In preparation for the constitutive formulations in later chapters, this short chapter will derive how different tensors behave under superimposed rigid body motion. This will enable us, for example, to show that is a function of in the beginning of the next chapter. The idea of “work conjugate” pairs will be introduced in this chapter. We will see that the Cauchy stress tensor, , for example, changes with rigid body rotation, while does not. Thus, and are not work-conjugate, and therefore it would not be appropriate to use them as a pair when forming a constitutive relationship. This was first recognized by Hill [Hill].
Let’s define the vector as follows:
, where is a superimposed rigid body motion, sometimes called a Euclidean transformation (see Fig).
We know that
What about “”?
Consider the following square body (see Fig), paying careful attention to the heavily bolded edge in order to track the motion of the body.
Clearly, if (which is what we’re going to want), and, say, ,
then (this should be clear from the above Fig).
Since we want to find such that , we can conclude that .
What we do know is the following:
So, , which gives:
Eq. 1 is what we wanted to find, and it should be expected. Recall that is physically understood to be a deformation (axial strains and shear), , followed by a rotation, . Thus, is the total deformation + rotation, , followed by an explicit rigid body rotation, , – i.e. a rotation that is superimposed on .
Now, we can see how different measures of strain and stress behave under superimposed rigid body motion, by substituting the above expression for .
Thus, is invariant to rigid body motion.
note: always ; since is an orthogonal tensor
informal proof: ;
NOT = which does not match either of the above expressions for
(or which is similarly )
matches expression for
matches expression for
formal proof: see Appendix A.1
What about , , and ?
Insert into the above expression
So, ; ( is symmetric ; is orthogonal)
Also, (skipped work)
How about rates?
note: ( since is orthogonal)
note: is antisymmetric, and indeed, the two terms that make up are each antisymmetric
How about stress?
Since , we can say that (1)
We also know the following three expressions to be true:
(4) (1) (3) (5)
(2) (5) ()
note: Since and , we can say that and are a “work-conjugate pair.” A more rigorous proof, starting with the “balance of mechanical energy” can be found in [Holzapfel].
How about the nominal stress and the Second Piola-Kirchhoff Stress?
Again, we will need to use
Since depends on , we will need . It turns out, though, that
note: the determinant of an orthogonal tensor is “1”
How about ?
is a good measure of stress when there are small (infinitesimal) strains. It will also be used as a “work-conjugate pair” with in the following chapters on constitutive relationships.