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.     

Use of Fracture Mirrors to Interpret Impact Fractures in Brittle Materials

The impact resistances and fracture mirror radii ( r_m ) of rods of several ceramic materials were measured. The fracture stresses (σ _f ) were determined from σ _f vs r_m^−1/2 curves obtained from fiexural strength tests. An analysis, based on the assumption that the principal factor contributing to the impact energy absorbed is the energy ( U_e ) required to deflect the specimen to the fracture stress, indicated that the impact energy absorbed ( U ) per unit of specimen cross-sectional area ( A ) increased in proportion to the square of the maximum stress. The analysis also indicated that the slopes of the curves of U/A vs σ _f ² are proportional to the reciprocal of Young's modulus. Experimental data for several materials are consistent with this analysis.     

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.     

Fracture of an Aluminosilicate Fiberboard

Mode-I fracture of aluminosilicate fiberboard that is used in large mirror casting molds was studied. The material was idealized as a transversely isotropic, layered composite that was composed of planar sheets of crosslinked fibers. Elastic constants, the toughness ( K_R ) curve, and the fracture work were measured at room temperature. The observed rising K_R behavior was attributed to crack bridging. Experimental measurements of the bridging stress were made using a specimen-renotching technique. Relationships between the bridging stress, K _R , and fracture work were explored and shown to be consistent.     

Fracture of A12O3 in Combined Tension/Torsion: I, Experiments

Combined tension/torsion tests were performed on solid A1₂O₃ rods at principal stress ratios σ₂/σ₁ of 0 (pure tension), -0.17, -0.38, -0.78, and -1.0 (pure torsion). The tensile principal stress at fracture σ₁ increased with increasing compressive principal stress σ₂, resulting in a higher strength in torsion than uniaxial tension. The ratio σ₁ (torsion)σ₁ (tension) was 1.31 for A1₂O₃, in general agreement with limited torsional data for brittle materials in the literature. Brittle fracture data in the tension-compression quadrant of principal stress space show an interesting dichotomy since strengthening is observed in torsional investigations, whereas weakening is observed for pressurized-tube studies. This difference may be either a pressurized-tube test artifact or a real effect due to the presence of stress gradients.     

FREQUENCY DOMAIN TESTS OF MULTIVARIATE GAUSSIANITY AND LINEARITY

A stationary multivariate time series { X _t } is defined as linear if it can be written in the form X _t = ∑^∞ _{j =−∞} A _j e _{t − j} where A _j are square matrices and e _t are independent and identically distributed random vectors. If the e _t } are normally distributed, then { X _t is a multivariate Gaussian linear process. This paper is concerned with the testing of departures of a vector stationary process from multivariate Gaussianity and linearity using the bispectral approach. First the definition and properties of cumulants of random matrices are used to obtain the expressions for the higher-order cumulant and spectral vectors of a linear vector process as defined above. Then it is shown that linearity of a vector process implies constancy of the modulus square of its normalized higher-order spectra whereas the component of such a vector process does not necessarily have a linear representation. Finally, statistics for the testing of multivariate Gaussianity and linearity are proposed.     

Mirror Constant for Tyranno® Silicon-Titanium-Carbon-Oxygen Fibers Measured in Situ in a Three-Dimensional Woven Silicon Carbide/Silicon Carbide Composite

The strength, S , of ceramic and glass fibers often can be estimated from fractographic investigation using the fracture mirror radius, r _m, and the relationship S = A _m/( r _m)^1/2, where A _mis the "mirror constant." The present work estimates the value of A _mfor Tyranno^® Si-Ti-C-O fibers in situ in a three-dimensional woven SiC/SiC-based composite to be 2.50 ± 0.09 MPa·m^1/2. This value is within the range of 2–2.51 MPa·m^1/2 previously obtained for nominally similar Nicalon^® Si-C-O fibers.     

Choice of thresholds for wavelet shrinkage estimate of the spectrum

We study the problem of estimating the log-spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coefficients. We propose the use of thresholds t _{j , n} depending on sample size n , wavelet basis ψ and resolution level j . At fine resolution levels ( j = 1, 2, ...) we propose
t _{j , n} = α _j log n
where {α _j } are level-dependent constants and at coarse levels ( j ≫ 1)
t _{j , n} = (π/√3)(log n )^1/2.
The purpose of this thresholding level is to make the reconstructed log-spectrum as nearly noise-free as possible. In addition to being pleasant from a visual point of view, the noise-free character leads to attractive theoretical properties over a wide range of smoothness assumptions. Previous proposals set much smaller thresholds and did not enjoy these properties.     

Effect of a Transverse Tensile Stress on the Electric-Field-Induced Domain Reorientation in Soft PZT: In Situ XRD Study

The effect of a transverse tensile stress on the electric-field-induced 90°-domain reorientation in tetragonal lead zirconate titanate (PZT) near the morphotropic phase boundary was investigated in situ using X-ray diffraction (XRD). The XRD intensity ratio, I ₍₀₀₂₎/ I ₍₂₀₀₎, which represents the ratio of the volume of the c -domains to that of the a -domains on the PZT surface, was examined as a function of the electric field at various stress levels. It was found that a transverse tensile stress changes the electric-field dependence of I ₍₀₀₂₎/ I ₍₂₀₀₎, especially at higher electric fields. Without a transverse tensile stress, I ₍₀₀₂₎/ I ₍₂₀₀₎ began to saturate at E ≈ 800 kV/m. With a transverse tensile stress of 75 MPa, I ₍₀₀₂₎/ I ₍₂₀₀₎ increased with an upward curvature with the electric field, indicating that the transverse tensile stress enhanced the field-induced 90°-domain reorientation, and increased the effective piezoelectric coefficients at larger electric fields. At E = 900 kV/m, the estimated d _31,domain changed from −200 × 10⁻¹² V/m at zero stress, to −350 × 10⁻¹² V/m at 75 MPa.     

Fracture of Al2O3 in Combined Tension/Torsion: II, Weibull Theory

The Weibull statistical fracture theory for multiaxial stresses has been extended to conditions of combined tension/torsion loading. At fracture, a tensile principal stress ratio σ₁ (tension/torsion) σ₁ (uniaxial tension) greater than one is predicted which is dependent on stress state, Weibull modulus, and fracture location. Comparison to experimental tension/torsion data for Al₂O₃ shows that the Weibull theory, although predicting correct trends, generally underestimates strengthening effects of the compressive principal stress, thus providing a conservative failure prediction. This discrepancy may be related to influences of stress state on crack-tip "process zone" behavior.     

Relationship between Fractal Geometry and Fractography

Fractal geometry has been used to describe irregular fracture surfaces in a quantitative way. The fractal dimensional increment has been related to the fracture toughness of the material through the elastic modulus and a characteristic structure parameter, a ₀. The study of fractography has shown the relationship between the flaw/mirror size ratio and the fracture toughness. An experimental observation has shown that the fracture toughness is related to the elastic modulus through another structure parameter, b ₀. Combining all of these relationships leads to the conclusion that the fractal dimensional increment, D ^*, is directly related to the flaw/mirror size ratio. Experimental measurements of the fractal dimension and the flaw/mirror size ratio on glasses, a glass-ceramic, polycrystalline ceramics, and a single crystal all agree with the prediction. The implication of this finding is that there is a linear scaling law in operation at fracture between the energy of crack initiation and of microbranching and is reflected in the features on the fracture surface.     

Calculations and Measurements of the Spatial Piezoelectric Response and of the d15 Parameter of PZT5A Ceramic

In the analytical section of this paper the magnitudes of the polarization vectors ( Pi ) of lead zirconate titanate are shown as they vary with the angle of an applied uniaxial stress. This is done by applying a directional cosine matrix to the stress tensor σ _jk in Pi = d _ijk σ _jk . In addition the polarization in the direction of a rotating stress is derived from rotating the d _ijk tensor. Two-dimensional as well as three-dimensional isometric plots of the results are shown. The experimental section describes the preparation of disks poled uniformly at 90° to their thickness and the use of the Berlincourt d ₃₃ meter to measure d ₃₃₃ as a function of angle. Very good agreement with the calculated results was obtained. The effect of electrodes on the measurement of d ₃₃ and the use of a d ₃₁ parameter to permit a d ₁₅ determination are also demonstrated. The d ₃₃₃ measurements provided a d ₁₅ value with a resolution of ± 8%. Its average value was within 5% of d ₁₅ values derived with other techniques.     

Theory of Filtration of Ceramics: II, Slip Casting on Radial Surfaces

The theory of filtration of slip casting of incompressible beds of ceramic materials in planar molds is extended to deposition on internal and external cylindrical surfaces. Formulas are developed for (a) calculating the variation of the liquid pressure in both cake and mold as a function of the radius and (b) determining the time to produce a given thickness of a consolidated body. Assuming that the mold permeability K _m and capillary suction P _cap are related by an inverse parabolic relation, K _m P _cap ²= J , there is a mold with a specific permeability that produces a maximum rate of deposition from the slip. The theory presented in this paper strictly applies only to incompressible cakes and should be used cautiously with moderately compactible beds. For highly flocculated slurries yielding highly compressible bodies, the equations would not be expected to yield reliable results. Radial geometry leads to mathematical complexities not encountered in analyses of planar molds.     

Phase Relations in the Garnet Region of the System Y2O3–Fe2O3–FeO–Al2O3

Liquidus equilibrium relations for the air isobaric section of the system Y₂O₃–Fe₂O₃–FeO–Al₂O₃ are presented. A Complete solid-solution series is found between yttrium iron garnet and yttrium aluminum garnet as well as extensive solid solutions in the spinel, hematite, orthoferrite, and corundum phases. Minimum melting temperatures are raised progressively with the addition of alumina from 1469°C in the system Y–Fe–O to a quaternary isobaric peritectic at 1547°C and composition Y _0.22 Fe _1.08 Al _0.70 O _2.83* Liquidus temperatures increase rapidly with alumina substitutions beyond this point. The thermal stability of the garnet phase is increased with alumina substitution to the extent that above composition Y _0.75 Fe _0.65 Al _0.60 O ₃ garnet melts directly to oxide liquid without the intrusion of the orthoferrite phase. Garnet solid solutions between Y _0.75 Fe _1.25 O ₃ and Y _0.75 Fe _0.32- Al _0.93 O ₃ can be crystallized from oxide liquids at minimum temperatures ranging from 1469° to 1547°C, respectively. During equilibrium crystallization of the garnet phase, large changes in composition occur through reaction with the liquid. Unless care is taken to minimize temperature fluctuations and unless growth proceeds very slowly, the crystals may show extensive compositional variation from core to exterior.     

Robustness of the autoregressive spectral estimate for linear processes with infinite variance

Consider a discrete-time linear process { x _t }, a one-sided moving average of independent identically distributed random variables {ε _t }, with the common distribution in the domain of attraction of a symmetric stable law of index δ∈ (0, 2) and the moving-average coefficients b ( j ) such that ε _t is invertible in terms of the present and possibly infinite past values of { x _t }. By treating { x _t } as if it is second-order stationary, a normalized spectral density function f (μ) is defined in terms of the b ( j ) and, having observed x ₁, ..., x _T , an autoregression of order k is fitted by the well-known Yule–Walker and least squares methods and the normalized autoregressive spectral estimators are constructed. On letting k ←∞ as T ←∞, but sufficiently slowly, these estimators are shown to be uniformly consistent for f (μ), the convergence rate being T ^−1/φ, φ > δ. The finite sample behaviour is investigated by a simulation study which also examines possible effects of considering 'non-invertible' models.     

Correlation between Radiative Transition Probabilities of Nd3+ and Composition in Silicate, Borate, and Phosphate Glasses

Compositional dependence of spontaneous emission probabilities between initial ⁴ F _3/2 and terminal ⁴ I _J J = 9/2, 11/2, 13/2, 15/2) levels of Nd³⁺ were studied for about 90 samples of silicate, borate, and phosphate glasses using the Judd–Ofelt theory. The effect of the covalency of the Nd–O bond on the magnitude of intensity parameters was estimated from the variation of spectral profiles of the ⁴ I _9/2→⁴ G _5/2, ² G _7/2 and ⁴ F _7/2, ⁴ S _3/2 transitions. Intensity parameters Ω₄ and Ω₆ and the spontaneous emission probabilities were strongly affected by the ionic packing ratio of the glass host. The results were discussed in terms of the site selectivity of Nd³⁺ ions in glasses.     

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.     

Inclusions of Nanometer-Sized Al2O3 Particles in a Crystalline (Sc,Lu)2(WO4)3 Matrix

Nanometer-sized Al₂O₃ particles were successfully synthesized as crystalline inclusions by mixing both components to form the nanometer-sized particles and the (Sc,Lu)₂(WO₄)₃ matrices in a crystal lattice by preparing a solid solution of (Sc,Lu)₂(WO₄)₃ and Al₂(MoO₄)₃ and then decomposing the solid solution. The particles were dispersed uniformly and without agglomeration, which is commonly observed with conventional preparation techniques. The average particle size of the Al₂O₃ was 3.5 nm, and the standard deviation was estimated to be 1.1 nm.     

TIME SERIES RESIDUALS WITH APPLICATION TO PROBABILITY DENSITY ESTIMATION

Abstract. A linear stationary and invertible process y _t models the second-order properties of T observations on a discrete time series, up to finitely many unknown parameters θ. Two estimators of the residuals or innovations ɛ _t of y _t are presented, based on a θ estimator which is root- T consistent with respect to a wide class of ɛ _t distributions, such as a Gaussian estimator. One sets unobserved y _t equal to their mean, the other treats y _t as a circulant and may be best computed via two passes of the fast Fourier transform. The convergence of both estimators to ɛ _t is investigated. We apply the estimated ɛ _t to estimate the probability density function of ɛ _t . Kernel density estimators are shown to converge uniformly in probability to the true density. A new sub-class of linear time series models is motivated.     

Preparation and Properties of Alumina/Nickel-Cobalt Alloy Nanocomposites

High-density nickel-cobalt alloy-dispersed Al₂O₃ (Al₂O₃/Ni-Co alloy) composites were obtained via the hydrogen reduction and hot pressing of Al₂O₃, Ni(NO₃)₂6H₂O, and Co(NO₃)₂6H₂O powder mixtures. Microstructural investigations revealed that nanometer-sized alloy particles were dispersed homogeneously at the matrix grain boundaries, forming the intergranular-type nanocomposite. High strength (>1 GPa) was registered for the Al₂O₃/10 wt% Ni-Co alloy composite. An inverse magnetostrictive response to applied stress was observed, because of the Ni-Co alloy dispersions, which indicates promise for incorporating new functions such as stress and fracture sensing into the structural ceramics without any loss of mechanical properties.