The upper triangular decomposition of the deformation gradient: possible decompositions of the distortion tensor

In the upper triangular decomposition, the deformation gradient is multiplicatively decomposed into a product of a rotation tensor and an upper triangular tensor called the distortion tensor. In this paper, it is shown that the upper triangular decomposition can be viewed as an extended polar decomposition. The six components of the distortion tensor can be directly related to pure stretch and simple shear deformations. Also, it is demonstrated that the distortion tensor can be non-uniquely decomposed into a product of matrices for one triaxial stretch and two simple shear deformations or for one triaxial stretch and three simple shear deformations. There are six possible decompositions for the former and 24 possible decompositions for the latter. Only one of these 30 possible decompositions was examined earlier. In addition, the distortion tensor is shown to be frame-invariant and can therefore be used as an independent kinematic variable to construct strain energy density functions.