ABSTRACT
In classical elasticity a fourth order tensor defined and named “Hook tensor” which relates stress and strain tensors in small deformations. The question arises whether or not such expression exists in large deformations. We have answered to this question in Section 4.1, where the Eq. (4.1.16) expresses the relationship between the stress and deformation (not strain). In this section we try to find some relationship between stress and strain. If we combine the Eqs. (4.1.5), (4.1.6) and (4.1.7) we can easily find that:
τ ε ε
ij jiJ w w
= ∂ ∂
+ ∂ ∂
1 2
(5.1.1)
In Section 2.2, we showed that Green strain tensor ε ij versus the deformation gradient tensor F can be defined via left Cauchy-Green tensor via Eq. (2.2.9), that is,
ε ij = −( ) = −( ) 1 2
1 2
F F I c IT c F FT= (5.1.2)
According to differentiation rules we have:
∂ ∂
= ∂ ∂
∂ ∂
w w
ij ijε εc c
From Eq. (5.1.2) we have, c I= +2 ε ij and therefore we can write:
∂ ∂
= ⇒ ∂ ∂
= ∂ ∂
c cε εij ij
w w2 2 (5.1.3)
Furthermore, in Section 3.1, from Eq. (3.1.34) we had:
τ ij ij J
s= 1 (5.1.4)
If we substitute Eq. (5.1.4) in the left side and Eq. (5.1.3) in the right side of Eq. (5.1.1) and then simplify we have:
s c
= ∂ ∂
2 w (5.1.5)
In Eq. (5.1.5) the w is a scalar but c is a matrix, now derivative of a scalar value versus a matrix in Eq. (5.1.5) seems to be ambiguous, but in fact it means that:
s w c
= ∂ ∂
2 (5.1.6)
Also Eq. (3.1.36) can be re-expressed by:
s F F T= ( )− −1 Jσ (5.1.7)
Equation (3.2.22) was the definition of nominal Kirchhoff stress, that is,
τ σ' = J (5.1.8)
Now we substitute Eq. (5.1.8) into Eq. (5.1.7) and the result into Eq. (5.1.5) then we have:
F F c
T− − = ∂ ∂
1 'τ 2 w
If we post multiply the above expression into FT and also pre multiply by F then we have:
τ ' = ∂ ∂
2F c FTw (5.1.9)
Equations (5.1.5) and (5.1.9) are two equations that both relate nominal Kirchhoff stress to left Cauchy Green tensor c, but is not a direct relationship. The complexity arises from this fact that we face both initial (reference) configuration and deformed configuration. The “push forward” operator Fφ* relates and maps the initial configuration dX to deformed configuration dx, that is,
d dX x⇒ push forward
d dx F X= φ*
While the “pull back” operator F φ* −1 relates and maps the deformed con-
figuration dx to un-deformed configuration dX , that is,
d dx X⇒ pull back
d dX F x= − φ* 1
Due to the duality of the transformations described above, we cannot discuss about one type of stress only. In reference configuration we defined 2nd Piola Kirchhoff stress s but in deformed configuration we defined nominal Kirchhoff τ '.
