An elementary approach to gap theorems

Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension >= 3. Suppose that the sectional curvature K satisfies -1-s(r) <= K <= -1, where r denotes distance to a fixed point in M. If lim(r ->infinity) e(2r) s(r) = 0, then (M, g) has to be isometric to H-n.The same proof also yields that if K satisfies -s(r) <= K <= 0 where lim(r ->infinity) r(2) s(r) = 0, then (M, g) is isometric to R-n, a result due to Greene and Wu.Our second result is a local one: Let (M, g) be any Riemannian manifold. For a E R, if K < a on a geodesic ball Bp (R) in M and K = a on partial derivative B-p (R), then K = a on B-p (R).

Re-entrant Phase Behavior of a Concentrated Anionic Surfactant System with Strongly Binding Counterions

The phase behavior of the anionic surfactant sodium dodecyl sulfate (SDS) in the presence of the strongly binding counterion p-toluidine hydrochloride (PTHC) has been examined using small-angle X-ray diffraction and polarizing microscopy. A hexagonal-to-lamellar transition on varying the PTHC to SDS molar ratio (alpha) occurs through a nematic phase of rodlike micelles (N-C) -> isotropic (I) -> nematic of disklike micelles (N-D) at a fixed surfactant concentration (phi). The lamellar phase is found to coexist with an isotropic phase (l') over a large region of the phase diagram. Deuterium nuclear magnetic resonance investigations of the phase behavior at phi = 0.4 confirm the transition from N-C to N-D on varying alpha. The viscoelastic and flow behaviors of the different phases were examined. A decrease in the steady shear viscosity across the different phases with increasing alpha suggests a decrease in the aspect ratio of the micellar aggregates. From the transient shear stress response of the N-C and N-D nematic phases in step shear experiments, they were characterized to be tumbling and now aligning, respectively. Our studies reveal that by tuning the morphology of the surfactant micelles strongly binding counterions modify the phase behavior and rheological properties of concentrated surfactant solutions.

Finite element analysis of laminated anisotropic thin-walled open-section beams

A finite element analysis of thin-walled open-section laminated anisotropic beams is presented herein. A two-noded, 8 degrees of freedom per node thin-walled open-section laminated anisotropic beam finite element has been developed and used. The displacements of the element reference axes are expressed in terms of one-dimensional first order Hermite interpolation polynomials and line member assumptions are invoked in the formulation of the stiffness matrix. The problems of: 1. (a) an isotropic material Z section straight cantilever beam, and 2. (b) a single-layer (0°) composite Z section straight cantilever beam, for which continuum solutions (exact/approximate) are possible, have been solved in order to evaluate the performance of the finite element. Its applicability has been shown by solving the following problems: 3. (c) a two-layer (45°/−45°) composite Z section straight cantilever beam, 4. (d) a three-layer (0°/45°/0°) composite Z section straight cantilever beam.

A hybrid clustering procedure for concentric and chain-like clusters

K-means algorithm is a well known nonhierarchical method for clustering data. The most important limitations of this algorithm are that: (1) it gives final clusters on the basis of the cluster centroids or the seed points chosen initially, and (2) it is appropriate for data sets having fairly isotropic clusters. But this algorithm has the advantage of low computation and storage requirements. On the other hand, hierarchical agglomerative clustering algorithm, which can cluster nonisotropic (chain-like and concentric) clusters, requires high storage and computation requirements. This paper suggests a new method for selecting the initial seed points, so that theK-means algorithm gives the same results for any input data order. This paper also describes a hybrid clustering algorithm, based on the concepts of multilevel theory, which is nonhierarchical at the first level and hierarchical from second level onwards, to cluster data sets having (i) chain-like clusters and (ii) concentric clusters. It is observed that this hybrid clustering algorithm gives the same results as the hierarchical clustering algorithm, with less computation and storage requirements.

Coupled amplitude theory of nonlinear surface acoustic waves

The nonlinear propagation characteristics of surface acoustic waves on an isotropic elastic solid have been studied in this paper. The solution of the harmonic boundary value problem for Rayleigh waves is obtained as a generalized Fourier series whose coefficients are proportional to the slowly varying amplitudes of the various harmonics. The infinite set of coupled equations for the amplitudes when solved exhibit an oscillatory slow variation signifying a continuous transfer of energy back and forth among the various harmonics. A conservation relation is derived among all the harmonic amplitudes.

On the propagation of a multi-dimensional shock of arbitrary strength

In this paper the kinematics of a curved shock of arbitrary strength has been discussed using the theory of generalised functions. This is the extension of Moslov’s work where he has considered isentropic flow even across the shock. The condition for a nontrivial jump in the flow variables gives the shock manifold equation (sme). An equation for the rate of change of shock strength along the shock rays (defined as the characteristics of the sme) has been obtained. This exact result is then compared with the approximate result of shock dynamics derived by Whitham. The comparison shows that the approximate equations of shock dynamics deviate considerably from the exact equations derived here. In the last section we have derived the conservation form of our shock dynamic equations. These conservation forms would be very useful in numerical computations as it would allow us to derive difference schemes for which it would not be necessary to fit the shock-shock explicitly.

Finite element analysis of moving contact in mechanically fastened joints

The Finite Element Method (FEM) has made a number of otherwise intractable problems solvable. An important aspect for achieving an economical and accurate solution through FEM is matching the formulation and the computational organisation to the problem. This was realised forcefully in the present case of the solution of a class of moving contact boundary value problems of fastener joints. This paper deals with the problem of changing contact at the pin-hole interface of a fastener joint. Due to moving contact, the stresses and displacements are nonlinear with load. This would, in general, need an interactive-incremental approach for solution. However, by posing the problem in an inverse way, a solution is sought for obtaining loads to suit given contact configuration. Numerical results are given for typical isotropic and composite plates with rigid pins. Two cases of loading are considered: (i) load applied only at the edges of the plate and (ii) load applied at the pin and reacted at a part of the edge of the plate. Load-contact relationships, compliance and stress-patterns are investigated. This paper clearly demonstrates the simplification achieved by a suitable formulation of the problem. The results are of significance to the design and analysis of fastener joints.

Finite element analysis of moving contact in mechanically fastened joints

The Finite Element Method (FEM) has made a number of otherwise intractable problems solvable. An important aspect for achieving an economical and accurate solution through FEM is matching the formulation and the computational organisation to the problem. This was realised forcefully in the present case of the solution of a class of moving contact boundary value problems of fastener joints. This paper deals with the problem of changing contact at the pin-hole interface of a fastener joint. Due to moving contact, the stresses and displacements are nonlinear with load. This would, in general, need an interactive-incremental approach for solution. However, by posing the problem in an inverse way, a solution is sought for obtaining loads to suit given contact configuration. Numerical results are given for typical isotropic and composite plates with rigid pins. Two cases of loading are considered: (i) load applied only at the edges of the plate and (ii) load applied at the pin and reacted at a part of the edge of the plate. Load-contact relationships, compliance and stress-patterns are investigated. This paper clearly demonstrates the simplification achieved by a suitable formulation of the problem. The results are of significance to the design and analysis of fastener joints.

Vibrations of laminated beams

Governing equations in the form of simultaneous ordinary differential equations have been derived for natural vibration analysis of isotropic laminated beams. This formulation includes significant secondary effects such as transverse shear and rotatory inetia. Through a numerical example, the influence of these secondary effects has been studied.

Natural frequencies of polar orthotropic annular plates

The classical Rayleigh-Ritz method with simple polynomials as admissible functions has been used for obtaining natural frequencies of transversely vibrating polar orthotropic annular plates. The method in conjunction with transformations introduced in the analysis has been found to be quite effective, particularly for large hole sizes. Estimates of natural frequencies corresponding to modes with one as well as two nodal diameters are obtained for the nine combinations of clamped, simply supported and free edge conditions and for various values of rigidity ratio and hole sizes. Based on the variation of eigenvalue parameter with rigidity ratio, the frequencies of these modes as well as those of axisymmetric modes have been expressed by means of simple formulae in terms of rigidity ratio and the frequencies of corresponding modes in the isotropic case. These formulae have been used in determining the fundamental frequencies of orthotropic plates.

ESR study of copper diethyldithiophosphate

ESR investigations are reported in single crystals of copper diethyldithiophosphate, magnetically diluted with the corresponding diamagnetic nickel complex. The spectrum at normal gain shows hyperfine components from 63Cu, 65Cu, and 31P nuclei. At much higher gain, hyperfine interaction from 33S nuclei in the ligand is detected. The spin Hamiltonian parameters relating to copper show tetragonal symmetry. The measured parameters are g = 2.085, g =2.025, A63Cu = 149.6 × 10−4 cm−1, A65Cu = 160.8 × 10−4 cm−1, BCu = 32.5 × 10−4 cm−1 and QCu 5.5 × 10−4cm−1. The 31P interaction is isotropic with a coupling constant AP = 9.6 × 10−4 cm−1. Angular variation of the 33S lines shows two different hyperfine tensors indicating the presence of two chemically inequivalent Cu S bonds. The experimentally determined hyperfine constants are A =34.9×10−4 cm−1, B =26.1×10−4 cm−1, A =60.4×10−4 cm−1, B =55.5×10−4 cm−1. The hyperfine parameters show that the hybridization of the ligand orbitals is very sensitive to the symmetry around the ligand. The g values and Cu hyperfine parameters are not much affected by the distortions occurring in the ligand. The energies of the d-d transitions are determined by optical absorption measurements on Cu diethyldithiophosphate in solution. Using the spin Hamiltonian parameters together with optical absorption results, the MO parameters for the complex are calculated. It is found that in addition to the bond, the bonds are also strongly covalent. ©1973 The American Institute of Physics

Rapid distortion of axisymmetric turbulence

A generalization of the isotropic theory of Batchelor & Proudman (1954) is developed to estimate the effect of sudden but arbitrary three-dimensional distortion on homogeneous, initially axisymmetric turbulence. The energy changes due to distortion are expressed in terms of the Fourier coefficients of an expansion in zonal harmonics of the two independent scalar functions that describe the axisymmetric spectral tensor. However, for two special but non-trivial forms of this tensor, which represent possibly the simplest kinds of non-isotropic turbulence and specify the angular distribution but not the wavenumber dependence, the energy ratios have been determined in closed form. The deviation of the ratio from its isotropic value is the product of a factor containing R, the initial value of the ratio of the longitudinal to the transverse energy component, and another factor depending only on the geometry of the distortion. It is found that, in axisymmetric and large two-dimensional contractions, the isotropic theory gives nearly the correct longitudinal energy, but (when R > 1) over-estimates the increase in the transverse energy; the product of the two intensities varies little unless the distortion is very large, thus accounting for the stress-freezing observed in rapidly accelerated shear flows.Comparisons with available experimental data for the spectra and for the energy ratios show reasonable agreement. The different ansatzes predict results in broad qualitative agreement with a simple strategem suggested by Reynolds & Tucker (1975), but the quantitative differences are not always negligible.

A numerical method for separation of stresses in photo-orthotropic elasticity

A numerical method is suggested for separation of stresses in photo-orthotropic elasticity using the numerical solution of compatibility equation for orthotropic case. The compatibility equation is written in terms of a stress parameter S analogous to the sum of principal stresses in two-dimensional isotropic case. The solution of this equation provides a relation between the normal stresses. The photoelastic data give the shear stress and another relation between the two normal stresses. The accuracy of the numerical method and its application to practical problems are illustrated with examples.

Depressed weir with two unequal sheetpiles in anisotropic soil

An analytical solution is presented, making use of the Schwartz-Christoffel transformation, for determining the seepage characteristics for the problem of flow under a weir having two unequal sheetpiles at the ends and embedded in an anisotropic porous medium of finite thickness. Results for several particular cases of simple hydraulic structures can be obtained from the general solution presented. Numerical results in nondimensional form have been given for quantity of seepage and exit gradient distribution for various conditions in the equivalent transformed isotropic section and, by making use of the physical parameters in the actual anisotropic plane and the set of transformation relations given, these quantities (seepage loss, exit gradient) can be interpreted in the actual anisotropic physical plane.

Flexural Strength of Modified Square Footings

The ultimate flexural strength behavior of isolated square tapered and beam-slab reinforced footings are presented. Yield line solutions are developed for generalized contact pressure distributions and the influence of taper, beam size, fillet size, negative moment capacity, and contact pressure distribution on the collapse load is brought out. In beam-slab footings the optimum relative beam capacity required to make the beam rigid is indicated. Results of experimental investigations on footings resting on sand reveal that tapered (with isotropic as well as with alternative reinforcement patterns) and beam-slab footings exhibit superior structural behavior in terms of normalized first crack load, collapse load, relative rigidity, relative efficiency, and failure mechanism.