Finite Strain Theory - Polar Decomposition of The Deformation Gradient Tensor

Polar Decomposition of The Deformation Gradient Tensor

The deformation gradient, like any second-order tensor, can be decomposed, using the polar decomposition theorem, into a product of two second-order tensors (Truesdell and Noll, 1965): an orthogonal tensor and a positive definite symmetric tensor, i.e.

where the tensor is a proper orthogonal tensor, i.e. and, representing a rotation; the tensor is the right stretch tensor; and the left stretch tensor. The terms right and left means that they are to the right and left of the rotation tensor, respectively. and are both positive definite, i.e. and, and symmetric tensors, i.e. and, of second order.

This decomposition implies that the deformation of a line element in the undeformed configuration onto in the deformed configuration, i.e., may be obtained either by first stretching the element by, i.e., followed by a rotation, i.e. ; or equivalently, by applying a rigid rotation first, i.e., followed later by a stretching, i.e. (See Figure 3).

It can be shown that,

so that and have the same eigenvalues or principal stretches, but different eigenvectors or principal directions and, respectively. The principal directions are related by

This polar decomposition is unique as is non-symmetric.