First Piola-Kirchhoff tensor

Comparison with (6.2) shows that second and first Piola-Kirchhoff tensors are... 

In Chap. 3 it was shown that the stress tensor T,j is symmetric, which allows to interchange indices. This symmetry also pertains to Iab while the first Piola-Kirchhoff tensor does not have this symmetry, a fact that is underlined by the choice of lower- and upper-case indices for spatial and material frame coordinates arrd quantities. 

Removing the rigid body component from F , it is possible to construct the strain tensor = F — A , which conjugated to the asymmetric First Piola Kirchhoff (FPK) stress tensor P = Pi toi referred to the curved reference beam, Simo (1985). P, is the FPK stress vector acting on the deformed face in the current beam corresponding to the normal ioi in the curved reference configuration. The spatial strain vector acting on the current beam cross section is obtained as e = e ioi-... 

Since the first Piola-Kirchhoff stress II is not symmetric as understood by (2.110), we introduce a symmetrized tensor T, called the second Piola-Kirchhoff stress, and the Euler stress t, which is the transformed tensor of T, into the deformed body using the rotation tensor R ... 

It may first be noted that the referential symmetric Piola-Kirchhoff stress tensor S and the spatial Cauchy stress tensor s are related by (A.39). Again with the back stress in mind, it will be assumed in this section that the set of internal state variables is comprised of a single second-order tensor whose referential and spatial forms are related by a similar equation, i.e., by... 

The symmetric stress tensor S was first used by Piola and Kirchhoff. In component form... 

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