Alternative stress measures

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.

The quantity,

${\displaystyle {\boldsymbol {\tau }}=J~{\boldsymbol {\sigma }}}$

is called the Kirchhoff stress tensor, with ${\displaystyle J}$ the determinant of ${\displaystyle {\boldsymbol {F}}}$. It is used widely in numerical algorithms in metal plasticity (where there is no change in volume during plastic deformation). It can be called weighted Cauchy stress tensor as well.

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

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

or

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

This stress is unsymmetric and is a two-point tensor like the deformation gradient.
The asymmetry derives from the fact that, as a tensor, it has one index attached to the reference configuration and one to the deformed configuration.[4]

If we pull back ${\displaystyle d\mathbf {f} }$ to the reference configuration we obtain the traction acting on that surface before the deformation ${\displaystyle d\mathbf {f} _{0}}$ assuming it behaves like a generic vector belonging to the deformation. In particular 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,

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

The Biot stress is useful because it is energy conjugate to the right stretch tensor ${\displaystyle {\boldsymbol {U}}}$. The Biot stress is defined as the symmetric part of the tensor ${\displaystyle {\boldsymbol {P}}^{T}\cdot {\boldsymbol {R}}}$ where ${\displaystyle {\boldsymbol {R}}}$ is the rotation tensor obtained from a polar decomposition of the deformation gradient. Therefore, the Biot stress tensor is defined as

${\displaystyle {\boldsymbol {T}}={\tfrac {1}{2}}({\boldsymbol {R}}^{T}\cdot {\boldsymbol {P}}+{\boldsymbol {P}}^{T}\cdot {\boldsymbol {R}})~.}$

The Biot stress is also called the Jaumann stress.

The quantity ${\displaystyle {\boldsymbol {T}}}$ does not have any physical interpretation. However, the unsymmetrized Biot stress has the interpretation

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

From 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,

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

or,

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

In index notation,

${\displaystyle N_{Ij}=J~F_{Ik}^{-1}~\sigma _{kj}\qquad {\text{and}}\qquad P_{iJ}=J~\sigma _{ki}~F_{Jk}^{-1}}$

Therefore,

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

Note that ${\displaystyle {\boldsymbol {N}}}$ and ${\displaystyle {\boldsymbol {P}}}$ are (generally) not symmetric because ${\displaystyle {\boldsymbol {F}}}$ is (generally) not symmetric.

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,

${\displaystyle N_{Ij}=S_{IK}~F_{jK}^{T}\qquad {\text{and}}\qquad P_{iJ}=F_{iK}~S_{KJ}}$

Alternatively, we can write

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

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 2nd 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 ${\displaystyle {\boldsymbol {S}}}$.

Key:

$${\displaystyle J=\det \left({\boldsymbol {F}}\right),\quad {\boldsymbol {C}}={\boldsymbol {F}}^{T}{\boldsymbol {F}}={\boldsymbol {U}}^{2},\quad {\boldsymbol {F}}={\boldsymbol {R}}{\boldsymbol {U}},\quad {\boldsymbol {R}}^{T}={\boldsymbol {R}}^{-1},}$$

$${\displaystyle {\boldsymbol {P}}=J{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T},\quad {\boldsymbol {\tau }}=J{\boldsymbol {\sigma }},\quad {\boldsymbol {S}}=J{\boldsymbol {F}}^{-1}{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T},\quad {\boldsymbol {T}}={\boldsymbol {R}}^{T}{\boldsymbol {P}},\quad {\boldsymbol {M}}={\boldsymbol {C}}{\boldsymbol {S}}}$$

Conversion formulae

Equation for	${\displaystyle {\boldsymbol {\sigma }}}$	${\displaystyle {\boldsymbol {\tau }}}$	${\displaystyle {\boldsymbol {P}}}$	${\displaystyle {\boldsymbol {S}}}$	${\displaystyle {\boldsymbol {T}}}$	${\displaystyle {\boldsymbol {M}}}$
${\displaystyle {\boldsymbol {\sigma }}=\,}$	${\displaystyle {\boldsymbol {\sigma }}}$	${\displaystyle J^{-1}{\boldsymbol {\tau }}}$	${\displaystyle J^{-1}{\boldsymbol {P}}{\boldsymbol {F}}^{T}}$	${\displaystyle J^{-1}{\boldsymbol {F}}{\boldsymbol {S}}{\boldsymbol {F}}^{T}}$	${\displaystyle J^{-1}{\boldsymbol {R}}{\boldsymbol {T}}{\boldsymbol {F}}^{T}}$	${\displaystyle J^{-1}{\boldsymbol {F}}^{-T}{\boldsymbol {M}}{\boldsymbol {F}}^{T}}$ (non isotropy)
${\displaystyle {\boldsymbol {\tau }}=\,}$	${\displaystyle J{\boldsymbol {\sigma }}}$	${\displaystyle {\boldsymbol {\tau }}}$	${\displaystyle {\boldsymbol {P}}{\boldsymbol {F}}^{T}}$	${\displaystyle {\boldsymbol {F}}{\boldsymbol {S}}{\boldsymbol {F}}^{T}}$	${\displaystyle {\boldsymbol {R}}{\boldsymbol {T}}{\boldsymbol {F}}^{T}}$	${\displaystyle {\boldsymbol {F}}^{-T}{\boldsymbol {M}}{\boldsymbol {F}}^{T}}$ (non isotropy)
${\displaystyle {\boldsymbol {P}}=\,}$	${\displaystyle J{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T}}$	${\displaystyle {\boldsymbol {\tau }}{\boldsymbol {F}}^{-T}}$	${\displaystyle {\boldsymbol {P}}}$	${\displaystyle {\boldsymbol {F}}{\boldsymbol {S}}}$	${\displaystyle {\boldsymbol {R}}{\boldsymbol {T}}}$	${\displaystyle {\boldsymbol {F}}^{-T}{\boldsymbol {M}}}$
${\displaystyle {\boldsymbol {S}}=\,}$	${\displaystyle J{\boldsymbol {F}}^{-1}{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T}}$	${\displaystyle {\boldsymbol {F}}^{-1}{\boldsymbol {\tau }}{\boldsymbol {F}}^{-T}}$	${\displaystyle {\boldsymbol {F}}^{-1}{\boldsymbol {P}}}$	${\displaystyle {\boldsymbol {S}}}$	${\displaystyle {\boldsymbol {U}}^{-1}{\boldsymbol {T}}}$	${\displaystyle {\boldsymbol {C}}^{-1}{\boldsymbol {M}}}$
${\displaystyle {\boldsymbol {T}}=\,}$	${\displaystyle J{\boldsymbol {R}}^{T}{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T}}$	${\displaystyle {\boldsymbol {R}}^{T}{\boldsymbol {\tau }}{\boldsymbol {F}}^{-T}}$	${\displaystyle {\boldsymbol {R}}^{T}{\boldsymbol {P}}}$	${\displaystyle {\boldsymbol {U}}{\boldsymbol {S}}}$	${\displaystyle {\boldsymbol {T}}}$	${\displaystyle {\boldsymbol {U}}^{-1}{\boldsymbol {M}}}$
${\displaystyle {\boldsymbol {M}}=\,}$	${\displaystyle J{\boldsymbol {F}}^{T}{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T}}$ (non isotropy)	${\displaystyle {\boldsymbol {F}}^{T}{\boldsymbol {\tau }}{\boldsymbol {F}}^{-T}}$ (non isotropy)	${\displaystyle {\boldsymbol {F}}^{T}{\boldsymbol {P}}}$	${\displaystyle {\boldsymbol {C}}{\boldsymbol {S}}}$	${\displaystyle {\boldsymbol {U}}{\boldsymbol {T}}}$	${\displaystyle {\boldsymbol {M}}}$