On Advantages of the Kelvin Mapping in Finite Element Implementations of Deformation Processes

TL;DR

This paper advocates for the Kelvin mapping in finite element methods, highlighting its advantages in preserving tensor character and simplifying numerical implementation compared to the traditional Voigt mapping.

Contribution

It demonstrates the benefits of Kelvin mapping over Voigt mapping in finite element software, emphasizing improved accuracy and simplicity with minimal implementation effort.

Findings

Kelvin mapping preserves tensor character in numerical models.

It simplifies the transformation of tensor equations into matrix form.

Implementation changes are minimal, involving only scalar factors.

Abstract

Classical continuum mechanical theories operate on three-dimensional Eu-clidian space using scalar, vector, and tensor-valued quantities usually up to the order of four. For their numerical treatment, it is common practice to transform the relations into a matrix-vector format. This transformation is usually performed using the so-called Voigt mapping. This mapping does not preserve tensor character leaving significant room for error as stress and strain quantities follow from different mappings and thus have to be treated differently in certain mathematical operations. Despite its conceptual and notational difficulties having been pointed out, the Voigt mapping remains the foundation of most current finite element programmes. An alternative is the so-called Kelvin mapping which has recently gained recognition in studies of theoretical mechanics. This article is concerned with benefits of the Kelvin mapping in numerical modelling tools such as finite element software. The decisive difference to the Voigt mapping is that Kelvin's method preserves tensor character, and thus the numerical matrix notation directly corresponds to the original tensor notation. Further benefits in numerical implementations are that tensor norms are calculated identically without distinguishing stress or strain-type quantities and tensor equations can be directly transformed into matrix equations without additional considerations. The only implementational changes are related to a scalar factor in certain finite element matrices and hence, harvesting the mentioned benefits comes at very little cost.