REIDEMEISTER TORSION AND ANALYTIC TORSION OF SPHERES

We provide a simple topological derivation of a formula for the Reidemeister and the analytic torsion of spheres.

The analytic torsion of a disc

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

The analytic torsion of a disc

In this article, we study the Reidemeister torsion and the analytic torsion of the m dimensional disc, with the Ray and Singer homology basis (Adv Math 7:145-210, 1971). We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger-Muller theorem. We use a formula proved by Bruning and Ma (GAFA 16:767-873, 2006) that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary (Luck, J Diff Geom 37:263-322, 1993). Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the two sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang (Asian J Math 4:695-714, 2000), and we compare the results. The results of these work were announced in the study of Hartmann et al. (BUMI 2:529-533, 2009).

Knots and links in lens spaces

The aim of this dissertation is to improve the knowledge of knots and links in lens spaces. If the lens space L(p,q) is defined as a 3-ball with suitable boundary identifications, then a link in L(p,q) can be represented by a disk diagram, i.e. a regular projection of the link on a disk. In this contest, we obtain a complete finite set of Reidemeister-type moves establishing equivalence, up to ambient isotopy. Moreover, the connections of this new diagram with both grid and band diagrams for links in lens spaces are shown. A Wirtinger-type presentation for the group of the link and a diagrammatic method giving the first homology group are described. A class of twisted Alexander polynomials for links in lens spaces is computed, showing its correlation with Reidemeister torsion. One of the most important geometric invariants of links in lens spaces is the lift in 3-sphere of a link L in L(p,q), that is the counterimage of L under the universal covering of L(p,q). Starting from the disk diagram of the link, we obtain a diagram of the lift in the 3-sphere. Using this construction it is possible to find different knots and links in L(p,q) having equivalent lifts, hence we cannot distinguish different links in lens spaces only from their lift. The two final chapters investigate whether several existing invariants for links in lens spaces are essential, i.e. whether they may assume different values on links with equivalent lift. Namely, we consider the fundamental quandle, the group of the link, the twisted Alexander polynomials, the Kauffman Bracket Skein Module and an HOMFLY-PT-type invariant.

Numerical exploration of the defect’s effect on mechanical properties of nanowires under torsion

Molecular dynamics (MD) simulations have been carried out to investigate the defect’s effect on the mechanical properties of single-crystal copper nanowire with different surface defects, under torsion deformation. The torsional rigidity is found insensitive to the surface defects and the critical angle appears an obvious decrease due to the surface defects, the largest decrease is found for the nanowire with surface horizon defect. The deformation mechanism appears different degrees of influence due to surface defects. The surface defects play a role of dislocation sources. Comparing with single intrinsic stacking faults formation for the perfect nanowire, much affluent deformation processes have been activated because of surface defects, for instance, we find the twins formation for the nanowire with a surface 45o defect.

Ultimate load behaviour of box-columns under combined loading of axial compression and torsion

This paper describes a study of the theoretical and experimental behaviour of box-columns of varying b/t ratios under loadings of axial compression and torsion and their combinations. Details of the testing rigs and the testing methods, the results obtained such as the load-deflection curves and the interaction diagrams, and experimental observations regarding the behaviour of box-models and the types of local plastic mechanisms associated with each type of loading are presented. A simplified rigid-plastic analysis is carried out to study the collapse behaviour of box-columns under these loadings, based on the observed plastic mechanisms, and the results are compared with those of experiments.

On extension and torsion of a compressible elastic circular cylinder

In this paper we examine the combined extension and torsion of a compressible isotropic elastic cylinder of finite extent. The equilibrium equations are formulated in terms of the principal stretches and then applied to the special case of pure torsion superimposed on a uniform extension (an isochoric deformation). Explicit necessary and sufficient conditions on the strain-energy function for the material to support this deformation with vanishing traction on the lateral surfaces of the cylinder are obtained. Some strain-energy functions satisfying these conditions are considered, existing results are recovered as special cases and new results are obtained. We also point out how the strain-energy functions generated from the considered isochoric deformation considered (of a compressible material) can be used to generate energy functions and corresponding solutions for the incompressible theory.

Behaviour and strength of hollow flange channel sections under torsion and bending

Hollow flange channel section is a cold-formed high-strength and thin-walled steel section with a unique shape including two rectangular hollow flanges and a slender web. Due to its mono-symmetric characteristics, it will also be subjected to torsion when subjected to transverse loads in practical applications. Past research on steel beams subject to torsion has concentrated on open sections while very few steel design standards give suitable design rules for torsion design. Since the hollow flange channel section is different from conventional open sections, its torsional behaviour remains unknown to researchers. Therefore the elastic behaviour of hollow flange channel sections subject to uniform and non-uniform torsion, and combined torsion and bending was investigated using the solutions of appropriate differential equilibrium equations. The section torsion shear flow, warping normal stress distribution, and section constants including torsion constant and warping constant were obtained. The results were compared with those from finite element analyses that verified the accuracy of analytical solutions. Parametric studies were undertaken for simply supported beams subject to a uniformly distributed torque and a uniformly distributed transverse load applied away from the shear centre. This paper presents the details of this research into the elastic behaviour and strength of hollow flange channel sections subject to torsion and bending and the results.

Bending and torsion of hollow flange channel beams

The LiteSteel beam (LSB) is a cold-formed high strength steel channel section made of two torsionally rigid closed flanges and a slender web. Due to its mono-symmetric characteristics, its centroid and shear centre do not coincide. The LSBs can be used in floor systems as joists or bearers and in these applications they are often subjected to transverse loads that are applied away from the shear centre. Hence they are often subjected to combined bending and torsion actions. Previous researches on LSBs have concentrated on their bending or shear behaviour and strengths, and only limited research has been undertaken on their combined bending and torsion behaviour. Therefore in this research a series of nine experiments was first conducted on LSBs subject to combined bending and torsion. Three LSB sections were tested to failure under eccentric loading at mid-span, and appropriate results were obtained from seven tests. A special test rig was used to simulate two different eccentricities and to provide accurate simple boundary conditions at the supports. Finite element models of tested LSBs were developed using ANSYS, and the ultimate strengths, failure modes, and load–displacement curves were obtained and compared with corresponding test results. Finite element analyses agreed well with test results and hence the developed models were used in a parametric study to investigate the effects of load locations, eccentricities, and spans on the combined bending and torsion behaviour of LSBs. The interaction between the ultimate bending and torsional moment capacities was studied and a simple design rule was proposed. This paper presents the details of the tests, finite element analyses, and parametric study of LSBs subject to combined bending and torsion, and the results.

Determination of stress-intensity factors for cracks in tubes under torsion

An experimental investigation by two-dimensional photoelastic technique is carried out to study the stress distribution and to determine the stress-intensity factors for arbitrarily oriented cracks in thin cylindrical shells subjected to torsion. A new method is employed to evaluate the pure and mixed-mode SIF's.

Influence of electroslag refining on the hot ductility of AISI 4340 steel in torsion

Electroslag refining is a useful remelting process by which clean steels can be produced for sophisticated applications. In this investigation, AISI 4340 steel has been electroslag refined and the improvement in its hot ductility has been assessed using hot torsion tests; electroslag refining has improved the hot ductility considerably. The temperature at which peak ductility is obtained has also increased — from 1473 K in the unrefined steel to 1573 K in ESR steel. Results indicate that it should be possible to subject the ESR ingot to much higher strains per unit operation during industrial hot working processes such as forging, which would result in a considerable saving of power. The improvement in hot ductility in ESR steel has been attributed primarily to the removal of non-metallic inclusions and the reduction in sulphur content. From the apparent activation energy estimated from the hot torsion data, the dynamic recrystallization process is identified as the mechanism controlling the rate of hot deformation.

Mixed-mode stress intensity factors for arbitrarily oriented cracks in cylindrical shells under axial pull and torsion

Results of photoelastic investigations conducted on cylindrical tubes (made of Araldite material) containing cracks oriented at 0°, 30°, 45°, 60° and 90° to the axis of the tube and subjected to axial and torsional loads are reported. The stress-intensity factors (SIFs) were determined by analysing the crack-tip stress fields. Smith and Smith's method [Engng Fracture Mech.4, 357–366 (1972)] and a new method developed by the authors by modifying Rakesh et al.'s method [Proc. 26th Congress of ISTAM, India (1981)] were employed to evaluate the mixed-mode SIFs.

Bond length distortions associated with torsion potentials; ab-initio studies on methanedio

The 4-31G basis set is used to study the bond length variations as functions of dihedral angels in methanediol. This study is compared with O---C---O bond angle optimization studies by Gorenstein and Kar and the possible reason for bond length shorteing in the trans---trans configuration is analysed.

Influence of state of stress on the processing map for hot working of stainless steel type AISI 304L: compression vs. torsion

Processing maps for hot working of stainless steel of type AISI 304L have been developed on the basis of the flow stress data generated by compression and torsion in the temperature range 600–1200 °C and strain rate range 0.1–100 s−1. The efficiency of power dissipation given by 2m/(m+1) where m is the strain rate sensitivity is plotted as a function of temperature and strain rate to obtain a processing map, which is interpreted on the basis of the Dynamic Materials Model. The maps obtained by compression as well as torsion exhibited a domain of dynamic recrystallization with its peak efficiency occurring at 1200 °C and 0.1 s−1. These are the optimum hot-working parameters which may be obtained by either of the test techniques. The peak efficiency for the dynamic recrystallization is apparently higher (64%) than that obtained in constant-true-strain-rate compression (41%) and the difference in explained on the basis of strain rate variations occurring across the section of solid torsion bar. A region of flow instability has occurred at lower temperatures (below 1000 °C) and higher strain rates (above 1 s−1) and is wider in torsion than in compression. To achieve complete microstructure control in a component, the state of stress will have to be considered.