Error Bounds for Approximate Solutions for Bending of Unsymmetrically Laminated Plates

Abstract

Certain types of heterogeneous plates exhibit coupling between membrane and flexural effects in their constitutive relations. Such a situation commonly occurs in unsymmetrically laminated plates and in reinforced concrete slabs after cracking. Approximate solutions, principally the “reduced bending stiffness” approximation, have been proposed in the past. The accuracy of this approximation has been examined for several specific cases, but no general investigations have been reported. This paper presents a method for determining bounds on the relative mean square error of approximate solutions to general coupled plate bending problems.     

An analytical theory for diffusion of fluids in crystalline nanoporous materials

An analytical theory for diffusion of fluids in zeolites and other nanoporous materials has been developed. The theory incorporates molecular level information about the nanoporous material, which is obtainable from an energy minimization and does not require molecular dynamics computer simulations. The theory is statistical mechanical in nature and assumes a lattice composed of adsorption sites. The theory yields a self-diffusion coefficient, which is a function of (i) temperature, (ii) adsorbate density, (iii) adsorbate size, (iv) adsorbate-adsorbate energetic interaction and (v) adsorbate-pore energetic interaction. The theory is generalized and is applicable to nanoporous materials with three-dimensional porous networks (e.g. faujusite) and one-dimensional porous networks (e.g. A1P0₄-5).

The theory is self-contained and incorporates no fitting parameters. The theory does not require computational effort beyond a few seconds on a standard personal computer.     

Interatomic interactions and the Cotton—Mouton effect for helium

We report calculations of the interaction-induced polarizability (δα^anis), magnetizability (δξ^anis;) and hypermagnetizability (δη^anis) anisotropies for the helium gas as a function of the interatomic separation. From these data we determine the virial coefficients for the Cotton—Mouton effect and the hypermagnetizability anisotropy of helium. We also find the mean polarizability and magnetizability as a function of the interatomic separation and the virial coefficients for these properties. The results for the Cotton—Mouton effect indicate that pressure affects the Cotton—Mouton constant to the same extent as it does the second hyperpolarizability (γ) and the virial coefficient b^CME(ω, T) lies in the range of ?1.6 to ?1.8 cm³ mol^?1. This means that pressure effects for the Cotton-Mouton constant could be detected with modern experimental techniques. All calculations were carried out using the full configuration interaction technique and large basis sets of London atomic orbitals. The polarizability calculations were performed both for relevant optical frequencies as well as the static case.     

Statistical geometry of hard sphere systems: exact relations for first-order phase transitions in multicomponent systems

Statistical geometry provides exact relations between the thermophysical properties of hard sphere systems and various geometric quantities. Utilizing recently derived relations that describe multicomponent hard sphere systems, we derive two new expressions which place rigorous constraints on the behaviour of geometric quantities across a first-order phase transition. Some implications of these constraints for a binary mixture of additive hard spheres are discussed.     

The internal coordinate path Hamiltonian; application to methanol and malonaldehyde

The internal coordinate path Hamiltonian is introduced for the study of the vibrations of molecules which have one large amplitude motion. The Hamiltonian is represented in terms of a one path coordinate and 3N—7 normal coordinates. The variational method is used to solve the Schrödinger equation. The molecules studied are methanol and malonaldehyde. For methanol the internal coordinate is a dihedral angle, for malonaldehyde it is the difference in the distances between the migrating hydrogen and the neighbouring oxygen atoms. For methanol there is little coupling between the path and the normal coordinates and so no complications were encountered in the calculations which used harmonic surfaces generated by density functional and M?ller—Plesset theory. Fundamental frequencies were predicted. Malonaldehyde is a different story. There is substantial coupling between the path coordinate and several of the normal coordinates. This introduces many complications: an anharmonic surface is essential and large variational configuration interaction calculations are essential for convergence. Furthermore, because the Coriolis terms require the evaluation of derivatives of both the nuclear coordinates and the normal coordinate eigenvectors along the path, great care must be taken with these numerical procedures. B3LYP predicts too low a transition state which overemphasizes the large Coriolis terms near the transition state. This may be one of the reasons why our fundamental vibrations are in poor agreement with observation. It is most encouraging that the tunnelling splitting is 58 cm^?1 (obs. 21.56 cm^?1), obtained with our quartic density functional surface.     

Monte Carlo and cell model calculations for the solid—fluid phase behaviour of the triangle-well model

Monte Carlo simulations and cell model calculations are reported for the vapour-liquid and solid-liquid phase behaviour of the triangle-well model system. The behaviour is examined as a function of the range of the triangle-well attraction, from 1.05 to 2.5 times the diameter of the hard core of the potential. Cell model calculations indicate that the stable solid is almost always face-centred cubic (fcc), except for a small set of conditions where hexagonal close-packed (hcp) is favoured. This outcome differs markedly from a much earlier study performed for the square-well model potential, where a much richer phase diagram was observed, with significant regions of stability for hep and body-centred cubic (bcc) phases. Monte Carlo simulations indicate that the cell model calculations represent well the true phase behaviour for this model system. The differing behaviour between the triangle-well and square-well models indicates an important role for the flatness of the potential well in governing the stability of hcp and bcc phases relative to the fcc phase.     

A simulation study of the effects of adsorbent energy on the diffusion of molecules in slit pores: 1 commensurate slit widths

Equilibrium molecular dynamics simulation has been used to study the self-diffusion coefficients (from correlation of the molecular velocity) and the collective, or centre of mass diffusion coefficients (from correlation of the streaming velocity) of a Lennard-Jones fluid in model slit pores. The slit widths were chosen to be integer multiples of the Lennard-Jones adsorbate diameter, and therefore are close to being commensurate with layered adsorbate structures. Slits of reduced width H* = 3 and 5 were examined at a reduced temperature of T* = 1.0. The adsorbate densities ranged from 0.3 to 0.9 in reduced units. The adsorbent adsorbate interaction was modelled as a simple potential with inverse 4th power attraction plus hard wall repulsion, and systems with reduced parameter u₀* ranging from ?5 to +5 were studied. Molecule-wall scattering was represented by a diffuse reflection algorithm. The density distributions show strong layering in the attractive system, but this is absent in the most repulsive slits, except at very high densities. Self-diffusion is only weakly dependent on u₀* and slit width at high densities, but a strong dependence on u₀* appears at low densities. The collective diffusion coefficient is less easy to calculate with high accuracy; nevertheless, it is clear that there is a strong dependence of this property on u₀* Trajectory plots show zones in which the particles are more or less strongly localized, but undergo irregular oscillatory motion corresponding to regions of high density in the single-particle distributions.     

Low-Loss Many-to-One Fiber Couplers with Few or Single-Moded Inputs and a Multi-Mode Output

The coupling of light from a number of few or single-moded fibers into a single multi-mode fiber is analyzed using geometric optics, and simple results demonstrating mode conservation are derived. Coupling from multiple single-mode fibers into a multi-mode fiber is investigated in detail using the overlap integral to determine coupling into each mode of the output fiber as a function of the light phase in the inputs. As well as results with practical relevance to fiber tapped delay-line filters and optical CDMA, the analysis provides pedagogic insight into light propagation and the light-gathering properties of fiber.