Rank-one connection

Fig. 10.31. Schematic illustration of deformation gradients in different wells for the cubic to tetragonal transformation that satisfy the rank-one connection condition (adapted from Ball and James (1992)).

Rank-One Connection for Model Deformations Examine the deformation mapping given by... 

We say that X, is d-connected if X, consists of one connected component (note that the word connected is reserved for componentwise connectedness). If X, is locally noetherian, then a connected component of X, is a closed open subdiagram of schemes in a natural way. If this is the case, the rank function i, x) i—> ranko y, of a locally free sheaf IF is constant on... 

This is how we rank the two carbon atoms for each carbon atom, we write a list of three atoms it is connected to (other than the stereocenter). Let s do the example above to see how this works. The carbon atom on the left side of the stereocenter has four bonds one to the stereocenter, one to another carbon atom, and then two hydrogen atoms. So, other than the stereocenter, it has three bonds (C, H, and H). Now let s look at the carbon atom on the right side of the stereocenter. It has four bonds one to the stereocenter and then three hydrogen atoms. So, other than the stereocenter, it has three bonds (H, H, and H). We compare the two lists and look for the hrst point of difference ... 

The set of internal variables is usually determined when considering a particular system in more detail. For concentrated solutions and melts of polymers, for example, a set of relaxation equation for internal variables were determined in the previous chapter. One can see that all the internal variables for the entangled systems are tensors of the second rank, while, to describe viscoelasticity of weakly entangled systems, one needs in a set of conformational variables xfk which characterise the deviations of the form and size of macromolecular coils from the equilibrium values. To describe behaviour of strongly entangled systems, one needs both in the set of conformational variables and in the other set of orientational variables w fc which are connected with the mean orientation of the segments of the macromolecules. 

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