Alternative stress measures#Second Piola–Kirchhoff stress

Consider the situation shown in the following figure. The following definitions use the notations shown in the figure.

In the reference configuration $\backslash Omega\_0$, the outward normal to a surface element $d\backslash Gamma\_0$ is $\backslash mathbf\{N\}\; \backslash equiv\; \backslash mathbf\{n\}\_0$ and the traction acting on that surface (assuming it deforms like a generic vector belonging to the deformation) is $\backslash mathbf\{t\}\_0$ leading to a force vector $d\backslash mathbf\{f\}\_0$. In the deformed configuration $\backslash Omega$, the surface element changes to $d\backslash Gamma$ with outward normal $\backslash mathbf\{n\}$ and traction vector $\backslash mathbf\{t\}$ leading to a force $d\backslash mathbf\{f\}$. Note that this surface can either be a hypothetical cut inside the body or an actual surface. The quantity $\backslash boldsymbol\{F\}$ is the deformation gradient tensor, $J$ is its determinant.

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

The quantity,

:$$

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

is called the Kirchhoff stress tensor, with $J$ the determinant of $\backslash boldsymbol\{F\}$. 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.

{{main|Piola–Kirchhoff stress tensor}}

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

:$$

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

:$$

\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.{{cite book|title=Three-Dimensional Elasticity|url=https://books.google.com/books?id=tlGCC3w27iIC|date=1 April 1988|publisher=Elsevier|isbn=978-0-08-087541-5}}

If we pull back $d\backslash mathbf\{f\}$ to the reference configuration we obtain the traction acting on that surface before the deformation $d\backslash mathbf\{f\}\_0$ assuming it behaves like a generic vector belonging to the deformation. In particular 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 ($\backslash boldsymbol\{S\}$) 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

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

:$$

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

:$$

\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:

:$$

\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}^T\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}^T\cdot\boldsymbol{F}^{-T}

In index notation,

:$$

N_{Ij} = J~F_{Ik}^{-1}~\sigma_{kj} \qquad \text{and} \qquad

P_{iJ} = J~\sigma_{ki}~F^{-1}_{Jk}

Therefore,

:$$

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

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

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 $\backslash boldsymbol\{S\}$),

:$$

\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^T_{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}

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 2nd 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, $\backslash boldsymbol\{S\}$ is the pull back of $\backslash boldsymbol\{\backslash tau\}$ by $\backslash boldsymbol\{F\}$ and $\backslash boldsymbol\{\backslash tau\}$ is the push forward of $\backslash boldsymbol\{S\}$.

Key: $$J=\backslash det\backslash left(\backslash boldsymbol\{F\}\backslash right),\backslash quad\backslash boldsymbol\{C\}=\backslash boldsymbol\{F\}^\{T\}\backslash boldsymbol\{F\}=\backslash boldsymbol\{U\}^\{2\},\backslash quad\backslash boldsymbol\{F\}=\backslash boldsymbol\{R\}\backslash boldsymbol\{U\},\backslash quad\; \backslash boldsymbol\{R\}^T=\backslash boldsymbol\{R\}^\{-1\},$$ $$\backslash boldsymbol\{P\}=J\backslash boldsymbol\{\backslash sigma\}\backslash boldsymbol\{F\}^\{-T\},\backslash quad\backslash boldsymbol\{\backslash tau\}=J\backslash boldsymbol\{\backslash sigma\},\backslash quad$$

\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}

• Stress (physics)
• Finite strain theory
• Continuum mechanics
• Hyperelastic material
• Cauchy elastic material
• Critical plane analysis

Category:Solid mechanics

Category:Continuum mechanics

Category:Gustav Kirchhoff

Category:Tensor physical quantities