Stress measures

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The most commonly used measure of stress is the Cauchy stress tensor, often called simply the stress tensor or "true 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:

  1. The Kirchhoff stress (\boldsymbol{\tau}).
  2. The Nominal stress (\boldsymbol{N}).
  3. The first Piola-Kirchhoff stress (\boldsymbol{P}). This stress tensor is the transpose of the nominal stress (\boldsymbol{P} = \boldsymbol{N}^T).
  4. The second Piola-Kirchhoff stress or PK2 stress (\boldsymbol{S}).
  5. The Biot stress (\boldsymbol{T})

Definitions of stress measures[edit]

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

Quantities used in the definition of stress measures

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. The quantity \boldsymbol{F} is the deformation gradient tensor.

Cauchy stress[edit]

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[edit]

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[edit]

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[edit]

If we pull back d\mathbf{f} to the reference configuration, we have

  d\mathbf{f}_0 = \boldsymbol{F}^{-1}\cdot d\mathbf{f}


  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


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

Biot stress[edit]

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} + \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[edit]

Relations between Cauchy stress and nominal stress[edit]

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


  \boldsymbol{\sigma}^T\cdot\mathbf{n}~d\Gamma = d\mathbf{f} = \boldsymbol{N}^T\cdot\mathbf{n}_0~d\Gamma_0


  \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


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


  \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}


  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[edit]

Recall that

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


  d\mathbf{f} = \boldsymbol{F}\cdot d\mathbf{f}_0 = \boldsymbol{F} \cdot (\boldsymbol{S}^T \cdot \mathbf{n}_0~d\Gamma_0)


  \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^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}

Relations between Cauchy stress and second P-K stress[edit]

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}


  \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


  \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}


  \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}.

See also[edit]


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