Stress Used in Lagrangian Equation for Solid Mechanics

[bcl]

Bathe (reference below) outlines the updated Lagrangian (UL) and total Lagrangian (TL) approaches using the second Piola Kirchhoff (PK2) stress. Others (i.e., Ji, et al. and Abaqus) define the UL and TL formulations in terms of the Kirchhoff or the Cauchy stress in rate form. This form requires consideration of an objective rate (since the rate of the Kirchhoff or Cauchy stress is not objective).

Why even mess with the Kirchhoff or Cauchy stress in the governing equation? Why not just cast the equation as Bathe did in terms of the PK2 stress, since the rate of the PK2 stress is objective? If your constitutive equation is in terms of the Kirchhoff or Cauchy stress could you not just transform it through the pullback operator to cast it in terms of the PK2 stress?

Bathe, http://web.mit.edu/kjb/www/Books/FEP_2nd_Edition_4th_Printing.pdf )
Ji, et al., https://pdfs.semanticscholar.org/d915/d7bb83fb2b5a9f3d41f699751eb8ac557e3d.pdf
Abaqus 6.11 Theory Manual, Section 1.5.1, http://130.149.89.49:2080/v6.11/books/stm/default.htm

 

[afreiden]

I had the same question when I was first introduced to the UL approach and objective rates...

In a high-fidelity finite element analysis (e.x. a mesh of continuum elements in ADINA or ABAQUS), two things need to happen:
1) In order for the elements to talk to each other, they need a common frame of reference.
2) In order to do dynamics, the continuum mechanics must be done in rate form.

Regarding 1, the common frame of reference that makes the most sense is the global coordinate system. Most call this the "spatial" reference frame, but I'm not sure what terminology Bathe uses. In the spatial reference frame, we ought to see a change of the stress tensor for any element that undergoes a rotation. Since the PK2 stress tensor (and its work-conjugate Green-Lagrange strain) is defined in the "material" coordinate system, it does not change at all under rigid body rotations (Bathe says "invariant"). There's so much emphasis on the Cauchy stress (and its work-conjugate Almansi strain) because it's defined in the spatial reference frame.

Regarding 2, the time derivative of the Cauchy stress is really important for time-stepping, but since it is not work-conjugate with anything (e.x. the time rate of change of the Green-Lagrange strain, nor the velocity strain tensor, nor the Cauchy stress itself), you have to instead use an "objective" rate (e.x. Jaumann rate). The objective rate should be work-conjugate with the velocity strain tensor. The computer simulation will update the "objective" stress tensor for that particular element, for that particular time-step. The "objective" stress tensors lack physical meaning, but then the FEA will use them to find the Cauchy stresses, and perform equilibrium for the entire FEA, then do it all over again for the next time-step.