A representation of Beltrami's strain compatibility condition (Q2714561)

Considered is an isotropic homogeneous elastic material occupying some domain \(V\) of three-dimensional space. The author sets out the Beltrami deformation compatibility condition when the volume forces are absent NEWLINE\[NEWLINE \Delta\bigg[\hat T + \frac{1}{2(1+\nu)}\,\nabla(R\sigma)\bigg] = 0,\tag{1} NEWLINE\]NEWLINE where \(\hat T\) is the stress tensor, \(\sigma\) is the first invariant of the stress tensor, \(\nu\) is the Poisson ratio, \(\Delta\) is the Laplace operator, \(\nabla\) is the nabla-operator, \(R\) is the vector radius. In terms of (1) a representation of the harmonic tensor is obtained via the stress tensor. This allows to find out all components of the stress tensor for certain volume expansion.