The equivalent elastic crack: 2. X-Y equivalences and asymptotic analysis

The use of crack growth resistance curves (R-a) to predict the behaviour of cracked specimens is a well-established practice for ceramics and materials. In Part 1 of this work, the authors showed that the use of R-a curves may imply a certain elastic equivalence between an actual specimen governed by a cohesive crack model and a virtual equivalent linear elastic specimen. Part 1 included the analysis of certain classes of equivalences (P-Y equivalences) where the loads acting on the actual and on the equivalent specimen were forced to be equal. This paper analyzes more general equivalences in which two arbitrary variables X and Y are forced to be identical in the actual and in the equivalent specimens. In particular, the J-CTOD equivalence and the size-effect-based equivalence put forward by Bazant are analyzed. The first part of the paper deals with the bases and general applicability of these equivalences. The second part presents the results of asymptotic analyses intended to assess the applicability of the equivalences to specimens of relatively large size.     

A simple model to explain the geometry dependence of J−Δa-curves

Very different patterns of geometry effects may occur if in a bend type specimen the ligament length is increased at constant specimen thickness. These geometry effects can be explained by a model of the total crack growth resistance R which is defined as the sum of the non-reversible energy which must be put into a body to produce an increment of crack area. The relationship between R and the commonly used J–a-curves is worked out. The model shows that a wider-lower pattern, i.e. the case where an increase of the ligament length causes a flatter J–a-curve, appears for fracture under lefm conditions or for contained yielding where R is independent of the geometry. A wider-no effect or a wider-higher pattern may be observed if the crack extension occurs at the plastic limit load and if either plane strain or plane stress conditions prevail.     

On mode I and mode II energy release rates of an interface crack

The opening (mode I) and sliding (mode II) components of the energy that is released during an incremental extension of an interface crack between two different elastic materials are evaluated by the Irwin's crack closure method. Each component of the energies (G _I and G _II ) is expressed in terms of the functions of the length of the incremental crack extension (a) and the real and imaginary part of the complex stress intensity factor defined by Malyshev and Salganik. It is found that values of G _I /a and G _II /a oscillate violently when a approaches zero and that, hence, in contrast with the case for homogeneous materials, each energy release rate should be defined as G _I /a and G _II /a for an actual crack growth step size.     

On the driving force for fatigue crack formation from inclusions and voids in a cast A356 aluminum alloy

Monotonic and cyclic finite element simulations are conducted on linear-elastic inclusions and voids embedded in an elasto-plastic matrix material. The elasto-plastic material is modeled with both kinematic and isotropic hardening laws cast in a hardening minus recovery format. Three loading amplitudes (/2=0.10%, 0.15, 0.20%) and three load ratios (R=–1, 0, 0.5) are considered. From a continuum standpoint, the primary driving force for fatigue crack formation is assumed to be the local maximum plastic shear strain range, _max, with respect to all possible shear strain planes. For certain inhomogeneities, the _max was as high as ten times the far field strains. Bonded inclusions have _max values two orders of magnitude smaller than voids, cracked, or debonded inclusions. A cracked inclusion facilitates extremely large local stresses in the broken particle halves, which will invariably facilitate the debonding of a cracked particle. Based on these two observations, debonded inclusions and voids are asserted to be the critical inhomogeneities for fatigue crack formation. Furthermore, for voids and debonded inclusions, shape has a negligible effect on fatigue crack formation compared to other significant effects such as inhomogeneity size and reversed loading conditions (R ratio). Increasing the size of an inclusion by a factor of four increases _max by about a factor of two. At low R ratios (–1) equivalent sized voids and debonded inclusions have comparable _max values. At higher R ratios (0, 0.5) debonded inclusions have _max values twice that of voids.     

The unzipping model of fatigue crack growth and its application to a two-phase steel

The unzipping analysis, based on the alternate shear deformation process of two intersecting shear planes at a crack tip, is extended to study fatigue crack growth in a two-phase martensitic-ferritic steel. The unzipping crack increment a _uz is directly related to K and J in the case of small scale yielding. It is preferrable to use a _uz is directly related to K and J in the case of small scale yielding. It is preferable to use a _uz as a physical parameter to correlate with the growth rates of micro-cracks and fatigue cracks in a multi-phase material. In the case of micro-cracks, K is often not applicable because of extensive plastic deformation; and in the case of multi-phase material, neither K nor J is applicable because of material inhomogeneity. The effective K, K _eff, is defined in terms of a _uz. The relations between the endurance limit of a two-phase steel and crack nucleus size, ferrite layer thickness, the constraint by the strong martensite on crack tip deformation in the ferrite domain, and K _th's of the martensite and ferrite are analyzed.

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Résumé Une analyse de rupture progressive et continue des liaisons, basée sur un processus de déformation de cisaillement alterné de deux plants de cisaillement s'intersectant à l'extrémité d'une fissure, a été étendue à l'analyse de la propagation des fissures de fatigue dans un acier martensito-ferritique à deux phases. L'accroissement de la fissure a est directement en relation avec K et J dans le cas de déformation plastique de faible étendue. II est préférable d'utiliser a comme paramètre physique en corrélation avec les vitesses de croissance de microfissures et des fissures de fatigue dans un matériau à phases multiples. Dans le cas de microfissures, K n'est souvent pas applicable en raison de la déformation plastique importante. Dans le cas de matériau multiphase ni K ni J ne sont applicables en raison de l'inhomogénéité du matériau. La valeur effective K _eff est définie en terme de a. Les relations entre la limite d'endurance d'un acier à deux phases et la taille du nodule de fissuration, l'épaisseur de la couche de ferrite, la contrainte qu'exerce une zone martensitique dure sur le domaine ferritique, sur la déformation à l'extrémité de la fissure en domaine ferritique, et les valeurs de K de la martensite et de la ferrite sont analysées.

     

The use of linear elastic fracture mechanics for particulate solids

The fracture of model porous particulate solids consisting of sand and low volume fractions of a polymeric binder has been investigated. In order to apply a linear elastic fracture mechanics analysis it was necessary to use an effective crack length comprising the sum of the notch depth and a quantity a. The behaviour of the model solids was intermediate between that exhibited by ceramics and polymers. A possible interpretation of a is that it represents a process zone size where the energy dissipation mechanism could involve either microcracking or yielding of the interparticle junctions in the zone. The former would correspond to ceramic-like behaviour while the latter is characteristic of polymers.     

Determination of da/dN-ΔK 1 curves for small cracks in alumina in alternating bending tests

Crack-growth relations under cyclic fatigue conditions are mostly determined for long cracks. In order to determine da/dN-K curves for small cracks from lifetimes under cyclic load a procedure has been derived which is based on a method usually applied to subcritical crack growth. To prove the cyclic effect and to demonstrate the procedure in detail, measurements were carried out on an Al₂O₃-ceramic in bending with anR-ratio ofR=–1 and two types of relatively small cracks, namely natural cracks and Knoop-cracks. It was found that both crack types exhibit the same da/dN-K relation. The exponent of the Paris law for fatigue crack growth is significantly different from the exponent of the power law for subcritical crack growth.     

The significance of RPG load for fatigue crack propagation and the development of a compliance measuring system

In can be postulated that fatigue crack does not grow if no damage occurs in the vicinity of the crack tip. Damage may occur beyond the Re-tensile Plastic zone's Generated load (RPG load) in the vicinity of a crack tip under loading process. We propose an effective stress intensity factor range ( K _RP) corresponding to the period in which the re-tensile plastic zone appears, in place of K _eff proposed by Elber [1], for a fatigue crack propagation parameter.We then consider the small change of compliance for a cracked body under cyclic loading, for the purpose of measuring RPG load as well as crack opening load and crack closing load. Moreover a subtraction circuit which can measure the small change of compliance during fatigue test is developed and an automatic controlled system which can control the adequate values of resistance in the circuit and the output voltage range from strain amplifiers for minimizing relative noise level is also developed. Then fatigue crack propagation tests of CT specimens were carried out with various stress ratios of constant amplitude loadings. Moreover K _th tests with the conditions of constant stress ratio and constant maximum load with increasing stepwise minimum load were also carried out. It becomes clear that the logarithmic curve of K _RP—crack propagation rate appears to be linear in a wide range from the region of very slow growth rate to the region of stable growth rate. On the other hand, threshold phenomenon appears only circumstantially due to the particular loading pattern on K _eff based on the crack opening load and K _eff ^cl based on the crack closing load. Moreover K _RP gives the quantitative effect of stress ratio on fatigue crack propagation rate.     

An investigation of the micromechanisms of fatigue crack growth in structural gas turbine engine alloys

This paper presents an overview of fatigue fracture modes in selected structural alloys employed in gas turbine engines. These include the mechanisms of fatigue crack growth in the near-threshold, Paris and high-K regimes obtained from Ti-6Al-4V, Inconel 718 and PWA 1472 (a single crystal nickel-based superalloy of similar chemical composition to Inconel 718). Fatigue fracture modes in these materials are shown to be strong functions of the stress intensity factor range, K, and the maximum stress intensity factor, K _max. Fatigue mechanism maps are also presented to show the parametric ranges of K and K _max corresponding to the different fatigue fracture modes.     

Fatigue crack closure over the range of stress ratios from −1 to 0.8 down to stress intensity threshold level in HT80 steel and SUS304 stainless steel

The closure phenomena of fatigue cracks were investigated with a 1 mm gage length extensometer over the range of stress ratio, R, from –1 to 0.8. Plate specimens with a center slot of HT80 steel and SUS304 stainless steel were fatigued under push-pull loading, and the crack propagation rate, da/dn, was measured. The stress ratio, R, was found to influence da/dn in both materials. The crack opening stress intensity factor, K _op, was determined from the relationship between the crack tip extensometer displacement and the load. The effective stress intensity range ratio, U(=K_eff/K), decreases with the decrease of the stress intensity amplitude, K/2. As for the data which show the crack closure phenomena (R0.4), the relationship between log(da/dn) and log(K _eff/2) falls on a straight line near the stress intensity threshold level. For R=0.8 where the crack tip is fully open over the whole range of loading, the data show a discrepancy from the same line. The strain at the crack tip was also measured with the Moiré fringe multiplication method. A large amount of plastic strain at the crack tip was observed even below the crack opening load for R=–1 in HT80 steel. These phenomena show that fatigue damage still exists when the crack is closed. These also show that the crack closure cannot fully account for the effect of R on da/dn.

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Résumé Les phénomènes de fermeture de fissures de fatigue ont été étudiés à l'aide d'un extensomètre d'une longueur de référence d'un millimètre sur une étendue des rapports de contrainte R variant de –1 à +0,8. Les éprouvettes plates comportant un trou central oblong, étaient en acier HT80 et en acier inoxydable SUS304. Elles ont été soumises à fatigue sous des conditions de traction-compression, et la vitesse de propagation de la fissure, da/dn, a été mesurée. Le rapport des tensions R est apparu avoir une influence sur da/dn, dans le cas des deux matériaux. Le facteur d'intensité des contraintes correspondant à l'ouverture de la fissure K _op a été déterminé à partir de la relation entre le déplacement de l'extensomètre à l'extrémité de la fissure et la charge. Le rapport effectif de la variation de l'intensité de la contrainte U(=K_eff/K) décroit lorsque décroit l'amplitude de l'intensité de contrainte K/2. En ce qui concerne les données montrant le phénomène de fermeture de la fissure (pour R0,4) la relation entre log(da/dn) et log(K _eff/2) tombe sur une ligne droite au voisinage du niveau de seuil de l'intensité des contraintes. Pour R=0,8 qui correspond à une extrémité d'entaille complètement ouverte quelle que soit la charge, les données montrent une déviation par rapport à cette ligne. La déformation à l'extrémité de la fissure a également été mesurée à l'aide d'une méthode de multiplication des franges de Moiré. On a observé une grande quantité de déformations plastiques à l'extrémité de la fissure, même en dessous de la charge d'ouverture de la fissure correspondant à R=–1 pour l'acier HT80. Ces phénomènes montrent que le dommage par fatigue existe encore lorsque la fissure est refermée. Ils montrent également que la fermeture de la fissure ne peut pas tenir complètement compte de l'effet de R sur da/dn.

     

The anomalous thermal boundary resistance in superfluid4He near Tλ

The anomalous thermal boundary resistance R of superfluid⁴He near the lambda point T was studied in a cell with parallel copper faces and with various fluid layer thicknesses. The study was made as a function of the heat current Q and reduced temperature (T — T)/T = . In all cases, R tends to a maximum value R_itmax at T = T. This value is independent of Q, and is reproducible for various experiments. This contrasts with the regular boundary resistance which can vary considerably between successive experiments. Near T, the limiting slope dR/d¦¦ is found to be proportional to Q^–1, and this leads to a scaled representation of the data. This analysis is extended to data of similar experiments with gold plated copper surfaces by Duncan and Ahlers and by Zhong et al. The measurements of R over the whole range of¦¦ where it is observable are discussed and compared with previous experiments.     

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

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

Subgap peak and Tomash-McMillan-Anderson oscillations in the density of states of SNS Bridges

The quasiparticle spectra and the densities of states of superconducting-normal-superconducting junctions are computed from the WKBJ transformed Bogoliubov-de Gennes Equations (BdGE), which are solved by Picard iteration and numerical integration. It is shown that the influence of the proximity effect on the bound states can be modeled by a rectangular pair potential well of effective normal layer thickness 2a*= _–L ^L [1–(z)/]dz, where (z) is the pair potential of the junction, is its asymptotic constant value, and 2L is the total length of the sample. The density of states exhibits a subgap peak at energies less than besides the BCS peak atE=; forE> there are geometrical resonances which are due to electron-hole interferences in finiteS layers of thicknessL-a*.     

Superconductivity of small metallic particles. I

The ground state energy and orbital magnetic susceptibility of a small-particle superconductor are calculated with the use of perturbation theory, which is expected to give correct results in the limit of small particle size, i.e., when the average one-electron energy level spacing is much larger than the energy gap of the bulk metal. The method is applicable up to particles of intermediate size for which / = 1/4–1/2 (e.g., aluminum particles of radius 25–30 Å). The magnetic susceptibility does not undergo any drastic change in the region ~ when the particle size is varied, and quantum size effects do not show up in the susceptibility.Supported by a Grant-in-aid from the Japanese Ministry of Education.     

Thermodynamic properties of liquid3He-4He mixtures near the tricritical point. II. Data analysis by the scaling-field method

The scaling theory for tricritical phenomena by Riedel is applied to the analysis of thermodynamic properties of liquid³He-⁴He mixtures near the tricritical point. Within this theory experimental data for the phase diagram, the³He molar concentrationX, and the concentration susceptibility (X/) _T are discussed in terms of two scaling fields that are functions of the temperatureT and the difference = ₃ – ₄ of the chemical potentials of the two helium isotopes. The quantitiesX and (X/) _T in terms of thefields T and as independent variables are obtained for the intervals –0.1<T – T _t<0.53 K and –9< – _t <0.8 J/mole, from vapor pressure and calorimetric data described in a previous paper by Goellner, Behringer, and Meyer. The transformed data are analyzed to yield the tricritical exponents, amplitudes, scaling fields, and scaling functions. The values of the tricritical exponents are found to agree with those predicted by the renormalization-group theory of Riedel and Wegner. (Logarithmic corrections are beyond the precision of the present experiment.) Relations between amplitudes are derived and tested experimentally. The (linear) scaling fields are determined by using their relationship to geometrical features of the phase diagram. The data forX and (X/) _T are found to scale in terms of these generalized scaling variables. The sizes of the tricritical scaling regions in the normal and superfluid phases are estimated; the range of apparent tricritical scaling is found to be appreciably larger in the normal-fluid phase than in the superfluid phase. The tricritical scaling function for the concentration susceptibility is compared with the analogous scaling function for the compressibility of pure³He near thecritical gas—liquid phase transition. Finally, when the critical line near the tricritical point is approached along a path of constant < _t , the experimental data are found to exhibit the onset of the crossover from tricritical to critical behavior in qualitative agreement with crossover scaling.Work supported in part by the National Science Foundation through Grant No. GH-36882 and Grant No. GH-32007, and by a grant of the Army Research Office (Durham).     

Changes in surface characteristics of dental resin composites after polishing

The objectives of this study were (1) to determine in vitro changes in surface roughness and color of dental resin composites after application of three finishing and polishing systems; (2) to evaluate the difference in color stability after immersion in a dye solution after polishing; and (3) to evaluate the effects of surface condition, especially roughness, on measured color depending on the color measuring geometries of specular component excluded (SCE) and specular component included (SCI). Color and surface roughness (R_a) of resin composites of four brands of A2 shade and one brand of Yellow Enamel shade were measured after polymerization, after polishing with Enhance (Dentsply), Sof-Lex (3M ESPE), or Super-Snap (Shofu) composite finishing and polishing systems. Color was also measured after immersion in 2% methylene blue solution. Color was measured according to the CIELAB color scale. Color changes (E*_ab) after polishing/staining and by the measuring geometry were calculated by the equation; E*_ab = [(L*)² + (a*)² + (b*)²]^1/2. Ra value was measured with a surface roughness tester. E*_ab and L* values after polishing and after staining varied among polishing systems when measured with SCE geometry. Composites polished with Super-Snap and Sof-Lex systems showed higher E*_ab and L* values than those polished with Enhance polishing system with SCE geometry. E*_ab and L* values between specimens with different surface conditions measured with SCE geometry were significantly higher than those with SCI (p < 0.01). Changes in R_a value after polishing was insignificant in most cases.     

Influence of nonmetallic inclusions on fatigue crack growth in a structural steel

A study of the fatigue behaviour of a hardened and tempered steel, at two inclusion levels, has been carried out according to the linear elastic fracture mechanics criteria. The influence of inclusions on the fatigue crack growth rate has turned out to be a function of the local stress intensity factor range,K _I, at which fracture propagates. At lowK _I values, to which are related crack growth rates less than 10^–5 mm cycle^–1, the crack growth rate in the steel with higher inclusion content is lower than in the steel with lower inclusion content. AsK _I increases, an inversion in the difference between the two rates occurs. In the dirtier steel, the higherK _I, the higher the growth rate than in the other steel. The difference between the two rates becomes nil just below the fast propagationK _Ic level. By fractographic analysis, it has been possible to find out how inclusions affect fatigue behaviour.     

General features of fatigue fracture diagrams of metals

In this paper the results of investigations of fatigue fracture diagrams (v-K diagrams) of materials or products (depicting the relationship between the fatigue crack growth rate, v, and stress intensity factor range in a cycle, K) performed in the Physico-mechanical Institute are summarized. Typical diagrams are described and their main features in normalized coordinates are clearly demonstrated. On this basis the defining parameters of crack growth rate curves (v-K curves), K_th, K_fc, m, and K^*, characterizing the crack extension resistance of material are validated. Convenient expressions for analytical approximating v-K curves by splines are proposed. The main observed deviations of v-K curves from typical ones are shown.Paper presented at the XIIth International Colloquium on Mechanical Fatigue (Miskolc, Hungary, 10–12 March, 1994).Published in Problemy Prochnosti, No. 1, pp. 30–35, January, 1996.