Higher Order Tensors

Abstract

Higher order tensors, as the name implies, are objects which are generalizations of the mathematical entities encountered in now-classical tensor analysis. In a continuum-mechanical context these objects arise naturally in the development of a theory for the description of the mechanical response of inhomogeneous, materially uniform, nonsimple ($ \equiv $higher order), gross bodies, where it becomes mandatory to study fields with values in tensor products of tangent spaces of “higher order contact.” As in the case of classical tensors, herein we regard higher order tensors as objects worthy of study per se. Generalizing upon the classical case, a (repeated) index-lable (double) summation convention is devised for higher order tensors. Further compactness is achieved by introducing an index-block summation convention. After establishing the relation of higher order tensors to r-jet theory, r-vectors and r-covectors, we first investigate some of their algebraic properties and then consider fields of such objects. Certain properties of intrinsic higher order tensors are determined. Succeeding a study of higher order metric tensors, with the aid of r-jet bundle theory, we construct a theory for the covariant differentiation of higher order tensors and illustrate it by giving several examples. Finally, we exhibit a continuum-mechanical application of the preceding development.