Denoting by P the 1st Piola-Kirchhoff (PK) stress tensor, and by S the 2nd PK stress tensor, these are related through the formula

P=FS,

where F=I+Grad(u) is the deformation gradient tensor, with I the identity tensor and u the displacement vector. Using this formula, the momentum balance equation written in terms of the 1st PK stress can be re-written equivalently in terms of the 2nd PK stress. Note that the 2nd PK stress tensor is symmetric, and, for homogeneous isotropic hyperelastic materials, is coaxial with the right Cauchy-Green deformation tensor C, and hence also with the Green-Lagrange strain tensor E=(C-I)/2.

Thank you so much L. Angela Mihai

I agree with Angela's clear answer but not with the misleading last sentence. The 2nd PK stress tensor is coaxial with the right Cauchy-Green deformation tensor C, and hence also with the Green-Lagrange strain tensor E=(C-I)/2 for isotropic (hyperelastic) material. But for a general material we do not have such a simple observation. Only if the loading is in the direction of material symmetry axes, then stress and strain are coaxial in anisotropic materials.

I completely agree with Dr. Staat's description. In fact, neither coaxiality nor symmetry can be generalized for every material, and we must understand the assumptions used in the derivation of these quantities. For example, Cauchy stresses in complex fluids such as nematic LCs, as well as those near dislocations and phase boundaries have been shown to be asymmetric (for example, refer Article Asymmetry of the atomic-level stress tensor in homogeneous a...

Thank you so muchManfred Staat Kshitiz Upadhyay

Hi Ksthitz,

you can find the response for your question in the book of Malavern (1969) on continuum mechanics; I have read a recent paper in which the Lagrangian equilibrium equation in terms of PK2 has been taken form this book.

Kind Regards,

Hocine

Thanks Hocine Bechir