Any matrix can be decomposed into a sum of symmetric and antisymmetric matrices, but can be decomposed into a *product* of two matrices (one symmetric and one orthogonal)

(1)

and are called the Right Stretch Tensor and Left Stretch Tensor due to their respective positions (relative to ) in eq. 1.

(symmetric)

(symmetric)

(orthogonal)

A proof of the orthogonality of is given in **Appendix A.3**.

(2)

(3)

is a *rigid body* rotation, while or each stretch and rotate (they each contain both normal and shear deformations, typically)

and have the same eigenvalues. Eigenvalues do not have an “order,” *per se*, but since and typically have different eigenvectors, one should be cautious when assuming equivalency of eigenvalues. In the principal directions of or , or contains no shear deformation (see Fig). We’ll look at actual stresses, later.

To find the principal directions of and , we must solve the eigenvalue problem:

(pictured)

e.x. Polar Decomposition

The deformed equilibrium configuration of a body defined by the deformation mapping:

, ,

Determine:

a) and

b) Eigenvalues and eigenvectors of

c) and

d)

e)

a) =

note: will typically be a function of , , and we’ll be interested in the value of at a particular point. Here, it doesn’t matter.

b) ;

; ;

*i.e.* for ,

Choose arbitrary ; solve for from either equation ; normalize:

c)

(symmetric)

where

It’s important to keep in mind that , , and have the same eigenvectors and that these eigenvectors (and any other strain direction or magnitude) are generally dependent on the particular point in space of interest (*e.g.* , , ). For this simple example, is independent of , , and (*i.e.* the deformation is the same everywhere in the body – a “homogenous” deformation), but this would generally not be the case.

The eigenvalues occur in the direction of the eigenvectors, thus, ; ; . It makes sense that if we only know the eigenvectors and want to *undo* this transformation to bring us back to , then we need a transformation matrix that involves . We can see that is the opposite of the usual , and without going through the rigorous derivation of why , , , it at least makes some sense intuitively.

d) (orthogonal ;

e)