Linear displacement geometry

A conscious choice of such elements can be made but in general the equilibrium distribution of stress cannot be found except for particular geometries. The assumptions of uniform strain throughout the assembly or of uniform stress were respectively made by Voigt and by Reuss. Returning to the structures actually perceivable in polymers one may consider the spherulite in a semi crystalline polymer as being unsuitable as a RVE because the boundary is not included. However, an assembly of spherulites would be acceptable, since it would contain sufficient to make it entirely typical of the bulk and because such an assembly would have moduli independent of the surface tractions and displacements. The linear size of such a representative volume element of spherulites would be perhaps several hundred microns. 

This geometry of this problem inspires a description in terms of cylindrical coordinates, within which a plausible form for the displacements is the assumption that, like a helical ramp in a parking garage, the displacements increase linearly in the winding angle, 6, yielding the solution... 

The double torsion test specimen has many advantages over other fracture toughness specimen geometries. Since it is a linear compliance test piece, the crack length is not required in the calculation. The crack propagates at constant velocity which is determined by the crosshead displacement rate. Several readings of the critical load required for crack propagation can be made on each specimen. 

For +2 cations such as zinc(ll) and cadmiunXIl) each metallothionem molecule contains up to seven metal atoms. X-ray studies indicate that the metal atoms are in approximately tetrahedral sites bound to the cysteine sulfur atoms. The soft mer-cury(JI) ion has a higher affinity for sulfur and will displace cadmium from mefallothio-nein. At first the mercury ions occupy tetrahedral sites but as the number increases, the geometries of the metal sites and protein change until about nine Hg(II) atoms are bound in a linear (S—Hg—S) fashion.92 Up to twelve + I cations such as copper(l) find silverfUcan bind per molecule, indicating fi coordination number lower than four, probably three (see Problem 12.34). 

Chemical model systems, in particular the picket fence 33 (Figure 10.13), have been particularly useful in the studies on hemoglobin. The small chemical models may be crystallized and their structures determined with far higher precision than the structures of the actual proteins can be. Binding measurements may be made on the models without the complications that arise from the protein structure. The precise displacement of the iron atom from the heme and the geometry of iron-ligand bonds were first measured in the models.29,33,34 The Fe—02 bond is bent, whereas the Fe—CO bond is linear (structures 10.15). The Fe—02 bond is bent at 156° in hemoglobin.35... 

Consider expanding the energy of a molecule in displacements from some reference geometry. We denote these displacements by 7o, where a runs over nonredundant displacements (3 N — 6 for a nonlinear reference geometry and 3 N — 5 for the linear case). The energy expansion is thus... 

We find that the eg distortions derived from the Ham quenching and the intensity distribution in the progression differ by less than ten per cent, thus confirming the soundness of our analytical procedure. In order to get the actual displacements in Cr-X bond lengths, A(Cr-X)e j and A(Cr-X)ax, for the equilibrium geometry of the luminescent T2 state, the AQ values have to be linearly transformed (17). For the > component of T2g the values in the last two columns of Table II are obtained. The result for the Cs2NaYClg lattice is visualized in Figure 7. 

---