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.     

Determination of nonlinear energy toughness values for cyclic loading applications

For several years the nonlinear energy method proposed by Liebowitz and Eftis has been examined as a failure criterion for static testing of center-cracked and compact tension specimens. Since the method appears to be valid under conditions of crack-tip plasticity, subcritical crack growth and load relaxation, tests have been conducted to ascertain the merit of this method as a failure criterion under cyclic loading conditions. The nonlinear energy toughness for cyclic loading, $\text{G}\text{?}{}_{\mathrm{fc}}$, is obtained from an envelope of the cyclic load-displacement record, which naturally imposes some restrictions on the loading program.The cyclic toughness parameter, $\text{G}\text{?}{}_{\mathrm{fc}}$, has been evaluated for thin, center-cracked sheets of 2024-T3 and 7075-T6 aluminum alloys. The specimen dimensions were held constant and the load parameters were varied so that a significant variation of the cyclic life was obtained. Both alloys exhibited a significant reduction of $\text{G}\text{?}{}_{\mathrm{fc}}$ with increasing cyclic life in a manner similar to the classical $\text{S-N}$ diagram. For example, the ratio of cyclic to static toughness, $\text{G}\text{?}{}_{\mathrm{fc}}\text{/G}{}_{\mathrm{c}}$, was found to be about 0.8 when failure occurred after approximately 150 cycles. There appeared to be a tendency for the curve to level off at this point, which suggests that these curves may represent compressed S-N curves. It is felt that this method may serve the design process by allowing the establishment of a fracture toughness parameter capable of including the effects of the entire loading history of a structure into the fracture toughness requirements.     

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 of thin sections containing surface cracks

Surface-cracked specimens of several thicknesses of 7075-T651 and 7075-T6 aluminum were tested in uniaxial tension. For thicknesses t less than 0.25 in., the gross fracture stress σ_f of 7075-T651 Al was empirically related to flaw size by the following expression:

$\text{δ}{}_{\mathrm{f}}\text{σ}{}_{\mathrm{ult}}\text{= 1 + S(}\text{a}\text{φ}{}^2\text{.t}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$

where σ_ult is the ultimate strength, a the crack depth, φ a function of crack shape, and S a proportionality constant equal to ?1.7 in.^{$\text{?1}\text{2}$}. For 0.25-in. thick 7075-T651 aluminum, σ_f was found to obey this relationship only when $\text{a}\text{φ}{}^2$ is less than 0.065 in.; for larger flaws, such that $\text{0.065 <}\text{a}\text{φ}{}^2\text{< 0.11, σ}{}_{\mathrm{f}}$ is better predicted by Irwin's surface-crack equation with an apparent K_IC value of $\text{32.2}\text{ksi-in.}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.Fracture data for thin sections of 2014-T6 and 2014-T651 Al tested at ?423°F are analyzed in terms of the empirical relationship above and are found to be in good agreement. For these alloys, S has a value of ?2.6 in.^{$\text{?1}\text{2}$}.Applicability of the empirical relationship and Irwin's surface-crack analysis to the fracture of thin sections is discussed in terms of crack size, section thickness, and plastic zone size.     

Corresponding states treatment of refrigerants and cryogenic fluidsLe traitement des états correspondants des frigorigènes et des cryofluides

An analysis of temperature dependence of saturated liquid density along the liquid vapour coexistence curve shows that a plot of reduced properties T_r(=T/T_c) against T_r?_r (=T?/T_c?c), goes through a maximum at a particular T_r for all the following liquids: Ne, Ar, Kr, Xe, N₂, O₂, CH₄, C₂H₆, C₂H₄, CO₂, R11, 12, 12B1, 13, 13B1, 14, 21, 22, 23, 32, 40, 113, 114, 115, H₂O and NH₃. For all these liquids, except NH₃, T_r is ≈ 0.82, whether they are polar, non-polar, monoatomic or polyatomic. This characteristic temperature, $\text{T}{}^1\text{(=T}{}_{\mathrm{<mtext>c</mtext>}}\text{T}{}_{\mathrm{<mtext>r</mtext>}}\text{)}$, has been used to evaluate the energy parameter and the corresponding molar volume has been used to evaluate the length parameters for calculation of reduced critical properties. $\text{T}{}^1$ is also found to be a point of inflexion on the saturated liquid viscosity versus temperature plot.     

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.     

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.     

Thermal expansion of corundum structure Rh2O3

The directional thermal expansion coefficients of the corundum structure form of Rh₂O₃ were determined from room temperature to 850°C by x-ray diffraction methods. Rh₂O₃ has a lower thermal expansion and is less anisotropic in thermal expansion than alumina. The directional thermal expansion coefficients of Rh₂O₃ expressed in second degree polynominal form are: $\text{“α}{}_{\mathrm{<mtext>a</mtext>}}\text{” = 5.350 ×10}{}^{\mathrm{?6}}\text{+ 1.281 ×10}{}^{\mathrm{?9}}\text{T}\text{? 1.133 ×10}{}^{\mathrm{?14}}\text{T}{}^2\text{/°}\text{C}$ and $\text{“α}{}_{\mathrm{<mtext>c</mtext>}}\text{” = 5.246 ×10}{}^{\mathrm{?6}}\text{+ 6.369 ×10}{}^{\mathrm{?9}}\text{T}\text{? 7.480 ×10}{}^{\mathrm{?14}}\text{T}{}^2\text{/°}\text{C}$.     

Copper doped zinc oxide Y2BaZnO5: ERS and optical investigation

In copper doped Y₂BaZnO₅ oxides, copper exhibits a distorted square pyramidal coordination which is consistant with the values of g and A tensors obtained from O band ERS spectrum for a sample containing about 1 % Cu. Three values for g and A are observed, g₁ = 2.049₅, g₂ = 2.051₅, g₃ = 2.275, $\text{¦}\text{A}{}_1\text{¦ = 13 10}{}^{\mathrm{?4}}\text{cm}{}^{\mathrm{?1}}$, $\text{¦}\text{A}{}_2\text{¦ = 10 10}{}^{\mathrm{?4}}\text{cm}{}^{\mathrm{?1}}$ and $\text{¦}\text{A}{}_3\text{¦ = 147.5 10}{}^{\mathrm{?4}}\text{cm}{}^{\mathrm{?1}}$. Since $\text{g}{}_1\text{?}\text{g}{}_2$ an approximate C_4v point symmetry can be assumed for copper. The electronic spectrum shows three bands at 11700, 14500 and 20500 cm^?1 which can be assigned to the transitions A₁ → B₁, B₂ → B₁ and E → B₁ respectively. The orbital reduction parameters are calculated and the bonding covalency is discussed.     

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.     

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.     

Effect of loading history on stress corrosion cracking in high strength steel

The effect of preloading on crack nucleation time was examined with compact tension specimens having various notch radius in 0.1N-H²SO⁴ aqueous solution for 200°C tempered AISI 4340 steel. Crack nucleation time t_n increases by preloading for a given apparent stress intensity factor K_p2. The curve K_?2 vs. t_n deviates upward from the curve for the non preloading case. A linear relationship between the crack nucleation time and parameter is seen in semi-log diagram, where is taken as the value at t_n=α due to preloading. The apparent threshold stress intensity factor increases with K_?2 which is the apparent stress intensity factor of preloading. A detached crack is nucleated at some distance from the notch root and extends in a form of circle. This distance increases with increasing K_?2. The effect of load reduction during crack growth was examined. When the K-value was reduced from K₁ to K₂, an incubation time was observed before the crack started growing under the K₂-value. The incubation time t_m tends to increase with increasing ΔK = K₁-K₂. The threshold stress intensity factor was also found to increase for high load reduction.In order to explain these experimental results, a new dislocation model is proposed on the basis of stress induced diffusion of hydrogen in high stress region ahead of the notch root or a crack. This model suggests that the change in the crack nucleation time and the increase of the incubation time due to preloading or load reduction are caused by reducing the hydrostatic pressure and by spreading the hydrogen saturated region which requires more time for the hydrogen accumulation due to preloading or load reduction. The theory predicts the experimentally observed relations between and t_n and between log t_in and ΔK.     

Magnetohydrodynamic stratified flow past a sphere

This paper considers the uniform, steady, horizontal flow of a vertically stratified, electrically conducting, non-diffusive fluid over a non-conducting sphere in the presence of a uniform magnetic field. The force exerted on the sphere is investigated on the basis of the method of matched asymptotic expansions, for small values of a stratification parameter α, $\text{R}{}_{\mathrm{e}}\text{? ¦α¦}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$, $\text{F}{}_{\mathrm{r}}{}^2\text{? ¦α¦}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>3</mtext>}}$ and for $\text{M}{}^2\text{= 0(α}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. Up to the first order of calculations the drag is computed for a few typical values of magnetic interaction parameter when, (a) The applied magnetic field lies in the vertical plane and inclined with the ambient flow direction. (b) The applied magnetic field lies in the horizontal plane and perpendicular to the flow direction. Further it is shown that the sphere has no tendency to rotate nor it experiences a lift force upto the order of calculations which we have made. The drag experienced by the sphere is found to be increased due to the combined effects of stratification and magnetic field.     

The synthesis of the first stage graphite salt C8+AsF6− and its relationship to the first stage graphite/AsF5 intercalate

Oxidation of graphite with excess O₂⁺AsF₆^?, in suspension in SO₂C1F, produces the blue first-stage graphite salt of composition C₈AsF₆, which X-ray single crystal photographs show is hexagonal with a = 4.92(2), c = 8.10(2), $\text{V}\text{= 170}\text{A}\text{?}{}^3$. The blue first-stage material of approximate composition C₁₀AsF₅ obtained from graphite and AsF₅ has a related pseudo cell. Arsenic X-ray absorption-edge spectra show that C₈AsF₆ contains AsF₆^? alone, and that the graphite/AsF₅ intercalte contains AsF₆^? and AsF₃ in accord with the AsF₅ reduction:

$\text{3 AsF}{}_5\text{+ 2 e}{}^{\mathrm{?}}\text{→ 2 AsF}{}_6{}^{\mathrm{?}}\text{+ AsF}{}_3$

Treatment of the graphite/AsF₅ compound with F₂ gas results in conversion of all of the intercalated arsenic to AsF₆^?.     

Prior-to-failure extension of flaws under monotonic and pulsating loadings

An equation governing the prior to failure crack propagation is proposed. For a rate-sensitive solid containing two-dimensional crack and subject to the tensile mode of fracture the differential equations are integrated numerically for the loads increasing monotonically in time. The resulting integral curves gs = σ(l) and l= l(t), i.e. load vs crack length and length vs time, indicate that the growth of cracks in the subcritical range is strongly rate dependent.The fatigue growth, viewed as a sequence of slow growth periods, is simulated on EAI 380 analogue computer. The fourth power law proposed by Paris is confirmed only within certain range of high-cycle fatigue propagation and for a rate-insensitive solid. Otherwise, that is for a more pronounced rate dependency induced by viscosity of a solid and/or in the proximity of the final instability point the growth is markedly enhanced. For sufficiently small ratios of the applied stress intensity range ΔK to the toughness K_c, the suggested fatigue growth law consists of two terms, i.e.

$\text{d}\text{l}\text{d}\text{n}\text{=}\text{l}{}_1\text{12}\text{4}\text{ΔK}\text{K}{}_{\mathrm{c}}{}^4\text{+Cf}{}^{\mathrm{?1}}\text{ΔK}\text{K}{}_{\mathrm{c}}{}^2\text{, l}{}_1\text{=}\text{πK}{}^2{}_{\mathrm{c}}\text{8Y}{}^2$

First term is the familiar Paris expression while the second one accounts for the rate-dependent contribution; f denotes frequency and Y is the yield strength. Rate-sensitivity C is defined by eq. (1.13).     

Critical behaviour of homogeneous reacting systems with large activation energy

An exothermic chemical reaction which has a large activation energy takes place within a well-stirred fluid. Different criteria for critical reactions, e.g. the appearance of a point of inflexion in the graph of temperature against time, are discussed. It is found that the critical value of the initial concentration depends on the criterion chosen. However, for small ?, where 1/? is the dimensionless activation energy, these critical values differ from each other by only exponentially small amounts, i.e. the differences are only exp(?K/?) for some positive K. The leading terms in the asymptotic expansions of the critical initial concentration C_cr all take the form $\text{C}{}_{\mathrm{cr}}\text{C}\text{?}{}_0\text{+ ?}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{C}{}_{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{+ (? l}\text{n}\text{?)C}{}_1\text{+ 0(?) for ? → 0}$, thereby providing a correction to formulae that appear in the literature.     

Electronic and spin configurations of Co3+ and Ni3+ ions in oxides of K2NiF4 structure: A magnetic susceptibility study

In La₄LiCoO₈, Li⁺ and Co³⁺ ions are ordered in two dimensions and Co³⁺ ions undergo transitions from the low-spin to the intermediate as well as the high-spin states. Both Sr₄TaCoO₈ and Sr₄NbCoO₈ exhibit low to intermediate-spin state transitions of Co³⁺ ions. In the system LaSr_1?xBa_xNiO₄, the e_g electrons are essentially in extended states forming a band. With increase in x, the band width decreases accompanying an increase in unit cell volume; high-spin Ni³⁺ ions are formed to a small extent with increasing x, but there is no spin-state transition. In LaSrAl_1?xNi_xO₄, at small x, there is a small proportion of high-spin Ni³⁺; when x ≈ 0.6, there is an abrupt decrease in the $\text{c}$/$\text{a}$ ratio, signalling the formation of the band. In LnSrNiO₄, the $\text{c}$/$\text{a}$ ratio decreases sharply between Ln = La and Nd; this is likely to be accompanied by a broadening of the band.     

n-PbTep+−Pb1−xSnxTe heterojunctions prepared by a modified hot wall technique

$\text{n-PbTe}\text{p}{}^{\mathrm{+}}\text{?}\text{Pb}{}_{\mathrm{1?x}}\text{Sn}{}_{\mathrm{x}}\text{Te}$ heterojunctions with a long wavelength spectral cutoff (λ_c ≈ 6 μm) were prepared using the double-channel hot wall technique. The electrical and photoelectrical properties of the heterojunctions at 77, 197 and 300 K were investigated. Detectors with R_oA equal to 170 Ω cm² and a quantum efficiency of 25–40% were obtained. Reasons for the shift of the long wavelength spectral cutoff of the heterojunctions towards shorter wavelengths are given.     

A model for fatigue crack growth in a threshold region

The prediction of fatigue crack growth at very low ΔK values, and in particular for the threshold region, is important in design and in many engineering applications. A simple model for cyclic crack propagation in ductile materials is discussed and the expression

$\text{da}\text{dN}\text{=}\text{2}{}^{\mathrm{1+n}}\text{(1?2v)(ΔK}{}^2{}_{\mathrm{eff}}\text{?ΔK}{}^2{}_{\mathrm{c,eff}}\text{)}\text{4(1+n)π σ}{}^{\mathrm{1?n}}{}_{\mathrm{yc}}\text{E}{}^{\mathrm{1+n}}\text{?}{}^{\mathrm{1+n}}{}_{\mathrm{f}}$

developed. Here, n is the cyclic strain hardening exponent, σ_yc is cyclic yield, and ε_f is the true fracture strain. The model is successfully used in the analysis of fatigue data BS 4360-50D steel.