Deformation approach to the evaluation of the period of initiation and growth of fatigue macrocracks

We performed experimental investigation of the opening displacements of the contours of stress concentrators (notches and cracks) for various amplitudes of cyclic loading. On the basis of experimental results, we propose a new deformation parameter _t which is a function of the notch (crack) tip opening displacement , namely, _t /(+d*), where is the radius of the tip of the notch andd* is the characteristic size of the prefracture zone. It is shown that this parameter uniquely determines the number of cyclesN _l to the initiation of a fatigue macrocrack independently of the geometry of the specimens and stress concentrators in elastic and elastoplastic materials, i.e., the dependence of _t onN ₁ is a characteristic of the material. It is experimentally demonstrated that this dependence enables one to quantitatively describe the process of fatigue fracture both in the stage of initiation of macrocracks and their propagation.Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 5, pp. 7–21, September – October, 1995.     

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.

---

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.

     

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.     

The equivalent elastic crack: 1. Load-Y equivalences

The use of crack growth resistance curves (R–a) to predict the behaviour of cracked specimens is a well established practice for ceramic and cementitious materials. When the cohesive crack model can be applied to these materials it is shown that the use of R–a curves is equivalent to setting up an elastic equivalence that greatly simplifies the computations. This equivalence has its drawbacks and limitations. This paper analyzes a class of equivalences—the load (P) versus another variable—between cracked cohesive materials and linear elastic ones, ascertaining its applicability to the interpretations of fracture data.     

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.     

Electron fractography and fatigue crack propagation in 7075-T6 aluminum sheet

A centrally slotted thick sheet of 7075-T6 aluminum alloy was cyclically loaded. Striation spacings and crack propagation rates on the specimen surface were measured and compared and the fractographs were examined. The average striation spacing is found proportional to K ^1.8; while the surface crack propagation rate is proportional to K^{2 8}, where K is stress intensity factor range. Cleavage fractures of brittle particles appear to cause a difference between the overall surface crack propagation rate and striation spacings.In the lower K region, there are fewer cleavage fractures, the striations are more distinct and the directions of the striations do not deviate much from the normal to the direction of the overall crack propagation. In the higher K region, there are more cleavage fractures. In this region striations run in divergent directions and become less evident. The differences in fracture surface features result from the cleavage fracture of brittle particles.

---

Zusammenfassung Ein zentral geschlitztes Blech aus Aluminium-legierung wurde zyklischen Beanspruchungen unterworfen.Die Abstände zwischen den Verformungslinien sowie die Rißfortpflanzungsgeschwindigkeit an der Proben-oberfläche wurden gemessen und miteinander verglichen; die Mikrofiaktogramme wurden untersucht.Es ergab sick daß der mittlere Abstand zwischen zwei Verformungslinien K ^1.8 proportional ist, während die an der Oberfläche gemessene Rißfortpflanzungsgeschwindigkeit K ^2.8 proportional ist, wobei K der Schwankungsbereich den Span nungsintensitätsfaktors darstellt.Dieses unterschiedliche Verhalten wird durch das Auftreten von Spaltbruchen in den spröden Materialteilen erklärt. Im Bereich kleiner K-Werte gibt es nur wenig Spaltbrüche; die Verformungslinien sind gut ausgebildet and ihre Richtung weicht nur geringfügig von der Normalen zur allgemeinen RiBfortpflanzungsrichtung ab.Im Bereich hoher K-Werte sind die Spaltbrüche zahlreicher und die Verformungslinien entwickeln sich nach verschiedenen Richtungen ; sie Bind auch weniger ausgeprägt. Die Unterschiede in der Ausbildung der Bruchoberfläche stammen von den Spaltbruchen spröder Teilchen her.

---

Résumé Une tôle de d'alliage d'aluminium comportant une entaille en son centre a été soumisse à sollicitations cyclicques.On a mesuré et comparé les espacements entre les stries, et les vitesses de propagation de la fissure en surface. Des microfractographies ont également été effectuées. On a trouvé que la distance moyenne séparant deux stries était proportionnelle à K ^1.8, tandis que la vitesse de propagation de la fissure, mesureé en surface, était proportionnelle à K ^2.8, K désignant l'intervalle de variation du facteur d'intensité des contraintes.Des ruptures par clivage de portions fragiles apparaissent être la cause des differences rencontrées.Dans la zone des faibles valeurs de K, il n'y a que peu de ruptures par clivage, les stries sont plus distinctes, et leur orientation ne dévie pas beaucoup de la normale à la direction générale de la propagation de la fissure.Dans la zone des valeurs élevées de K, les ruptures par clivage sent plus nombreuses; les stries se développent selon des orientations divergentes; elles deviennent moins visibles.Les différences qui caractérisent les aspects des ruptures precedent de la rupture par clivage de portions fragiles.

     

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.     

Localized single-particle excitations and the density of states for superconducting composites

The problem of localized single-particle excitations and the density of states (DOS) for an inhomogeneous system consisting of a spherical superconductor (with radius a and order parameter ₁) embedded in another superconductor (order parameter ₂) of infinite size is considered. With the assumption of constant values of ₁ and ₂, the Bogoliubov equations are solved for general values of l (the orbital angular momentum quantum number). For a fixed value of ₁/₂ and different values of ₂/E _F, the dependence of the excitation energy (l=0)/₂ on the particle sizek _F a is shown (k _F is the Fermi wave vector andE _F is the Fermi energy). Fork _F a=300, 450, and 800 and a fixed value of ₂/E _F, the variations in the DOS by changing ₁/₂ are also shown.     

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*.     

Localized spin fluctuations

Scaling arguments and renormalization group techniques are used in the Anderson model Hamiltonian for the nonmagnetic limit of a magnetic impurity. The range of validity of the theory is limited toU/ 1, where the electron-electron collisions can already compete strongly with the electron-hole collisions. A parquet approximation is developed; it gives for the static susceptibility in the symmetric case = ()^–1 exp (+U/). There are some hints that the exchange force might play a crucial role in the formation of localized magnetic moment. These considerations are consistent with the electron-hole symmetry in the symmetric Anderson model.     

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.     

Fatigue fracture toughness and crack propagation rate

Fatigue crack growth rate, da/dN, of two high strength steels were examined in a laboratory air at different stress ratios, covering almost the entire range of stress intensity, K, from nearly threshold value, Kth, to final fracture. The fatigue fracture toughness, Kfc, corresponding to the final fracture in fatigue, was also determined. The lower the Kfc, the higher da/dN and reduced Kth are revealed.This correlation was analyzed quantitatively based on the four parameter Weibull function. And the stress ratio dependency of the fatigue crack propagation curve can be cleared in a successful manner.The fatigue characteristic stress intensities, Ke and Kv, are proposed to define the transition behaviour in fatigue crack growth curve, from so called region 1 to 2, and from region 2 to 3, respectively. Especially the Kv valua can be specified to be the 0.63Kfc.

---

Résumé On a étudié la vitesse de propagation de fissure en fatigue da/dN de deux aciers à haute résistance dans un atmosphère de laboratoire sous des sollicitations couvrant toute la gamme des intensités de contraintes variables K, depuis une valeur voisine de la valeur du seuil Kth jusqu'à celle correspondant à rupture finale.La ténacité à la rupture par fatigue Kfe correspondant à la rupture finale par fatigue a été également déterminée. II s'avère que plus Kfe est faible, plus élevée est da/dN et plus Kth est réduite. Cette correlation est analysée quantitativement en se basant sur la fonction de Weibull à quatre paramètres. On peut ainsi clarifier la manière dont le rapport de contraintes influe les courbes de propagation des fissures de fatigue.On propose de définir pas les facteurs caractéristiques d'intensité de contrainte Ke et Kv les comportements de transition de la courbe de vitesse de propagation de la fissure entre respectivement les régions dénommées 1 et 2, et 2 et 3.En particulier, on peut spécifier que la valeur Kv vaut 0,63 Kfe.

     

Interaction between a dislocation and various divalent impurity in KCl single crystals for the Fleischer's model taking account of the Friedel relation

The interaction between a dislocation and the impurity in KCl : Mg²⁺ (0.035 mol% in the melt), KCl : Ca²⁺ (0.035 and 0.065 mol% in the melt) and KCl : Ba²⁺ (0.050 and 0.065 mol% in the melt) was investigated from the strain-rate cycling test during the Blaha effect measurement. This was carried out at 77–254 K. As a result, it was found that the Fleischer's model taking account of the Friedel relation seems to be suitable for KCl : Ca²⁺ and KCl : Ba²⁺. However, it was not appropriate for KCl : Mg²⁺. Furthermore, the values of T _c, H(T _c) and G ₀ were obtained for the specimens. T _c is the critical temperature at which effective stress is zero. H(T _c) and G ₀ are the enthalpy and the Gibbs free energy of activation for the breakaway of the dislocation from the impurity, respectively. H(T _c) was almost the same for the specimens except KCl : Mg²⁺. G ₀ increased with increasing the divalent cation size. In addition, the tetragonality around the divalent ion-positive ion vacancy pair was estimated on the basis of G ₀ for the each specimen.     

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.     

ΔK ol level effects on fatigue crack propagation

In order to determine the effects of K _ol level on fatigue life, a single peak load was applied at distinct K levels of 7.8×10.3 and 9.8×10³ p.s.i. in^1/2. Here the K _ol level was defined to be a K level at which overload was applied. Three different overload ratios of 1.5, 2.0, and 2.5 were used to determine the overload ratio effect on the recovery factor. The result showed that the recovery factor, Z, was linearly related to K as Z = qK+Z _o, where q was a function of overload ratio. The value of q decreased as the overload ratio increased in a given K _ol level and seemed to be an important factor as well as retardation cycles in determining the fatigue life. For the same overload ratio, specimens that underwent overload at a smaller K _ol level showed more improved fatigue life.Nomenclature a Crack length - a ^* Overload affected zone size - B Specimen thickness - (da/dN)_ca Crack growth rate due to constant amplitude fatigue load - (da/dN)_ol Crack growth rate after overload is applied - E Young's modulus - K Stress intensity factor - K _min Minimum stress intensity factor - K _max Maximum stress intensity factor - K _ol K level at which overload is applied - N Number of cycles - N _D Number of delayed cycles - N _f Number of cycles needed for a specimen to be completely fractured - r _p Assumed plastic zone size - S Load - _ys Yield stress - W Width - Z Recovery factor     

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).     

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.     

Low endurance fatigue in metals and polymers

The fatigue behaviour of commercially pure aluminium and of nylon under sequentially varying strain amplitudes is compared with a damage law of the type suggested by Miner. Aluminium obeys such a law for both cyclic and uniaxial prestrains but the behaviour of nylon is significantly affected by microcracking, which produces a marked effect of loading sequence.Appendix N Number of strain cycles at a given time - N _f Value of N at failure - True tensile stress - True stress range for a strain cycled specimen - _h Value of at half the life of the specimen - True tensile strain - Total true strain range - _p True plastic strain range (= the breadth of the hysteresis loop at = 0) - _d True diametral strain range - E Young's modulus - Linear strain hardening rate when tested at a particular value of _p - D Damage due to cycling - D _p Damage due to prestrain - _p Prestrain. C, K, K₁, , are constants     

Delayed retardation phenomena of fatigue crack growth in various steels and alloys

The delayed retardation phenomena of fatigue crack growth resulting from a single application of overload were investigated for five steels, two aluminium alloys and a titanium alloy. As long as the small scale yielding condition was satisfied at the overloaded crack tips, the retardation behaviour of these materials was expressed consistently by four parameters; the peak/baseline stress ratio, r, the exponent in the Paris equation, m, the overload-affected zone size, _D, and the crack distance at the minimum rate of crack growth, _B. Then the parameters, _B and _D, characterizing the retardation phenomena for these materials were determined. The retardation of aluminium alloys was stronger than that of the other materials. This is attributed to the lower value of _B/ _D in aluminium alloys than in the other materials. In the case of r=2, the overload-affected zone sizes, _D, were nearly equal to 1.5 ₀ in HT80 steel and aluminium alloys, slightly lower than 1.5 ₀ in SNCM8 steel and much larger than 1.5 ₀ in A553 steel and the titanium alloy, where ₀ is the monotonic plastic zone size calculated according to the Dugdale model. The dependence of retardation on baseline stress intensity, K ₁, appeared somewhat complicated. In the cases of A553 steel and A2017 aluminium alloy the amount of retardation increased with increasing K ₁ value, while in the cases of HT80 steel and Ti-6A1-4V titanium alloy the tendency appeared in the reverse direction. The former behaviour was related to the change in the stress state from plane strain to plane stress at the overloaded crack tips and the latter was related to the threshold of stress intensity.