Compatibility of the Infinitesimal Deformation Tensor

Abstract

Our next main task will be to investigate how the displacement field can be recovered if the infinitesimal stress or strain tensors are known. Although the scope of this question turns out to be rather broad, the gist of what we are about to do is contained in the following basic scenario. Assume that (forumala presented). is an open set and (forumala presented). are given scalar fields ((forumala presented).); we are interested in finding a new scalar field (forumala presented). such that (forumala presented). and (forumala presented). in D. Of course, if such a field exists then (forumala presented).. This latter condition is necessary for the existence of a (forumala presented). having the foregoing stated properties. To put it differently, the obtained condition ensures the compatibility (or consistency) of the two equations satisfied by (forumala presented).. It is fairly straightforward to show that under certain circumstances the condition is also sufficient. We note in passing that if (forumala presented)., the compatibility condition in this basic case can be cast as (forumala presented).. The extension of these naive calculations to vector and tensor fields (as explained in the next sections) leads naturally to a discussion of the Beltrami–Michell equations and the concept of Weingarten-Volterra dislocation in multiply connected linearly elastic bodies.