Tensor decompositions¶

tmech provides tensor decompositions that expose not only the decomposition factors but also their derivative tensors, which are required for consistent linearization when constitutive models are formulated in terms of stretch or rotation tensors.

Polar decomposition¶

The polar decomposition factors a deformation gradient as

$\mathbf{F} = \mathbf{R}\,\mathbf{U} = \mathbf{V}\,\mathbf{R}$

into a rotation $\mathbf{R}$ and the right and left stretch tensors $\mathbf{U}$ and $\mathbf{V}$ . It is computed via Newton iteration or eigenvalue-based methods. The decomposition exposes the derivative tensors $\partial \mathbf{R}/\partial \mathbf{F}$ , $\partial \mathbf{U}/\partial \mathbf{F}$ , and $\partial \mathbf{V}/\partial \mathbf{F}$ .

Eigenvalue decomposition¶

Eigenvalue (spectral) decomposition is supported for 2D and 3D symmetric tensors,

$\mathbf{A} = \sum_{i} \lambda_i\, \mathbf{E}_i$

exposing the eigenvalues $\lambda_i$ , eigenvectors, and the eigenbasis $\mathbf{E}_i$ , together with their derivatives.