# Advanced elasticity/Stress measures
## Stress measures

Three stress measures are widely used in continuum mechanics (particularly in the computational context). These are
* The Cauchy stress (${\boldsymbol {\sigma }}$) or true stress.
* The Nominal stress (${\boldsymbol {N}}$) (which is the transpose of the first Piola-Kirchhoff stress (${\boldsymbol {P}}={\boldsymbol {N}}^{T}$).
* The second Piola-Kirchhoff stress or PK2 stress (${\boldsymbol {S}}$).
Consider the situation shown the following figure.
* Quantities used in the definition of stress measures
The following definitions use the information in the figure. In the reference configuration $\Omega _{0}$, the outward normal to a surface element $d\Gamma _{0}$ is $\mathbf {N} \equiv \mathbf {n} _{0}$ and the traction acting on that surface is $\mathbf {t} _{0}$ leading to a force vector $d\mathbf {f} _{0}$. In the deformed configuration $\Omega$, the surface element changes to $d\Gamma$ with outward normal $\mathbf {n}$ and traction vector $\mathbf {t}$ leading to a force $d\mathbf {f}$. Note that this surface can either be a hypothetical cut inside the body or an actual surface.

### Cauchy stress

The Cauchy stress (or true stress) is a measure of the force acting on an element of area in the deformed configuration. This tensor is symmetric and is defined via
    $d\mathbf {f} =\mathbf {t} ~d\Gamma ={\boldsymbol {\sigma }}^{T}\cdot \mathbf {n} ~d\Gamma$
or
    $\mathbf {t} ={\boldsymbol {\sigma }}^{T}\cdot \mathbf {n}$
where $\mathbf {t}$ is the traction and $\mathbf {n}$ is the normal to the surface on which the traction acts.

### Nominal stress/First Piola-Kirchhoff stress

The nominal stress (${\boldsymbol {N}}={\boldsymbol {P}}^{T}$) is the transpose of the first Piola-Kirchhoff stress (PK1 stress) (${\boldsymbol {P}}$) and is defined via
    $d\mathbf {f} =\mathbf {t} _{0}~d\Gamma _{0}={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}={\boldsymbol {P}}\cdot \mathbf {n} _{0}~d\Gamma _{0}$
or
    $\mathbf {t} _{0}={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}={\boldsymbol {P}}\cdot \mathbf {n} _{0}$
This stress is unsymmetric and is a two point tensor like the deformation gradient. This is because it relates the force in the deformed configuration to an oriented area vector in the reference configuration.

### 2nd Piola Kirchhoff stress

If we pull back $d\mathbf {f}$ to the reference configuration, we have
    $d\mathbf {f} _{0}={\boldsymbol {F}}^{-1}\cdot d\mathbf {f}$
or,
    $d\mathbf {f} _{0}={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}={\boldsymbol {F}}^{-1}\cdot \mathbf {t} _{0}~d\Gamma _{0}$
The PK2 stress (${\boldsymbol {S}}$) is symmetric and is defined via the relation
    $d\mathbf {f} _{0}={\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}={\boldsymbol {F}}^{-1}\cdot \mathbf {t} _{0}~d\Gamma _{0}$
Therefore,
    ${\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}={\boldsymbol {F}}^{-1}\cdot \mathbf {t} _{0}$

### Relations between Cauchy stress and nominal stress

Recall Nanson's formula relating areas in the reference and deformed configurations:
    $\mathbf {n} ~d\Gamma =J~{\boldsymbol {F}}^{-T}\cdot \mathbf {n} _{0}~d\Gamma _{0}$
Now,
    ${\boldsymbol {\sigma }}^{T}\cdot \mathbf {n} ~d\Gamma =d\mathbf {f} ={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}$
Hence,
    ${\boldsymbol {\sigma }}^{T}\cdot (J~{\boldsymbol {F}}^{-T}\cdot \mathbf {n} _{0}~d\Gamma _{0})={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}$
or,
    ${\boldsymbol {N}}^{T}=J~({\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }})^{T}=J~{\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}$
or,
    ${\boldsymbol {N}}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\qquad {\text{and}}\qquad {\boldsymbol {N}}^{T}={\boldsymbol {P}}=J~{\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}$
In index notation,
    $N_{ij}=J~F_{ik}^{-1}~\sigma _{kj}\qquad {\text{and}}\qquad P_{ij}=J~\sigma _{ik}~F_{jk}^{-1}$
Therefore,
    $J~{\boldsymbol {\sigma }}={\boldsymbol {F}}\cdot {\boldsymbol {N}}={\boldsymbol {P}}\cdot {\boldsymbol {F}}^{T}~.$
The quantity ${\boldsymbol {\tau }}=J~{\boldsymbol {\sigma }}$ is called the Kirchhoff stress tensor and is used widely in numerical algorithms in metal plasticity (where there is no change in volume during plastic deformation).
Note that ${\boldsymbol {N}}$ and ${\boldsymbol {P}}$ are not symmetric because ${\boldsymbol {F}}$ is not symmetric.

### Relations between nominal stress and second P-K stress

Recall that
    ${\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}=d\mathbf {f}$
and
    $d\mathbf {f} ={\boldsymbol {F}}\cdot d\mathbf {f} _{0}={\boldsymbol {F}}\cdot ({\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0})$
Therefore,
    ${\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}={\boldsymbol {F}}\cdot {\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}$
or (using the symmetry of ${\boldsymbol {S}}$),
    ${\boldsymbol {N}}={\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}\qquad {\text{and}}\qquad {\boldsymbol {P}}={\boldsymbol {F}}\cdot {\boldsymbol {S}}$
In index notation,
    $N_{ij}=S_{ik}~F_{jk}\qquad {\text{and}}\qquad P_{ij}=F_{ik}~S_{kj}$
Alternatively, we can write
    ${\boldsymbol {S}}={\boldsymbol {N}}\cdot {\boldsymbol {F}}^{-T}\qquad {\text{and}}\qquad {\boldsymbol {S}}={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {P}}$

### Relations between Cauchy stress and second P-K stress

Recall that
    ${\boldsymbol {N}}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}$
In terms of the 2nd PK stress, we have
    ${\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}$
Therefore,
    ${\boldsymbol {S}}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\tau }}\cdot {\boldsymbol {F}}^{-T}$
In index notation,
    $S_{ij}=F_{ik}^{-1}~\tau _{kl}~F_{jl}^{-1}$
Since the Cauchy stress (and hence the Kirchhoff stress) is symmetric, the 2n PK stress is also symmetric.
Alternatively, we can write
    ${\boldsymbol {\sigma }}=J^{-1}~{\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}$
or,
    ${\boldsymbol {\tau }}={\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}~.$
Clearly, from definition of the push-forward and pull-back operations, we have
    ${\boldsymbol {S}}=\varphi ^{*}[{\boldsymbol {\tau }}]={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\tau }}\cdot {\boldsymbol {F}}^{-T}$
and
    ${\boldsymbol {\tau }}=\varphi _{*}[{\boldsymbol {S}}]={\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}~.$
Therefore, ${\boldsymbol {S}}$ is the pull back of ${\boldsymbol {\tau }}$ by ${\boldsymbol {F}}$ and ${\boldsymbol {\tau }}$ is the push forward of ${\boldsymbol {S}}$.