I'm interested in representing elastic deformations (e.g. stretching) using Lie groups. There are a few references to using $GL(3,\mathbf{R})$ but I'm wondering if possible to use subgroups of $GL(3,\mathbf{R})$. For example, deformation gradient ($F$) can be decomposed into rotation $R$ and symmetric positive definite stretch $U$ as in $F = RU$. $U$ then decomposable by SVD into $U = PEP^T$ where $P =$ matrix of eigenvectors of $U$ and $E =$ diag. matrix of eigenvalues of $U$. Such diag. matrices are Lie subgroup of $GL(3,\mathbf{R})$ and represent pure stretches along orthogonal stretch axes. Similarly, 3x3 identity matrices with off-diag. positive entries represent shears.

So, my question is:

Can elastic deformations be represented by subgroups of $GL(3,\mathbf{R})$ like those mentioned above?