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 = RtV2R, B = V2 = RU2Rt . (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 = CijEi⊗Ej = FaiFajEi ⊗ Ej
2. The left Cauchy-Green deformation tensor depends only on the coordinate basis used in the current configuration, i.e., B = Bijei ⊗ ej = FiaFjaei ⊗ ej
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 εa its principal value that is
UNa = εaNa, (3.78)
CNa = U2Na = ε2aNa. (3.79)
Now let na = RNa. Then
Bna = RU2RtRNa = ε2aRNa = ε2ana. (3.80)
Thus, we have shown that ε2a is the principal value for both C and B tensors and Na and RNa are their principal directions respectively