Elastic Properties of Polycrystalline Yttrium Oxide, Dysprosium Oxide, Holmium Oxide, and Erbium Oxide: Room Temperature Measurements

The Young's and shear moduli of polycrystalline yttrium oxide, dysprosium oxide, holmium oxide, and erbium oxide were determined at room temperature as a function of volume fraction porosity using the sonic resonance technique. Linear relations empirically described the data for Y₂O₃, Hr₂O₃, and Er₂O₃. The Young's and shear moduli data for Dy₂O₃ were empirically described by Hasselman's and Spriggs'equations, respectively. The empirical curves which best fit the data were compared to theoretical expressions and agreed closely with a modified form of Mackenzie's equation. Values for bulk modulus, Poisson's ratio, and the Debye temperature were computed for each oxide, and the bulk modulus-volume relation was determined and compared to that of other oxides.     

Elastic and Anelastic Response of Polycrystalline UO2-PuO2

A sonic resonance technique was used to investigate the room-temperature elastic and anelastic properties of physically mixed U_0.8PU_0.2O₂ as a function of density, stoichiometry, and cation homogeneity. The effect of porosity on the elastic moduli was linear and is described by E =2102.7 (1–2.03 P )± 13.5 Kbars for the Young's modulus, G =823.5(1–2.05 P )± 9.1 kbars for the shear modulus, and B = 1584.8(1–1.89 P )± 59.1 kbars for the bulk modulus, where P is the volume fraction porosity. Poisson's ratio was 0.28 and was not a function of porosity. The Debye temperature of U_0.8Pu_0.2O₂ computed from the Young's and shear moduli for theoretically dense specimens was 379°K. Variation of the O/M ratio from 1.968 to 2.006 produced no significant change in either the damping capacity or the elastic moduli of single-phase 80%UO₂-20% PuO₂ solid solutions. An approximate 24% decrease of the room-temperature Young's and shear moduli and an approximate increase by a factor of 14 in the internal friction were observed with gross modifications of plutonium cation homogeneity. Preliminary results suggest that internal friction measurements might be used to assay the homogeneity of UO₂-PuO₂ solid solutions.     

Some Physical Properties of High-Density Thorium Dioxide

The values for a number of physical properties are reported for a very high density form of thorium dioxide. When specimens of a mixture of 99½% ThO² and ½% CaO, by weight, were hydrostatically pressed at 30,000 lb. per sq. in. and heat-treated for 1 hour at 1800°C., they attained 99.0% of theoretical density. All the test specimens were extremely brittle. Physical-property values determined at room tempera- ture were the following: lattice constant; bulk and theoretical (X-ray) densities; compressive and impact strengths; Knoop hardness; modulus of rupture and Young's modulus, determined by a static method; Young's modulus and the shear modulus, determined by a dynamic method; Poisson's ratio and the bulk modulus, calculated from the dynamic-test data; and the velocity of sound through the material. The properties determined at elevated temperatures were the following : linear thermal expansion modulus of rupture and Young's modulus, determined by a static method; Young's modulus and the shear modulus, determined by a dynamic method; and Poisson's ratio, calculated from the elevated-temperature dynamic-test data. "Martin's diameter" grain counts were taken for the material both before and after heat-treatment.     

Elastic Moduli of Glasses at Elevated Temperatures by a Dynamic Method

The elastic constants of thirteen glasses were measured by a dynamic method at elevated temperatures. Both Young's modulus and the shear modulus were determined and from these Pois-son's ratio was calculated as a function of temperature. Fused silica, Vycor-brand glass, and Pyrex-brand glass had positive temperature coefficients of elastic moduli, whereas all the other glasses showed negative coefficients. Poisson's ratio was found to rise with temperature in all thirteen glasses. This is interpreted as an indication of an approach to the liquid state.     

High-Temperature Elasticity and Expansivity of Forsterite and Steatite

The elastic constants and coefficients of thermal expansion of polycrystalline forsterite (Mg₂SiO₄) and steatite (MgSiO₃) were determined from room temperature to 1000°K. Two elastic moduli, the adiabatic bulk modulus, B_s , and the shear modulus, G, decrease linearly with temperature above 500°K. The Grüneisen constant γ and a parameter δ, defined as — (dB_s/dT)/αB_s, calculated from the present data were virtually independent of temperature at the high-temperature range. Poisson's ratio, δ, rises linearly with temperature over the range of measurement; the slope is highest for materials with the lowest room-temperature value of σ.     

Temperature Dependence of Elastic Constants of Vitreous Silica

The relation between the elastic moduli and temperature from room temperature to about 1300°C was determined for a group of vitreous silica specimens by a dynamic resonance method. All the curves were approximately parabolic in shape, reaching a maximum near 1050° to 1200°C. At the maximum value, Young's modulus was more than 11% higher and the shear modulus was about 9% higher than their room-temperature values. Poisson's ratio was then computed to rise from about 1/6 at room temperature to about 1/5 at the temperature of maximum elastic modulus. Small but significant differences were observed in the temperature-modulus curves for specimens from different sources. These differences were found to be related to differences in the infrared transmission curves.     

Elastic Properties of Mullite

Using ultrasonic methods, we determined the ambient-temperature elastic constants of dense (99.7%) hot-pressed polycrystalline 3:2 mullite (3Al₂O₃2SiO₂). We report the usual polycrystal elastic constants: Young's, shear, and bulk moduli, and the Poisson's ratio (nu). The Poisson's ratio (nu = 0.280) suggests interatomic bonding that is very different from that of either alumina or silica. Temperature dependences of the elastic constants show a strong irregularity; for example, nu decreases as the temperature increases. This irregularity suggests an internal-state change that is perhaps related to the incommensurate structural modulation.     

Damage Development and Moduli Reductions in Nicalon—Calcium Aluminosilicate Composites under Static Fatigue and Cyclic Fatigue

Unidirectional and cross-ply Nicalon fiber-reinforced calcium aluminosilicate (CAS) glass-ceramic composite specimens were subjected to tension–tension cyclic fatigue and static fatigue loadings. Microcrack densities, longitudinal Young's modulus, and major Poisson's ratio were measured at regular intervals of load cycles and load time. The matrix crack (0° plies) density and transverse crack (90° plies) density increased gradually with fatigue cycles and load time. The crack growth is environmentally driven and depends on the maximum load and time. Young's modulus and Poisson's ratio decreased gradually with fatigue cycles and load time. The saturation crack densities under fatigue loadings were found to be comparable to those under monotonic loading. A matrix crack growth limit strain exists, below which matrix cracks do not grow significantly under fatigue loading. This limit coincides with the matrix crack initiation strain. Linear correlations between crack density and moduli reductions obtained from quasi-static data can predict the moduli reductions under cyclic loading, using experimentally measured crack densities. A logarithmic correlation can predict the Young's modulus reduction in a limited stress range. A fatigue crack growth model is proposed to explain the presence of two distinct regimes of crack growth and Young's modulus reduction.     

Elastic Properties of Polycrystalline Ytterbium Oxide

The sonic resonance technique was used to determine the elastic moduli of polycrystalline Yb₂O₃ samples of rectangular cross section ranging in pore fraction from 0.058 to 0.27 at temperatures from 25° to 1000°C. The data showed that the Young's and shear moduli are best related to pore fraction by a linear equation and to temperature by the Wachtman equation. The Debye temperature for Yb₂O₃ was calculated from the Young's and shear moduli to be 385°K.     

Relation of Single-Crystal Elastic Constants to Polycrystalline Isotropic Elastic Moduli of MgO: II, Temperature Dependence

The technical adiabatic elastic moduli E_[hkl] and G_hkl of single crystals of magnesium oxide were measured over the temperature range 298° to about 1600°K by a Förster-type resonance method. These data were compared with the low-temperature values (80° to 560°K) of the principal elastic constants c_ij and coefficients Sy reported by Durand. Combining Durand's data and the present data, the elastic moduli for single-crystal magnesium oxide were evaluated for the temperature range 80° to 1600°K. Young's modulus and the shear modulus of densely formed isotropic polycrystalline magnesium oxide were measured over the temperature range 298° to 1600°K. The data on the elastic constants of the single crystals were compared with the measured elastic moduli of the isotropic polycrystalline magnesium oxide on the basis of the Voigt-Reuss-Hill approximation. The temperature dependence of the elastic moduli was fitted into the expression M = M_c— BT exp (—T_c/T) suggested by Wachtman et al. ; mean deviations were less than 0.4% for the temperature range considered. The significance of the present data is discussed with particular emphasis on the following points: (1) the temperature variation of the elastic modulus is a function of thermal expansion, (2) the temperature dependence of the elastic modulus can be well described by the foregoing expression for a wide range of temperature, (3) the expression gives a value of the elastic modulus at 0°K, and (4) it may be possible to make use of measurements on the elastic properties of a densely sintered polycrystalline material to obtain information heretofore obtainable only from the corresponding single-crystal data.     

Elastic Moduli of Transparent Yttria

The elastic moduli of yttria (Y₂O₃) samples that were made from powders with various particle morphologies were studied by means of ultrasonic measurements. The soundwave velocities in the longitudinal and transverse modes were measured. The elastic moduli were calculated from the sound velocities and density. For the high-purity, high-density (>5000 kg/m³) Y₂O₃ that was prepared in the present study, the average density and elastic moduli (and their standard deviations) were as follows: density (ρ) of 5020 ± 18 kg/m³, Young's modulus ( E ) of 179.8 ± 4.8 GPa, shear modulus ( G ) of 69.2 ± 2.0 GPa, bulk modulus ( B ) of 148.9 ± 3.0 GPa, and Poisson's ratio (ν) of 0.299 ± 0.004. The average longitudinal and transverse soundwave velocities ( V _l and V _t, respectively) were 6931 ± 65 and 3712 ± 49 m/s, respectively. The elastic moduli of lanthana-strengthened yttria (LSY) were ∼6% lower than those of high-purity Y₂O₃, and the nu value for LSY was ∼0.304. It has been argued that soundwave velocity is better than density, in regard to predicting the elastic moduli of fully dense and slightly porous materials. A linear equation that describes the change of the elastic moduli with soundwave velocity alone has been suggested. This equation was applicable to a relative elastic moduli range of 0.75–1.02.     

Elastic Constants of Erbia Single Crystals

The elastic properties of single-crystalline erbia (Er₂O₃) at room temperature have been investigated using resonant ultrasound spectroscopy. The three independent stiffness constants of anisotropic Er₂O₃ cubic type-C crystals have been measured. The values of the stiffness constants were c ₁₁= 256.4 GPa, c ₁₂= 146.8 GPa, and c ₄₄= 75.2 GPa. From the stiffness constants, the estimated values for dense polycrystalline erbia for Young's modulus, the shear modulus, the bulk modulus, and Poisson's ratio at room temperature were 179 GPa, 67 GPa, 183.3 GPa, and 0.337, respectively. The value of Young's modulus is a minimum along [001] and a maximum along [111]. The value of the shear modulus is independent of the direction in the (001) and (111) planes, whereas it decreases in (11¯0) from 75 GPa along [001] to 55 GPa along [110].     

Elastic Properties of Silica Xerogels

To calculate elastic constants, longitudinal and tranverse acoustic wave velocities were measured for silica xerogels as a function of relative humidity (rh). The silica xerogels studied are microporous with open porosity of 53 vol%. The longitudinal wave velocity exhibits a minimum at about 35% rh. The transverse wave velocity decreases to a constant value for 35% rh. Consequently, Young's modulus is a minimum at about 35% rh, whereas the shear modulus decreases to a constant value at 35% rh. The bulk modulus and Poisson's ratio exhibit minimum values at about 15% rh. Young's modulus decreases from 4.91 to 3.42 GPa at 35% rh and then increases to 3.60 GPa at 55% rh. Poisson's ratio decreases from 0.184 to 0.164 at 15% rh and then increases to 0.272 at 55% rh. Below 35% rh, silica xerogels adsorb a monolayer of hydroxyls, whereas above 35% rh silica xerogels show pore filling.     

Elastic Moduli of Porous Sintered Materials as Modeled by a Variable-Aspect-Ratio Self-Consistent Oblate-Spheroidal-Inclusion Theory

If the porosity of sintered materials is modeled by oblate spheroids, self-consistent elastic moduli theory can be used to determine an effective aspect ratio for the spheroids from experimental data on elastic moduli vs porosity. The resultant effective aspect ratio serves as an additional parameter to describe observed elastic behavior. With the exception of one case in the seven investigated, this single parameter satisfactorily describes both Young's modulus and shear modulus vs porosity data. The fit to these data sets is tested by comparing theoretical and experimental values for bulk moduli and Poisson ratios. An effective aspect ratio is also shown to accurately describe a continuous distribution of aspect ratios varying between spheres and disks, implying that a continuous distribution of oblate spheroids may be a useful model for the elastic behavior of sintered materials.     

Comparison of Mechanical Properties of Silicon Nitrides with Controlled Porosities Produced by Different Fabrication Routes

Three different series of porous silicon nitride ceramics with volume fraction porosities in the range 0–0.5 were fabricated using different preparation routes: (i) partial sintering, (ii) the addition of fugitive inclusions, and (iii) partial hot pressing. The use of different sintering additives and firing conditions, depending on the preparation route, gives rise to different materials within a certain porosity range with variations in terms of microstructure and grain boundary phase. Mechanical properties, elastic moduli, and strength have been evaluated separately for each series of materials. Porosity dependences of Young's modulus, shear modulus, Poisson's ratio, and fracture strength have been assessed and a comparison of the different materials is made and discussed in relation to their microstructural features.     

Bulk Modulus and Young's Modulus of Nanocrystalline γ-Alumina

Compression measurements were performed for the first time on nanocrystalline γ-alumina utilizing a diamond anvil cell (DAC) and the energy dispersive X-ray diffraction method. The cubic unit cell ( a = 0.7924 nm) for γ-alumina was found to have a volume compression of about 2.4% over the pressure range from ambient to 3.8 GPa at room temperature under both hydrostatic and nonhydrostatic conditions. Using the first-order Bridgman equation and the Birch equation of state, the isothermal bulk modulus ( B ₀) was determined to be 162 ± 14 GPa and Young's modulus ( E ) was estimated to be 253 ± 22 GPa assuming a Poisson's ratio for γ-alumina of 0.24 ± 0.2.     

Elastic Behavior of Polymer-Impregnated Porous Ceramics

The effect of polymer impregnation on the elastic behavior of porous ceramics was investigated. Large increases in the Young's, shear, and bulk moduli and Poisson's ratio were observed. On the basis of a two-dimensional model containing elliptical inclusions, it was shown that the primary role of the polymer on impregnation is to significantly offset the original reduction in elastic moduli caused by the pore phase. The relative effect of polymer impregnation on elastic behavior increases with increasing pore volume fraction and pore ellipticity. These conclusions are also expected to explain the elastic behavior of polymer-modified cements and concretes.     

Mechanical Properties of Chemically Strengthened Glasses at High Temperatures

Young's modulus, shear modulus, and modulus of rapture for two chemically strengthened glasses were determined at high temperatures. The Young's modulus and shear modulus decreased with increasing temperature, with a sharp inflection slightly above room temperature. The region of inflection indicated an internal friction peak. For comparison Young's modulus and shear modulus were determined as a function of temperature on a thermally tempered soda-lime-silica glass and on a semitempered borosilicate glass. Curves of these moduli, in contrast to those for the chemically strengthened glasses, did not reveal regions of inflection. The modulus of rupture is not affected by short exposure to heat up to 260°C., but decreases appreciably when exposed to temperatures above 204°C for 200 hr or more. Deflection measurements at room temperature showed that the two chemically strengthened glasses had about five times as much delayed elasticity as did thermally tempered soda-lime-silica glass.     

Elastic Properties of Nickel-Based Anodes for Solid Oxide Fuel Cells as a Function of the Fraction of Reduced NiO

The changes in porosity and elastic moduli of YSZ-containing nickel-based anode materials for solid oxide fuel cells were studied as a function of the fraction of reduced NiO. Anode samples were reduced in a gas mixture of 4% hydrogen and 96% argon for different periods of time at 800°C and their Young's and shear moduli were determined afterward at room temperature using resonant ultrasound spectroscopy and impulse excitation. It was found that the magnitude of Young's and shear moduli decreased significantly with increasing fraction of reduced NiO and that the magnitude of the elastic moduli of a fully reduced Ni–YSZ anode was ∼45% lower than that of unreduced NiO–YSZ. Because the elastic moduli of NiO are close to those of Ni, the observed decrease in the magnitude of the elastic moduli was found to be caused mainly by the significant increase in the porosity of the sample as a result of NiO reduction. Expressions are presented for the amount of porosity and the magnitude of the elastic moduli as a function of the fraction of reduced NiO.     

Elastic Properties of Model Porous Ceramics

The finite-element method (FEM) is used to study the influence of porosity and pore shape on the elastic properties of model porous ceramics. Young's modulus of each model is practically independent of the solid Poisson's ratio. At a sufficiently high porosity, Poisson's ratio of the porous models converges to a fixed value independent of the solid Poisson's ratio. Young's modulus of the models is in good agreement with experimental data. We provide simple formulas that can be used to predict the elastic properties of ceramics and allow the accurate interpretation of empirical property–porosity relations in terms of pore shape and structure.