Showing papers on "Isotropy published in 1972"

Large Deformation Isotropic Elasticity—On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids

Abstract: A method of approach to the correlation of theory and experiment for incompressible isotropic elastic solids under finite strain was developed in a previous paper (Ogden 1972) Here, the results of that work are extended to incorporate the effects of compressibility (under isothermal conditions) The strain-energy function constructed for incompressible materials is augmented by a function of the density ratio with the result that experimental data on the compressibility of rubberlike materials are adequately accounted for At the same time the good fit of the strain-energy function arising in the incompressibility theory to the data in simple tension, pure shear and equibiaxial tension is maintained in the compressible theory without any change in the values of the material constants A full discussion of inequalities which may reasonably be imposed upon the material parameters occurring in the compressible theory is included

The computer study of transport processes under extreme conditions

Abstract: A method is developed to investigate the behaviour of a liquid under the action of a very high shearing force using computer simulated molecular dynamics. Values for the viscosity are calculated but these require more extensive computation for statistical accuracy. A short calculation is, however, sufficient to establish the nature of the processes involved, and to plot the velocity distribution curve; these graphs are presented. The method abandons the usual homogeneous isotropic cyclic conditions used so far in calculation of transport coefficients and can be applied to arbitrarily large shear.

On a class of conservation laws in linearized and finite elastostatics

Abstract: Several years ago ESHELBY [1] (1956), in a paper devoted to the continuum theory of lattice defects, deduced a surface-integral representation for the "force on an elastic singularity or inhomogeneity", which-in the absence of such defects-gives rise to a conservation law for regular elastostatic fields appropriate to homogeneous but not necessarily isotropic solids in the presence of infinitesimal deformations. Morevoer, ESHIELBY noted that his result, when suitably interpreted, remains strictly valid for finite deformations of elastic solids.

Crack propagation in an elastic solid subjected to general loading—II. Non-uniform rate of extension

Abstract: The stress intensity factor of a half-plane crack extending non-uniformly in an isotropic elastic solid subjected to general loading is determined. The loading is applied in such a way that a state of plane strain exists and that crack extension takes place in Mode I. The crack tip is initially at rest and then moves in an arbitrary way in the plane of the crack. In the process of obtaining the stress intensity factor, the complete elastic field is determined for a crack which starts from some initial position, extends at a constant rate for some time, and then suddenly stops. Once the stress intensity factor is known for arbitrary motion of the crack tip, the Griffith fracture criterion is applied to obtain an equation of motion for the crack tip which is consistent with the assumptions of this criterion. Numerical results are included for the stress intensity factor and for the velocity-dependent term in the equation of motion.

An Experimental and Theoretical Investigation of Elastic Wave Propagation in a Cylinder

Abstract: A high‐speed computer was used to investigate the problem of wave propagation in an isotropic elastic cylinder. Dispersion curves corresponding to real, imaginary, and complex propagation constants for the symmetric and the first four antisymmetric modes of propagation are given. The radial distributions of axial and radial displacements and of shear and normal stresses are given for the symmetric mode. By using a finite number of modes of propagation, an approximate solution is found for the problem of the L(0,1) mode impinging on a traction‐free interface. The reflection coefficient is determined in this way and the accompanying generation of higher order modes at the interface is shown to cause a high‐amplitude end resonance. Experimental results obtained by using the resonance method in conjunction with a long rod are presented to substantiate the calculated reflection coefficient and the frequency of end resonance. Phase velocities, based on measurements of the wavelength of standing waves and resonance frequencies, were obtained for the symmetric and first two antisymmetric modes. These measurements extend into the frequency range of more than one propagating mode. The rms deviation between theoretical and experimental results is in general less than 0.2% with the exception of the dispersion curve for the L(0,2) mode which deviates by 0.7%.

Governing Equations for Vibrating Constrained-Layer Damping Sandwich Plates and Beams

Abstract: A simple differential equation is derived to describe constrained-layer damping in nonsymmetric sandwich plates and beams composed of isotropic and homogeneous layers. The natural boundary conditions related to this equation are determined and some typical numerical results obtained by this equation are given. The equation is valid within the linear theories of elasticity and viscoelasticity in the absence of any constraints on thicknesses, positions, symmetries, and densities of the layers.

Central crack in plane orthotropic rectangular sheet

Abstract: The plane problem of a central crack in a rectangular sheet of orthotropic material is considered. The solution is found by an extension of the modified mapping-collocation technique, originally formulated for plane isotropic analysis. Application of the technique outlined in this paper for plane orthotropic problems to a wider class of geometries and loading is evident. The numerical results indicate a dependence of the orthotropic stress intensity factors on both geometric and elastic constants over a certain parameter range.

Effective Stiffness of Randomly Oriented Fibre Composites

Abstract: This paper is concerned with the theoretical derivation of the effective stiffness properties for two types of randomly oriented fibre systems:(a) A system with completely three-dimensional random orientation of fibres in an isotropic matrix phase;(b) A system with two-dimensional (planar) random orientation of fibres in an isotropic matrix phase, appropriate to thin sections in a state of plane stress.The formulas derived predict the effective isotropic properties of the composite in terms of the elastic properties of each phase and the volume concentration of the fibre phase. These results provide the theoretical limits with which experimental results can be compared to assess the effectiveness of bonds, dispersion of fibres and any other factors which may affect the performance. In the last section a comparison is made with experimental results. In addition, the relation of the present study to pre vious work on this topic is discussed.

Neutron inelastic scattering study of the lattice dynamics of strontium titanate: harmonic models

Abstract: Extensive neutron inelastic scattering measurements of the frequencies of normal modes in strontium titanate propagating in the crystallographic directions (0, 0, zeta ), ( zeta , zeta , 0), ( zeta , zeta , zeta ), (1/2, 1/2, zeta ) and ( zeta , zeta , 1/2) are reported. The temperature dependence of the 'soft' modes at points Gamma and R have been investigated with particular emphasis on the mode of symmetry Delta 2, propagating in the direction (1/2, 1/2, zeta ); quantitative estimates of the isotropy of the dispersion surface at Gamma and R are given. Rigid ion, rigid shell, and deformable shell models of the crystal dynamics have been developed. Based on these, the frequency distribution, specific heat, one-phonon X ray scattering intensity. Debye-Waller factor, and two-phonon Raman spectrum have been calculated; where possible, these are compared with the available experimental measurements.

A simple constitutive theory for a fluid-saturated porous solid

Abstract: A mixture theory attributing distinct velocity fields to the separate constituents is adopted to describe the deformations and motions of a fluid-saturated porous solid. Constitutive laws for the partial stresses are related to the response of the respective constituents as single continuums in terms of effective stress and effective deformation. A simple multiplicative decomposition of the deformation gradient tensor with emphasis on finite deformation is introduced; this decomposition allows a definition of effective dilatation by appropriate scaling while leaving the isochoric (shear measure) part unchanged. The interrelation between the constitutive laws for the different constituents arises in the scaling functions. This description is theoretically possible for mixtures of any simple materials, but the concepts have most appeal when one constituent is a freely diffusing fluid. The cases of water-saturated elastic and elastic-plastic solids are illustrated, and the uniaxial strain response is examined. The extent to which a single interaction scaling function can be determined by isotropic pressure data is shown, and in illustration the scheme is applied to data for a saturated tuff.

Numerical simulation of turbulence

Abstract: Numerical simulations of three-dimensional homogeneous, isotropic turbulence at windtunnel Reynolds numbers (Rλ 20–40) are reported. The results of the simulations are compared with the predictions of turbulence theories.

Experiment on the geometry of the fine-structure regions in fully turbulent fluid

Abstract: An attempt has been made to identify the geometric character of the random fine-structure regions dispersed in fully turbulent fluid. The technique was measure-ment of two-position coincidence functions for the presence of velocity fine-structure. In this primitive approach, we tried to distinguish among only three possible categories for the random shapes: (a) ‘blobs’, (b) ‘rods’ and (c) 'slabs’, depending on whether three mean orthogonal dimensions of the domains were such that (a) all were of the same order, (b) one was an order larger than the other two, or (c) one was an order smaller than the other two.Highly idealized paradigms for these three categories were studied analytically: the two-position coincidence functions were computed for the cases of (a) spheres, (b) circular cylinders and (c) plane slabs, each field containing randomly sized elements distributed randomly in space with a homogeneous and isotropic distribution. Comparison of the measured coincidence functions with these three paradigms suggests that the fine-structure regions are more nearly of ‘rod-like’ geometry than like either of the other two. No attempt was made to distinguish shapes which might be called 'strips’.

Interaction of a shear wall with the soil for incident plane SH waves

Abstract: The closed-form solution of the dynamic interaction of a shear wall and the isotropic homogeneous and elastic half-space, previously studied only for vertically-incident SH waves, is generalized to any angle of incidence. It is shown that the interaction equation is independent of the incidence angle, while the surface-ground displacements heavily depend on it. For the two-dimensional model studied, it is demonstrated that disturbances generated by waves scattering and diffracting around the rigid foundation mass are not a local phenomenon but extend to large distances relative to the characteristic foundation length.

Elasticity and anisotropy of dunite and bronzitite

Abstract: Compressional and shear wave velocities in 13 directions at pressures to 10 kb were measured in dunite from the Twin Sisters peaks, Washington, and in bronzitite from the Stillwater Complex, Montana. Dunite is characterized by a strong preferred orientation of olivine grains, leading to anisotropy of velocity of 15% at 10 kb. The bronzitite is more nearly isotropic, the greatest difference of P wave velocities reaching 6%. The comparison of elasticity diagrams, obtained at pressures of 0.5 kb and 10 kb, shows that the system of pores, more open at low pressure, is of minor influence; anisotropy is controlled mainly by the preferred orientation of olivine and bronzite in both rocks. Most of the compressional and shear wave velocities, calculated according to Voigt's scheme on the basis of petrofabric determinations and published single-crystal data for olivine and bronzitite, are in good agreement with the ultrasonic velocities measured at 10 kb. The results confirm the possibility of using this procedure to determine anisotropic elasticity of simple aggregates and suggest a possibility of using ultrasonic measurements in petrostructural analysis.

Decay of Isotropic Turbulence Generated by a Mechanically Agitated Grid

Abstract: Experimental study of weak isotropic turbulence, created by a mechanically agitated grid, has indicated that in the absence of large linear‐momentum wakes the energy of turbulence relaxes very quickly into a stable self‐preserving structure, which, depending on the initial Reynolds number of turbulence, decays at different constant inverse powers of time. Both the longitudinal correlation coefficients and the corresponding spectral distributions, except for the difference in the parametric constants, are of the same functional type as those found previously for a passive grid.

Transfer Matrix Approach to Layered Systems

Abstract: The stress and displacement distribution in a finite layered medium, statically loaded at the surface, is found by means of transfer matrices. Each layer is of infinite extent in the horizontal direction, of constant depth, and is considered to be linearly elastic, homogeneous and isotropic. The method developed has the built-in advantage of enforcing interface continuity conditions automatically. Its flexibility is shown by applying the technique to layered composites, in which case it predicts local as well as global behavior with equal ease.

A note on linear thermoelasticity

Abstract: A uniqueness theorem is obtained for a theory of linear thermoelasticity which allows for“second sound”. Propagation of acceleration waves in an isotropic material is also discussed.

On Saint-Venant's principle in plane anisotropic elasticity

Abstract: Methods involving energy-decay inequalities are applied in investigating Saint-Venant's principle for the planproblem of linear elastostatics for a wide class of anisotropic media. A lower bound (in terms of the elastic constants) is obtained for the rate of exponential decay of stresses and this is compared with the known result for the isotropic case.

Particle creation in isotropic cosmologies

Abstract: The simplest covariant generalization of the scalar wave equation leads to significant pion creation and annihilation processes near an isotropic Friedmann-type singularity (such processes are negligible for particles of nonzero spin). Estimates for a plausible initial state yield pion creation of the same order of magnitude as obtained by Zeldovich near an anisotropic Kasner-type singularity.

Depolarized rayleigh scattering and the mean-squared optical anisotropies of n-alkanes in solution

Abstract: A procedure has been developed whereby the mean-squared optical anisotropy 〈γ2〉 associated with isolated solute molecules can be obtained from measurements of the depolarized Rayleigh scattering by their solutions in optically isotropic and geometrically symmetric solvents. Depolarized Rayleigh ratios RHV for light scattered at an angle of 90° by carbon tetrachloride solutions of 7 n-alkanes from C5 to C22 have been measured using a He-Ne laser (λ= 632.8 nm) as the light source. Intensity due to Raman scattering was eliminated by a narrow bandpass filter of 3 nm width. The scattered intensity due to collision induced anisotropy was estimated from the decrease in RHV when a 0.3 nm filter was used, and a procedure is described for suppressing the contribution to RHV from this source for the n-alkane solutions. Values of 〈γ2〉 for the n-alkanes obtained by extrapolating the present results to infinite dilution are considerably smaller than those reported by other workers. The dependence of 〈γ2〉 on chain length is well reproduced by calculations carried out using the valence optical scheme and rotational isomeric state theory, gauche states being assigned an energy of 500 ± 100 cal mol–1 relative to trans in conformity with other evidence. Coincidence between calculated and observed optical anisotropies requires assignment of a value of 0.54 A3 to the parameter Γ, which is related to the bond anisotropies by Γ=ΔαCC–1.95 ΔαCH according to numerical calculations for a ∠ CCC angle of 112°.

Metallic State of the Electron-Hole Liquid, Particularly in Germanium

Abstract: We have calculated the ground-state energy of an electron-hole liquid. The kinetic and exchange energies are included exactly, and the correlation energy is estimated using Hubbard's modification of the random-phase approximation. In an isotropic electron-hole liquid, the metallic state is not bound relative to free excitons. In Ge the anisotropic band structure leads to a substantial binding of the metallic state. Application of a large $〈111〉$ strain to Ge reduces the situation to one resembling the isotropic case.

A Survey of Failure Theories of Isotropic and Anisotropic Materials

Abstract: : A survey of various theories of strength for isotropic and anisotropic materials is presented. For anisotropic materials theories of failure are broadly classified as theories with or without distinct failure modes. The former class includes the criteria based upon maximum stress and strain while the latter are quadratic or biquadratic representatives in which the transition from one failure mode to another is gradual.

Nonlinear Evolution of Whistler Instabilities.

Abstract: Theoretical and computer simulation studies of whistler instabilities in anisotropic collisionless plasmas are presented. Initial bi‐Maxwellian, Maxwellian with loss cone, and hot Maxwellian superimposed on a more dense cold isotropic background electron distributions were used. Many of the observed features are common to all cases. Initially, the total wave magnetic energy grows; the average growth rate is in good agreement with linear theory; and the electron distribution isotropizes rapidly with a concomitant switching off of high k number, initially unstable waves. Then, the total wave magnetic energy saturates and at this time there is a residual kinetic energy anisotropy. This anisotropy persists after saturation, although there is a further tendency toward isotropy, together with a further switching off of the higher mode numbers for the remainder of the simulation. Throughout the simulation experiments the perpendicular and parallel energy constants are conserved to good accuracy.

Calculation of Plane‐Wave Propagation in Anisotropic Elastic‐Plastic Solids

Abstract: Wave‐propagation effects induced in isotropic materials by either explosive or impact loading uniformly over a planar surface are much simpler than those produced in anisotropic solids in which significant transverse particle motion may occur. Such behavior in linearly elastic anisotropic solids has been investigated previously, and in the present work the analysis has been extended to include elastic‐plastic behavior. A general mathematical description of both elastically and plastically anisotropic materials is presented, and then several specific applications are considered in detail. For the case of an elastically isotropic solid that is plastically anisotropic with a single slip plane and direction, it is found that for certain orientations two plastic waves can be propagated, one of which is quasilongitudinal and the other is quasitransverse. For more general orientations there may be two quasitransverse waves and a single quasilongitudinal disturbance. These results approximate the behavior of a single‐crystal material of hexagonal symmetry, such as beryllium, for which effects of elastic anisotropy are small and slip occurs predominantly on the basal plane. Since there is a single spatial coordinate involved in planar wave propagation, the general equations describing elastic‐plastic wave propagation can be written in one‐dimensional finite‐difference form and solved numerically for arbitrary crystal structure and orientation. A number of specific examples are considered and compared with analytical solutions.