High-Rank Deformators and the Kröner Incompatibility Tensors with Two-Dimensional Structure of Indices

Abstract

In n-dimensional space (multidimensional continuum), the compatibility equations ( $$m \geqslant 1$$ , $$n \geqslant 2$$ ) are derived for the components of the generalized strains of rank m associated with the generalized displacements of rank m – 1 by the analogs of the Cauchy kinematic relations. The compatibility conditions may be written in the form of vanishing of all components of the incompatibility tensor of rank $$m(n - 2)$$ or of the generalized Riemann–Christoffel tensor of rank 2m, which is dual to it. The number of independent components of these tensors coinciding with the number of compatibility equations in terms of generalized strains is determined.