On the Helmholtz decomposition for polyadics

Abstract

Abstract.A polyadic field of rank n is the tensor product of n vector fields. Helmholtz showed that a vector field, which is a polyadic field of rank 1, is nonuniquely decomposable into the gradient of a scalar function plus the rotation of a vector function. We show here that a polyadic field of rank n is, again nonuniquely, decomposable into a term consisting of n successive applications of the gradient to a scalar function, plus a term that consists of (n -1) successive applications of the gradient to the rotation of a vector function, plus a term that consists of (n -2) successive applications of the gradient to the rotation of the rotation of a dyadic function and so on, until the last (n + l)th term, which consists of n successive applications of the rotation operator to a polyadic function of rank n. Obviously, the n = 1 case recovers the Helmholtz decomposition theorem. For dyadic fields a more symmetric representation is provided and formulae that provide the potential representation functions are given. The special cases of symmetric and antisymmetric dyadics are discussed in detail. Finally, the multidivergence type relations, which reduce the number of independent scalar representation functions to n2, are presented.

Frontiers in growth and remodeling

“…A general field representation of a second-order tensor includes further contributions in addition to the gradient form present in the definition of F in equation (2). To derive this general form, we consider the Helmholtz representation of a vector field, $$\mathit{h}(\mathit{X})={bold\nabla }_{\mathrm{bold-italicX}}a(\mathit{X})+{bold\nabla }_{\mathrm{bold-italicX}}^{\mathrm{normalt}}\times \mathit{b}(\mathit{X})+{\mathrm{bold-italich}}_c,$$ where ${bold\nabla }_{\mathrm{bold-italicX}}^{\mathrm{normalt}}\times \{\mathrm{o}\}$ denotes the curl operation and h c is a constant vector [25]. A second-order tensor can be represented as H ( X ) = h i ( X ) ⊗ e i , wherein e i is a Cartesian base system, so that the application of equation (8) results in $$\begin{array}{cccc}\mathit{H}(\mathit{X})\hfill & ={bold\nabla }_{\mathrm{bold-italicX}}[a_i(\mathit{X}){\mathrm{bold-italice}}_i]\hfill & +{bold\nabla }_{\mathrm{bold-italicX}}^{\mathrm{normalt}}\times [{\mathrm{bold-italicb}}_i(\mathit{X})\otimes {\mathrm{bold-italice}}_i]\hfill & +{\mathrm{bold-italicH}}_c\hfill \\ \hfill & ={bold\nabla }_{\mathrm{bold-italicX}}\mathit{a}(\mathit{X})\hfill & +{bold\nabla }_{\mathrm{bold-italicX}}^{\mathrm{normalt}}\times \mathit{B}(\mathit{X})\hfill & +{\mathrm{bold-italicH}}_c\hfill \end{array}$$ with H c = const.…”

Geometrical Foundations of Continuum Mechanics

“…The extension of the Helmholtz decomposition to tensor fields of arbitrary order has been advocated in [49]. The extension of the Helmholtz decomposition to tensor fields of arbitrary order has been advocated in [49].…”

Transient dynamic analysis of 3-D gradient elastic solids by BEM

“…According to the Helmholtz decomposition applied to dyadic fields [20], the fundamental solutionŨ Ã ðrÞ can be decomposed into irrotational and solenoidal parts as…”