Stress measures

The most commonly used measure of stress is the Cauchy stress. However, several other measures of stress can be defined.^[1]^[2]^[3] Some such stress measures that are widely used in continuum mechanics, particularly in the computational context, are:

The Cauchy stress ( $\boldsymbol{\sigma}$ ) or true stress.
The Kirchhoff stress ( $\boldsymbol{\tau}$ ).
The Nominal stress ( $\boldsymbol{N}$ ).
The first Piola-Kirchhoff stress ( $\boldsymbol{P}$ ). This stress tensor is the transpose of the nominal stress ( $\boldsymbol{P} = \boldsymbol{N}^T$ ).
The second Piola-Kirchhoff stress or PK2 stress ( $\boldsymbol{S}$ ).
The Biot stress ( $\boldsymbol{T}$ )

1 Definitions of stress measures
2 Relations between stress measures
3 See also
4 References

Definitions of stress measures

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

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

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

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

Kirchhoff stress

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).

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

$d\mathbf{f} = \mathbf{t}_0~d\Gamma_0 = \boldsymbol{N}^T\cdot\mathbf{n}_0~d\Gamma_0 = \boldsymbol{P}\cdot\mathbf{n}_0~d\Gamma_0$

$\mathbf{t}_0 = \boldsymbol{N}^T\cdot\mathbf{n}_0 = \boldsymbol{P}\cdot\mathbf{n}_0$

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.

Second 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,

$\boldsymbol{S}^T\cdot\mathbf{n}_0 = \boldsymbol{F}^{-1}\cdot\mathbf{t}_0$

Biot stress

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

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

The Biot stress is also called the Jaumann stress.

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

$\boldsymbol{R}^T~d\mathbf{f} = (\boldsymbol{P}^T\cdot\boldsymbol{R})^T\cdot\mathbf{n}_0~d\Gamma_0$

Relations between stress measures

Relations between Cauchy stress and nominal stress

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,

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

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,

$N_{Ij} = J~F_{Ik}^{-1}~\sigma_{kj} \qquad \text{and} \qquad P_{iJ} = J~\sigma_{ik}~F^{-1}_{Jk}$

Therefore,

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

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 $\boldsymbol{S}$ .

References

^ J. Bonet and R. W. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press.
^ R. W. Ogden, 1984, Non-linear Elastic Deformations, Dover.
^ L. D. Landau, E. M. Lifshitz, Theory of Elasticity, third edition

Stress measures

Contents

Definitions of stress measures

Cauchy stress

Kirchhoff stress

Nominal stress/First Piola-Kirchhoff stress

Second Piola-Kirchhoff stress

Biot stress

Relations between stress measures

Relations between Cauchy stress and nominal stress

Relations between nominal stress and second P-K stress

Relations between Cauchy stress and second P-K stress

See also

References