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 $D\subset \mathbb {E}^2$ is an open set and $g_j\in C^1(D)$ are given scalar fields ($j=1,2$); we are interested in finding a new scalar field $\phi \in C^2(D)$ such that $\phi _{\,,\,1} = g_1$ and $\phi _{\,,\,2}=g_2$ in D. Of course, if such a field exists then $\phi _{\,,\,12}=\phi _{\,,\,21}$ or $g_{1,\,2}=g_{2,\,1}$. This latter condition is necessary for the existence of a $\phi$ having the foregoing stated properties. To put it differently, the obtained condition ensures the compatibility (or consistency) of the two equations satisfied by $\phi$. It is fairly straightforward to show that under certain circumstances the condition is also sufficient. We note in passing that if $\varvec{g}:=(g_1,\,g_2,\,0)$, the compatibility condition in this basic case can be cast as $\varvec{\nabla }\wedge \varvec{g}= \varvec{0}$. 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.