable crack advance - a ₁ length of chevron notch at specimen surface (distance from line of load application to point of chevron emergence at specimen surface) - a ₀ initial crack length (distance from line of load application to tip of chevron) - a _R crack length at ending of stable crack extension (conversely, crack length at onset of abrupt, unstable crack advance) - B specimen thickness - H specimen half-height - K _1A stress intensity factor at arrest of unstable crack advance - K _IR stress intensity factor at end of stable crack extension (crack growth resistance) - K _IC plane strain fracture toughness - P _max maximum applied load in fracture toughness test - W specimen width - Y ^* dimensionless stress intensity factor coefficient for chevron-notched specimen - Y ^* _m minimum value ofY ^* as a function of - a/W - ₀ a ₀/W - ₁ a ₁/W     

DYNAMIC CRACK PROPAGATION AND CRACK ARREST INVESTIGATED WITH A NEW SPECIMEN GEOMETRY: PART II: EXPERIMENTAL STUDY ON A LOW-ALLOY FERRITIC STEEL

The novel experiment developed in our Institute to investigate crack initiation, rapid crack propagation and crack arrest with one specimen, the ring test, was applied to a ferritic HSLA pipeline steel. The maximum crack speed achieved in these experiments was between 230 m/s and 1165 m/s. The fracture toughness at crack arrest, K_Ia, was determined by a static analysis of this specific test. In all cases, it was found that K_Ia was much lower than K_Ic. The values of K_Ia decrease when the maximum crack speed increases, the results being largely scattered. The fracture toughness at crack arrest is therefore not an intrinsic parameter of the material for a given temperature. Cleavage fracture obtained under these conditions is characterized by the existence of numerous cleavage microcracks, mechanical twins and unbroken ligaments. The decrease in fracture toughness when crack speed increases is related, using the Beremin or the RKR model, to the high strain rates at the tip of a rapid propagating crack. A model which takes into account the effect of unbroken ligaments left in the wake of a propagating crack is developed to account for the large values of K_Ia which were occasionally measured.     

Fracture-induced photon emission of aluminium oxide and its application to fracture mechanical investigations

Mechanically-stimulated luminescence is generated during sub-critical crack growth prior to macroscopic bending fracture of sintered alumina, fusion-cast Al₂O₃, sapphire, and ruby. At similar toughness, K _Ic, the measured intensity increases with the hardness of the tested specimens, it does not depend on the macroscopic fracture strength.     

Fracture toughness of concretes at high temperature

The fracture toughness of ordinary and refractory concretes in the range of 20–1300C was investigated, and the stress intensity factor, K _Ic, on three-point bent specimens (according to ASTM E-399 recommendation) determined. With an increase in testing temperature, the stress intensity factor decreases for both concretes. The values of K _Ic at 20C for both concretes are comparable, being equal to 0.64 MNm^–3/2 for ordinary concrete, and 0.72 MNm^–3/2 for refractory concrete, respectively. At 1100C, K _Ic has a value of 0.043 MNm^–3/2 for ordinary concrete, and for the refractory concrete at 1300C, K _Ic=0.34 MNm^–3/2. The method presented for predicting the behaviour of concrete at high temperature may be used in engineering practice.     

The effect of binder thickness and residual stresses on the fracture toughness of cemented carbides

Much of the data on WC-Co cermets show that the fracture toughness,K _Ic, increases with increasing tungsten carbide grain size at fixed volume fraction of the cobalt binder phase. It is shown that the origin of this effect can be explained on the basis of the plane stress fracture of constrained cobalt phase and the periodic internal stresses arising due to differential thermal contraction of the two phases. Quantitative models have been derived which take these two effects into account. The effect of macroscopic residual stresses, such as those generated by milling WC-Co drilling inserts, on the apparent toughness has also been analysed. It is shown that for the chevron-notched type specimen the macroscopic residual stress affects not only the maximum load but also the length of the crack at which the maximum occurs. A graphical method is presented which permits the evaluation of the true K_Ic.     

Relationship between KIc and plane-strain tensile ductility and microscopic mode of fracture

In most structural steels, the critical plane-strain stress-intensity factor, K_Ic, increases markedly with increasing test temperature. Because of this transition behavior with temperature and the inherently high fracture toughness of many steels, very thick specimens must be tested to determine valid K_Ic values. The large size of these specimens and the cost of conducting the tests minimize the usefulness of this procedure as a research tool for analyzing the fracture behavior of steels under plane-strain conditions. Therefore, as part of a long-range program to obtain K_Ic values from small specimens and to extend linear-elastic fracture mechanics to the region of elastic-plastic fracture mechanics, the Research Laboratory investigated the relationship between K_Ic and ordinary tensile material properties for four steels ranging in yield strength from 80 to 250 ksi (552–1720 MN/m²).The results showed that, for these steels, the variation of K_Ic with temperature was similar to the variation of the plane-strain tensile ductility with temperature. Scanning electron micrographs showed that the increase in the plane-strain stress-intensity factor for unstable crack extension, K_Ic, with increasing temperature could be related to changes in the microscopic mode of fracture at the crack tip. That is, at temperatures below the fracture-toughness transition temperature, the mode of fracture was cleavage, whereas at temperatures well above the transition-temperature region, the fracture mode was ductile tear. In the transition-temperature region, a gradual change in fracture mode from cleavage to ductile tear occurred at the tip of the fatigue crack in the K_Ic specimens.Scanning electron micrographs of the fracture-initiation region in the plane-strain tensile-ductility specimens showed that the increase in plane-strain tensile ductility with increasing temperature for steels ranging in yield strength from 80 to 250 ksi was accompanied by a change in the microscopic mode of fracture. The change in the microscopic mode of fracture in the plane-strain tensile-ductility specimens was similar to the change observed in the crack-initiation region in the K_Ic specimens. That is, the microscopic mode of fracture in the plane-strain tensile-ductility specimens gradually changed from cleavage at cryogenic temperatures to ductile tear at room temperature. Thus, it is suggested that the increase in K_Ic with increasing temperature is caused by an increase in the plane-strain tensile ductility with increasing temperature and that this increase in ductility is related to a change in the microscopic mode of fracture from cleavage to ductile tear.     

大初始缝高比混凝土试件双K断裂参数的试验研究

双K 断裂准则能够定量描述混凝土裂缝的起裂、稳定扩展和失稳断裂。该文采用混凝土三点弯曲梁试件,通过在试验中测得的起裂荷载P_ini、最大荷载P_max及临界裂缝口张开位移CMOD_C计算了初始缝高比为0.3~0.9共7组试件的起裂断裂韧度K_ICini 和失稳断裂韧度K_ICun 。结果表明,当初始缝高比为0.3~0.7时,混凝土裂缝扩展经历起裂、稳定扩展和失稳破坏3 个阶段,双K 参数均是与初始缝高比无关的材料参数;当初始缝高比大于或等于0.8 时,混凝土裂缝起裂后便进入失稳扩展阶段,起裂荷载即为最大荷载,且计算得到的K_ICini 仍与初始缝高比无关。因此,在确定K_ICini 时,仅需测得初始缝高比大于或等于0.8试件的P_max,将P_max作为P_ini直接计算得到K_ICini。同以往的试验方法相比,其结果更为准确且试验方法简单。     

混凝土软化本构关系对双K断裂参数的影响

混凝土裂缝扩展的双K 断裂准则，用于描述混凝土结构裂缝的起裂、稳定扩展和失稳断裂。其相 应的双K 断裂参数ini c K I 、un c K I 可通过简便的实验和基于虚拟裂缝扩展粘聚力的解析方法确定。已取得的 研究结果表明，起裂韧度ini c K I 的解析解与所确定的混凝土软化本构关系有关。对于简单实用的双线性软化 本构曲线而言，参数s s 、s w 、0 w 的取值直接影响软化本构关系的确定。根据本课题组新近所做楔入劈 拉实验，经分析对比得到了与实测结果相符合、为确定双K 断裂参数所使用的双线性软化本构曲线的相应 参数。