fruitkiwi said:
i believe that there should be some reason why these two tensor coexist.anyone to share with me?
It's a good question.
In finite (large strain, nonlinear) "hyper"elasticity, which is where you are presumably headed, you will find that some authors develop a stress strain relationship using pair #1 and others use pair #2.
Regardless, you will find that you need to switch back and forth between the two pairs at least once when performing the derivation.
Let me briefly elaborate:
Consider that the whole point of this branch of continuum mechanics is to develop the equations used in finite element simulations.
These equations primarily consist of:
a) some relationship between stress and strain that depends only on the material (F=kx, essentially -- but we're dealing with complicated materials like rubber)
b) some equilibrium relationship involving stress (F=ma, essentially)
Since our reference frame might be rotating (rigid body rotation is occurring) in our finite element simulation, the derivation of "a" begins, initially, within a coordinate system that rotates as well. This is because we need the stress to be a function of material strain and not a function of rigid body rotation. So, we
always begin, initially, with stress strain relationship #1, since this relationship is in fact invariant under (rotating with) rigid body rotation.
However: Since we are talking about finite elements, we need adjacent elements to interact with each other properly. In other words, we need a COMMON frame of reference for the purpose of looking at stress equilibrium (Newton's second and third laws, essentially). If the need for a common frame of reference isn't clear, it may be helpful to consider two separate objects that are impacting each other, which is something that is, in fact, commonly considered in computer simulations. Hopefully you can then see why we use the Cauchy Stress for our treatment of equilibrium (i.e.
stress strain relationship #2 will be unavoidable when equilibrium between elements is considered).
If you have other questions, shoot