Piola tensors

Piola tensor plane strain plane stress Poisson s ratio... 

The tensor E E, called the Piola tensor (Astarita and Marrucci 1974), is closely related to B. In an extensional deformation, E E is exactly equal to B. B, a symmetric tensor, contains information about the orientation of the three principal axes of stretch and about the magnitudes of the three principal stretch ratios, but no information about rotations of material lines that occurred during that deformation. Thus, for example, from the Finger tensor alone, one could not determine whether a deformation was a simple shear (which has rotation of material lines) or a planar extensional deformation (which does not). The Finger tensor B(r, f) describes the change in shape of a small material element between times t and t, not whether it was rotated during this time interval. 

Another generalization uses referential (material) symmetric Piola-Kirchhoff stress and Green strain tensors in place of the stress and strain tensors used in the small deformation theory. These tensors have components relative to a fixed reference configuration, and the theory of Section 5.2 carries over intact when small deformation quantities are replaced by their referential counterparts. The referential formulation has the advantage that tensor components do not change with relative rotation between the coordinate frame and the material, and it is relatively easy to construct specific constitutive functions for specific materials, even when they are anisotropic. 

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... 

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 ... 

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 ... 

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