First Piola-Kirchhoff stress

We introduce a relationship between the Cauchy stress a defined in the deformed body with its basis e, and the first Piola-Kirchhoff stress II defined in the undeformed body with its basis Ej as follows ... 

The transpose of the first Piola-Kirchhoff stress S = II is known as the nominal stress ... 

Note that in some books, e.g., Kitagawa (1987) pp. 33, the first Piola-Kirchhoff stress and the nominal stress are defined in an opposite sense. 

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 ... 

The specific first Piola-Kirchhoff stress 11, specific second Piola-Kirchhoff stress and specific EuIct stress t are defined by... 

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... 

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-... 

A linearized constitutive relation of the interface has been given by Gurtin and Murdoch [32, 33]. For an isotropic interface relative to the reference configuration, the linearized constitutive relation for the interfacial Piola-Kirchhoff stress of the first kind can be written as... 

In the nonlinear analysis of solids, there are two kinds of nonlinearities - the material nonlinearity and the geometric nonlinearity. The material nonlinearity is basically due to the existence of a nonlinear relation toween the stresses and the strains. The geometric nonlinearity implies that the strains involved are very large so that all the stress measures (Cauchy stress, Kirchhoff stress, first and second order Piola-Kirchhoff stresses, etc.) and the strain measures (engineering strain, natural strain, Green-Lagrange strain, etc.) are very much different in meaning and in numerical values. 

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

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