Understanding Objective Derivatives in Continuum Mechanics: A Simplified Guide

[muzialis]

Hi there,

I am trying to refine my continuum mechanics, which I learned as an enginner.
I need to get a better undertstanding of the differences between upper and lowerc onvected derivative, and Jaumann derivative, as well as Lie derivative. I am not far, hopefully, from having gained a decent operational understanding but I would really benefit from some simple, heuristic, explanations. I am wondering if anybody could advise a text with a non exessive technical exposition. Can alternatively anybody diret me to simple examples allowing me to capture the essence of the matter? Many thanks

 

[afreiden]

Blast from the past!
Good questions should not go unanswered!
Keeping in mind that the purpose of this branch of continuum mechanics is to develop the equations that are used in FEA software, I will say the following:

The Jaumann rate, for example, has a different physical interpretation, depending on whether you are applying it to the Cauchy stress or linear stress. In general though, all of these "objective" rates (btw, Truesdell, who is one of the fathers of modern solid mechanics, did not like the word "objective" ... just sayin') are necessary for the following reason(s):

1) Under a rigid body rotation, we need our chosen measure of stress (e.g. linear, infinitesimal) and strain (e.g. Lagrangian) to behave the same under rigid body rotation. This doesn't mean that they have to be invariant to rigid body rotation (not necessarily.. though, linear stress and Lagrangian strain happen to both be invariant); but if the stress changes, then the associated strain must change, similarly. This must be the case, otherwise we can't define a stress-strain relationship for the material that will be applicable under rigid body rotations!

The RATES of stress and strain must also behave the same under rigid body rotation. So, stress, stress rate, strain, and strain rate must all behave the same under rigid body rotation. Sometimes "objective" rates are required in order to accomplish this. For example, the direct time derivative of Cauchy stress, $\dot{\sigma}$, transforms in an entirely different manner than the Cauchy stress, $\sigma$. The Jaumann rate of Cauchy stress, $\mathring{\sigma}$, for example, transforms in the SAME way as $\sigma$.

2) We want our final solution for stress to be in a spatial (global) frame-of-reference. Why? One potential reason: we want to be able to simulate multiple objects impacting each other in FEA - all of these "objects" need to have a common frame of reference! So, let's say that we begin to apply some external force (for example) in increments of time, and obtain the stress (rate), from strain (rate). To do this we must ensure that the consistencies addressed in "1" were met. So, once we've done this for the first increment in time, then, we need to transform this stress (rate) to the global coordinate system before moving on to the next time step and continuing to apply our external forces. Suffice to say, in order to revert to the spatial frame of reference, we need to go back to our rate formula (e.x. the formula for the Jaumann rate), rearrange it to solve for the appropriate quantity that we know to have a "spatial" physical interpretation - can't really go into more detail than that and continue to speak in broad terms.

So, the "objective" rate formulation, such as the Jaumann rate, can pop up in either "1" and/or "2."

In linear infinitesimal elasticity, I believe the Lie derivative that you mention (a.k.a. "Truesdell Rate"?) is exact, whereas the Jaumann neglects some shear deformation. I believe the Jaumann rate is used in some advanced FEA codes due to its simplicity and it may have some advantages in plasticity. The convected rate is derived in Asaro/Lubarda's book, and it's final form is similar to the Truesdell rate. I'm not 100% sure, but I think that LS-DYNA uses the convected rate, at least for some applications.