I'm interested in representing elastic deformations (e.g. stretching) as Lie groups. There are a few references to using GL(3,R) but I'm wondering if possible to use GL(3,R) subgroups. For example, deformation gradient (F) can be decomposed into rotation R & 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 & E = diag. matrix of eigenvalues of U. Such diag. matrices are Lie subgroup of GL(3,R) & 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 GL(3,R) subgroups like those mentioned above ?