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**Properties of the deformation tensors **

Since det(F) > 0, using the polar decomposition theorem (2.116) the deformation gradient can be represented as:**F = RU = VR, ** (3.76)

where R is a proper orthogonal tensor, U and V are the right and left stretch tensors respectively. (These stretch tensors are called right and left because they are on the right and left of the orthogonal tensor, R.) These stretch tensors are unique, positive definite and symmetric.

Substituting (3.76) in (3.59) and (3.63) we obtain**C = U ^{2} = R^{t}V^{2}R, B = V^{2} = RU^{2}R^{t} . ** (3.77)

Next, we record certain properties of the right and left Cauchy-Green deformation tensors.

1. The right Cauchy-Green deformation tensor,

2. The left Cauchy-Green deformation tensor depends only on the coordinate basis used in the current configuration, i.e.,

3. It is easy to see that both the tensors

4. Since the deformation tensors are symmetric, they have three real principal values and their principal directions are orthonormal.

5. Both these tensors have the same eigen or principal values but different eigen or principal directions. To see this, let Na be the principal direction of the tensor U and Îµ

then

Now let

Thus, we have shown that Îµ

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