The overall objective of this study is to develop methods for providing a statistical summary of material fatigue stress-life (S-N) data for engineering design purposes. Specific goals are: (1) Development of an analytical model for characterizing fatigue strength. This model would include: (a) a description of the trend of the data (e.g., the median curve through the data), (b) a description of the scatter of the data (e.g., the standard deviation of N as a function of S), and (c) the statistical distribution of N given S or S given N. (2) Development of an algorithm for constructing a design curve from the data. The curve should be on the safe side of the data and should reflect uncertainties in the physical process as well as statistical uncertainty associated with small sample sizes. (3) Development of a statistical model that can be applied in a structural reliability analysis in which all design factors are treated as random variables. Significant achievements are: (1) Demonstration, using representative fatigue data sets, that the bilinear model seems to provide a consistently adequate description of the trend of fatigue data. (2) Demonstration, using representative fatigue data sets, that the pure X error source model seems to provide a consistently adequate description of the uncertainties observed in heteroscedastic fatigue data. The pure X error source model is based on recognition of the uncertainties in local fatigue stress. (3) Development of a procedure for constructing a design curve using the tolerance limit concept developed by D. B. Owen. A more practical simplified or approximate Owen curve was shown to have a minimum loss of confidence level, relative to exact Owen theory, under fairly general conditions. (4) Recommendations for methods of developing a statistical model for reliability analysis. A comprehensive study of this issue was not pursued.

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In this work I model the liquid crystal polymers as worms and explore the Spheroidal approach to examine their statistical mechanics. Several models are presented in this work to describe main chain-, side chain-polymers, polymer networks and gels in their nematic state. In the case of main chain nematic polymers, the worm flexibility, favouring disorder, and the nematic potential, tending to align segments to be parallel to each other, compete to determine the properties of polymers. I predict the temperature dependence of order parameter and phase transition behaviour for different lengths of the polymers, and the dimensions as well. Subsequently, I examine the critical features of the nematic polymer when an electrical field is applied. Side chain polymers with semi-flexible backbone and stiff nematogenic pendants form interesting nematic phases, largely as a consequence of competition between backbone entropy and pendant order. I classify them into three categories: NI, NII, and NIII phase, according to volume fractions, temperature, nematic coupling constants, and stiffness. In these phases the backbone and pendants have orders different in magnitude and/or in sign in order to achieve a stable state. Phase diagrams are given. In addition, I predict unusual properties such as anomalous temperature variation of optical anisotropy and molecular conformational changes. Crosslinks confine polymer chains in a network so that their strands have a shape different from their natural ones. Such constraints shift phase transition temperature. The other effect is that crosslinks give the system rubber elasticity. Combining rubber elasticity with liquid crystal features, networks exhibit unusual phenomena, such as discontinuous stress-strain relations, spontaneous shape changes, non-linear stress-optical laws and deviations from classical behaviour of conventional elastomers. It is proposed that residual nematic interaction is responsible for deviations found in classical elastomers. The nematic networks swollen by isotropic solvent form nematic gels. At low temperatures a nematic gel coexists with excess solvent, at high temperatures the coexisting gel is isotropic. In addition, coexistence is predicted between nematic and isotropic gels. There is an associated triple point. There are possible elastic problems associated with different phases coexisting in one gel sample. Main chain nematic polymers have been modelled either as homogeneous worms, or as jointed rods by others. In reality the polymers are composed of the mesogens linked by semi-flexible spacers. One must expect that the spacers have an order differing from the mesogens. The consecutive mesogens are not decoupled and the spacers are able to talk to each other via the mesogens in between. The model presented takes account of molecular parameters, such as length of the mesogen and spacer, and their interactions. The nematic order of the two components, the nematic-isotropic transition, and dimensions of the polymers are addressed. Finally, I examine both worm and jointed rod models, to see when each is applicable. Accordingly an elastically jointed rod model is presented. Hairpins, found naturally in the worm problem, also exist for jointed systems but their scaling is quite different. Comparisons of these results with experiments are accordingly made and are found to be satisfactory.

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