Additional Measures of Stress — Lesson 2

In this video measures of stress, aside from the previously derived Cauchy definition, are derived. An incompressible cylinder is used as an initial example. The discussion then moves to a more general case. Using the more general case and reintroducing Nanson’s formula, the first Piola-Kirchhoff stress tensor is defined. Lastly, we show you how to rewrite the first Piola-Kirchhoff stress in coordinate notation.

This video begins with some properties of the first Piola-Kirchhoff stress tensor, which leads to the definition of the Kirchhoff stress tensor. Additional matrix operations are used and lead to the definition of the second Piola-Kirchhoff stress tensor. As in the prior video, the coordinate notation of the second Piola-Kirchhoff stress is developed to discuss why it is not a physically measurable value, but that it is useful in computational mechanics.

Answers to Frequently Asked Questions:

  • Though we see that the first Piola-Kirchhoff stress exists in both the deformed and reference configuration using coordinate notation, physically how does this translate to the force definition?

  • Are P & S in a uniaxial deformation the same?