On Finite Deformations of an Elastic Isotropic Material (Classic Reprint)

Excerpt from On Finite Deformations of an Elastic Isotropic Material
The proof of (2) is based on the realization that there exist subsets called cores of the domain D which is under going the transformation, in which the relative change in distance of any two points is at most 6. In fact, any convex subset of D whose distance from the boundary of D exceeds a certain amount o 5(e, d) is such a core. Moreover, 5 tends to 0 for e - a>o. This implies that for convex D and small 6 there are cores of D which fill all of D except for a thin boundary layer. Since in a core the mutual distance between any two points changes by a small amount for small 6, the trans formation of a, core is essentially rigid, which leads to the desired result. Similar estimates can doubtlessly be obtained for D which have shapes differing from square plates.
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