Strained coordinates

The high enantioselectivity again can be rationalized by enantioface-selective alkene coordination in 63 (Fig. 35). The olefin moiety is expected to bind trans to the upper imidazoline moiety [70,73] thereby releasing the catalyst strain. Coordination at this position may, in principal, afford four different isomers assuming the stereoelectronically preferred perpendicular orientation of the alkene and the Pt(II) square plane. In the coordination mode shown, steric repulsion between both olefin substituents and the ferrocene moiety is minimized. Outer-sphere attack of the indole core results in the formation of the product s stereocenter. 

Alternative methods of analysis have been examined and evaluated. Shokoohi and Elrod[533] solved the Navier-Stokes equations numerically in the axisymmetric form. Bogy15271 used the Cosserat theory developed by Green.[534] Ibrahim and Linl535 conducted a weakly nonlinear instability analysis. The method of strained coordinates was also examined. In spite of the mathematical or computational elegance, all of these methods suffer from inherent complexity. Lee15361 developed a 1 -D, nonlinear direct-simulation technique that proved to be a simple and practical method for investigating the nonlinear instability of a liquid j et. Lee s direct-simulation approach formed the... 

In this chapter an effective Hamiltonian for a static cooperative Jahn-Teller effect is proposed. This Hamiltonian acts in the space of local active distortions only and possesses extrema points of the potential energy equivalent to those of the full microscopic Hamiltonian, defined in the space of all phonon and uniform strain coordinates. First we present the derivation of this effective Hamiltonian for a general case and then apply the theory to the investigation of the structure of Jahn-Teller hexagonal perovskites. 

The additional comment that the high aflSnity of metalloenzymes for their metals as "compared with the stability of chelates which use the same ligands, argues against a thermodynamically strained coordination is similarly not relevant and based upon a misinterpretation of the entatic site hypothesis. Entasis implies that the difference in energy between the ground state and transition state for the enzymatic reaction is reduced, not that the metal-enzyme complex is thermodynamically less stable, as was inferred. Indeed, there is no reason to suppose that the distorted environment of a metal ion in an enzyme as opposed to a simple metal complex leads necessarily to an increase in free energy. The studies of alkaline phosphatase just presented certainly seem consistent with the entatic state hypothesis. 

If X, is set to zero, the strained coordinates method will become the traditional regular perturbation method. 

Further exposition of this method of strained coordinates can be found in Nayfeh (1973) and Aziz and Na (1984). 

Since the potential flow solution (Eq. 20) does not satisfy the no-slip condition on the solid surface, we must consider a thin layer (called the Stokes layer) adjacent to the surface n = 0, where n refers to the local coordinate normal to the wall. In this thin layer, we use the strained coordinate... 

The material damping, or mechanical hysteresis, is related with the energy dissipation in the volume of a macro-continuous medium. This phenomenon takes place when a more or less homogeneous volume of material is subjected to cyclic stresses. As a result of inelasticity, a phase shift occurs between strain and stress and in the stress-strain coordinates, the geometrical locus of operational point becomes a loop, known as hysteresis loop, which evolves with the number of stress cycles and offer useful information upon the material state. 

This step is called kinematic or geometrical linearization. For the following development we will employ the coordinates Su of this infinitesimal strain tensor as a measure of the local strain of the body in the immediate vicinity of material point X at position x in the actual configuration. Usually they will simply be referred to as strain coordinates. As a further important consequence of the geometrical linearization the fact should be noted that derivatives with respect to material coordinates Xk may be replaced by the corresponding derivatives with respect to spatial coordinates Xk. Within the limits of applicabiUty of the linear theory of elasticity relations (3.26) and (3.28) are also simplified. The former reduces to... 

Sometimes it is necessary to express the coordinates of the strain terrsor Sy in terms of the stresses Tu. This reqitires the solution of Eq. (3.51). If the various symmetries involved are taken into account, this turns out to form a system of six linear equations for the six strain coordinates. A solution exists if the determinant of the coefficients (containing only the elastic stiffnesses) does not vanish. The result takes the form... 

The second derivatives of H2 or G2 with respect to the strain coordinates still are functions of all the variables and might be called stiffness functions since their values at the reference state are per definition the second-order stiffness constants. In this sense the third-order stiffnesses are a measure of the strain dependence of the stiffnesses. The symbol A stands for a or and the symbol + for E or Dio indicate the thermal and electrical constraints in the manner used in Table 4.3. 

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