Relation Between Power and Exponential Laws of Slow Crack Growth

The coefficients ₀, and N of the power law of slow crack growth, =₀ exp [–/(RT)](K/K_C)^N, are evaluated in terms of the fundamental parameters V₀. Q₀. and B of the exponential law, V=V₀ exp [(-Q₀+BK)/(RT)]. It is shown that N =θ(BK_C)/(RT),=₀−θBK_C, and ₀=V₀θ^−N, where θ is a dimensionless coefficient with a value ranging from 0.2858 to 1.0, depending on the ratio of stress intensity at the fatigue limit to the critical stress intensity factor.     

Use of Crack Branching Data for Measuring Near-Surface Residual Stresses in Tempered Glass

An analytical procedure based on fracture mechanics is used to obtain the amount of residual stress in glass from measurements on the fracture surface. The technique utilizes the measurement of microcrack branching distances, known as the mirror — mist boundary, which occur at a critical crack branching stress intensity (K _m ) value. This procedurre shows that σ _A r _m ^1/2 Y _F (θ) =σ _R r ^1/2 _m +Ψ₀, where σ _A is the applied stress, r _m is the microcrack branching radius, σ _R is the residual stress, Y _F ( θ ) is the crack-border correction factor, and Ψ₀ is a material constant based on K _m . Thus, the equation is that of a straight line with the slope equal to the magnitude of the residual stress. Data for tempered glass from the literature are used to demonstrate the applicability of the technique.     

Interpretation of the Parabolic and Nonparabolic Oxidation Behavior of Silicon Oxynitride

The oxidation process of Si₂N₂O, prepared by a hot isostatic pressing technique, has been studied by the thermogravimetric method. The oxidation has been performed in oxygen for 20 h in the temperature range 1300° to 1600°C, producing oxide scales of amorphous SiO₂ and α-cristobalite. The weight gain for T 1350°C does not begin to follow a parabolic rate law, until a certain time, t ₀. The A ₀ parameter in the parabolic rate law, (Δ w / A ₀)²= K _p t + B , represents the cross section area, A , through which the oxygen diffuses; in the derivation of this law A is assumed to be constant during the experiment. If crystallization occurs during the oxidation process, A will decrease with time. A function, A ( t ), describing the time dependence, has been developed and incorporated into the parabolic rate law, yielding a new rate law, which reads Δ W/A ₀= a arctan √ bt + c √ t . This new rate law is valid in the time interval t < t ₀, whereas, for t > t ₀, the oxidation process follows the equation (Δ w/A ₀)²= K °_p t + B ₀. The relation of the latter equation to the common parabolic rate law is described. All of the oxidation curves are described by these equations. The activation energy of the oxygen diffusion (and of the oxidation ( K _p)) is found to be 245 ± 25 kJ/mol, which is consistent with literature values reported for oxygen diffusion.     

Kinking of a Crack out of an Interface: Role of In-Plane Stress

A crack lying in the interface between two brittle elastic solids can advance either by continued growth in the interface or by kinking out of the interface into one of the adjoining materials. This competition can be assessed by comparing the ratio of the energy release rates for interface cracking and for kinking out of the interface to the ratio of interface toughness to substrate toughness. The stress parallel to the interface, σ₀, influences the energy release rate of the kinked crack and can significantly alter the conditions for interface cracking over substrate cracking if sufficiently large. This paper provides the dependence of the energy release rate ratio on the in-plane stress. The nondimensional stress parameter which emerges is, σ₀( a / E _* T_i)^1/2, where a is the initial length of the kink into the substrate, E _* is a modulus quantity, and T_i is the fracture energy of the interface. An experimental observation of the cracking of reaction product layers in bonds between Ti(Ta) and Al₂O₃ is rationalized by the theory.     

Slow Crack Growth Behavior in Transformation-Toughened Al2O3-ZrO2(Y2O3) Ceramics

Significant increases in the critical fracture toughness (K _IC ) over that of alumina are obtained by the stress-induced phase transformation in partially stabilized ZrO₂ particles which are dispersed in alumina. More importantly, improved slow crack growth resistance is observed in the alumina ceramics containing partially stabilized ZrO₂ particles when the stress-induced phase transformation occurs. Thus, increasing the contribution of the ZrO₂ phase transformation by tailoring the Y₂O₃ stabilizer content not only increases the critical fracture toughness (K_IC) but also the K _Ia to initiate slow crack growth. For example, crack velocities ( v )≥10^–9 m/s are obtained only at K _Ia≥5 MPa.m^1/2 in transformation-toughened ( K _IC=8.5 MPa.m^1/2) composites vs K _Ia≥2.7 MPa.m^1/2 for comparable velocities in composites where the transformation does not occur ( K _IC=4.5 MPa.m^1/2). This behavior is a result of crack-tip shielding by the dissipation of strain energy in the transformation zone surrounding the crack. The stress corrosion parameter n is lower and A greater in these fine-grained composite materials than in fine-grained aluminas. This is a result of the residual tensile stresses associated with larger (≥1 μm) monoclinic ZrO₂ particles which reside along the intergranular crack path.     

Mode I Fracture Toughness of a Small-Grained Silicon Nitride: Orientation, Temperature, and Crack Length Effects

The Mode I fracture toughness ( K _{I C} ) of a small-grained Si₃N₄ was determined as a function of hot-pressing orientation, temperature, testing atmosphere, and crack length using the single-edge precracked beam method. The diameter of the Si₃N₄ grains was <0.4 µm, with aspect ratios of 2–8. K _{I C} at 25°C was 6.6 ± 0.2 and 5.9 ± 0.1 MPa·m^1/2 for the T–S and T–L orientations, respectively. This difference was attributed to the amount of elongated grains in the plane of crack growth. For both orientations, a continual decrease in K _IC was observed through 1200°C, to ∼4.1 MPa·m^1/2, before increasing rapidly to 7.5–8 MPa·m^1/2 at 1300°C. The decrease in K _IC through 1200°C was a result of grain-boundary glassy phase softening. At 1300°C, reorientation of elongated grains in the direction of the applied load was suggested to explain the large increase in K _IC. Crack healing was observed in specimens annealed in air. No R -curve behavior was observed for crack lengths as short as 300 µm at either 25° or 1000°C.     

Fracture Toughness and High-Temperature Slow Crack Growth inSiC

The subcritical growth of macroscopic cracks in hot-pressed and in sintered SiC was examined at 900° to 1100°C in oxygen partial pressures of 10⁻⁴ and 10⁻⁸ atm. The K ₁-V data for the hot-pressed SiC were generated using the double torsion specimen in the incremental displacement rate mode. These data, in addition to earlier results, included three distinct regions in the plot of log V vs log K ₁,: a low slope region, N∼20, of substantial slow crack growth which was insensitive to testing temperature and environment; a plateau region which shifted to higher velocities with increasing temperature; and a region of very high slope with K values near K _IC The results may be described qualitatively by viscous flow of the gram-boundary phase. No slow crack growth could be detected in the sintered SiC when tested hi the environments described.     

Effect of Micmcracking on the Fracture Toughness and Fracture Surface Fractal Dimension of Lithia-Based Glass-Ceramics

The effect of thermally induced microcracks on the fracture toughness and fractal dimension of fully crystalline lithia disilicate glass-ceramics was studied. The fracture toughness, K _IC, for the nonmicrocracked lithia disilicate, 3.02 ± 0.12 MPa·m^1/2, was significantly greater than the value of 1.31 ± 0.05 MPa·m^1/2 for the microcracked specimens. The fractal dimensional increment, D *, was 0.24 ± 0.01 for nonmicrocracked lithia disilicate specimens compared with a value of 0.18 ± 0.01 for the microcracked specimens. The relationship between K _IC and D * implies that the two materials exhibit dissimilar fracture behavior because of microstructural differences. Estimates of the characteristic length involved in the fracture process, a ₀, indicate that the materials have an identical fracture process at the atomic level. This apparent contradiction may be explained by the scale on which the measurements were taken. It is suggested that fractal analysis at the atomic level would yield equivalent D * values for the two different microstructures.     

Communications of the American Ceramic Society Estimation of Thermochemical Data for Calcium Silicate Hydrate (C-S-H)

Calcium silicate hydrate (C-S-H) can be viewed as a solid solution, 0.833Ca(OH)₂.SiO₂.0.917H₂O-xCa(OH)₂, at equilibrium at 30°C. On this basis, the change in Gibbsfree energy (ΔG_r) in the solid-solution reaction was calculated from solubility duta for C-S-H in water. The change in ΔG_r with real ratio decreased notably for the higher calcium contents (CaO/Si0₂1.7; ×0.867). Thermochemical values for C-S-H (CaO/SiO₂=1.7) were estimated to be ΔH°=-2890 kJ/mol, ΔG°=-2630 kJ/mol, and S°=200 J1/mol.K at 298 K .     

Phase Relations in the System Bi2O3-WO3

Phase relations in the system Bi₂O₃-WO₃ were studied from 500° to 1100°C. Four intermediate phases, 7Bi₂O₃· WO₃, 7Bi₂O₃· 2WO₃, Bi₂O₃· WO₃, and Bi₂O₃· 2WO₃, were found. The 7B₂O · WO₃ phase is tetragonal with a ₀= 5.52 Å and c ₀= 17.39 Å and transforms to the fcc structure at 784°C; 7Bi₂O₃· 2WO₃ has the fcc structure and forms an extensive range of solid solutions in the system. Both Bi₂O₃· WO₃ and Bi₂O₃· 2WO₃ are orthorhombic with (in Å) a ₀= 5.45, b ₀=5.46, c ₀= 16.42 and a ₀= 5.42, b ₀= 5.41, c ₀= 23.7, respectively. Two eutectic points and one peritectic exist in the system at, respectively, 905°± 3°C and 64 mol% WO₃, 907°± 3°C and 70 mol% WO₃, and 965°± 5°C and 10 mol% WO₃.     

Ammonium Iodate—A Potentially Useful Nonlinear Optical Material

The space group of NH₄IO₃ was determined, from precession photographs and observations of symmetry-related characteristics, to be Pna 2₁. The cell constants are a ₀= 6.3740 ± 0.0005, b ₀= 6.4115±0.0005, and c ₀=9.1706±0.0005 Å. The crystal is piezoelectric along all three axes and pyroelectric with dP/dT ⋍(3±1)×10⁻⁹ C/cm²°C along the c ₀ axis at room temperature. At 85°C a first-order transition to a nonpolar piezoelectric phase is observed.     

Fractographic Criteria for Subcritical Crack Growth Boundaries in 96% Al2O3

The percent intergranular fracture (PIF) was measured along radii extending from fracture origins in 96% A1₂O₃ specimens, fractured at various loading rates and temperatures, and plotted vs estimates of stress intensity factors ( K ₁) at the corresponding crack lengths. Two types of curves were observed. The first was similar to curves previously observed for hot-pressed alumina. In this case the subcritical crack-growth boundary was located approximately where the minimum in the PIF occurred near K ₁=4MPa·m^½, as was also the case for hotpressed alumina. Therefore, the location of this minimum or the projecting grams formed by intergranular fracture as the crack velocity increased can be used as criteria for locating the subcritical crack-growth boundary. The second type of curve lacks the minima in PIF characteristic of the first type and is characterized by a gradual trend toward higher PIF beginning at K ₁=3MPa·m^½. This type of curve may be caused by acceleration of the crack to high crack velocities at values of K ₁ approximately equal to or slightly greater than those necessary to cause critical crack growth on the lower fracture-energy planes in sapphire. Assuming that this is the case, the K ₁ at which the trend toward higher PIF begins can be used to calculate the radius to the critical flaw boundary for this type of fracture.     

Preparation of Silicon Carbide/Aluminum Nitride Ceramics Using Organometallic Precursors

Solid solutions of 2H -SiC/AlN can be prepared at temperatures less than 1600°C by rapid pyrolysis ("hot drop") of mixtures of [(Me₃Si)_0.80((CH₂=CH)MeSi)_1.0(MeHSi)_0.35] _n (VPS) or [MeHSiCH₂] _n (MPCS) with [R₂AlNH₂]₃, where R=Et, i -Bu or simply by slow pyrolysis of the precursor mixture in the case of [Et₂AlNH₂]₃. In contrast, slow pyrolysis of mixtures of VPS or MPCS with [ i -Bu₂AlNH₂]₃ yields a composite of 2 H -AlN and 3 C -SiC at 1600°C, which transforms into a single 2 H -SiC/AlN solid solution on heating to 2000°C. The influences of the nature of the precursor and processing conditions on the structure, composition, and purity of the SiC/AlN materials are discussed.     

Cell Constants for Dicalcium Silicate-Alkali Fluoride Complexes

X-ray diffraction powder patterns were used to determine the cell constants for the complexes 2CaO·SiO_2·LiF and 2CaO·SiO₂·KF by Lipson's method of indexing, assuming orthorhombic structures. The resulting cell constants are α₀= 18.2, b₀=20.6, and c₀=7.2 Å for the Li complex and α₀=18.7, b₀=20.5, and c₀6.9 Å for the K complex.     

Alkali Strontium-Barium-Lead Niobate Systems with a Tungsten Bronze Structure: Crystallographic Properties and Curie Points

Crystallographic parameters, Curie points, and freezing temperatures are presented for a series of A⁺A₂²⁺Nb₅O₁₅ compositions with tungsten bronze-related structures, where A⁺=Na, K, or Rb, and A²⁺=Sr or Ba. Results are also reported for bronze solid solutions in the systems KNbO₃-PbNb₂O₆, K(Sr-Ba)₂Nb₅O₁₅, K(Ba-Pb)₂Nb₅O₁₅, K(Sr-Pb)₂Nb₅O₁₅, and (KSr₂-K₂La)Nb₅O₁₅. On the basis of these data, an empirical relation is shown between the ferroelectric transition temperature, T _c, and axial ratio °10 c _o/ a ₀. The T _c vs °10 c ₀/ a ₀ curve for compositions with polar axes perpendicular to [00l] has a negative slope, whereas that for compositions with polar axes parallel to [00l] has a positive slope of about the same magnitude. These results are compared with measurements of the temperature dependence of c ₀/ a ₀ for KSrPbNb₅O₁₅, and KBa₂Nb₅O₁₅; the latter is discussed in terms of the Devonshire free energy theory.     

Modified Lead Calcium Titanate Ceramics with a Relatively Large Dielectric Constant for Hydrophone Applications

Modified lead titanate of the base composition (Pb,Ca)Ti-(Co,W)O₃+ 0.01 mol% MnO was doped with 2 to 6 mol% strontium and barium to increase the dielectric constant, K , while preserving a high piezoelectric anisotropy ( d ₃₃ /d ₃₁). K was increased above 500 and a d ₃₃ /d ₃₁ ratio of 34/1 was achieved when the composition Pb_0.65Ca_0.31Sr_0.04Ti_0.09-(CO_0.5W_0.5)_0.06O₃+ 0.01 mol% MnO was fabricated. The large d ₃₃ /d ₃₁ resulted in an impressive hydrostatic piezoelectric coefficient ( d _h= 81 × 10_—12 m₂/V). The relatively low transition temperature ( T _c= 120°C) did not result in an increased aging rate when samples were tested at 40°C and 99% humidity for 120 h after poling. A small tetragonality ratio (1.008) enabled saturation of ceramic properties at poling fields as low as 30 kV/cm at 100°C.     

Preparation of Undoped Lead Titanate Ceramics via Sol-Gel Processing

Crack-free, undoped PbTiO₃ ceramics were fabricated successfully using sol–gel-synthesized powder prepared from chelated titanium alkoxide and lead acetate. The sintered ceramics, 8.3 mm in diameter and 6–8 mm thick, were 96% dense. In the present study, PbTiO₃ ceramics with excess lead (Pb:Ti = 1.1:1.0) had large grains, averaging 14.3 μm. Lower-lead ceramics (Pb:Ti = 1.0:1.0 and 0.9:1.0) had smaller grains, averaging 1.8 μm. The PbTiO₃ ceramics with a high lead content cracked during sintering at 1150°C, whereas the other ceramics did not crack. Excess lead, in a more-than-stoichiometric ratio, promoted grain growth and caused disintegration of the ceramics. Therefore, uncracked PbTiO₃ ceramics apparently can be fabricated by avoiding excess lead, possibly because restricted grain growth in low-lead ceramics causes low residual stress over many small grains during transition. The electrical properties measured in the present study for PbTiO₃ ceramics with a Pb:Ti ratio of 1:1 are d ₃₃= 35 pC/N, K ₃= 64, k _p= 0.59, and k _t≈ 0.     

A note on L1 density estimation for linear processes

In this paper, we consider the L ₁ performance of a kernel estimator, f^_n of the density of a linear process X_t ∑^∞ _{k =0} a _k Z _t−k , a ₀ = 1, where { Z _t } is a sequence of independent and identically distributed (i.i.d.) random variables with E | Z ₁|^ε< ∞, for some ε > 1, and { a_k } is a sequence of reals converging to zero at a certain rate. Asymptotic minimizations of the integrated L ₁ risk of f_n and its upper bounds are considered. This paper extends the earlier results for the i.i.d. case by Devroye and Gyorfi ( Nonparametric Density Estimation: The L₁ View. New York: Wiley, 1985) and by Hall and Wand (Minimizing the L ₁ distance in nonparametric density estimation, J. Multivariate Anal. 26 (1988), 59–88) to the linear process case. Numerical examples to illustrate the performance of f_n are also presented.     

Mechanical Properties of Partially Dense Alumina Produced from Powder Compacts

The elastic modulus ( E ), the critical strain energy release rate ( G _c), and the flexural strength (σ) have been determined for two partially dense alumina bodies produced from the same powder but with different initial densities. The mechanical properties were measured for specimens fabricated at four different relative densities. The measured elastic modulus, critical strain energy release rate, and a calculated critical stress intensity factor ( K _c) were observed to be linearly related to (ρ–ρ₀)/(1 –ρ₀), where ρ is the current relative density and ρ₀ is the initial relative density of the powder compact. With the observed linear relations for E, G _c (or K _c), and the assumption that the crack length responsible for failure was present in the initial powder compact and shrunk in proportion to the relative density change, a Griffith equation was constructed to estimate the strength at any relative density. This relation was in good agreement with measurements.     

Structure and Microwave Dielectric Characteristics of Ca1+xNd1−xAl1−xTixO4 Ceramics

CaNdAlO₄ microwave dielectric ceramics were modified by Ca/Ti co-substitution, and their dielectric characteristics were evaluated along with their structure and microstructures. Ca_{1+ x} Nd_{1− x} Al_{1− x} Ti _x O₄ ( x =0, 0.025, 0.05, 0.10, 0.15, 0.20) ceramics with the relative density of over 95% theoretical density were obtained by sintering at 1400°–1450°C in air for 3 h, where the K₂NiF₄-type solid solution single phase was determined from the compositions of x <0.20, while a small amount of CaTiO₃ secondary phase was detected for x =0.20. With Ca/Ti co-substitution in CaNdAlO₄ ceramics, the dielectric constant (ɛ_r) increased with increasing x , and the temperature coefficient of resonant frequency (τ_f) was adjusted from negative to positive, while the Q × f ₀ value increased significantly at first and reached an extreme value at x =0.025 and the maximum at x =0.15. The best combination of microwave dielectric characteristics were achieved at x =0.15 (ɛ_r=19.5, Q × f ₀=93 400 GHz, τ_f=−2 ppm/°C). The improvement of the Q × f ₀ value primarily originated from the reduced interlayer polarization with Ca/Ti co-substitution, while the decreased tolerance factor, the subsequent increased interlayer stress, and the appearance of CaTiO₃ secondary phase brought negative effects upon the Q × f ₀ value.