Fracture toughness characterization of several aluminum alloys in semi-brittle fracture

A method has recently been developed for determining a nonlinear fracture toughness parameter defined by the relation $\text{G}\text{?}{}_{\mathrm{c}}\text{=}\text{C}\text{?}\text{G}{}_{\mathrm{c}}$ where G_c is the critical elastic strain energy rate as defined by Irwin. The $\text{C}\text{?}$ term is a function of the nonlinearity of the load-displacement test record and has been evaluated using the three parameter Ramberg-Osgood approach, although other curve fitting techniques could be applied as well. The method is quite straightforward and is applicable to plane stress, plane strain and mixed mode testing although only plane stress conditions are considered in this paper. For the case of a linear load-displacement record $\text{C}\text{?}\text{→ 1}$ and $\text{G}\text{?}{}_{\mathrm{c}}$ reduces to the linear elastic result.The toughness parameter $\text{G}\text{?}{}_{\mathrm{c}}$ has been evaluated for a number of high strength aluminum alloys and compared with published G_c values for these materials. The tests were conducted on center-cracked sheets of 2014-T6, 2024-T81, 7075-T6 and 7475-T61 aluminum alloys under conditions of varying specimen geometry and displacement gage length. It was found that the values of $\text{G}\text{?}{}_{\mathrm{c}}$ obtained from displacement readings with a gage length of 2 in. generally agreed with published values of $\text{G}{}_{\mathrm{c}}\text{=}\text{K}{}_{\mathrm{c}}{}^2\text{E}$. The $\text{G}\text{?}{}_{\mathrm{c}}$ values were found to vary inversely with gage length and a/w ratios. The variation in values for $\text{G}\text{?}{}_{\mathrm{c}}$ is of the same order of magnitude as the scatter in published values for G_c. However, $\text{G}\text{?}{}_{\mathrm{c}}$ appears to be less sensitive than G_c to changes in a/w.     

G̃c and R-curve fracture toughness values for aluminum alloys under plane stress conditions

The effect of specimen geometry and subcritical crack growth on the nonlinear energy fracture toughness, $\text{G}\text{?}{}_{\mathrm{c}}$, has been examined for thin, center-cracked sheets of 2024-T3 and 7075-T6 aluminum alloys. The procedure followed was to independently vary the specimen length, L, width, w, andd crack length-to-specimen width ratio and to determine the toughness both at the onset of subcritical crack growth and at the initiation of unstable fracture. Comparisons were also made with the R-curve toughness, G_R, evaluated at unstable fracture from which it was found that both $\text{G}\text{?}{}_{\mathrm{c}}$ and G_R displayed the same trend of change with geometrical variables, with $\text{G}\text{?}{}_{\mathrm{c}}$ consistently higher than G_R. When the nonlinear energy fracture toughness was evaluated at the onset of subcritical crack growth, it was found that the geometry dependence essentially disappeared.Scanning electron microscopic examination of some typical fracture surfaces showed that stable crack growth was accompanied by a gradual change of fracture mode from plane strain to plane stress. An analysis of possible errors in the experimental procedure showed that the scatter observed in $\text{G}\text{?}{}_{\mathrm{c}}$ values was not due to experimental errors, but apparently due to inhomogeneities in the materials. Several techniques were also introduced for the purpose of more directly incorporating crack growth into the $\text{G}\text{?}{}_{\mathrm{c}}$ determination, but it was found that they did not cause significant variation in the toughness values.     

A comparison of the geometry dependence of several nonlinear fracture toughness parameters

A number of fracture toughness tests on compact tension specimens have been performed for the purpose of comparing several nonlinear fracture toughness methods; including the nonlinear energy ($\text{G}\text{?}{}_{\mathrm{I}}$), J-integral ($\text{J}{}_{\mathrm{I}}$), COD ($\text{G}{}_{\mathrm{\delta }}$), and linear (–$\text{G}{}_{\mathrm{I}}$) approaches. The effect of variations in specimen thickness ($\text{B}$) and width ($\text{w}$) on the fracture toughness was examined for 7075-T651, 2124-T851, 2048-T35I, and 2048-T851 aluminum alloys, Ti-6Al-4V, and 4340 steel. Fracture toughness values were evaluated at both the initiation of stable crack growth and the onset of unstable fracture (peak load).It was found that the peak load toughness values are quite geometry sensitive at thicknesses below the requirement for plane strain fracture. At the initiation of stable crack growth, the toughness values are constant over a much larger range of specimen thickness. However, the nonlinearity of the load displacement curve is quite limited at this point and the associated fracture toughness is only 30–50% of the peak-load values.     

Fracture of round bars loaded in mode III and a procedure for KIIIc determination

The available literature reports few studies on Mode $\text{III}$ fracture. In fact, even though the Mode $\text{III}$ fracture toughness, $\text{K}{}_{\mathrm{IIIc}}$ is an important factor in design and analysis of fractures under torsional loading, the procedure for the measurement of $\text{K}{}_{\mathrm{III}}$ is yet to be standardized.In this paper, the results of a simple analysis, which determines the displacement produced by the size dependent plasticity and the growth of crack in a circumferentially cracked round bar subjected to Mode $\text{III}$ loading, are presented. It is shown that plastic zone size decreases as bar diameter increases. This implies that whereas a small diameter bar can fail in a ductile manner, a large diameter bar may undergo a predominantly linear, elastic and brittle fracture even though the material is ductile.The analytical results are verified by experimental measurements on circumferentially cracked specimens loaded in Mode $\text{III}$. Based on this, one can determine the limiting value of the bar diameter, which must be tested to obtain a $\text{K}{}_{\mathrm{IIIc}}$ in a predominantly linear elastic condition. Further, a new and simple procedure for the determination of $\text{K}{}_{\mathrm{IIIc}}$ is proposed. This procedure enables one to determine the stress intensity factor at which a crack starts extension in the specimen under Mode $\text{III}$ loading.     

A study of the size-effect on plastic energy dissipation and toughness in 2024-T3 aluminum alloy sheets

The plastic energy dissipation before crack growth initiation and during stable crack growth was determined in centercracked thin sheet specimens of 2024-T3 aluminum alloy with different width and crack length-to-width ratios. The plastic energy dissipation rate versus stable crack growth curve was found to be approximately linear, but the slope decreased considerably with increase in crack length. No correlation was observed between plastic energy dissipation rate and the linear toughness ($\text{G}\text{?}{}_{\mathrm{c}}$), the nonlinear energy toughness ($\text{G}\text{?}{}_{\mathrm{c}}$) or the $\text{R-}\text{curve}$ toughness ($\text{G}{}_{\mathrm{R}}$). The role of net section yielding on the decrease in stable crack growth and toughness values in small specimens is discussed.     

On fracture toughness in the nonlinear range

A general definition of fracture toughness, designated by $\text{G}\text{?}{}_{\mathrm{c}}$, is developed which is appropriate to situations of subcritical crack growth and/or large-scale crack border plastic yield. The theoretical basis as well as comparisons with other proposed measures of fracture toughness are also discussed. A simple method is given for evaluating $\text{G}\text{?}{}_{\mathrm{c}}$ which is based on use of the load-displacement test record.     

Fracture toughness of H-11 steel and its susceptibility to stress corrosion cracking

The plane strain fracture toughness, $\text{K}{}_{\mathrm{IC}}$, was measured for H-11 steel using compact tension specimens after two separate heat treatments; one to provide good mechanical properties for normal engineering applications. Rc 46, 195 ksi yield strength, and the ofter heat treatment to provide maximum strength, Rc 56, 245 ksi yield strength. The Rc 46 samples exhibited no subcritical crack growth in either steam distilled water or hydraulic oil at $\text{K}{}_{\mathrm{I}}$ levels approaching 95% of the specimens' $\text{K}{}_{\mathrm{IC}}$ with times exceeding 20 hr. The Rc 56 samples under constant displacement loading exhibited subcritical crack growth in steam distilled water with a measured $\text{K}{}_{\mathrm{Iacc}}$ of 17.8 ksi √(in). The specimens were subjected to the environment just prior to loading, subcritical crack growth commenced without an incubation period, and both the $\text{K}{}_{\mathrm{I}}$ and crack growth increased with time. No subcritical crack growth was encountered in hydraulic oil at $\text{K}{}_{\mathrm{I}}$ levels approaching 98% of the specimens' $\text{K}{}_{\mathrm{IC}}$ with times exceeding 24 hr.     

Fracture toughness and fracture energy measurements on aluminium alloys

Methods of conducting and analysing instrumented Charpy impact tests have been discussed and applied in measuring the initiation fracture toughness, $\text{K}{}_{\mathrm{Ic}}$, of two precipitation hardened Al-alloys.For full speed impact tests a method for indirectly deriving fracture load from “system stiffness” and “time to fracture” has been found to be the most suitable. In the lower speed impact tests measured fracture load has been used directly to calculate $\text{K}{}_{\mathrm{Ic}}$ In these tests an energy method superficially resembling “specific surface energy” has also been used to calculate $\text{K}{}_{\mathrm{Ic}}$.     

A modified thickness criterion for fracture toughness testing

A modified criterion is developed on an empirical basis for the minimum thickness $\text{B}{}_{\min }$ of a plane strain fracture toughness test specimen: $\text{B}{}_{\mathrm{<mtext>min</mtext>}}\text{= 400}\text{K}{}_{\mathrm{Ic}}{}^2\text{Eδ}{}_{\mathrm{Y}}$ where $\text{K}{}_{\mathrm{Ic}}$ is the plane strain fracture toughness, $\text{E}$ is the Young's modulus and δ_Y is the yield stress of the material. The modified criterion is tested alongside the ASTM thickness criterion against published data on the variation of $\text{K}{}_{\mathrm{c}}$ with thickness, and shows significantly the better agreement with observed values of $\text{B}{}_{\mathrm{<mtext>min</mtext>}}$ for a wide range of materials.An attempt has been made to rationalise this criterion. The expression is considered to take into account two major factors which determine $\text{B}{}_{\mathrm{<mtext>min</mtext>}}$, the attainment of plane strain in the specimen interior ahead of the crack tip, and the role of microstructure in determining how far the quasi-plane strain fracture (square fracture) extends beyond the region of true plane strain.     

Incipient fracture angle,fracture loci and critical stress for mixed mode loading

A prediction of the direction of incipient crack growth in brittle-like materials and the associated fracture loci under mixed mode loading is proposed. It is postulated that the direction of unstable crack propagation is determined by the “weakest” near-tip element defined as the one which would relax maximum potential energy upon prospective crack extension. Starting from the energy rate principle of crack extension (Eshelby energy-momentum tensor and Rice $\text{J-}\text{internal}$ vector) it is deduced that a crack will extent in the direction along which the following stress criterion is satisfied, $\text{(δ}{}_{\mathrm{\theta \theta }}{}^2\text{? δ}{}_{\mathrm{rr}}{}^2\text{) →}\text{maximum (for}\text{δ}{}_{\mathrm{\theta \theta \; >\; 0)}}$ The fracture angle in pure Mode II (70.4° away from the original straight path) is shown to be unstable in the sense that any slight tension along the crack (non-singular at the crack tip) affects considerably (up to 22%) the directionality of crack extension. It appears to be sensitive to the extent of the near-tip zone ($\text{r}{}_0$) in which linear elasticity does not hold and the non-singular stress term (squared).The fracture loci in mixed mode loading (generated by projecting the $\text{J-}\text{integral}$ vector along the prospective fracture path and letting this scalar function attain a critical value) is quadratic in $\text{K}{}_1$ and $\text{K}{}_2$ with an interactive cross product term $\text{K}{}_1\text{× K}{}_2$.The suggested criterion with its implication in predicting critical fracture load, exhibits behavior which is consistent with experimental observations collected from several sources. The common and uncommon features with respect to other known criteria are compared and discussed.     

Criterion of crack initiation and spreading

The area of the true stress/true strain diagram determined under tensile test conditions is equal to the energy absorbed per unit volume at the point of fracture. In the case of tensile as well as other loading tests like compression, low cycle fatigue, etc. a spreading crack will be initiated when the critical specific fracture energy ($\text{W}{}_{\mathrm{c}}$) characteristic of the material has been absorbed whereby the proportionality law of notched specimens can then be derived. The specific fracture energy of notched tensile test specimens as expressed in the function of temperature will describe the brittle fracture sensitivity of the material. From the specific fracture energy of notched specimens the fracture toughness and the critical energy release rate can be determined. The method described herein is also adaptable for the determination of the brittle fracture sensitivity of welded joints.     

A correlation between dynamic initiation toughness and crack arrest toughness

Dynamic initiation toughness values were obtained from testing small 3-point bend specimens at ?50, ?25, 0, +23°C. Two specimen orientations were tested which showed no marked difference in critical dynamic initiation toughness. The obtained $\text{K}{}_{\mathrm{Id}}$ data were correlated with the crack arrest toughness $\text{K}{}_{\mathrm{Im}}$ and $\text{K}{}_{\mathrm{Ia}}$. The value of $\text{K}{}_{\mathrm{Id}}$ is in the range 0.93 to 1.29 times the corresponding fast fracture toughness, $\text{k}{}_{\mathrm{im}}$ while the ratio $\text{K}{}_{\mathrm{Id}}\text{/K}{}_{\mathrm{Ia}}$ varies between 1.27 and 1.60.     

The condition of fatigue crack growth in mixed mode condition

The condition of the initiation of fatigue crack growth in mixed mode conditions has been investigated by using precracked low carbon steel specimens.It is pointed out that, firstly, the critical condition of crack growth should be defined with regard to the modes of fatigue crack growth, i.e. shear mode and tensile mode. Secondly, it is proposed that the critical condition of fatigue crack growth is given by the local tensile stress and shearing stress at the notch tip determined by stress intensity factors $\text{K}{}_{\mathrm{I}}$ and $\text{K}{}_{\mathrm{II}}$, and that this criterion is generally applicable to in-plane-loading conditions, i.e. Mode $\text{I}$, Mode $\text{II}$ and Mixed Mode conditions.     

An experimental study of crack tip plastic flow in mild steel

The paper deals with an experimental study on the nature of plastic flow at the root of a crack in mild steel beams, for the non-valid $\text{K}{}_{\mathrm{IC}}$ test regime, under three point hending loads. Photoelastic coating technique has been used to measure the plasticity spread ahead of the tip in relation to the load-COD record. It is observed that in all cases there is a sudden increase in specimen compliance near the maximum linear load due to an abrupt increase in plastic zone size on some preferential planes ahead of the crack tip. This abrupt increase in plastic flow was seen to occur along the 45° planes (with respect to the plane of the crack) for thicker beams and/or with longer cracks. In contrast, the plastic zone extended more on the plane of the crack for thin section beams with relatively shorter cracks. The stress intensity factor required to cause this sudden loss of resistance to localized deformation is found to be remaining constant beyond a certain crack length for a given specimen thickness. These observations suggest that a critical stress intensity factor () concept can be introduced to describe the abrupt flow localization ahead of the crack tip. This () can be taken as a new parameter in addition to those commonly used in characterising the overall “fracture” behaviour of large scale yielding materials like mild steel, especially in the non-valid $\text{K}{}_{\mathrm{IC}}$ test regime.     

Kinetics of subcritical cracking under the chloride and sulphide environments in a low alloy steel

The effects of two aqueous environments, namely chloride and sulphide have been investigated using fracture mechanics approaches in a Ni-Cr-Mo alloy, tempered between 200–600°C temperatures after quenching. The experimental investigation included tensile and fracture toughness tests in the ambient condition, environmental tests to determine the threshold, K_ISCC and the crack growth rate values $\text{da}\text{dt}$ and fracture surface studies. An attempt has been made to substantiate the role of microstructure and the source of hydrogen on the susceptibility to failure by computing $\text{C}{}_{\mathrm{c}}\text{C}{}_{\mathrm{o}}$ ratios for the hydrogen induced cracking process. A crack growth rate expression of the type, $\text{da}\text{dt}\text{= c'(K)}{}^{\mathrm{n}}$ is proposed for Stages I and II to account for the discrepancy between the theoretically calculated and the experimental $\text{da}\text{dt}$ data. The experimental values of the constants c' and n are determined. For all the tempering conditions investigated, the H₂S environment appears to be more hostile than the NaCl medium. However, the susceptibility to both the environments is more pronounced for yield strength values greater than 1500 MPa. The $\text{K}{}_{\mathrm{If}}\text{K}{}_{\mathrm{IC}}$ ratio is bound to be less than 1 under the H₂S, and greater than 1 under the NaCl solution.     

Studies on crack growth rate under high temperature creep,fatigue and creep-fatigue interaction—I: On the experimental studies on crack growth rate as affected by aσg,σg and temperature

Many experimental studies have been reported on the measurements of crack growth rate and the observation of crack growth behaviour under high temperature creep, fatigue and creep-fatigue interaction in literatures. However, many of them have been done in air atmosphere. Furthermore, in many of them the measurements of the crack growth rate have been carried out by interrupting intermittently the running of the testing machine. In such experiments the complex effects due to the atmosphere, the interruption period and the corresponding unloading operation for the crack length measurement might have been involved.In the present paper in order to eliminate such effects, series of experimental studies on the crack growth behaviour under creep, fatigue and creep-fatigue interaction conditions on 304 stainless steel have been carried out by using high temperature microscope and observing the crack length continuously during running the test without interruption in vacuum of 10^?5mm Hg.Among the results, it was found that crack growth rates on a time basis, $\text{da/dt}$, under high temperature creep and creep-fatigue interaction conditions can not be described in terms of solely elastic stress intensity factor $\text{k}{}_{\mathrm{i}}$ or only net section stress σ_net, both independent of gross section stress σ_g. The relation between crack growth rate and stress intensity factor under high temperature fatigue condition changes with some trend according to gross section stress at lower $\text{K}{}_{\mathrm{I}}$ level and it can be approximately described in terms of stress intensity factor $\text{K}{}_{\mathrm{I}}$ only, at higher $\text{K}{}_{\mathrm{I}}$ level. The threshold stress intensity factor and the threshold net section stress under high temperature creep, fatigue and creep-fatigue interaction conditions appears to be almost independent of temperature.     

Third-order bounds on the effective bulk and shear modulus of a dispersion of fully penetrable spheres

We evaluate the third-order Beran-Molyneux bounds on the effective bulk modulus K_e and the third-order McCoy bounds on the effective shear modulus $\text{μ}{}_{\mathrm{e}}$ of a model of a two-phase composite in which one of the phases consists of spherical inclusions (or voids), with bulk and shear moduli, K₂ and $\text{μ}{}_2$, respectively, and volume fraction $\text{φ}{}_2$, dispersed randomly throughout a matrix phase, with bulk and shear moduli, K₁ and $\text{μ}{}_1$, respectively, and volume fraction $\text{φ}{}_1$. We tabulate the two fundamental microstructural parameters I₁ and L₁ required to evaluate the bounds, which depend upon the three-point matrix probability function of the model, for the aforementioned fully-penetrablesphere model. We compare the third-order bounds on K_e and $\text{μ}{}_{\mathrm{e}}$ to the second-order bounds due to Hashin and Shtrikman and to Walpole. We find that the third-order bounds for our model are always more restrictive than the corresponding second-order bounds. When the moduli of the phases differ by an order of magnitude, the third-order bounds are sharp enough to provide quantitatively useful estimates of K_e and $\text{μ}{}_{\mathrm{e}}$ for all $\text{φ}{}_2$. The third-order bounds are very restrictive at low $\text{φ}{}_2$ values (e.g., $\text{φ}{}_2\text{= 0.1}$) where they remain useful for cases in which the moduli of the phases differ by two orders of magnitude. Experimental values of $\text{μ}{}_{\mathrm{e}}$ measured by Corson for a tungstenlead composite are found to lie within the McCoy bounds for our model, with the lower bound giving a good estimate of the data.     

The use of small specimen strength ratio for measuring fracture toughness

The specimen strength ratio ($\text{R}{}_{\mathrm{s}}$), determined from small specimen tests was correlated with plane strain fracture toughness ($\text{K}{}_{\mathrm{Ic}}$) values for many heats of A533B-1 steel. A variety of loading rate and specimen size results suggest that $\text{K}{}_{\mathrm{Ic}}$ can be predicted from the small specimen strength ratio up to values of $\text{R}{}_{\mathrm{s}}$ near 2.0. Also, conservative estimates of cleavage-initiated, elastic-plastic fracture toughness can extend beyond $\text{R}{}_{\mathrm{s}}$ values of 2.0. The ASTM E399 size criterion appears to be too restrictive for the class of steel studied, and a more appropriate requirement would reduce the ASTM criterion by a factor of four.     

Crack arrest in steels

This paper examines 3 theories that have been used to characterize the arrest capabilities of steels and structures: (1) The static analysis, arrest toughness ($\text{K}{}_{\mathrm{Ia}}$) theory; (2) The dynamically loaded/stationary crack toughness ($\text{K}{}_{\mathrm{Id}}$) theory, and (3) The dynamic analysis, propagating crack energy or toughness ($\text{R}{}_{\mathrm{ID}}$ or $\text{K}{}_{\mathrm{ID}}$) theory. These three concepts are examined in the light of measurements of unstable fracture and crack arrest in wedge-loaded DCB test pieces together with a fully dynamic analysis of the experiments.     

Some problems in experimental fracture mechanics

A standard procedure for the determination of fracture toughness K_IC is discussed. The insufficiency of the existing K_ic determination confidence criteria is stressed and the following criteria are proposed instead: φ_max ? 1.5%; $\text{σ}{}_{\mathrm{fr}}{}^{\mathrm{net}}\text{σ}{}_{\mathrm{0.2\; ?\; 0.8}}$, in conjunction with the old criterion $\text{P}{}_{\max }\text{P}{}_{\mathrm{Q}}\text{? 1.1}$. Determination of K_IC from P_max should be used instead of from P_Q.A method for the determination of a point on the “force-displacement” diagram corresponding to crack growth initiation is set forth. The method is based on specimen compliance tests under repeated load-relief cycles. The crack growth initiation point is used to determine both the critical crack opening and plane strain fracture toughness. The indefinite effect of the growing crack (in the ease of crack opening or Cherepanov-Rice integral calculations) is thereby eliminated. Necessity is emphasized to determine the share of the J-integral which contributes to fracture process. A method for plotting the elastic displacement diagram is proposed which allows on the basis of preliminary estimates to determine fracture toughness of small-sized specimens without using special setups. The area ratio between the plastic and elastic strain diagrams is proposed to be adopted as fracture type criterion. Certain experiments to determine crack resistance of material specimens are described and discussed.