Studies on crack growth rate under high temperature creep,fatigue and creep-fatigue interaction—II: On the role of fracture mechanics parameter aeffσg

For high temperature creep, fatigue and creep-fatigue interaction, several authors have recently attempted to express crack growth rate in terms of stress intensity factor $\text{K}{}_{\mathrm{I}}\text{= α}\text{aσ}{}_{\mathrm{g}}$, where a is the equivalent crack length as the sum of the initial notch length a₀ and the actual crack length $\text{a}{}^1$, that is, $\text{a = a}{}_0\text{+ a}{}^1$. On the other hand, it has been shown by Yokobori and Konosu that under the large scale yielding condition, the local stress distribution near the notch tip is given by the fracture mechanics parameter of $\text{aσ}{}_{\mathrm{g}}\text{?(σ}{}_{\mathrm{g}}$), where a is the cycloidal notch length, σ_g is the gross section stress and $\text{?(σ}{}_{\mathrm{g}}$) is a function of σ_g. Furthermore, when the crack growth from the initial notch is concerned, it is more reasonable to use the effective crack length a_eff taking into account of the effect of the initial notch instead of the equivalent crack length a. Thus we believe mathematical formula for the crack growth rate under high temperature creep, fatigue and creep-fatigue interaction conditions may be expressed at least in principle as function of $\text{a}{}_{\mathrm{eff}}\text{σ}{}_{\mathrm{g}}\text{, σ}{}_{\mathrm{g}}$ and temperature.In the present paper, the geometrical change of notch shape from the instant of load application was continuously observed during the tests without interruption under high temperature creep, fatigue and creep-fatigue interaction conditions. Also, the effective crack length a_eff was calculated by the finite element method for the accurate estimation of local stress distribution near the tip of the crack initiated from the initial notch root. Furthermore, experimental data on crack growth rates previously obtained are analysed in terms of the parameter of $\text{a}{}_{\mathrm{eff}}\text{σ}{}_{\mathrm{g}}$ with gross section stresses and temperatures as parameters, respectively.     

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.     

Crack tip closure and environmental crack propagation

Crack propagation rate, da/dN, and crack tip closure stress, σ_cc, in part-through crack fatigue specimens of aluminum alloys are drastically affected by gaseous environments. The present studies indicate that the crack closure reflects the influence of the environment on the plastic deformation at the crack tip, and, therefore, on the crack propagation rates. Postulating that da/dN is mainly determined by $\text{ΔK}{}_{\mathrm{eff}}\text{∝ (σ}{}_{\max }\text{?σ}{}_{\mathrm{cc}}\text{)}$ (instead of $\text{ΔK ∝ (σ}{}_{\max }\text{?σ}{}_{\min }\text{)}$, as is done traditionally) leads to the relationship $\text{da/dN = A(ΔK}{}_{\mathrm{eff}}\text{)}{}^{\mathrm{n}}$ in which $\text{A}$ and $\text{n}$ are virtually independent of the gaseous environment. The exponents are $\text{n ≈ 3.3}$ for Al 7075 T651 and $\text{n ≈ 3.1}$ for Al 2024 T351, respectively.     

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.     

Cumulative damage in creep

The effect of two step stress variation and intermittent loading on the creep behaviour of commercially pure aluminium and stainless steel has been investigated. In the two step stress variation, the first stress σ₁ was applied for a given time t₁ and the stress level was switched over to σ₂. The resultant creep rate ?_s2⁺ and the failure time t₂ have been observed. Under the intermittent loading programme, the stress cycle was applied in the order σ₁-zero-σ₁ and the average creep rate on each reloading has been observed. The experimental data appear to give a cumulative damage rule in the form

$\text{t}{}_1\text{t}{}_{\mathrm{r1}}\text{+}\text{t}{}_2\text{t}{}_{\mathrm{r2}}\text{e}\text{?}{}_{\mathrm{s2}}{}^{\mathrm{+}}\text{e}\text{?}{}_{\mathrm{s2}}\text{=1}$

where t_r is the creep rupture time corresponding to a given stress and ?_s2 is the creep rate under the second stress in normal creep.     

Fatigue crack propagation work coefficient—a material constant giving degree of resistance to fatigue crack growth

Study on fatigue crack growth in steels was carried out from energetic point of view, i.e. taking account of plastic work around the fatigue crack. Based on the examination of the relation between fatigue crack growth rate (da/dN) and the plastic work around the fatigue crack tip (W_0.02 in SUS304, Fe-3Si and HT 60 steels, a material constant-fatigue crack propagation work coefficient-Q_0.02 is proposed. It is the ratio of W_0.02 to da/dN and means the degree of the resistance to fatigue crack growth. Numerical expression of Q_0.02 by mechanical properties was derived, which is given by

$\text{Q}{}_{0.02}\text{=}\text{9.3x10}{}^1\text{(σ}{}_{\mathrm{y}}\text{+σ}{}_{0.2}\text{)}\text{σ}{}_{\mathrm{y}}{}^{1.3}$

Comparison of Q_0.02 of various steels showed that Q_0.02 of high strength steels is very small compared with that of low strength steels. Graphical representation of the relation between Q_0.02 and da/dN at various values of ΔK/σ_y for steels revealed that da/dN at given value of ΔK/σ_y increase with decreasing Q_0.02. It is shown that fatigue crack growth behaviour of a steel (da/dN-ΔK relation) can be obtained from the Q_0.02-da/dN diagram by knowing the mechanical properties. Discussion on design stress level of the steels is also given.     

Some formulas for the crack opening stress level

Crack growth data for 2024-T3 sheet material were analysed with different formulas for $\text{ΔK}{}_{\mathrm{eff}}$ as a function fo the stress ratio $\text{R}$. The data covered $\text{R}$ values from ?1.0 to 0.54. A good correlation was obtained for $\text{ΔK}{}_{\mathrm{eff}}$/$\text{ΔK}$ = 0.55 + 0.33$\text{R}$ + 0.12$\text{R}{}^2$ The relation between log $\text{d}\text{a/}\text{d}\text{n}$ and log $\text{ΔK}{}_{\mathrm{eff}}$ was non-linear for high crack rates (> 1 μm/c).     

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

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

     

Kationen-anionen-abstände in kupfer- und chrom-thiospinellen

The Cu- and Cr-thiospinels are divided in three groups on account of the cation-anion-distances. I. MeCr₂S₄ (Me = Cd, Co, Fe, Hg, Mn, Zn). II. CuMe₂S₄ (Me = Cr; Ti; V; Zr; Cr, Hf; Cr, Sn; Cr, Ti; Cr, Zr). III. CuMe₂S₄ (Me = Co, Rh). According to the model of Lotgering and van Stapele, these groups correspond to the valence distribution

$\text{+2+3+3}$

$\text{+1+3+4}$

$\text{+1+3+3?2?1}$

$\text{I. Me(CrCr)S}{}_4$

$\text{II. Cu(MeMe)S}{}_4$

$\text{III. Cu(MeMe)S}{}_3\text{S}$

     

The plastic zone and growth of fatigue cracks in magnesium MA12 alloy at room and low temperatures

The plastic zone formed at the fatigue crack tip and the fracture topography in MA12 magnesium alloy samples, tested at 293 and 140 K in air and in vacuum, were analysed. It was found that the plastic zone formed in vacuum is characterized by a greater size ($\text{h}$) and degree of plastic strain that in air, and the crack growth rate ($\text{d}\text{l/}\text{d}\text{N}$) is lower. Temperature reduction leads to a decrease in $\text{h}$, while $\text{d}\text{l/}\text{d}\text{N}$ and the fracture mechanism are affected by temperature ambiguously, depending on the alloy microstructure and the $\text{K}{}_{\mathrm{<mtext>max</mtext>}}$ value. It was established that the size of the plastic zone can be described by the equation:

$\text{h=A}\text{(}\text{K}{}_{\max }\text{σ}{}_{\mathrm{0.2ps}}\text{)}{}^2$

where $\text{A}$ is a coefficient dependent on the alloy structural state, environment and test temperature. Evaluation of the cyclic plastic zone size at $\text{K}{}_{\mathrm{<mtext>max</mtext>}}$, corresponding to the transition from a low temperature region to a ‘Paris’ region, showed that this transition occurred when the cyclic plastic zon reached the structural parameter of the material.     

Effect of mechanical properties of material on rate of fatigue crack propagation

Many experimental and analytical equations on a rate of a fatigue crack propagation have been proposed. However, it seems that they can not fully express its complex behavior. There are still many problems remaining to be solved in order to clarify its mechanism. One of them is to clarify the relation between the rate of the crack propagation and the mechanical properties of material. In this paper, the rate of the crack propagation is analysed to clarify this problem. This analysis is based on the observation results of the fatigue crack propagation behavior previously by the authors. The analytical result is compared with the experimental one to make sure that they agree with each other. The conclusion obtained is; the rate of fatigue crack propagation is expressed by using the stress intensity factors as

$\text{dl}\text{dN}\text{= {}\text{c}\text{[Y}{}^2\text{F}{}^{\mathrm{a}}\text{E}{}^{\mathrm{a(1?n)}}\text{]}\text{} (K}{}_{\max }\text{)}{}^2\text{(K}{}_{\mathrm{a}}\text{)}{}^{\mathrm{a(2?n)}}$

. where C is a constant; E, Young's modulus; F, plastic coefficient; Y, yield stress; K_max and K_a, maximum and amplitude of the stress intensity factor, and α and n, exponents of the Manson-Coffin's law and work-hardening.     

Fatigue crack propagation from a crack inclined to the cyclic tensile axis

Cyclic stresses with stress ratio R = 0.65 were applied to sheet specimens of aluminium which have an initial crack inclined to the tensile axis at angles of 30°, 45°, 72° or 90°. The threshold condition for the non-propagation of the initial crack was found to be given by a quadratic form of the ranges of the stress intensity factors of modes I and II. The direction of fatigue crack extension from the inclined crack was roughly perpendicular to the tensile axis at stress ranges just above the threshold value for non-propagation. On the other hand, at stress ranges 1.6 times higher than the threshold values the crack grew in the direction of the initial crack. The rate of crack growth in the initial crack direction was found to be expressed by the following function of stress intensity factor ranges of mode I, K₁, and mode II, K₂: $\text{dc}\text{dN}\text{= C(K}{}_{\mathrm{eff}}\text{)sum}$, where $\text{K}{}_{\mathrm{eff}}\text{= [K}{}_1{}^4\text{+ 8K}{}_2{}^4\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$. This law was derived on the basis of the fatigue crack propagation model proposed by Weertman.     

Experimental research on surface crack propagation laws for low alloy steel

Surface crack propagation experiments were performed for low alloy steel. The testing result shows that the data for the crack propagation rate in the surface in the direction of the width may be treated by using Paris and Erdogan formula [1] for the crack propagation rate, and Shah and Kobayashi's formula for the stress intensity factor[2]. For the crack propagation rate in the direction of the depth, the data obtained cannot be treated in this way.It has been found that the data can also be treated by using the experimental formula suggested by Kawahara et al.[3].In addition, a test for investigating through-thickness crack propagation was made. It was found that the propagation rate of the through-thickness crack is much greater than those of the surface crack both in the direction of the width and in the direction of the depth. When $\text{ΔK(= 100 Kg/mm}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$) is the same, the propagation rate of through-thickness crack da/dN is five times as great as that of surface crack in the direction of the width.During the propagation of the crack, the relationship between the crack length b and the crack depth a is $\text{a}\text{b}\text{= A?B}\text{a}\text{h}$, where A = 0.97, B = 1.29.With db/dN determined, the prediction of fatigue life can be calculated by

$\text{N=}\text{∫}\text{a}{}_0\text{a}\text{A}\text{A?B}\text{a}\text{h}{}^2\text{d}\text{N}\text{d}\text{b}\text{d}\text{a}$

.     

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.     

Stability of layered bismuth compounds in relation to the structural mismatch

The relation between the stability and the structural mismatch in layered bismuth compounds, (Bi₂O₂)²⁺(A_n?1B_nO_3n+1)^2?, was formulated on the basis of an elastic model. The pseudo-tetragonal lattice parameter, $\text{a}$, of layered bismuth compounds was estimated from the following equation,

$\text{a}\text{=}\text{[}\text{a}{}_{\mathrm{<mtext>B</mtext>}}\text{′}{}^2\text{ap}\text{′}{}^2\text{(nK+1)}\text{(}\text{ap}\text{′}{}^2\text{+}\text{a}{}_{\mathrm{<mtext>B</mtext>}}\text{′}{}^2\text{nK}\text{)]}$

$\text{1}\text{2}$

where a_B′ is the lattice parameter of the unconstrained Bi₂O₂ unit, a_P′ the lattice parameter of the unconstrained perovskite-like unit, n the number of perovskite like layer in one structural unit, and K a constant. The change of the strain energy for ionic substitutions was estimated from the elastic relationships. It was found that the increase of n in certain component systems causes the increase of the lattice parameter, $\text{a}$, and the increase of the strain energy. This provides an explanation for the existence of maximum of n. New compounds, Pb₃Bi₄Ti₆O₂₁ ($\text{a}\text{=5.476}$, $\text{b}\text{a}\text{=1.000}$ and $\text{c}\text{=58.1}\text{A}\text{?}$) and Pb₄Bi₄Ti₇O₂₄ ($\text{a}\text{=5.485}$, $\text{b}\text{a}\text{=1.000}$ and $\text{c}\text{=66.2}\text{A}\text{?}$) were described.     

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 $\text{“α}{}_{\mathrm{<mtext>c</mtext>}}\text{” = 5.246 ×10}{}^{\mathrm{?6}}\text{+ 6.369 ×10}{}^{\mathrm{?9}}\text{T}\text{? 7.480 ×10}{}^{\mathrm{?14}}\text{T}{}^2\text{/°}\text{C}$.     

Temperature dependence of the surface diffusion distance of bismuth atoms adsorbed on mica,carbon and silicon monoxide surfaces

The mean distance of surface diffusion of bismuth adatoms on mica, carbon and silicon monoxide surfaces has been determined at different temperatures by measurement of the instantaneous sticking coefficient and the nucleus density.The surface diffusion distance has been found to increase with decreasing temperature in accordance with the formula

$\text{X=}\text{1}\text{2}\text{a}{}_0\text{v}{}_{\mathrm{0d}}\text{v}{}_{\mathrm{0a}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{exp}\text{E}{}_{\mathrm{a}}\text{? E}{}_{\mathrm{d}}\text{2RT}$

at temperatures above 413, 373 and 383 K for mica, carbon and silicon monoxide respectively. Here X is one-half of the diffusion distance, E_a is the adsorption energy, E_d the activation energy for surface diffusion, a₀ the diffusion jump distance and v_0a and v_0d the vibrational frequencies associated with re-evaporation and with surface diffusion respectively. Below these temperatures it has been found that the temperature dependence of the diffusion distance deviates from the above formula; this can be explained by the presence of residual gas molecules adsorbed on the surfaces.From the temperature dependence of the diffusion distance, the respective values of the pre-exponential term $\text{a}{}_0\text{(}\text{v}{}_{\mathrm{0d}}\text{v}{}_{\mathrm{0a}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and the difference of energies E_a?E_d have been estimated as 7.6 Å and 5.8 kcal mol^?1 for mica, 17 Å and 3.2 kcal mol^?1 for carbon and 58 Å and 1.3 kcal mol^?1 for silicon monoxide.     

Fracture toughness and fatigue crack propagation in high strength steel from room temperature to −180°c

Fracture toughness under tensile test and fatigue test on high strength steel at temperature ranging from room temperature to ?180°C were experimentally studied. The value of fracture toughness under fatigue test is considerably tower than that obtained under tensile test.Within the range from room temperature to ?100°C the following results were obtained: the power coefficient δ of the fatigue crack propagation rate $\text{[(dc)/(dN)] = AΔK}{}^5$ is related with [(1)/($\text{T}$)] as: $\text{δ = b}{}_1\text{+ [(a}{}_1\text{)/(kT)]}$. $\text{[(dc)/(dN)]}$ shows Arrhenius type, and, however, different equation from usual stress dependent rate process equation. The trend is in good agreement with the dislocation dynamics theory of fatigue crack propagation.     

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

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