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.

Creep and fatigue of a single crystal nickel base superalloy

The introduction of single crystal casting techniques has led to the development of existing nickel-base superalloys to produce materials with optimum mechanical properties in the single crystal condition. As single crystals are known to be anisotropic, a study is needed to determine the general mechanical properties of these materials, and determine the effects of crystal orientation upon them. A study has been carried out to identify the effect of orientation and temperature on the creep and fatigue properties of a development single crystal superalloy, SRR 99. Creep testing and crystal rotation experiments have been made on SRR 99 and an earlier development alloy, SRR 9. Fatigue experiments at elevated temperatures have been carried out on both notched and un-notched specimens of alloy SRR 99. To aid in this analysis, several analytical techniques have been employed including Laue x-ray orientation analysis, measurement of strain by photographic methods and microstructural examination. Crystal rotation experiments have indicated that shear of 1 precipitates by lbrace111rbrace< 112> slip systems is operative during primary creep deformation at temperatures of 750oC and 850oC. The effect of orientation variation obtained by standard casting practices was not found to be significant. Creep rupture was found to be associated with multiple crack initiation from micropores. Fatigue crack initiation in un-notched specimens was found to be related to microporosity and microstructural defects. Failure was predominantly by crystallographic crack growth on lbrace111rbrace planes. The use of linear elastic fracture mechanics to describe fatigue crack propagation in alloy SRR 99 was found to be acceptable at temperatures up to 850oC. Variation of temperature, frequency and crystal orientation was found to have only moderate effect upon crack propagation rates.

Application of the finite element method to problems in fracture mechanics

Numerical techniques have been finding increasing use in all aspects of fracture mechanics, and often provide the only means for analyzing fracture problems. The work presented here, is concerned with the application of the finite element method to cracked structures. The present work was directed towards the establishment of a comprehensive two-dimensional finite element, linear elastic, fracture analysis package. Significant progress has been made to this end, and features which can now be studied include multi-crack tip mixed-mode problems, involving partial crack closure. The crack tip core element was refined and special local crack tip elements were employed to reduce the element density in the neighbourhood of the core region. The work builds upon experience gained by previous research workers and, as part of the general development, the program was modified to incorporate the eight-node isoparametric quadrilateral element. Also. a more flexible solving routine was developed, and provided a very compact method of solving large sets of simultaneous equations, stored in a segmented form. To complement the finite element analysis programs, an automatic mesh generation program has been developed, which enables complex problems. involving fine element detail, to be investigated with a minimum of input data. The scheme has proven to be versati Ie and reasonably easy to implement. Numerous examples are given to demonstrate the accuracy and flexibility of the finite element technique.

Settlement of fill with particular reference to opencast coal mining sites

A study of information available on the settlement characteristics of backfill in restored opencast coal mining sites and other similar earthworks projects has been undertaken. In addition, the methods of opencast mining, compaction controls, monitoring and test methods have been reviewed. To consider and develop the methods of predicting the settlement of fill, three sites in the West Midlands have been examined; at each, the backfill had been placed in a controlled manner. In addition, use has been made of a finite element computer program to compare a simple two-dimensional linear elastic analysis with field observations of surface settlements in the vicinity of buried highwalls. On controlled backfill sites, settlement predictions have been accurately made, based on a linear relationship between settlement (expressed as a percentage of fill height) against logarithm of time. This `creep' settlement was found to be effectively complete within 18 months of restoration. A decrease of this percentage settlement was observed with increasing fill thickness; this is believed to be related to the speed with which the backfill is placed. A rising water table within the backfill is indicated to cause additional gradual settlement. A prediction method, based on settlement monitoring, has been developed and used to determine the pattern of settlement across highwalls and buried highwalls. The zone of appreciable differential settlement was found to be mainly limited to the highwall area, the magnitude was dictated by the highwall inclination. With a backfill cover of about 15 metres over a buried highwall the magnitude of differential settlement was negligible. Use has been made of the proposed settlement prediction method and monitoring to control the re-development of restored opencase sites. The specifications, tests and monitoring techniques developed in recent years have been used to aid this. Such techniques have been valuable in restoring land previously derelict due to past underground mining.

Fatigue crack initiation and propagation from defects in aluminium welds

Fatigue crack initiation and propagation in aluminium butt welds has been investigated. It is shown that the initiation of cracks from both buried defects and. from the weld reinforcement may be quantified by predictive laws based on either linear elastic fracture mechanics, or on Neuber's rule of stress and strain ooncentrations. The former is preferable on the grounds of theoretical models of crack tip plasticity, although either may be used as the basis of an effeotive design criteria against crack initiation. Fatigue lives fol1owing initiation were found to follow predictions based on the integration of a Paris type power law. The effect of residual stresses from the welding operation on both initiation and propagation was accounted for by a Forman type equation. This incorporated the notional stress ratio produced by the residual stresses after various heat treatments. A fracture mechanics analysis was found to be useful in describing the fatigue behaviour of the weldments at increased temperatures up to 300°C. It is pointed out, however, that the complex interaction of residual stresses, frequency, and changes in fracture mode necessitate great caution in the application of any general design criteria against crack initiation and growth at elevated. temperatures.

The relationship between fracture toughness and microstructure in titanium alloys

Linear Elastic Fracture Mechanics has been used to study the microstructural factors controlling the strength and toughness of two alpha-beta, titanium alloys. Fracture toughness was found to be independent of orientation for alloy Ti/6A1/4-V, but orientation dependent for IMI 700, bend and tension specimens giving similar toughness values. Increasing the solution temperature led to the usual inverse relationship between strength and toughness, with toughness becoming a minimum as the beta transus was approached. The production of a double heat treated microstructure led to a 100% increase in toughness in the high strength alloy and a 20% increase in alloy Ti/6A1/4V, with little decrease in strength. The double heat treated microstruoture was produced by cooling from the beta field into the alpha beta field, followed. by conventional solution treatment and ageing. Forging above the beta transus led to an increase in toughness over alpha beta forging in the high strength alloy, but had little effect on the toughness of Ti/6A1/4V. Light and electron microscopy showed that the increased toughness resulted from the alpha phase being changed from mainly continuous to a discontinuous platelet form in a transformed beta matrix. Void formation occurred at the alpha-beta interface and crack propagation was via the interface or across the platelet depending on which process required the least energy. Varying the solution treatment temperature produced a varying interplatelet spacing and platelet thickness. The finest interplatelet spacing was associated with the highest toughness, since a higher applied stress was required to give the necessary stress concentration to initiate void formation. The thickest alpha platelet size gave the highest toughness which could be interpreted in terms of Krafftt's "process zone size" and the critical crack tip displacement criterion by Hahn and Rosenfield from an analysis by Goodier and Field.

The fracture toughness of a range of cast steels

A range of plain carbon, carbon-manganese and low alloy cast steels were tested in order to determine their various fracture toughness values under elastic and elastic-plastic conditions. The main fracture toughness parameters which are considered are (1) Linear Elastic Fracture Mechanics (LEFM), (2) the J-Contour Integral, and (3) Crack Opening Displacement (COD). Results are obtained from fracture toughness specimens of various dimensions and the relevance of the validity criteria to cast steels is considered in some detail. In addition, the effect of casting position on specimen toughness values was noted. Valid KIC results according to LEFM, were obtained for three of the eight cast steels tested. Although KIC values from LEFM were not obtained from the remaining five steels, critical COD and J-integral values were determined. It is postulated that these values and particularly the critical J values can be used, with confidence for material selection or in defect tolerance calculations using these steels. Toughness values were found to vary with casting position in several of the steels tested and the possible reasons for such variations are discussed in the Thesis.

Structural optimization

The non-linear programming algorithms for the minimum weight design of structural frames are presented in this thesis. The first, which is applied to rigidly jointed and pin jointed plane frames subject to deflexion constraints, consists of a search in a feasible design space. Successive trial designs are developed so that the feasibility and the optimality of the designs are improved simultaneously. It is found that this method is restricted lo the design of structures with few unknown variables. The second non-linear programming algorithm is presented .in a general form. This consists of two types of search, one improving feasibility and the other optimality. The method speeds up the 'feasible direction' approach by obtaining a constant weight direction vector that is influenced by dominating constraints. For pin jointed plane and space frames this method is used to obtain a 'minimum weight' design which satisfies restrictions on stresses and deflexions. The matrix force method enables the design requirements to be expressed in a general form and the design problem is automatically formulated within the computer. Examples are given to explain the method and the design criteria are extended to include member buckling. Fundamental theorems are proposed and proved to confirm that structures are inter-related. These theorems are applicable to linear elastic structures and facilitate the prediction of the behaviour of one structure from the results of analysing another, more general, or related structure. It becomes possible to evaluate the significance of each member in the behaviour of a structure and the problem of minimum weight design is extended to include shape. A method is proposed to design structures of optimum shape with stress and deflexion limitations. Finally a detailed investigation is carried out into the design of structures to study the factors that influence their shape.

Spin transition of 1D, 2D and 3D iron(II) complex polymers - the tug-of-war between elastic interaction and a shock-absorber effect

The structures of linear chain Fe(II) spin-crossover compounds of α,β- and α,ω-bis (tetrazol-1-yl)alkane type ligands are described in relation to their magnetic properties. The first threefold interlocked 3-D catenane Fe(II) spin-transition system, [μ-tris(1,4-bis(tetrazol-1-yl)butane-N1,N1′) iron(II)] bis(perchlorate), will be discussed. An analysis is made among the structures and the cooperativity of the spin-crossover behaviour of polynuclear Fe(II) spin-transition materials.

An efficient, approximate path-following algorithm for elastic net based nonlinear spike enhancement

Unwanted spike noise in a digital signal is a common problem in digital filtering. However, sometimes the spikes are wanted and other, superimposed, signals are unwanted, and linear, time invariant (LTI) filtering is ineffective because the spikes are wideband - overlapping with independent noise in the frequency domain. So, no LTI filter can separate them, necessitating nonlinear filtering. However, there are applications in which the noise includes drift or smooth signals for which LTI filters are ideal. We describe a nonlinear filter formulated as the solution to an elastic net regularization problem, which attenuates band-limited signals and independent noise, while enhancing superimposed spikes. Making use of known analytic solutions a novel, approximate path-following algorithm is given that provides a good, filtered output with reduced computational effort by comparison to standard convex optimization methods. Accurate performance is shown on real, noisy electrophysiological recordings of neural spikes.