In [[continuum mechanics]], the most commonly used '''measure of stress''' is the [[Cauchy stress tensor]], often called simply ''the'' stress tensor or "true stress". However, several other tensor descriptions of stress can be defined. Some such stress tensors that are widely used in continuum mechanics, particularly in the computational context, are: *The [[Kirchhoff stress tensor]] ({$\boldsymbol{\tau}$}). *The [[Nominal stress tensor]] ({$\boldsymbol{N}$}). *The [[Nominal stress tensor|first Piola-Kirchhoff stress tensor]] ({$\boldsymbol{P}$}). This stress tensor is the transpose of the nominal stress ({$\boldsymbol{P} = \boldsymbol{N}^\top$}). *The [[second Piola-Kirchhoff stress tensor]] or PK2 stress ({$\boldsymbol{S}$}). *The [[Biot stress tensor]] ({$\boldsymbol{T}$}) == Definitions == The following notations will be used to define these stress measures: [[File:StressMeasures-noN.png|thumb|400px|Quantities used in the definition of stress measures]] The left half of the figure represents the reference (original) configuration {$\Omega_0$} of the object, and the right half represents the current (displaced) configuration {$\Omega$}. The two configurations are related by the displacement map {$\mathbf{F}$}: that is, the particle that was at some point {$p$} inside {$\Omega_0$} is at point {$F(p)$} inside {$\Omega$}. The displacement map {$\mathbf{F}$} is assumed to be [[continuous function|continuous]] and [[diferentiable function|differentiable]]. In the reference configuration {$\Omega_0$}, at left, the red quadrilateral represents a flat surface element with area {$d\Gamma_0$} and outward normal {$\mathbf{n}_0$}. In the displaced configuration, the same surface element has area {$d\Gamma$} and outward normal {$\mathbf{n}$}. (The original surface element, which may be either a hypothetical cut inside the body or an actual surface, is assumed to be so small that it remains essentially flat after mapping by {$F$}.) The traction force vector acting across that surface element is {$d\mathbf{f}$} so the stress vector is {$\mathbf{t} = d\mathbf{f}/d\Gamma$}. The force vector {$d\mathbf{f}$} corresponds in the reference configuration to the force vector {$d\mathbf{f}_0 = (\nabla F)^{-1}d\mathbf{f}$} acting over {$d\Gamma_0$}, where {$\nabla F$} is the [[Jacobian matrix]] of the map {$F$}. Therefore, the stress vector {$\mathbf{t}$} corresponds in the reference configuration to the stress vector {$\mathbf{t}_0 = d\mathbf{f}_0/d\Gamma_0$}. === Cauchy stress === The Cauchy stress tensor maps the normal {$\mathbf{n}$} in the current configuration to the stress vector {$\mathbf{t}$} in the current configuration. That is, :{$ \mathbf{t} = \boldsymbol{\sigma}^\top\cdot\mathbf{n} $} === Kirchhoff stress === The quantity {$\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}^\top$} is the transpose of the first Piola-Kirchhoff stress (PK1 stress) {$\boldsymbol{P}$} and is defined via :{$ d\mathbf{f} = \mathbf{t}_0~d\Gamma_0 = \boldsymbol{N}^\top\cdot\mathbf{n}_0~d\Gamma_0 = \boldsymbol{P}\cdot\mathbf{n}_0~d\Gamma_0 $} or :{$ \mathbf{t}_0 = \boldsymbol{N}^\top\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}^\top\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}^\top\cdot\mathbf{n}_0~d\Gamma_0 = \boldsymbol{F}^{-1}\cdot \mathbf{t}_0~d\Gamma_0 $} Therefore, :{$ \boldsymbol{S}^\top\cdot\mathbf{n}_0 = \boldsymbol{F}^{-1}\cdot\mathbf{t}_0 $} === Biot stress === The Biot stress is useful because it is [[energy conjugacy|energy conjugate]] to the [[Finite strain theory|right stretch tensor]] {$\boldsymbol{U}$}. The Biot stress is defined as the symmetric part of the tensor {$\boldsymbol{P}^\top\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}^\top\cdot\boldsymbol{P} + \boldsymbol{P}^\top\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}^\top~d\mathbf{f} = (\boldsymbol{P}^\top\cdot\boldsymbol{R})^\top\cdot\mathbf{n}_0~d\Gamma_0 $} == Relations between stress measures == ===Relations between Cauchy stress and nominal stress=== From [[Nanson formula|Nanson's formula]] relating areas in the reference and deformed configurations: :{$ \mathbf{n}~d\Gamma = J~\boldsymbol{F}^{-T}\cdot\mathbf{n}_0~d\Gamma_0 $} Now, :{$ \boldsymbol{\sigma}^\top\cdot\mathbf{n}~d\Gamma = d\mathbf{f} = \boldsymbol{N}^\top\cdot\mathbf{n}_0~d\Gamma_0 $} Hence, :{$ \boldsymbol{\sigma}^\top\cdot (J~\boldsymbol{F}^{-T}\cdot\mathbf{n}_0~d\Gamma_0) = \boldsymbol{N}^\top\cdot\mathbf{n}_0~d\Gamma_0 $} or, :{$ \boldsymbol{N}^\top = J~(\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma})^\top = J~\boldsymbol{\sigma}^\top\cdot\boldsymbol{F}^{-T} $} or, :{$ \boldsymbol{N} = J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma} \qquad \text{and} \qquad \boldsymbol{N}^\top = \boldsymbol{P} = J~\boldsymbol{\sigma}^\top\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{P}\cdot\boldsymbol{F}^\top~. $} Note that {$\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}^\top\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}^\top \cdot \mathbf{n}_0~d\Gamma_0) $} Therefore, :{$ \boldsymbol{N}^\top\cdot\mathbf{n}_0 = \boldsymbol{F}\cdot\boldsymbol{S}^\top\cdot\mathbf{n}_0 $} or (using the symmetry of {$\boldsymbol{S}$}), :{$ \boldsymbol{N} = \boldsymbol{S}\cdot\boldsymbol{F}^\top \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}^\top = 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}^\top $} or, :{$ \boldsymbol{\tau} = \boldsymbol{F}\cdot\boldsymbol{S}\cdot\boldsymbol{F}^\top ~. $} Clearly, from definition of the [[push-forward]] and [[pull-back]] operations, we have :{$ \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}^\top~. $} Therefore, {$\boldsymbol{S}$} is the pull back of {$\boldsymbol{\tau}$} by {$\boldsymbol{F}$} and {$\boldsymbol{\tau}$} is the push forward of {$\boldsymbol{S}$}. == See also == * [[Stress (physics)]] * [[Finite strain theory]] * [[Continuum mechanics]] * [[Hyperelastic material]] * [[Cauchy elastic material]] == 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 [[Category:Solid mechanics]] [[Category:Continuum mechanics]]