On the path independent integral-Ĵ

The path independent integral, $\text{J}\text{?}$, is presented as the rate of energy flux during crack extension. This integral is an extension of the J-integral proposed by Rice and includes the existence of a fracture process region and the effect of plastic deformations, body forces, thermal strains and inertial of material.It is shown that the $\text{J}\text{?}\text{-}\text{integral}$ can include as special cases other fracture mechanics parameters such as J by Rice, $\text{G}\text{?}$ by Eftis et al., J¹ by Blackburn or f by Strifors.A definition of the $\text{J}\text{?}\text{-}\text{integral}$ in a three-dimensional problem is presented and possibility of applying the $\text{J}\text{?}\text{-}\text{integral}$ as fracture criterion is discussed.     

A single specimen determination of J1C for aluminum alloys

A single specimen technique for measuring J_lC of aluminum alloys is presented and a formula,

$\text{J =}\text{P}{}^2\text{(23.12+34.68}\text{ΔC}\text{C}\text{)}\text{2EB}{}^2\text{w(1?}\text{a}\text{w)}{}^3$

for calculating J-integral is proposed. Experimental results show that the J_1C measured by this technique is in good agreement with that measured by using a multispecimen method. It is simple and effective for aluminum alloys. The results calculated from experimental data using the formula above are consistent with the results calculated using

$\text{J =}\text{1+α}\text{1+α}{}^2\text{2A}\text{B(w?a)}$

     

Crystal structure of K4[H2J2O10].8H20

Crystals of K₄ [H₂J₂O₁₀] 8H₂O belong to the triclinic system, space group P 1 with a = 7.161 (2) $\text{A}\text{?}\text{, b}\text{= 10,553}$ (5) $\text{A}\text{?}\text{, c}\text{= 7,081}$ (2) $\text{A}\text{?}\text{, α = 98°1′}$, β = 117°8′, γ = 90°6′ and Z = 1. The crystal structure has been determined on the basis of photographic data from 877 independent reflections, with the final R value of 6,6%. The iodine atoms are surrounded by a distorted octahedra consisting of five oxygen atoms and one OH group. The average J-O distance is 2.03 Å. There are 2 independent K atoms in the structure. K(1) has a coordination number of eight, while K(2) is surrounded by 6 (5 water molecules + one OH group) nearest neighbours.     

Bounds on the effective thermal conductivity of a dispersion of fully penetrable spheres

We evaluate upper and lower bounds on the effective thermal conductivity K_e of a model of two-phase composite materials in which one of the phases consists of spherical inclusions (or voids) of conductivity K₂ and volume fraction φ₂, dispersed randomly throughout a matrix phase of conductivity K₁ and volume fraction φ₁. Our evaluations compare third-order bounds of Beran and of Brown, which utilize the three-point matrix probability function of the model, with bounds of De Vera and Strieder (which apply only to the aforementioned “fully penetrable sphere” model) and of Hashin and Shtrikman. The comparisons are made over extended ranges of values of both φ₂ and $\text{α =}\text{K}{}_2\text{K}{}_1$ and reveal that the best bounds among those we have tested (generally those of Beran) are sharp enough to give quantitatively useful estimates of K_e for $\text{0.1 ≤}\text{K}{}_2\text{K}{}_1\text{≤ 10}$ over a wide range of φ₂ values. They are sharp at high φ₂ values (i.e., φ₂ = 0.9) and very sharp at low φ₂ values (e.g. φ₂ = 0.1) where they remain useful for $\text{K}{}_2\text{K}{}_1\text{≈ 100}$. They are less sharp at intermediate values (e.g. φ₂ = 0.5). As is well known, such results immediately translate into equivalent results for the electrical conductivity, dielectric constant, or magnetic permeability of composites.     

On the cyclic behavior of cast and extruded aluminum alloys. Part A: Fatigue crack propagation

The fatigue crack propagation (FCP) response of a cast and extruded aluminum alloy was examined as a function of mean stress and specimen orientation. The extruded alloy was tested in both the longitudinal and transverse orientation and no difference in FCP response was noted. FCP tests were conducted at R ratios of 0.1, 0.5, 0.65 and 0.8. In the threshold regime, it was seen that as R ratio increased, $\text{Δ}\text{K}{}_{\mathrm{TH}}$ decreased. In addition, ΔK_TH values determined for the cast alloy were superior to those determined for the extruded alloy at all R ratios examined. The threshold regime was also shown to be K_MAX rather than $\text{Δ}\text{K}$ dependent. At intermediate ΔK levels, a mean stress effect was seen for both alloys at R ratios less than 0.5. Crack closure was monitored during testing so that ΔK_EFF values could be determined. $\text{Δ}\text{K}{}_{\mathrm{EFF}}$ was seen to explain mean stress effects at intermediate $\text{Δ}\text{K}$ levels.     

Fracture mechanics approach to fatigue crack initiation from deep notches

Fracture mechanics approach is applied to fatigue crack initiation at the tips of deep, blunt notches including those with very small notch-tip radius. The theoretical relations between the stress intensity range ΔK_ρ and the notch-tip radius ρ for a fixed life for crack initiation were derived based on the models of dislocation-dipole accumulation and blocked slip-band. Those are approximated by a simpler equation: $\text{ΔK}{}_{\mathrm{\rho }}\text{ΔK}{}_{\mathrm{o}}\text{= (1 + ρ/ρ0)}\text{1}\text{2}$ where ΔK₀ and ρ0 are material constants which are related to the fatigue strength of smooth specimens Δρ0 as $\text{Δρ0 =}\text{2ΔK}{}_0\text{(πρ0)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. The results of experiments done with bluntly notched compact tension specimens of a structural low-carbon steel agree with the above relation between $\text{ΔK}{}_{\mathrm{gr}}\text{ΔK}{}_{\mathrm{o}}$ and ρ/ρ_o. The method to predict ΔK_o, ρ_o and Δρ_o from the fatigue data of cracked and smooth specimens is proposed.     

A mechanical model to predict elastic-plastic fracture toughness in high strength materials

A mechanical model of crack initiation and propagation, which is based on the actual mechanism of ductile fracture in high strength materials, is proposed. Assuming that a crack initiates when the equivalent stress at a distance ρ from the crack tip reaches a critical value $\text{}\text{?}$gsf, an equation for predicting fracture toughness J_IC is obtained. From comparison between the predicted values and the experimental results, it is found that the distance ρ corresponds to the spacing of micro-inclusions. The temperature dependence of fracture toughness J_IC estimated according to the derived equation is given in an Arrhenius form of equation and is nearly consistent with the experimental results.     

Nucleation mechanism of stress corrosion cracking from notches

Crack nucleation mechanism of hydrogen assisted cracking at notched cracks in aqueous solutions is investigated, using the compact type specimens with various notch radius in low-tempered 4340 steel. A detached crack initiates at some distance ahead of the notch root. The crack nucleation at the notched root is determined by the electrical potential method. When the crack initiates, the voltage difference starts to increase. The crack nucleation site is examined by SEM. The time for crack nucleation increases with the notch root radius, ρ, and decreases with the apparent stress intensity factor K_ρ. A linear relationship between the crack nucleation time, t_n, and the parameter ${}^2\text{K}{}_{\mathrm{\rho }}\text{/(πρ)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{-(2K}{}_{\mathrm{\rho }}\text{/(πρ)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}_{\mathrm{th}}\text{}}$ is seen in semi-log diagram, where $\text{(2K}{}_{\mathrm{\rho }}\text{/(πρ)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}_{\mathrm{th}}$ is almost equal to the yield shear strength.In order to explain these experimental results, a new model of micromechanics is proposed on the basis of stress induced diffusion of hydrogen in the high stress region ahead of the notch root. This model suggests that the detached crack initiates at the elasto-plastic boundary where the hydrogen concentration is from 2 to 5 times higher than that of the notch root surface. The theory agrees with experiments with respect to $\text{{2K}{}_{\mathrm{\rho }}\text{/(πρ)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{-(2K}{}_{\mathrm{\rho }}\text{/(πρ)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}_{\mathrm{th}}\text{}}$ vs t_n and t_n vs ρ.The empirical equation holds under constant t_n, K_ρ = K_o(ρ/ρ_eff)^m where K₀ is the stress intensity factor with ρ ≈ 0 under the present environment, ρeff is the effective notch radius and m is constant. The value of m is 0.25 for the crack nucleation time (t_n)_th corresponding to the threshold stress intensity factor (K_ρ)_th, 0.5 for t_n < (t_n)_th and 0 for ρ ≦ ρ_eff. The above equation agrees with the theoretical equation proposed by Tanaka and Mura for any t_n and ρ_eff.     

Etude magnetique et par spectroscopie Mössbauer de NH4FeF4

The NH₄FeF₄ lattice is constituted by sheets of (FeF₆^3? octahedra alternating with NH₄⁺ layers and may be described therefore as a two-dimensional material. At about 200 K the x^?1 vs. T curve shows actually a flat minimum typical of two-dimensional interactions. A calculation of the exchange integral has been performed using a high temperature series expansion technique and leads to $\text{J}\text{k}\text{= ? 26}\text{K}$. A Mössbauer spectroscopy investigation of the variation of the hyperfine field vs. T gives a Néel temperature T_N = 135 K and a critical exponent β = 0, 26. This last value is close to those obtained for RbFeF₄ and CsFeF₄ which have similar crystal and magnetic structures.     

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.     

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.     

A new method of determining the stress intensity factor K from isochromatic fringe loops

This paper describes the four parameter method of analysis for determining the stress intensity factor K. Dynamic photoelastic isochromatic fringe patterns associated with cracks propagating in centerpin loaded, eccentric-pin-loaded and crack-line-loaded SEN specimens of Homalite 100 were recorded. Data was obtained for tests over a range of crack velocities from arrest to the terminal velocity of 14,900 in. sec (378 m sec).Six measurements describing the size and shape of the experimental isochromatic loops were used to determine the stress intensity factor K by employing a comparison function to match analytical and experimental results. The analytical isochromatic loops were generated with a Westergaard stress function of the form $\text{Z(z) =}\text{K}\text{√2πz}\text{{1 + β(}\text{z}\text{a})}$ and a superimposed normal stress σ_Ox=αK|√2πz which acts parallel to the direction of crack extension. Results were obtained by the computer program (FRACTURE) for different values of the four parameters to give 8925 analytical fringe loops. Another computer program (SEARCH) was used to find a small group of solution which given very low values of the comparison function f_c. The final solution which contains the value of K was obtained from the small group by selecting the most consistent solution.The results obtained for Homalite 100 show that K_min and K_k are nearly the same and that ? increases abruptly from 0 to about 10,000 in./sec (254 msec) for modest increases in K above 400 psi √in. (4.4 × 10⁵$\text{Nm}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$). Further increases in crack velocity require significant increases in K until terminal velocity is     

A numerical and experimental investigation on the use of J-integral

Finite element analysis was used to determine numerically Rice's J-integral values in centrally notched plates of 4340 steel. These numerical values were compared with corresponding J-integral values using Dugdale model and antiplane strain model with a power law hardening of n = 0.3. J-integral was also computed for a crack extending into its own plastic yield region under constant loading. For increasing level of loading, σ, the rate of increase in J-integral decreases and J-integral remains almost constant at $\text{σ}\text{σ}{}_{\mathrm{YS}}\text{= 1}$ under such crack extension.A limited number of fracture tests were conducted with centrally notched 4340 steel specimens heat treated to yield strength levels of 150,180, 210 and 240 ksi. Fracture data showed that the critical J-integral, calculated and measured, is insensitive to crack tip sharpness for the lower strength 4340 material and thus the J_c fracture criteria appears suited in correlating fracture data.     

Oxonium-potassium substitution in potassium hexacyanoferrate (III). Two polytypic modifications

Clear evidence is presented for partial substitution of potassium ions by oxonium cations in K₃Fe(CN)₆. Two polytypic modifications have been characterized as 1M (one-layer monoclinic) and 12 0_γ (twelve-layer orthorhombic), the crystallographic data being determined with single crystals: $\text{a}\text{= 7.042}\text{A}\text{?}$, $\text{b}\text{= 10.402}\text{A}\text{?}$, $\text{c}\text{= 8.370}\text{A}\text{?}$, β = 107.32° for 1M; $\text{a}\text{= 80.67}\text{A}\text{?}$, $\text{b}\text{= 10.40}\text{A}\text{?}$, $\text{c}\text{= 8.37}\text{A}\text{?}$ for 120_r. The close relations with the basic K₃Fe(CN)₆ structural type are discussed.     

The preparation,crystal structures and properties of the potassium vanadium sulfides KXV5S8 (0.5≦x≦0.7)

The crystal structures of the potassium vanadium sulphides K_0.7V₅S₈ and K_0.5V₅S₈ (=KV₁₀S₁₆) have been determined. K_0.7V₅S₈ has a C-centered monoclinic unit cell of dimensions $\text{a}\text{=17.499(3)}\text{A}\text{?}$, $\text{b}\text{=3.2986(6)}\text{A}\text{?}$, $\text{c}\text{=8.489(1)}\text{A}\text{?}$, ß=103.98(1)°, spacegroup C2/m; isomorphous with TIV₅S₈; K_0.5V₅S₈ has essentially the same structure, but due to ordering of the K atoms, the monoclinic b-axis is doubled, thus forming a superstructure. The cell parameters are: $\text{a}\text{=17.462(4)}\text{A}\text{?}$, $\text{b}\text{=6.556(2)}\text{A}\text{?}$, $\text{c}\text{=8.4595(9)}\text{A}\text{?}$, ß=103.86(1)°, spacegroup P2. The structures are characterized by a three-dimensional framework of VS₆ octahedra with channels in which the K atoms are situated. Both compounds exhibit metallic behaviour.     

Comparison of various methods of measuring kic on small precracked bend specimens that fracture after general yield

An experimental investigation into the measurement of fracture toughness on small precracked bend specimens after general yield is described. Six parameters are compared on the basis of their accuracy and utility in predicting $\text{K}$_Ic. The parameters are: (i) the on-load crack tip opening displacement, COD, (ii) the lateral (notch root, crack tip) contraction, NRC, (iii) the angle of bend, (iv) the stretch zone, (v) the $\text{J}$ integral, and (vi) the equivalent energy. The materials used were. for the most part, A533B and 4340 steels.     

An investigation of the edge-sliding mode in fracture mechanics

A boundary collocation procedure has been applied to the Williams stress function to determine the elastic stress distribution for the crack tip region of a finite, edge-cracked plate subjected to mode II loading at the crack tips. The asymmetric specimen selected was particularly suitable for the determination of plane strain fracture toughness for mode II loading. Numerical solutions for stress intensity factors for the edge-sliding mode obtained by the boundary collocation method were in close agreement with values obtained from photoelastic experiments.Fracture tests of several compact shear specimens of 2024-T4 aluminum were conducted in order to experimentally investigate the behavior of the edge-sliding mode. In each case a brittle shear failure was observed and mode II fracture toughness values were obtained. The average value for K_IIc obtained from two tests was 39.5 ksi(in)^{$\text{1}\text{2}$}. No K_Ic. data for 2024-T4 were available for comparison purposes; however, K_Ic values for a similar alloy, 2024-T351, have been reported as 34ksi(in)^{$\text{1}\text{2}$} which is only about 15 per cent below the corresponding K_IIc value.     

A single test method for evaluation of the J integral as a fracture parameter

A semi-analytical method for the evaluation of the Rice fracture parameter $\text{J}$ is presented. The method is such that a value of $\text{J}$ may be determined from a single load-displacement test record, and procedures are set out, together with appropriate equations, for evaluating center cracked sheets, compact tension specimens and single edge notched beams. The method utilizes the elastic stress intensity relationships in conjunction with a modified form of the Irwin plastic zone size correction factor. Tests were carried out on center cracked sheets and compact tension specimens manufactured from a high strength aluminium alloy of a thickness (3.2 mm) for which conditions of plane stress prevailed. The $\text{J}$ values are shown to be dependent on specimen dimensions. The results are compared with those obtained by a graphical procedure. Some anomalies are noted and comparison of center cracked sheets with compact tension specimens discussed.     

Observations of electron-beam-induced thermal effects in amorphous Se80As20 and Se64As16Au20 thin films

Transmission and scanning electron microscopy have been employed to study the effect of localized 100 kV electron-beam heating on amorphous Se₈₀As₂₀ and Se₆₄As₁₆Au₂₀ glass films supported on carbon substrates. Crystallization, melting, vaporization and grain growth have been observed and related to the input electron-beam power density and estimated film temperature rise. These studies show that thin films of the former system are highly resistant to devitrification prior to their melting and vaporization. Films of the latter system, however, are more easily vaporized and undergo an apparent crystallization upon exposure at J ≈ 1.8 A cm^?2 to form α-AuSe and hexagonal arsenic. Exposure at current densities in excess of 2.4 A cm^?2 results in rapid vaporization of selenium and arsenic and the formation of an annulus of irregularly shaped twinned gold crystals at the hole periphery. Slow or prolonged heating of such peripheral zones was observed to result in the growth of crystallographically shaped nuclei consisting of assemblies of tetrahedra (hexagons, pentagons) together with trigonal platelets that exhibited non-integer (111) diffraction spots of type $\text{1}\text{3}\text{(422)}$. Exposures at J>3 A cm^?2 resulted in very rapid vaporization of Se and As and induced the growth of dendritic single-crystal gold films also exhibiting the anomalous (111) structure.     

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.