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