Tuesday, 2 January 2018

Symmetry of the 3times3 Cauchy Stress Tensor


When presenting the stress tensor (say in a non-relativistic context), it is shown to be a tensor in the sense that it is a linear vector transformation: it operates on a vector n (the normal to a surface), and returns a vector tn which is the traction vector. It is then shown that conservation of angular momentum leads to symmetry of the matrix.


However, tensors are more more naturally presented a multilinear functions. I wonder:



  • What type of tensor is the Cauchy stress tensor? Is n a vector or a co-vector? What about tn?

  • Is there a way to understand the symmetry when thinking of the stress tensor as a function of two vectors (or two co-vectors), under which it will seems intuitive why σ(A,B)=σ(B,A)?


Edit: To clarify, let's look, for example, at the 1st coordinate of the traction vector tn of an arbitrary normal n: This is $\left.Fromsymmetry,thisisequivalentto\lefttheinnerproductofnwiththetractionvectorofasurfaceorthogonaltoe_1$. Mathematically, I understand why this is correct. But is there any intuitive meaning as to why these two quantities are the same?





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