The mechanics of ultrasonic tube bending

The technology of precision bending of tubes has recently increased in importance and is widely demanded for many industrial applications. However, whilst attention has been concentrated on automation and increasing the production rate of the bending machines, it seems that with one exception very little work has been done in order to understand and therefore fundamentally improve the bending process. A new development for the process of draw-bending of tubes, in which the supporting mandrel is axially vibrated at an ultrasonic frequency, has been perfected. A research programme was undertaken to study the mechanics of tube• bending under both vibratory and non-vibratory conditions. For this purpose, a conventional tube-bending machine was modified and equipped with an oscillatory system. Thin-walled mild steel tubes of different diameter to thickness ratios were bent to mean bend radii having various values from 1.5 to 2.0 times the tube diameter. It was found that the application of ultrasonic vibration reduces the process forces and that the force reduction increases with increasing the vibration amplitude. A reduction in the bending torque of up to 30 per cent was recorded and a reduction in the maximum tube-wall thinning of about 15 per cent was observed. The friction vector reversal mechanism as well as a reduction in friction account for the changes of the forces and the strains. Monitoring the mandrel friction during bending showed, in some cases, that the axial vibration reverses the mandrel .mean force from tension to compression and, thus, the mandrel is assisting the tube motion instead of resisting it. A theory has been proposed to describe the mechanics of deformation during draw-bending of tubes, which embodies the conditions of both "with" and "without" mandrel axial vibration. A theoretical analysis, based on the equilibrium of forces approach, has been developed in which the basic process parameters were taken into consideration. The stresses, the strains and the bending torque were calculated utilising this new solution, and a specially written computer programme was used to perform the computations. It was shown that the theory is in good agreement with the measured values of the strains under vibratory and non-vibratory conditions. Also, the predicted bending 'torque showed a similar trend to that recorded experimentally.

A study of the micro-mechanics of granular media

The development of more realistic constitutive models for granular media, such as sand, requires ingredients which take into account the internal micro-mechanical response to deformation. Unfortunately, at present, very little is known about these mechanisms and therefore it is instructive to find out more about the internal nature of granular samples by conducting suitable tests. In contrast to physical testing the method of investigation used in this study employs the Distinct Element Method. This is a computer based, iterative, time-dependent technique that allows the deformation of granular assemblies to be numerically simulated. By making assumptions regarding contact stiffnesses each individual contact force can be measured and by resolution particle centroid forces can be calculated. Then by dividing particle forces by their respective mass, particle centroid velocities and displacements are obtained by numerical integration. The Distinct Element Method is incorporated into a computer program 'Ball'. This program is effectively a numerical apparatus which forms a logical housing for this method and allows data input and output, and also provides testing control. By using this numerical apparatus tests have been carried out on disc assemblies and many new interesting observations regarding the micromechanical behaviour are revealed. In order to relate the observed microscopic mechanisms of deformation to the flow of the granular system two separate approaches have been used. Firstly a constitutive model has been developed which describes the yield function, flow rule and translation rule for regular assemblies of spheres and discs when subjected to coaxial deformation. Secondly statistical analyses have been carried out using data which was extracted from the simulation tests. These analyses define and quantify granular structure and then show how the force and velocity distributions use the structure to produce the corresponding stress and strain-rate tensors.

The application of contact mechanics to the numerical simulation of particulate material

Particulate solids are complex redundant systems which consist of discrete particles. The interactions between the particles are complex and have been the subject of many theoretical and experimental investigations. Invetigations of particulate material have been restricted by the lack of quantitative information on the mechanisms occurring within an assembly. Laboratory experimentation is limited as information on the internal behaviour can only be inferred from measurements on the assembly boundary, or the use of intrusive measuring devices. In addition comparisons between test data are uncertain due to the difficulty in reproducing exact replicas of physical systems. Nevertheless, theoretical and technological advances require more detailed material information. However, numerical simulation affords access to information on every particle and hence the micro-mechanical behaviour within an assembly, and can replicate desired systems. To use a computer program to numerically simulate material behaviour accurately it is necessary to incorporte realistic interaction laws. This research programme used the finite difference simulation program `BALL', developed by Cundall (1971), which employed linear spring force-displacement laws. It was thus necessary to incorporate more realistic interaction laws. Therefore, this research programme was primarily concerned with the implementation of the normal force-displacement law of Hertz (1882) and the tangential force-displacement laws of Mindlin and Deresiewicz (1953). Within this thesis the contact mechanics theories employed in the program are developed and the adaptations which were necessary to incorporate these laws are detailed. Verification of the new contact force-displacement laws was achieved by simulating a quasi-static oblique contact and single particle oblique impact. Applications of the program to the simulation of large assemblies of particles is given, and the problems in undertaking quasi-static shear tests along with the results from two successful shear tests are described.

The mechanics of creep crack growth in a Magnox alloy

Magnox AL80 has been used for a study of creep crack propagation. A number of variables have been considered such as specimen geometry,notch root radius, material thickness, creep prestrain and stress level.The work has covered the material behaving under two values of the creep exponent, n=3.5 and n=7, according to the stress level. As well as observing initiation times and crack growth rates, scribed grids have been used to examine the near crack tip strain levels and distributions. It was shown that estimations of COD from notch flank opening can give misleading indications of material behaviour and that a more informative method was to monitor displacements in the material surrounding the crack tip. Strong evidence was found for crack advance being displacement controlled, however it was shown that the COD approach should be considered geometry dependant. The summation of ∈xx and ∈yy provided the most successful description of crack advance as it produced a single value that described propagation in all the cases concidered. The strain distributions indicates that σyy was related to distance from a point ahead of the crack tip by the exponent - (l/n+l) and that σxx is proportional to σyy. The constraint stresses arising in the DEN and CN specimens were evaluated. Initiation time was found to be principally affected by the stress level but was modified by the constraints arising from specimen geometry. Crack growth was found not to obey either the empirical K or σpett relationships but was reviewed in context of the observed strain behaviour.

Statistical mechanics of distribution networks

This thesis presents an analysis of the stability of complex distribution networks. We present a stability analysis against cascading failures. We propose a spin [binary] model, based on concepts of statistical mechanics. We test macroscopic properties of distribution networks with respect to various topological structures and distributions of microparameters. The equilibrium properties of the systems are obtained in a statistical mechanics framework by application of the replica method. We demonstrate the validity of our approach by comparing it with Monte Carlo simulations. We analyse the network properties in terms of phase diagrams and found both qualitative and quantitative dependence of the network properties on the network structure and macroparameters. The structure of the phase diagrams points at the existence of phase transition and the presence of stable and metastable states in the system. We also present an analysis of robustness against overloading in the distribution networks. We propose a model that describes a distribution process in a network. The model incorporates the currents between any connected hubs in the network, local constraints in the form of Kirchoff's law and a global optimizational criterion. The flow of currents in the system is driven by the consumption. We study two principal types of model: infinite and finite link capacity. The key properties are the distributions of currents in the system. We again use a statistical mechanics framework to describe the currents in the system in terms of macroscopic parameters. In order to obtain observable properties we apply the replica method. We are able to assess the criticality of the level of demand with respect to the available resources and the architecture of the network. Furthermore, the parts of the system, where critical currents may emerge, can be identified. This, in turn, provides us with the characteristic description of the spread of the overloading in the systems.

Composite CDMA—a statistical mechanics analysis

Code division multiple access (CDMA) in which the spreading code assignment to users contains a random element has recently become a cornerstone of CDMA research. The random element in the construction is particularly attractive as it provides robustness and flexibility in utilizing multiaccess channels, whilst not making significant sacrifices in terms of transmission power. Random codes are generated from some ensemble; here we consider the possibility of combining two standard paradigms, sparsely and densely spread codes, in a single composite code ensemble. The composite code analysis includes a replica symmetric calculation of performance in the large system limit, and investigation of finite systems through a composite belief propagation algorithm. A variety of codes are examined with a focus on the high multi-access interference regime. We demonstrate scenarios both in the large size limit and for finite systems in which the composite code has typical performance exceeding those of sparse and dense codes at equivalent signal to noise ratio.

Relaxation procedures for an iterative MFS algorithm for the stable reconstruction of elastic fields from Cauchy data in two-dimensional isotropic linear elasticity

We investigate two numerical procedures for the Cauchy problem in linear elasticity, involving the relaxation of either the given boundary displacements (Dirichlet data) or the prescribed boundary tractions (Neumann data) on the over-specified boundary, in the alternating iterative algorithm of Kozlov et al. (1991). The two mixed direct (well-posed) problems associated with each iteration are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method, while the optimal value of the regularization parameter is chosen via the generalized cross-validation (GCV) criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The MFS-based iterative algorithms with relaxation are tested for Cauchy problems for isotropic linear elastic materials in various geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the proposed method.