Transient Creep in Fe-Doped Polycrystalline MgO

The transient creep rate in polycrystalline Fe-doped MgO (grain size 4 to 19 μm) follows the exponential time relation     The initial strain rate,     , depends linearly on stress, and the relaxation time, τ, is thermally activated. Two mechanisms for the transient were considered, i.e. (1) viscous grain-boundary sliding with elastic accommodation and (2) transient diffusional creep. Although the results are not conclusive, transient diffusional creep appears to be the most promising mechanism.     

Deformation of Polycrystalline MgO Under Pressure

Polycrystalline MgO appears to deform ductilely when it is strained under confining pressures greater than 2 kbars at room temperature. Although the {110}〈110〉 slip systems in MgO do not provide the five independent slip systems required for homogeneous deformation, the lack is apparently met in part by the activity of additional slip systems and in part by microfracture. The strong pressure dependence of the stress-strain curves probably arises from the pressure sensitivity of microfracture, which becomes less important as the temperature is raised; high-temperature experiments at 2 and 5 kbars confining pressure show that the pressure dependence virtually disappears at 750°C and that there is a simultaneous trend toward wavy slip. High-temperature experiments on single crystals oriented for cube slip produced only kinking below 300°C; above 300° there is some cube slip, but it is not clear that this is a major deformation mechanism. The pressure dependence of the fracture and flow stresses is interpreted in terms of recent theories of crack propagation under pressure.     

Deformation of Polycrystalline MgO at Elevated Temperatures

Stress-strain curves for five types of polycrystalline MgO are presented as a function of temperature. All types were nominally dense and pure but differed in grain size, composition, and porosity. Above 1200°C deformation occurred by grain boundary shearing accompanied in some cases by slip; below 800°C, specimens fractured primarily by grain boundary parting with little permanent strain. Between 800° and 1200°C, one type deformed plastically by slip; the other four types were brittle. The observed behavior is analyzed in terms of the presence of mobile dislocations, the resistance to dislocation motion, and the strength of grain boundaries.     

Effect of Microstructure on Deformation of Polycrystalline MgO

Six types of polycrystalline MgO, one nominally fully dense and the others having porosities of ∼1 to 2%, were tested in compression up to 1400°C. At 1200°C and above, all materials deformed plastically; two of the materials, both porous, also exhibited plastic flow at temperatures down to 800°C, and a third at 1000°C. A qualitative analysis of the microstructures of these materials indicated that the differences in behavior arose primarily because of variations in the size and distribution of pores and in the concentration of impurities at the grain boundaries. It is suggested that the following factors aid plasticity below ∼1200°C: (1) Strong grain boundaries, in the absence of excessive impurities, permit buildup of stress concentrations with consequent nucleation of slip on the {100} system and the extension of slip across the boundaries and (2) clusters of very fine pores within the grains allow some mass accommodation.     

Deformation of Polycrystalline MgO at High Hydrostatic Pressure

Vacuum-hot-pressed MgO polycrystals were deformed in compression at hydrostatic pressures up to 1000 MPa in an effort to obtain gross plastic deformation at room temperature. Although transmission electron microscopy observations indicated an increase in dislocation density at pressures > 400 MPa, direct optical observations during deformation indicated that microfracturing persists as a mode of deformation throughout the stress-strain curves at even the highest test pressures. As in previous investigations, pressures > 400 MPa led to a transition from completely brittle to ductile behavior, as indicated by stress-strain plots, but direct optical observations suggested that the apparent transition is an effect of pressure on the mode of crack propagation.     

High-Temperature Creep and Dislocation Etch Pits in Polycrystalline Beryllium Oxide

Compressive creep of high-density polycrystalline beryllium oxide was investigated in the range 1850° to 2050°C. Creep rate was dependent on the applied stress to the 2.5 power, and the apparent activation energy for creep was 145 kcal/mole. Etch pit studies showed that the dislocation density in tested specimens was two orders of magnitude greater than that in assintered material. The diffusion process controlling creep was ascribed to volume diffusion of the anion. The deformation behavior was governed by dislocation motion.     

Influence of Surface Condition on the Strength of Polycrystalline MgO

The mechanical behavior of high-purity (greater than 99.9%) polycrystalline MgO, with relative density exceeding 9875, is described. By fracto-graphic techniques, it was determined that inter-granular cracks initiate fracture. These cracks originate from the extension of surface flaws into the MgO and are particularly severe in areas of high, localized, intergranular porosity. Ceramic machining techniques, such as diamond sawing and lapping with coarse abrasives, induce these surface flaws and limit the strength of MgO. The effectiveness of lapping, thermal conditioning, and chemical polishing techniques in removing these flaws and producing a substantial increase in mechanical strength is shown to be dependent on grain size and porosity.     

Dislocation Structures in Sapphire Deformed by Basal Slip

Dislocation structures in sapphire deformed by basal slip at 1200° to 1500°C were studied by transmission electron microscopy. Long dislocation dipoles of edge character are formed by interactions of dislocations on parallel basal planes ("edge-trapping"); screw dislocations annihilate by cross-slip. The accumulation of dislocation dipoles leads to work-hardening, but the dipoles also break up into smaller loops by climb, causing recovery. Eventually a steady state of zero work-hardening is reached, where the rate of accumulation of dipoles by edge-trapping is equal to their rate of annihilation by climb. From these observations, it is suggested that dislocation climb plays an important role in the deformation process for basal slip in the range 1200° to 1500°C.     

Modeling of Probabilistic Failure of Polycrystalline Silicon MEMS Structures

Microelectromechanical Systems (MEMS) devices typically need to be designed against a very low failure probability, which is on the order of or lower. Experimental determination of the target strength for such a low failure probability requires testing of tens of thousands of specimens, which can be cost prohibitive for the design process. Therefore, understanding the probabilistic failure of MEMS devices is of paramount importance for design. Currently available probabilistic models for predicting the strength statistics of MEMS structures are based on classical Weibull statistics. Significant advances in experimental techniques for measuring the strength of MEMS devices have produced data that have unambiguously demonstrated that the strength distributions consistently deviate from the Weibull distribution. Such deviations can be explained by the fact that the Weibull distributions are derived based on extreme value statistics, which is inapplicable to MEMS devices where the dimensions of the material microstructure are not negligible compared to the characteristic structural dimensions. This paper presents a robust probabilistic model for strength distribution of polycrystalline silicon (poly‐Si) MEMS structures that could be extended to other brittle materials at the microscale. The overall failure probability of the structure is related to the failure probability of each material element along its sidewalls through a weakest‐link statistical model. The failure statistics of the material element is determined by both the intrinsic random material strength as well as the random stress field induced by the sidewall geometry. Different from the classical Weibull statistics, the present model is designed to account for structures consisting of a finite number of material elements, and it predicts a scale effect on their failure statistics. It is shown that the model agrees well with the measured strength distributions of poly‐Si MEMS specimens of different sizes, and the calibrated mean strength of the material element is consistent with the theoretical strength of silicon. Meanwhile, it is shown that the two‐parameter and three‐parameter Weibull distributions cannot provide optimum and consistent fits of the observed size‐dependent strength distributions, and thus have very limited prediction capability. The present model explicitly relates the strength statistics to the size effect on the mean structural strength, and therefore provides an efficient means of determining the failure statistics of MEMS structures.     

Relation of Single-Crystal Elastic Constants to Polycrystalline Isotropic Elastic Moduli of MgO

The three elastic compliance coefficients of synthetic periclase were determined in the kilocycles per second frequency range by a resonance method. Polycrystalline elastic moduli on dense-formed MgO were measured. The calculated isotropic elastic moduli for polycrystalline MgO obtained from the single-crystal constants are in good agreement with experimental values measured on the dense MgO at the theoretical density point. The measured Young's modulus and the shear modulus of polycrystalline MgO are found to be 30.50 and 12.90 × 10¹¹ dynes per cm² respectively. The results of the present investigation are compared with the earlier work in the megacycles per second frequency range. A theoretical analysis is made to establish the validity of the present values. Schemes of averaging the single crystalline elastic constants for polycrystalline behavior are reviewed for crystals of cubic symmetry.     

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.     

Dislocation Structures of Low-Angle and Near-Σ3 Grain Boundaries in Alumina Bicrystals

Alumina bicrystals with low-angle and near-Σ3 <0001> tilt grain boundaries were fabricated using diffusion bonding to study the dislocation structures in alumina grain boundaries. The resulting grain-boundary structures were investigated using high-resolution transmission electron microscopy, and the grain-boundary energies were analyzed using theoretical calculations. It was found that partial dislocations with Burgers vectors of the type return ⅓<10[Onemacr]0> were periodically located in the boundaries and that a stacking fault between pairs of partials was formed in such boundaries. The length of the stacking fault decreased with increased misorientation angles, which was reasonably predicted by the theoretical calculation. In the case of a near-Σ3 grain boundary, an array of displacement shift complete dislocations with the Burgers vector of return ⅓<1[Onemacr]00> was periodically formed along the boundaries. These boundaries did not have stacking faults. The spacing between the dislocations decreased with increased deviation angle from the exact-Σ3 boundary with the tilt angle of 60°.     

Effect of Hot Extrusion, Other Constituents, and Temperature on the Strength and Fracture of Polycrystalline MgO

Improved agreement was confirmed between the Petch intercept and single-crystal yield stresses at 22°C. Hot-extruded MgO crystal specimens (recrystallized with no obvious grain boundary phases or residual porosity), stressed parallel with the resultant (100) axial texture (1) gave the highest and least-scattered strength–grain size results at 22°C, (2) showed direct fractographic evidence of microplastic initiated fracture at 22°C and showed macroscopic yield at 1315° and especially 1540°C, and (3) fractured entirely via transgranular cleavage, except for intergranular failure initiation from one or a few grain boundary surfaces exposed on the subsequent fracture surface, mainly at 1540°C. Hot-extruded, hot-pressed MgO billets gave comparable strength when fracture initiated transgranularly, but lower strength when fracture initiated from one or especially a few grain boundary surfaces exposed on the fracture (with residual pores). The extent and frequency of such boundary fracture increased with test temperature. While oxide additions of ≤5% or impurities in hot-pressed or hot-extruded MgO can make limited strength increases at larger grain sizes, those having limited solubility can limit strength at finer grain sizes, as can coarser surface finish (the latter especially at 22°C). Overall, MgO strength is seen as a balance between flaw and microplastic controlled failure, with several parameters shifting the balance.     

Surface Diffusion in MgO and Cr-Doped MgO

The surface diffusion of MgO and Cr-doped MgO was examined using grain-boundary grooving. Measurements were taken at T=1100° to 1400°C for a concentration range of 0 to 0.442 at.% Cr. The activation energy for surface diffusion of MgO was found to be 239± kj/mol. The surface diffusivity did not change markedly with chromium concentration. How-ever, an abrupt change in activation energy from ∼240 to ∼180 kj/mol occurred between 0.038 and 0.254 at % Cr, which may be associated with a change in surface structure.     

High-Temperature Elastic Properties of Polycrystalline MgO and Al2O3

Adiabatic bulk modulus, B_s , of polycrystalline MgO and Al₂O₃ was measured from 298° to 1473°K using the resonance technique. The Grüneisen constant, calculated from the measured bulk modulus, was constant over the whole temperature range (1.53 for MgO and 1.34 for Al₂O₃). Another important parameter,     , is constant at high temperature and is 3.1 for MgO and 3.6 for Al₂O₃. The Poisson's ratio increases linearly with temperature for MgO and Al₂O₃. To describe the change of bulk modulus with temperature a theoretical equation was verified by using the foregoing constants. A practical form of this theoretical equation is where B_s⁰ is the adiabatic bulk modulus at 0°K, δ is the quantity     , γ is the Grüneisen constant, H is the enthalpy. The experimental data are described very well by this equation, which is equivalent to the empirical equation suggested by Wachtman et al., B_s^T= B_s⁰ - CT exp (-T_c/T) , where C and T_c are empirical constants.     

Dislocation Structures in Single-Crystal Al2O3

Chemical polishing and etching techniques were used to reveal the dislocation structures of sapphire and ruby crystals grown by the flame-fusion and flux techniques. The average density of edge dislocations lying in prism planes was 3.0 × 10⁵ per cm², which could be changed only slightly by chromium additions and annealing at 2000°C. An average basal dislocation density of 2 × 10⁵ per cm² decreased 35 to 80% on annealing. Crystal orientation (i.e., angle between the c axis and the growth axis) showed no effect on dislocation density but a pronounced effect on subboundary arrangement and density. The substructure of 0° crystals was more complex than that of 90° crystals; 60° crystals possessed a structure intermediate between 0° and 90°. Principal observations included (1) prismatic and basal slip on all as-grown crystals; (2) profuse basal slip, readily polygonized on annealing; (3) dislocation densities of flux crystals lower than those of Verneuil crystals; and (4) a rare form of basal twinning, composition plane , on all flux crystals.     

Porosity Effects in Polycrystalline Graphite

Thermal and physical properties of a series of porous, polycrystalline graphites have been measured. The solid matrix material was characterized by X-ray diffraction measurements of crystallite size, interplanar spacing, and anisotropy. The pore structure was examined using electron micrography, porosimetry, permeability, and helium density measurements. Thermal and electrical conductivities and elastic modulus measurement results were correlated with the fractional porosities of the graphites.