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I'm just reading about the P-K stress tensors, while I've seen Cauchy stress tensor in a course a couple months ago. From a technical point of view, I think it's clear what they represent.

  • Cauchy stress tensor gives the force on the current configuration with respect to deformed areas on the current configuration. So if $da \hat{n}$ is the infinitesimal area vector in the current configuration, the force would be $df = T (da) \hat{n}$.

  • First P-K stress tensor gives the force in the current configuration but by means of the undeformed areas in the reference shape, so that $df= P (dA) \hat{N}$ (I denote with capital letters everything in the reference configuration)

  • Second P-K stress tensor does the same thing of the first P-K tensor, except it gives the pull-back of the force $df$ via the deformation to which the body is subject. So if the deformation is $F$, the second P-K tensor $S$ should be such that $S (dA) \hat{N}= F^{-1} df$

Now, the question is: why do I need them? When do I use the first, when the second, when the third? I'll try to answer myself, but please tell me if I am getting this right.

The first one is pretty clear: you have a deformed body, you measure the deformation and then find the stress forces applied to it. The problem is the other two: when/how to use them?

I have not an engeenering background, so sorry if this is a trivial question but I'd really like some practical, intuitive examples. (Also, if my understanding on the definitions of the three tensors is wrong, please tell me!)

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A common use case is Computational Solid Mechanics (CSM). The choice of stress measures ($P$, $T$ or $S$) depends on how you solve the mechanical equilibrium problem numerically. It all comes down to solving partial differential equations (PDEs) of one kind of stress tensor field. If the PDEs are formulated in the reference configuration, the First Piola-Kirchhoff stress tensor will be a natural choice.

For example, the FFT-based solvers (aka spectral solvers) require discretization of the solid material with a regular grid (mesh) in the reference configuration, where the grid points remain constant and equally-spaced. Common FFT algorithms simply do not work on a deformed grid, so it is impossible to use the current (deformed) configuration.

Therefore, you will see the mechanical equilibrium equation like: $$\mathbf \nabla \cdot P = 0$$ in every article of spectral solvers, where $\mathbf{P}$ is the first Piola-Kirchhoff stress tensor. Please refer to this paper for deeper understanding of the FFT methods in CSM. During simulation, the spectral solver calls the constitutive subroutine to calculate $\mathbf{P}$ at each grid point in the reference configuration at every time step and applies forward and backward FFT to $\mathbf{P}$.

In theoretical constitutive models, there are numerous scenarios where you need to formulate the constitutive equations in the reference configuration, using the first or second Piola-Kirchhoff stress tensors. I would quote another answer here:

It doesn't really make sense to talk about a measure of stress in isolation, without also considering how to measure strain, and how to model the constitutive equations that give the relationship between stress and strain.

For example 2-PK stress works nicely with Green strain, which is a convenient way to define the behavior of a body which has "small" elastic deformations superimposed on arbitrary large rigid body motions - in particular, large rigid body rotations, where approximations like $\sin\theta \approx \theta$ are not appropriate.

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