Imagine we're reviewing the strength of a glass table. I say, "This table can withstand a pushing force of 1000 N." You say, "I understand—a maximum force of 1000 N." I say, "Oh no, not for you. You're not standing directly over the table as I am, so your pushing force will be slightly angled, so you'll need to take the vector components, and then you'll need to incorporate the possibility of friction and your hand slipping, and..."
You'd say, "Stop, stop, stop. It's not useful for you to incorporate where I'm standing. When we talk about pushing on the table, it's most convenient if we always assume that the person is standing above the table, pushing down vertically on the center. You might think that your approach of considering where we're individually standing right now might seem more objective, but it's ultimately introducing more complexity. It's neither what I'd expect or what I'd want to use."
This is what the authors mean when they say, for example, "It is much more plausible that you want to see the stress in the fiber direction, even if the component is rotated." In other words, for convenience, we often want our stress tensor to rotate with our object, especially if our material is anisotropic and we're interested on the shear stress applied to laminate interfaces, for instance. The Second Piola-Kirchhoff stress tensor (with appropriate initial alignment) would always contain a component that describes this exact shear stress; with the least little bit of rotation, the Cauchy stress tensor would not. Does this make sense?
