Isotropic Aggregates Research Articles

Theoretical analyses of 〈111〉 pencil glide in b.c.c. crystals

A new theoretical analysis of 〈111〉 pencil glide in b.c.c. crystals is proposed. Complete sets of equations are given for one, two, three or four active slip systems. The yield surface for b.c.c. single crystals deforming by pencil glide is described, and the closed subspaces are listed. The cases where stress or shear rate and rotation ambiguities occur are also detailed. A correspondence is shown to exist between the plastic deformation rates that are induced by the classical eight solutions in pencil glide, and those associated with the Bishop and Hill vertices for {110}〈111〉 slip. Each six-fold vertex in groups B and D is related to a particular three-system pencil glide solution, and each eight-fold vertex in groups A, C, and E with one four-system solution as well as four three-system solutions. The analysis shows that the cones of normals of the Bishop and Hill vertices have very different extensions, the solid angle pertaining to the group C vertices being an order of magnitude greater than those of group A. The Taylor (prescribed strain) problem for b.c.c. crystals is treated in this way and it is shown that the necessary calculations are reduced by factors of three to six when compared with the previous methods. In particular the number of three-system solutions to be analyzed, which require a special numerical procedure, is reduced to a minimum. The Taylor factor for isotropic aggregates of b.c.c. crystals with pencil glide is also calculated for various imposed strain rates. Finally, the well known analyses of pencil glide available in the literature are compared; the correspondence between the notations is tabulated, and the approaches and sets of results obtained are evaluated and discussed in turn.

Rayleigh wave speeds in transversely isotropic materials

Equations from which the Rayleigh (surface) wave speed can be determined are given for propagation on a transversely isotropic, linear elastic half-space. These equations are obtained from the vanishing of the determinant of a real symmetric second-order tensor which arises from the six-dimensional formalism of Barnett and Lothe [Modern Problems in Elastic Wave Propagation, edited by J. Miklowitz and J. D. Achenbach (Wiley, New York, 1978)]. To evaluate the components of this tensor, 18 integrals are evaluated explicitly. The resulting equations are discussed for a transversely isotropic aluminum aggregate with an idealized 〈111〉 texture.

An Acoustoelastic Theory for Surface Waves in Anisotropic Media

A theory for surface waves in an anisotropic material is developed in the framework of acoustoelasticity in which the material’s strain energy density is taken to be a cubic function in the strain. In order to relate the surface wave speeds to the applied stress, a configuration is introduced in which the effect of the local rotation is removed. The development shows that the surface wave speed can be determined from the eigenvalues of a particular real symmetric 2×2 matrix. Numerical results are given for uniaxial loading applied to aluminum and copper single crystals and to an ideal transversely isotropic aggregate of aluminum.

Variations in the second- and third-order elastic constants in polycrystalline aggregates

Expressions for the expected variation in the effective second- and third-order elastic constants of statistically isotropic polycrystalline aggregates with a finite number of crystallites are developed. The analysis is based on the Voigt approach for the effective response of an aggregate and is valid for cubic crystals of the highest symmetry (m3m,432,4̄3m). In this approach, the effective stiffnesses are the averages of the stiffnesses in the constituent crystals. It is shown that the expected variance in the effective elastic stiffness is inversely proportional to the number of crystallites contributing to the average. Numerical evaluations for crystals of various materials show that the variations in the third-order stiffnesses are substantially larger than the variations in the second-order stiffnesses.

An Elastic Stress-Strain Relation for Sphere Arrays Undergoing Initial Stage Sintering

An analytical model for predicting the elastic constants for isotropic random sphere arrays is described. The model is based on the Hertz contact theory and accounts for the restructuring (neck growth) associated with initial stage sintering. The stress-strain relation is incremental since the stiffness depends on the loads at sphere contacts. As part of the model development, relationships are given for the elastic constants of an isotropic aggregate consisting of orthotropic parts. Predicted elastic constants are presented as functions of the neck size and hydrostatic compression, and comparisons are made with experimental data for cohesionless sphere arrays and fully restructured arrays approaching a porous body.

Effective elastic constants of polycrystalline aggregates

A method is presented for the determination of the effective elastic constants of a transversely isotropic aggregate of weakly anisotropic crystallites with cubic symmetry. The results obtained generalize those given in the literature for the second and third order elastic constants. In addition, the second moments and the binary angular correlations of the second order stiffnesses are obtained. It is also explained how these moments can be used to find the two-point correlations of the elastic constants.

Evaluation of the elastic moduli of a transversely isotropic aggregate of cubic crystals

The crystals and the aggregate have the same bulk modulus. The other three overall elastic moduli in the simplified stress-strain relations of Walpole (1985) are placed between upper and lower, Voigt and Reuss, bounds and some exact calculations are given for particular fibre textures.

Thermal expansion of polycrystalline aggregates: I. Exact analysis

An exact relation is developed between the thermal expansion coefficient and the bulk modulus of statistically isotropic polycrystalline aggregates composed of crystals of hexagonal, tetragonal or trigonal symmetry. This relation is exploited to derive simple close bounds for the thermal expansion coefficient in terms of single crystal properties. Comparison of bounds to experimentally obtained expansion coefficients shows fair to very good agreement.

Thermal expansion of polycrystalline aggregates: II. Self consistent approximation

T he S elf Consistent Scheme approximation method is applied to evaluate the thermal expansion coefficient of statistically isotropic polycrystalline aggregates, in terms of single crystal thermoelastic properties and polycrystal elastic properties. The case of orthorhombic crystals is considered in detail.

Acoustoelastic response of polycrystalline aggregates exhibiting transverse isotropy

Acoustoelasticity is an ultrasonic technique which has been used for the determination of active and residual stresses in common structural materials. This paper examines the effect of texture on the acoustoelastic response in polycrystalline bodies. In particular materials which are transversely isotropic aggregates of cubic crystals are studied. The second- and third-order elastic constants of the polycrystal are derived from the elastic properties of the constituent crystals, and the crystalline orientation relative to the body's symmetry axis. The acoustoelastic relations between velocity and deformation are then presented for the aggregate. Finally, evaluation of the acoustoelastic response for several ideal textures using data for aluminum single crystals shows that the response is highly dependent on the texture.

Seismic velocities of Lewisian metamorphic rocks at pressures to 8 kbar: relationship to crustal layering in North Britain

Summary P- and S-wave velocities on a set of rocks from the Archaean Lewisian metamorphic complex, north-west Britain, have been measured in the laboratory at confining pressures to 8 kbar. From estimates of velocities at crustal pressures and temperatures, it is concluded that the composition of the three crustal layers below the Scottish Caledonides are most likely to be, in order of increasing depth: (1) mixed metamorphic rocks in amphibolite or lower facies, plus granites, (2) mixed pyroxene-granulite facies rocks, of overall intermediate composition and (3) mafic garnet granulites. Poisson's ratios of isotropic aggregates of mafic amphibolites and serpentinized peridotites in the Lewisian are high (0.28-0.31) because of the presence of hydrous minerals. Only a highly preferred orientation of minerals in such rocks in the lower crust would lead to observed values of Poisson's ratio of about 0.25.

The elastic properties of (Mg x Fe1?x )O solid solutions

The velocities of compressional (V p) and shear (V s) waves in five well-characterized polycrystalline aggregates of (Mg x Fe1−x )O (0.85≧x>0.23) have been measured as functions of hydrostatic pressure (to 6 kbar) using the techniques of ultrasonic pulse transmission and super-position. Both velocities decrease systematically with increasing iron content. The elastic properties of stoichiometric FeO are inferred by inverting the solid solution data using a model in which the single crystal compliances (s ij) vary linearly with the volume fraction of either end member. This model, along with identification of the aggregate moduli with the arithmetic average of the Hashin-Shtrikman bounds, may be used to describe the variation of both bulk (K s) and shear modulus (μs) with composition for polycrystalline aggregates of phases of arbitrary symmetry. For the case of bulk modulus of an isotropic aggregate of crystals of cubic symmetry, the present model yields the formula first proposed by Liu (1968). The isotropic elastic moduli of FeO thus obtained (K s =1.82±0.05 Mbar, μ s =0.59±0.02 Mbar) are consistent with the wustite data of Mizutani et al. (1972) but inconsistent with bulk moduli derived from static compression data by Mao et al. (1969) and Rosenhauer et al. (1976). An alternative description of the variation of μ with x based on consideration of the solid solutions as two-phase aggregates is also presented. The observed differences in elastic moduli between MgO (K s=1.63 Mbar, μs=1.31 Mbar) and FeO (K s=1.82 Mbar, μs=0.59 Mbar) are consistent with the effects of Mg/Fe substitution in other oxide and silicate crystal structures.

Elasticity of single-crystal MgF2 (rutile structure) under pressure

The elastic moduli of single-crystal MgF 2 (rutile structure) have been measured ultrasonically to 7 kbars. Zero-pressure moduli are in good agreement with the results of others. The pressure derivative of the shear modulus c s = ( c 11 − c 12 )/2is−0.7 ± 0.1 which is consistent with the negative values also found in rutile-structure oxides. Estimates of the isotropic aggregate modulus derivatives are in good agreement with derivatives measured on polycrystalline aggregates by Rai and Manghnani. The calculated isotropic aggregate bulk modulus derivative is K ′ = 5.1 ± 0.2, which is lower than for the rutile-structure oxides, and comparable to that for other fluorides. More covalent bonding may increase the value of K ′, and hence the value for stishovite (SiO 2 , rutile) may be quite high.

Quadrupole Splittings in Deuterium NMR-Spectra of the Hexagonal Phase in the System Sodium Octanoate-d15-Water- Carbontetrachloride

Abstract The hexagonal phase of the system C7D15COONa-H2O-CCI4 has been studied by deuterium magnetic resonance. Order parameters for the different CD2-groups have been determined from the observed deuterium splittings in powder spectra. The order parameters range from 0.02 to 0.13. The order parameters decrease when CCI4 is solubilized, implying an increased motional freedom of the alkyl chain. For some samples a central, unsplit peak is observed originating from isotropic metastable aggregates.

Elastic moduli and anisotropy of dunite to 10 kilobars

Longitudinal and transverse wave velocities are reported for several directions at pressures to 10 kb in two samples of dunite from the Twin Sisters peaks, Washington. For both dunites, elastic-wave propagation is controlled to a large extent by olivine fabric. Dunite A, which has a strong concentration of olivine a crystallographic axes and girdles of b and c axes, is uniaxial in elastic properties. The longitudinal wave velocity at 10 kb for propagation parallel to the olivine a axes maximum is 8.76 km/sec. For propagation normal to the a axes maximum, longitudinal wave velocities are low (Vp=7.98 km/sec at 10 kb) and two transverse waves (Vs=4.41 and 4.69 km/sec at 10 kb) are clearly transmitted through the rock. Dunite B, with strong concentrations of all three olivine crystallographic axes, is similar in elastic properties to orthorhombic crystals with a high longitudinal wave velocity (9.15 km/sec at 10 kb) along the olivine a axes maximum and a low longitudinal wave velocity (7.83 km/sec at 10 kb) along the olivine b axes concentration. Elastic stiffnesses and compliances were computed from the velocities, and the physical properties of isotropic aggregates of the two dunites were calculated using the Voigt and Reuss averaging techniques. Primarily because of the presence of accessory minerals in the dunites, the Voigt and Reuss velocities are lower than values computed from olivine single-crystal data. The high pressure gradient (∂Vp/∂P=17.0 km sec−1 mb−1) observed for the longitudinal velocity of dunite B at 8 kb is interpreted as being due to the effect of grain boundary cracks.

Equation of state of polycrystalline and single-crystal MgO to 8 kilobars and 800°K

MgO has been measured again. Most measurements of the elastic properties of materials, interesting to geophysics, have been made as a function of pressure at room temperature or as a function of temperature at atmospheric pressure. A lapped seal between a buffer rod and sample has made it possible to use ultrasonic interferometry to 1000°K and 10 kb. The elastic constants of poly crystalline and single-crystal MgO were measured in a gas high-pressure system over a temperature range from 300°K to the Debye temperature of MgO. Data from the polycrystalline specimen indicated large effects of temperature on the pressure derivatives. These data did not agree with the results obtained from single-crystal measurements. Upon remeasuring the ceramic sample it becomes apparent that the data are not reproducible after the sample has been cycled to high temperature and pressure. Additional sintering, deformation of the individual grains, and recrystallization take place, which change the properties of the sample. These problems and the problems of sintering isotropic aggregates of theoretical density limit the usefulness of this widely used procedure. Order from the American Geophysical Union, Suite 435, 2100 Pennsylvania Ave., N.W., Washington D.C 20037. Document J70-001; $1.00.

Interaction of glassy carbon with alkali metal vapours

It is shown by X-ray diffraction techniques that potassium vapour intercalates into non- graphitizing carbons, such as glassy carbon, to the maximum extent in recognizable stages. The resultant anisotropic expansions of the component crystallites of an isotropic aggregate are shown to be sufficient to cause violent disruption in larger pieces. Sodium vapour also causes disruption, but the temperatures required are considerably higher than for potassium. The onset of disruption in potassium vapour is characterized by a delay time which depends on the holding temperature and an activation enthalpy of about 1.4 x 105J/mol.

The theoretical behavior of a polycrystalline solid as related to certain general concepts of continuum plasticity

This paper is directed toward establishing general characteristics of continuum stress-plastic strain relations from Schmid's Law of plastic slip in individual crystal grains. To this end certain theoretical results obtained by Lin are reaffirmed through a rigorous derivation, and the principle of maximum plastic work is extended to small elastic-plastic strains in an isotropic polycrystalline aggregate. It is shown that the macroscopic incremental plastic strain vector over a unit volume of a fine-grained solid is strictly normal to a yield surface in macrostress space only if the individual crystals are elastically isotropic. The resulting equations for polycrystalline solids are contrasted with those obtained from certain stability postulates and thermodynamic foundations in continuum plasticity, and general features of similarity are discussed.

Comments on the negative pressure dependence of the shear modulus found in some oxides

It is shown that the value of dμ/dP (μ being shear modulus) is small and possibly negative for oxide compounds that have small relative shear modulus and have structure in which the coordination of the atoms is small. This behavior is explained by means of an early theory of ZnS, which was originally proposed by Blackman. In Blackman's model the two crystalline shear constants decrease with pressure. Blackman's equations are extended for an isotropic aggregate of ZnS, with the result that dμ/dP is negative and dK/dP is normal (K being the bulk modulus). These results for the isotropic case of Blackman's ZnS model agree remarkably well with the measured values of ZnO.

Equation of State of Periclase and Birch's Relationship between Velocity and Density

BASED on systematic investigation of velocity of compressional waves in isotropic aggregates of oxides and silicates at high pressure, Birch1 observed that velocity is proportional to density for substances with similar mean atomic weight. He concluded that velocity in isotropic aggregates of oxides and silicates is a function of two principal variables—density and mean atomic weight—the variation of density for substances with the same mean atomic weight reflects differences of structure and composition. Birch also postulated that for a given substance, velocity and density may be related in much the same way when they are changed by compression, and this postulate was used in attempts to determine the density distribution in the Earth2. Later, McQueen et al.3 noticed that, for similar materials, the bulk sound velocity is also proportional to density. Many of Birch's determinations of the velocity of compressional waves Vp may now be combined with Simmons's4 determinations of the velocity of shear waves Vs to yield the bulk sound velocity C defined as for these determinations were made in the same laboratory with almost identical samples. Nai-hsien Mao (private communication) has confirmed these findings with many more data. This relationship may be expressed as where ρ is density and a and b are constant coefficients for substances with the same mean atomic weight; both coefficients may be functions of the mean atomic weight.