A materials characteristic length and microstructure

The preceding six sentences are about real materials. We now turn to the question if we wish to imagine an ideal material whose behavior follows equations and yet resembles real materials in selected ways, what properties should we assign to the imaginary material First, we should imagine it to be free of microstructure so that it obeys the same equations at every point, but, second, we should assume that might not be simply the product 

A separate complication is that the substrate may be anisotropic this possibility is discussed in Chapter 20. 

The following remarks use ideas that are introduced in Appendix 8C. We emphasize effects that occur or exist at a point that have no direction associated with them, as opposed to flows and gradients with respect to position, which are vectors. The purpose is to repeat the summary just given in another, more categorical, form. 

Appendix 8C describes a procedure for specifying the stress state at a point by means of three invariants rather than three principal stresses the invariants are totally free of any link to directions. The point is also made that the Laplacian operator has the same attribute any first derivative with respect to position, d/dx, is a vector, and a single second derivative d /8x is also linked to the direction x, but the Laplacian has no links to direction 

Laplacian functions of these are also possible direction-free influences. 

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