Andrew N. Norris

A differential scheme for the effective moduli of composites

Abstract A differential scheme to compute the effective moduli of composites is presented. The method is based on the idea of realizability, i.e. the composite is constructed explicitly from an initial material through a series of incremental additions. The construction process is uniquely specified by parametrizing the volume fractions of the included phases. The properties of the final composite depend upon the construction path taken and not just on the final volume fractions. Assuming the grain shapes are ellipsoidal, a system of ordinary differential equations for the moduli is obtained which is integrated along the path. The present method includes as special cases of paths or endpoints the differential scheme of Roscoe-Boucher and the self-consistent scheme of Kroner-Hill, respectively. The method includes a realization of the Hashin-Shtrikman bounds for a two-phase composite with K( 1 2- K 2 )(μ 1 – μ 2 ) ⪖ 0 . For example, the upper bounds are achieved by imbedding disks of the stiffer material in a matrix of the more compliant material.

Acoustic cloaking theory

An acoustic cloak is a compact region enclosing an object, such that sound incident from all directions passes through and around the cloak as though the object was not present. A theory of acoustic cloaking is developed using the transformation or change-of-variables method for mapping the cloaked region to a point with vanishing scattering strength. We show that the acoustical parameters in the cloak must be anisotropic: either the mass density or the mechanical stiffness or both. If the stiffness is isotropic, corresponding to a fluid with a single bulk modulus, then the inertial density must be infinite at the inner surface of the cloak. This requires an infinitely massive cloak. We show that perfect cloaking can be achieved with finite mass through the use of anisotropic stiffness. The generic class of anisotropic material required is known as a pentamode material (PM). If the transformation deformation gradient is symmetric then the PM parameters are explicit, otherwise its properties depend on a stress-like tensor that satisfies a static equilibrium equation. For a given transformation mapping, the material composition of the cloak is not uniquely defined, but the phase speed and wave velocity of the pseudo-acoustic waves in the cloak are unique. Examples are given from two and three dimensions.

Low-frequency dispersion and attenuation in partially saturated rocks

A theory is developed for the attenuation and dispersion of compressional waves in inhomogeneous fluid‐saturated materials. These effects are caused by material inhomogeneity on length scales of the order of centimeters and may be most significant at seismic wave frequencies, i.e., on the order of 100 Hz. The micromechanism involves diffusion of pore fluid between different regions, and is most effective in a partially saturated medium in which liquid can diffuse into regions occupied by gas. The local fluid flow effects can be replaced on the macroscopic scale by an effective viscoelastic medium, and the form of the viscoelastic creep function is illustrated for a compressional wave propagating normal to a layered medium. The wave speeds in the low‐ and high‐frequency limits are associated with conditions of uniform pressure and of uniform ‘‘no‐flow,’’ respectively. These correspond to the isothermal and isentropic wave speeds in a disordered thermoelastic medium.

Radiation from a point source and scattering theory in a fluid‐saturated porous solid

The time harmonic Green function for a point load in an unbounded fluid‐saturated porous solid is derived in the context of Biot’s theory. The solution contains the two compressional waves and one transverse wave that are predicted by the theory and have been observed in experiments. At low frequency, the slow compressional wave is diffusive and only the fast compressional and transverse waves radiate energy. At high frequency, the slow wave radiates, but with a decay radius which is on the order of cm in rocks. The general problem of scattering by an obstacle is considered. The point load solution may be used to obtain scattered fields in terms of the fields on the obstacle. Explicit expressions are presented for the scattering amplitudes of the three waves. Simple reciprocity relations between the scattering amplitudes for plane‐wave incidence are also given. These hold under the interchange of incident and observation directions and are completely general results. Finally, the point source solution is ...

Particle granular temperature in gas fluidized beds

Abstract In this paper we introduce and validate a novel non-intrusive probe of the average kinetic energy, or granular temperature, of the particles at the wall of a gas fluidized bed. We present data on the granular temperature of monodispersed glass spheres which span region B, and extend into region A, of the Geldart powder classification. The underlying physics of the measurement is the acoustic shot noise excitation of the surface of the fluid bed vessel by random particle impact. Quantitative determination of the average particle granular temperature is obtained through independent measurement of the wall transfer function determining the coupling between the acoustic shot noise excitation at one location and the response of an accelerometer at another location. We validate the concept and calibration of this acoustic shot noise probe in the frequency range 10–20 kHz, through a comprehensive series of laboratory measurements with gasses and cylinders of significantly different acoustic properties. We demonstrate its utility by presenting the first data on the dependence of the granular temperature on gas flow and particle diameter and make the first observation of a change in the character of the fluidization transition from first order (hysteretic and discontinuous) to second order (reversible and continuous) for Geldart B glass spheres as the A/B boundary is approached. We observe a striking difference in the dependence of the granular temperature on gas flow between Geldart B and A glass spheres, that suggests a fundamental difference in particle dynamics between spheres in the two Geldart regimes. Finally we use the vibrational probe to study the time dependence of the granular temperature under bed collapse conditions when fluidizing gas is withdrawn rapidly from the system. We show an exponential time dependence with a time constant of the order of 100 ms, and demonstrate the consistency of this result with a Langevin equation for the sphere velocity with a time constant derived from the sphere fluctuation velocity and a collisional coefficient of restitution of 0.9. From these results for the granular temperature and a kinetic model for a dense granular gas, we present estimates for the inertial pressure, velocity of sound, viscosity, and diffusion constant of the dense phase of a gas fluidized bed as a function of particle diameter and gas superficial velocity. The implication of these results for current models of gas fluidized beds, and the fundamental basis of the Geldart classification is discussed.

The Closest Elastic Tensor of Arbitrary Symmetry to an Elasticity Tensor of Lower Symmetry

The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest elasticity tensor of the specified symmetry. Solutions are presented for three distance functions, with particular attention to the Riemannian and log-Euclidean distances. These yield solutions that are invariant under inversion, i.e., the same whether elastic stiffness or compliance are considered. The Frobenius distance function, which corresponds to common notions of Euclidean length, is not invariant although it is simple to apply using projection operators. A complete description of the Euclidean projection method is presented. The three metrics are considered at a level of detail far greater than heretofore, as we develop the general framework to best fit a given set of moduli onto higher elastic symmetries. The procedures for finding the closest elasticity tensor are illustrated by application to a set of 21 moduli with no underlying symmetry.

An examination of the Mori-tanaka effective medium approximation for multiphase composites

The Mori-Tanaka method is considered in the context of both scalar thermal conductivity and anisotropic elasticity of multiphase composites, and some general properties are deduced. Particular attention is given to its relation to known general bounds, and to the differential scheme. It is shown that the moduli predicted by the method always satisfy the Hashin-Shtrikman and Hill-Hashin bounds for two-phase composites. The Mori-Tanaka approximation has to be used with caution in multiphase applications, but is on firmer ground for two-phase composites

A generalized differential effective medium theory

Abstract A generalization of the Differential Effective Medium approximation (DEM) is discussed. The new scheme is applied to the estimation of the effective permittivity of a two phase dielectric composite. Ordinary DEM corresponds to a realizable microgeometry in which the composite is built up incrementally through a process of homogenization, with one phase always in dilute suspension and the other phase associated with the percolating backbone. The generalization of DEM assumes a third phase which acts as a backbone. The other two phases are progressively added to the backbone such that each addition is in an effectively homogeneous medium. A canonical ordinary differential equation is derived which describes the change in material properties as a function of the volume concentration φ of the added phases in the composite. As φ→ 1, the Effective Medium Approximation (EMA) is obtained. For φ

Dynamic Green’s functions in anisotropic piezoelectric, thermoelastic and poroelastic solids

A procedure is described to generate fundamental solutions or Green’s functions for time harmonic point forces and sources. The linearity of the field equations permits the Green’s function to be represented as an integral over the surface of a unit sphere, where the integrand is the solution of a one-dimensional impulse response problem. The method is demonstrated for the theories of piezoelectricity, thermoelasticity, and poroelasticity. Time domain analogues are discussed and compared with known expressions for anisotropic elasticity.

On the correspondence between poroelasticity and thermoelasticity

An interesting and useful analogy can be drawn between the equations of static poroelasticity and the equations of thermoelasticity including entropy. The correspondence is of practical use in determining the effective parameters in an inhomogeneous poroelastic medium using known results from the literature on the effective thermal expansion coefficient and the effective heat capacity of a disordered thermoelastic continuum.