Am I mistaken or is there a mistake in the Wikipedia entry on Mechanical Stress? The index convention used in the text seems to be inconsistent with the illustrations (Figures 1.1, 2.3 and 2.4). In the figures, the second index refers to the direction of traction. But, in the text it is the other way around.
Specifically
The first Piola-Kirchoff stress should be P = J F-1 σ .
In reply to Specifically by Jerry Brown
Re: Definition of 1st P-K stress
This is another of the many confusing definitions in continuum mechanics (another familiar one is the Finger tensor). What you've defined above is often called the "nominal stress". The 1st P-K stress is the transpose of that quantity. See also
http://en.wikipedia.org/wiki/Stress_measures
-- Biswajit
In reply to Re: Definition of 1st P-K stress by Biswajit Banerjee
Depends on the index convention
You're right, provided the first index defines the direction of traction and the second index defines the direction of the normal to the surface it acts on. However, the Wikipedia entry I referenced had it the other way around and that leads to the definition P = J F-1 σ . See A. F. Bower's online text. He has a good discussion of this issue.
In reply to Re: Definition of 1st P-K stress by Biswajit Banerjee
I don't understand this
The Wikipedia Stress-measures document is confusing. In the section titled "Relations between Cauchy stress and nominal stress", at the 4th equation there is a dubious step involving the transpose of a product. The symmetry of the Cauchy stress doesn't justify it because the order of the indices comes out wrong.
I don't have all of the sources referenced in the article, but I do have Ogden. I don't see any connection with his derivations.
The definition of Cauchy stress at the beginning is in agreement with Bower (awkwardly so) and should lead to the definition of the first Piola-Kirchoff stress that I cited in my first post.
Fixed it
Fixed Wikipedia entry.