Lame approach

The second method is the Lame approach (Kashani and Young, 2008), which is based on displacement differential equations and is applicable to any cylindrical vessel with any diameter-to-wall-thickness ratio. The Lame method is often referred to as the solution for thick wall cylindrical pressure vessels. Equations for the hoop stress and radial stress in a thick-walled cylinder were developed by Lame in the early nineteenth century (Timoshenko and Goodier, 1969) ... 

Functionals which are related to this approach include among others the recent FT97 functional of Filatov and Thiel, 1997, the PW91 exchange functional (Perdew, 1991, and Burke, Perdew, and Wang, 1998) and the CAM(A) and CAM(B) functionals developed by Handy and coworkers (Laming, Termath, and Handy, 1993). 

In this approach, the horizontal coordinates x[t] and y[t] both satisfy linear, second order equations. The exact trajectories satisfy Lame s equation of order two, which is extremely difficult to analyse. The corresponding equation for the linear approximation is Mathieu s equation, which is known to have both periodic and aperiodic solutions. 

Ionization of liquid ammonia and water solutions.—Solutions of certain salts in liquid ammonia are good conductors of electricity so that liquid ammonia approaches water in its ionizing power. The effect, however, is largely due to the high speed at which the ions are supposed to travel in the solvent. For example, E. C. Franklin and H. P. Cady1 find that univalent ions travel, at —33°, nearly three lames as fast as in aq. soln. at 18°. Just as the solvent water, in the ionization theory of hydrolysis, is supposed to be ionized H20=0H -f-H, so in ammonolysis, the solvent ammonia is supposed to be ionized NH3==NH2-j-H . Sodamide, NaNH, furnishes sodium ions Na and amide ions NH 2 when dissolved in liquid ammonia, and it is to be considered as a base. It reddens phenolphthalein. The neutralization of this solution results in the union of H ions with NH2 ions to form ammonia molecules, just as the neutralization of bases is regarded as an effect of the union of H and OH ions. Acetamide, CH3.CO.NH2, ionizes in liquid ammonia in an analogous manner CH3.CO.NH2 CH3.CO.NH -f-H, and it thus behaves as an acid. 

Morals, S., Shterzer, N., Lamed, R., Bayer, E.A. et al. (2014) A combined cell-consortium approach for ligno-cellulose degradation by specialized Lactobacillus plantarum cells. Biotechnol Biofuels, 7, 112. 

These are the fluctuations observed in a structure, when a phase transition is approached, and which can be considered as precursors of the transformation. In general their local symmetry is closely related to that of the stucture to come. For instance, in the case of the solid/Iquid transition for which this term was proposed first , they are local perturbations of the crystalline order when the transition to the liquid is approached. Similar phenomena have indeed been observed in the case of the lame liar/cubic transformation in Cj2E0g/H20. They manifest themselves, on the X-ray pattern of the... 

For the analysis, only the positive internal pressure (Pj) is considered and a positive pressure gradient is maintained from the inner to outer radius. Hence, stress in the radial direction is always in compression and is insignificant compared to the magni-mde of hoop and axial stresses. As a result, hoop and axial stresses govern the design. Therefore, for approximate analysis, the assumption E = Ei = P3 is made and Lame s approach for an isotropic material can be used, wherein 3 is stiffness along the radial direction. 

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