Piola stress tensor

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... 

Without going into details we note also that sufficient conditions of minima of a may be discussed and further simpliflcations of these results using objectivity (3.124) and material symmetry may be obtained using Cauchy-Green tensors, Piola-Kirchhoff stress tensors, the known linearized constitutive equations of solids follow, e.g. Hook and Fourier laws with tensor (transport) coefficients which are reduced to scalars in isotropic solids (e.g. Cauchy law of deformation with Lame coefficients) [6, 7, 9, 13, 14]. 

Now we will define two new stress measures. One is the first Piola-Kirchoff stress tensor, T, and the other is the second Piola-Kirchoff stress tensor, S. Consider that a differential force dP is applied to the body. Obviously, the force does not change no matter it is from the undeformed body or the deformed body. This argument can be used to define the two new stress measures. Using the tensor note, we have... 

This is the first Piola-Kirchoff stress tensor. It means the force in the deformed configuration but in the undeformed area. It is noted that T is not symmetric so it is not easy to use. Similarly, we can obtain the second Piola-Kirchoff stress tensor, which is... 

Let us next look at the consequences for an appropriate formirlation of stress. It turns out that two additional stress tensors are needed to accommodate the requirements made above. The first of these, iA called first Piola-Kirehhoff tensor , is related to the stress tensor introduced in Chap. 3 by... 

The second additional stress tensor called second Piola-Kirehhoff tensor , is related to the stress Ty 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-... 

Second Piola-Kirchhoff stress tensor and related traction vector Lagrangian strain tensor and displacement vector... 

During the motion of the body, its volume, surface area, density, stresses, and strains change continuously. The stress measure that we shall use is the 2nd Piola-Kirchhofif stress tensor. The components of the 2nd Piola-Kirchhoff stress tensor in Cj will be denoted by To see the meaning of the 2nd Piola-KirchhofiF stress tensor, consider the force dF on surface dS in C2. The Cauchy stress tensor t is defined by... 

It is also important to note that the 2nd Piola-Kirchhoff stress tensor is energetically conjugate to the Green-Lagrange strain tensor and the Cauchy stress is energetically conjugate to the infinitesimal strain tensor. In other words, we have... 

Cartesian components of the 2nd Piola-Kirchhoff stress tensor Cartesian components of the Green-Lagrange strain tensor Components of the linear elasticity tensor Increment in the /th displacement component... 

The second Piola-Kirchhoff stress tensor represents a contact force density measured in the reference configuration per unit of reference area. 

According to Eq. (31) and definitions (28)-(30), the most general form of the second Piola-Kirchhoff stress tensor for an isotropic and hyperelastic material is ... 

All these constitutive choices for the free energy i/r lead to different expressions of the stress in terms of the deformation gradient. By applying the Coleman and Noll procedure [124], i.e., by restricting the form of the stress tensor in such a way that the Clausius-Plank inequality is verified for every admissible process, the Piola symmetric stress tensor is shown below... 

In the framework of nonlinear viscoelasticity, Fosdick and Yu [165] proposed their own constitutive equation. They assumed that the second Piola-Kirchhoff stress tensor is given by... 

A physical Lagrangian stress tensor is defined and established by applying vector transformation to the second Piola Kirchhoff stress tensor 11 components using equation (27), such that ... 

The Piola-Kirchlioff stress tensor with respect to K, denoted by tt, is defined by ... 

It is noted that within the Euler-Bemoulli theory context, it holds that 6 = v, 6y= — w (the cross section remains perpendicular to the deformed axis) hence, shear strain components Eqs. 3b and 3c vanish. Considering strains to be small and employing the second Piola-Kirchhoff stress tensor, the nonvanishing stress components are defined in terms of the strain ones as... 

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