Williams expansion

An alternative stress analysis approach, is based on the local stress field near a crack tip. Fig. 5. The solution of the boundary value problem for a semi-infinite, linear-elastic cracked body was found by Williams [48], and yields the Williams expansion of the stress field in a cracked body. The first term determines the local stress field in the vicinity of the crack tip ... 

The basic steam cycle for a steam turbine installation is called a Rankine cycle (named after Scottish engineer and physicist William John Macquorn Rankine). This cycle consists of a compression of liquid water, heating and evaporation in the heat source (a steam boiler or nuclear reactor), expansion of the... 

As has been shown by the X-ray diffraction method the parent metals (i.e. Pd or Ni), the a-phase, and /3-phase all have the same type of crystal lattice, namely face centered cubic of the NaCl type. However, the /9-phase exhibits a significant expansion of the lattice in comparison with the metal itself. Extensive X-ray structural studies of the Pd-H system have been carried out by Owen and Williams (14), and on the Ni-H system by Janko (8), Majchrzak (15), and Janko and Pielaszek (16). The relevant details arc to be found in the references cited. It should be emphasized here, however, that at moderate temperatures palladium and nickel hydrides have lattices of the NaCl type with parameters respectively 3.6% and 6% larger than those of the parent metals. Within the limits of the solid solution the metal lattice expands also with increased hydrogen concentration, but the lattice parameter does not depart significantly from that of the pure metal (for palladium at least up to about 100°C). 

The first analytical estimation of the value of the Markstein number in the presence of realistic gas expansion was given by Clavin and Williams [8]. In the approximation of a one-step Arrhenius reaction with a high activation energy P, they found fhe following expression for Ma ... 

P. Clavin and F.A. Williams. Effects of molecular diffusion and of thermal expansion on the structure and dynamics of premixed flames in turbulent flows of large scale and low intensity. Journal of Fluid Mechanics, 116 251-282,1982. 

B. Flaherty, S. Nahar, W. G. Overend, and N. R. Williams, Branched-chain sugars. XIV. Reactions of glycosulose derivatives with diazomethane. Ring expansion of glycosulose derivatives, J. Chem. Soc., Perkin Trans. I (1973) 632-638. 

Williams, N.A., Gugliemo, J. Thermal mechanical analysis of frozen solutions of mannitol and some related stereoiso-meres evidence of expansion during warming and correlation with vial breakage during lyophilization. PDA J. Parenteral Sd. Technol. 47,119-123, 1993... 

Andersen, O.K. (1971). Comments on the KKR wavefunctions extension of the spherical wave expansion beyond the muffin tins. In Computational Methods in Band Theory, eds. P.M. Marcus, J.F. Janak, A.R. Williams (Plenum, New York), 178-182. 

The Kelvin degree scale was originally introduced by William Thomson. It is a thermodynamic temperature scale based on the second law of thermodynamics and is identical to the absolute temperature scale based on the above volume expansion arguments. Substituting Equation (1.4) into Equation (1.3) gives ... 

This is identical in form with Eq. (48) and by equating the constant terms we find f(0,Ta) = 0.0490 and uf = 4.82 x 10 4 deg.-1. Williams et al. pointed out the fact that this value of af is, for many substances including non-polymeric ones such as glycerol, in reasonable agreement with the difference between the thermal expansion coefficients above and below the glass transition point. This suggests that the excess increase... 

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