The design of polyethylene (PE) pipelines for applications in the gas, water and chemical process industries has been based on data mainly obtained from stress rupture testing pipes only. In practice, installations are composed of both extruded pipe and injection moulded fittings which are joined by a fusion welding technique and are very often subjected to internal pressures of a fluctuating nature. Several makes of PE pipe systems were therefore obtained and work was undertaken to fully characterise mechanical performance in terms of internal pressure loadings. Butt-welded test specimens comprising pipe lengths and fittings were subjected to both static and fluctuating conditions at 80°C, at pressures resulting in brittle fractures (below the knee on stress rupture curves) and at frequencies not exceeding 7.5 cpm (0.125 Hz). Resulting fracture surfaces were examined to identify sources of crack initiation and mechanisms of failure. Mechanical behaviour of the PE pipe samples was found to be markedly influenced by the grade of plastics compound, the pipe system dimensions, mould designs and methods of processing. Fatigue loading was the most aggressive test method and significant reductions in lifetimes were observed in fittings or joints between pipes and fittings with only modest increases in the frequency of pressurisation. It was also demonstrated that improved stress rupture behaviour did not necessarily lead to better fatigue performance. For the square-wave loading profiles used, an idea of the relevant failure mechanisms in any given system was obtained by comparing experimental Nf values with those predicted from cumulative damage principles based on Nf=τSR/τmax. In all types of system, failure was initiated at a defect residual from processing or jointing. Over 95% of all small diameter pipe fractures originated from inclusions at or close to the inside wall. They were geometrically and elementally analysed and suggestions made as to their possible origin and means of elimination. For one PE a reasonable correlation was obtained, between lifetime under stress rupture or fatigue and the inclusion size as measured in the fracture plane.

In this paper our goal is to describe how to find the growth of Thompson's group F with generators a and b. Also, by studying elements through pipe systems, we describe how adding a third generator c affects geodesic length. We model the growth of Thompson's group F by producing a grammar for reduced pairs of trees based on Blake Fordham's tree structure. Then we change this grammar into a system of equations that describes the growth of Thompson's group F and simplify. To complete our second goal, we present and discuss a computer program that has led to some discoveries about how generators affect the pipe systems. We were able to find the growth function as a system of 11 equations for generators a and b.