# Continuum mechanics/Stress measures

## Stress measures

Three stress measures are widely used in continuum mechanics (particularly in the computational context). These are

1. The Cauchy stress (${\displaystyle {\boldsymbol {\sigma }}}$) or true stress.
2. The Nominal stress (${\displaystyle {\boldsymbol {N}}}$) (which is the transpose of the first Piola-Kirchhoff stress (${\displaystyle {\boldsymbol {P}}={\boldsymbol {N}}^{T}}$).
3. The second Piola-Kirchhoff stress or PK2 stress (${\displaystyle {\boldsymbol {S}}}$).

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 ${\displaystyle \Omega _{0}}$, the outward normal to a surface element ${\displaystyle d\Gamma _{0}}$ is ${\displaystyle \mathbf {N} \equiv \mathbf {n} _{0}}$ and the traction acting on that surface is ${\displaystyle \mathbf {t} _{0}}$ leading to a force vector ${\displaystyle d\mathbf {f} _{0}}$. In the deformed configuration ${\displaystyle \Omega }$, the surface element changes to ${\displaystyle d\Gamma }$ with outward normal ${\displaystyle \mathbf {n} }$ and traction vector ${\displaystyle \mathbf {t} }$ leading to a force ${\displaystyle d\mathbf {f} }$. 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

${\displaystyle d\mathbf {f} =\mathbf {t} ~d\Gamma ={\boldsymbol {\sigma }}^{T}\cdot \mathbf {n} ~d\Gamma }$

or

${\displaystyle \mathbf {t} ={\boldsymbol {\sigma }}^{T}\cdot \mathbf {n} }$

where ${\displaystyle \mathbf {t} }$ is the traction and ${\displaystyle \mathbf {n} }$ is the normal to the surface on which the traction acts.

### Nominal stress/First Piola-Kirchhoff stress

The nominal stress (${\displaystyle {\boldsymbol {N}}={\boldsymbol {P}}^{T}}$) is the transpose of the first Piola-Kirchhoff stress (PK1 stress) (${\displaystyle {\boldsymbol {P}}}$) and is defined via

${\displaystyle d\mathbf {f} =\mathbf {t} _{0}~d\Gamma _{0}={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}={\boldsymbol {P}}\cdot \mathbf {n} _{0}~d\Gamma _{0}}$

or

${\displaystyle \mathbf {t} _{0}={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}={\boldsymbol {P}}\cdot \mathbf {n} _{0}}$

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 ${\displaystyle d\mathbf {f} }$ to the reference configuration, we have

${\displaystyle d\mathbf {f} _{0}={\boldsymbol {F}}^{-1}\cdot d\mathbf {f} }$

or,

${\displaystyle d\mathbf {f} _{0}={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}={\boldsymbol {F}}^{-1}\cdot \mathbf {t} _{0}~d\Gamma _{0}}$

The PK2 stress (${\displaystyle {\boldsymbol {S}}}$) is symmetric and is defined via the relation

${\displaystyle d\mathbf {f} _{0}={\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}={\boldsymbol {F}}^{-1}\cdot \mathbf {t} _{0}~d\Gamma _{0}}$

Therefore,

${\displaystyle {\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}={\boldsymbol {F}}^{-1}\cdot \mathbf {t} _{0}}$

### Relations between Cauchy stress and nominal stress

Recall Nanson's formula relating areas in the reference and deformed configurations:

${\displaystyle \mathbf {n} ~d\Gamma =J~{\boldsymbol {F}}^{-T}\cdot \mathbf {n} _{0}~d\Gamma _{0}}$

Now,

${\displaystyle {\boldsymbol {\sigma }}^{T}\cdot \mathbf {n} ~d\Gamma =d\mathbf {f} ={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}}$

Hence,

${\displaystyle {\boldsymbol {\sigma }}^{T}\cdot (J~{\boldsymbol {F}}^{-T}\cdot \mathbf {n} _{0}~d\Gamma _{0})={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}}$

or,

${\displaystyle {\boldsymbol {N}}^{T}=J~({\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }})^{T}=J~{\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}}$

or,

${\displaystyle {\boldsymbol {N}}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\qquad {\text{and}}\qquad {\boldsymbol {N}}^{T}={\boldsymbol {P}}=J~{\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}}$

In index notation,

${\displaystyle N_{ij}=J~F_{ik}^{-1}~\sigma _{kj}\qquad {\text{and}}\qquad P_{ij}=J~\sigma _{ik}~F_{jk}^{-1}}$

Therefore,

${\displaystyle J~{\boldsymbol {\sigma }}={\boldsymbol {F}}\cdot {\boldsymbol {N}}={\boldsymbol {P}}\cdot {\boldsymbol {F}}^{T}~.}$

The quantity ${\displaystyle {\boldsymbol {\tau }}=J~{\boldsymbol {\sigma }}}$ 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 ${\displaystyle {\boldsymbol {N}}}$ and ${\displaystyle {\boldsymbol {P}}}$ are not symmetric because ${\displaystyle {\boldsymbol {F}}}$ is not symmetric.

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

Recall that

${\displaystyle {\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}=d\mathbf {f} }$

and

${\displaystyle d\mathbf {f} ={\boldsymbol {F}}\cdot d\mathbf {f} _{0}={\boldsymbol {F}}\cdot ({\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0})}$

Therefore,

${\displaystyle {\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}={\boldsymbol {F}}\cdot {\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}}$

or (using the symmetry of ${\displaystyle {\boldsymbol {S}}}$),

${\displaystyle {\boldsymbol {N}}={\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}\qquad {\text{and}}\qquad {\boldsymbol {P}}={\boldsymbol {F}}\cdot {\boldsymbol {S}}}$

In index notation,

${\displaystyle N_{ij}=S_{ik}~F_{jk}\qquad {\text{and}}\qquad P_{ij}=F_{ik}~S_{kj}}$

Alternatively, we can write

${\displaystyle {\boldsymbol {S}}={\boldsymbol {N}}\cdot {\boldsymbol {F}}^{-T}\qquad {\text{and}}\qquad {\boldsymbol {S}}={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {P}}}$

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

Recall that

${\displaystyle {\boldsymbol {N}}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}}$

In terms of the 2nd PK stress, we have

${\displaystyle {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}}$

Therefore,

${\displaystyle {\boldsymbol {S}}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\tau }}\cdot {\boldsymbol {F}}^{-T}}$

In index notation,

${\displaystyle S_{ij}=F_{ik}^{-1}~\tau _{kl}~F_{jl}^{-1}}$

Since the Cauchy stress (and hence the Kirchhoff stress) is symmetric, the 2n PK stress is also symmetric.

Alternatively, we can write

${\displaystyle {\boldsymbol {\sigma }}=J^{-1}~{\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}}$

or,

${\displaystyle {\boldsymbol {\tau }}={\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}~.}$

Clearly, from definition of the push-forward and pull-back operations, we have

${\displaystyle {\boldsymbol {S}}=\varphi ^{*}[{\boldsymbol {\tau }}]={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\tau }}\cdot {\boldsymbol {F}}^{-T}}$

and

${\displaystyle {\boldsymbol {\tau }}=\varphi _{*}[{\boldsymbol {S}}]={\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}~.}$

Therefore, ${\displaystyle {\boldsymbol {S}}}$ is the pull back of ${\displaystyle {\boldsymbol {\tau }}}$ by ${\displaystyle {\boldsymbol {F}}}$ and ${\displaystyle {\boldsymbol {\tau }}}$ is the push forward of ${\displaystyle {\boldsymbol {S}}}$.