Properties of the Deformation Tensors - Civil Engineering (CE) PDF Download

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 = U² = R^tV²R, B = V² = RU²R^t .                                                      (3.77)

Next, we record certain properties of the right and left Cauchy-Green deformation tensors.
1. The right Cauchy-Green deformation tensor, C depends only on the coordinate basis used in the reference configuration, i.e., C = C_ijE_i⊗E_j = F_aiF_ajE_i ⊗ E_j
2. The left Cauchy-Green deformation tensor depends only on the coordinate basis used in the current configuration, i.e., B = B_ije_i ⊗ e_j = F_iaF_jae_i ⊗ e_j 
3. It is easy to see that both the tensors C and B are symmetric and positive definite.
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 ε_a its principal value that is

UN_a = ε_aN_a,                           (3.78)

then

CN_a = U²N_a = ε²_aN_a.              (3.79)

Now let n_a = RN_a. Then

Bn_a = RU²R^tRN_a = ε²_aRN_a = ε²_an_a.          (3.80)

Thus, we have shown that ε²_a is the principal value for both C and B tensors and Na and RN_a are their principal directions respectively