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In [[continuum mechanics]], the most commonly used measure of [[stress (mechanics)|stress]] is the [[Cauchy stress tensor]], often called simply ''the'' stress tensor or "true stress". However, several alternative measures of stress can be defined:<ref>J. Bonet and R. W. Wood, ''Nonlinear Continuum Mechanics for Finite Element Analysis'', Cambridge University Press.</ref><ref>R. W. Ogden, 1984, ''Non-linear Elastic Deformations'', Dover.</ref><ref>L. D. Landau, E. M. Lifshitz, ''Theory of Elasticity'', third edition</ref> |
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#The Kirchhoff stress (<math>\boldsymbol{\tau}</math>). |
#The Kirchhoff stress (<math>\boldsymbol{\tau}</math>). |
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#The |
#The nominal stress (<math>\boldsymbol{N}</math>). |
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#The first |
#The [[Piola–Kirchhoff stress tensor]]s |
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##The first Piola–Kirchhoff stress (<math>\boldsymbol{P}</math>). This stress tensor is the transpose of the nominal stress (<math>\boldsymbol{P} = \boldsymbol{N}^T</math>). |
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#The second |
##The second Piola–Kirchhoff stress or PK2 stress (<math>\boldsymbol{S}</math>). |
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#The Biot stress (<math>\boldsymbol{T}</math>) |
#The Biot stress (<math>\boldsymbol{T}</math>) |
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== Definitions |
== Definitions == |
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Consider the situation shown in the following figure. The following definitions use the notations shown in the figure. |
Consider the situation shown in the following figure. The following definitions use the notations shown in the figure. |
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{|align="center" |
{|align="center" |
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|[[Image:StressMeasures.png|thumb|400px|Quantities used in the definition of stress measures]] |
|[[Image:StressMeasures.png|thumb|400px|Quantities used in the definition of stress measures]] |
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|} |
|} |
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In the reference configuration <math>\Omega_0</math>, the outward normal to a surface element <math>d\Gamma_0</math> is <math>\mathbf{N} \equiv \mathbf{n}_0</math> and the traction acting on that surface is <math>\mathbf{t}_0</math> leading to a force vector <math>d\mathbf{f}_0</math>. In the deformed configuration <math>\Omega</math>, the surface element changes to <math>d\Gamma</math> with outward normal <math>\mathbf{n}</math> and traction vector <math>\mathbf{t}</math> leading to a force <math>d\mathbf{f}</math>. Note that this surface can either be a hypothetical cut inside the body or an actual surface. The quantity <math>\boldsymbol{F}</math> is the [[Finite strain theory#Deformation gradient tensor|deformation gradient tensor]], <math>J</math> is its determinant. |
In the reference configuration <math>\Omega_0</math>, the outward normal to a surface element <math>d\Gamma_0</math> is <math>\mathbf{N} \equiv \mathbf{n}_0</math> and the traction acting on that surface (assuming it deforms like a generic vector belonging to the deformation) is <math>\mathbf{t}_0</math> leading to a force vector <math>d\mathbf{f}_0</math>. In the deformed configuration <math>\Omega</math>, the surface element changes to <math>d\Gamma</math> with outward normal <math>\mathbf{n}</math> and traction vector <math>\mathbf{t}</math> leading to a force <math>d\mathbf{f}</math>. Note that this surface can either be a hypothetical cut inside the body or an actual surface. The quantity <math>\boldsymbol{F}</math> is the [[Finite strain theory#Deformation gradient tensor|deformation gradient tensor]], <math>J</math> is its determinant. |
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=== Cauchy stress === |
=== Cauchy stress === |
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=== Kirchhoff stress === |
=== Kirchhoff stress === |
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The quantity, |
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⚫ | |||
:<math> |
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⚫ | |||
⚫ | |||
</math> |
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⚫ | |||
⚫ | |||
=== Piola–Kirchhoff stress === |
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{{main|Piola–Kirchhoff stress tensor}} |
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=== Nominal stress/First |
==== Nominal stress/First Piola–Kirchhoff stress ==== |
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The nominal stress <math>\boldsymbol{N}=\boldsymbol{P}^T</math> is the transpose of the first |
The nominal stress <math>\boldsymbol{N}=\boldsymbol{P}^T</math> is the transpose of the first Piola–Kirchhoff stress (PK1 stress, also called engineering stress) <math>\boldsymbol{P}</math> and is defined via |
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:<math> |
:<math> |
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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 |
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 |
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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.<ref>{{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}}</ref> |
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.<ref>{{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}}</ref> |
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=== Second |
==== Second Piola–Kirchhoff stress ==== |
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If we [[Pullback (differential geometry)|pull back]] <math>d\mathbf{f}</math> to the reference configuration |
If we [[Pullback (differential geometry)|pull back]] <math>d\mathbf{f}</math> to the reference configuration we obtain the traction acting on that surface before the deformation <math>d\mathbf{f}_0</math> assuming it behaves like a generic vector belonging to the deformation. In particular we have |
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:<math> |
:<math> |
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</math> |
</math> |
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== Relations |
== Relations == |
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===Relations between Cauchy stress and nominal stress=== |
===Relations between Cauchy stress and nominal stress=== |
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⚫ | |||
From [[Nanson h |
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⚫ | |||
:<math> |
:<math> |
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\mathbf{n}~d\Gamma = J~\boldsymbol{F}^{-T}\cdot\mathbf{n}_0~d\Gamma_0 |
\mathbf{n}~d\Gamma = J~\boldsymbol{F}^{-T}\cdot\mathbf{n}_0~d\Gamma_0 |
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</math> |
</math> |
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Note that <math>\boldsymbol{N}</math> and <math>\boldsymbol{P}</math> are not symmetric because <math>\boldsymbol{F}</math> is not symmetric. |
Note that <math>\boldsymbol{N}</math> and <math>\boldsymbol{P}</math> are (generally) not symmetric because <math>\boldsymbol{F}</math> is (generally) not symmetric. |
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===Relations between nominal stress and second |
===Relations between nominal stress and second P–K stress=== |
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Recall that |
Recall that |
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:<math> |
:<math> |
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</math> |
</math> |
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===Relations between Cauchy stress and second |
===Relations between Cauchy stress and second P–K stress=== |
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Recall that |
Recall that |
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:<math> |
:<math> |
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Therefore, <math>\boldsymbol{S}</math> is the pull back of <math>\boldsymbol{\tau}</math> by <math>\boldsymbol{F}</math> and <math>\boldsymbol{\tau}</math> is the push forward of <math>\boldsymbol{S}</math>. |
Therefore, <math>\boldsymbol{S}</math> is the pull back of <math>\boldsymbol{\tau}</math> by <math>\boldsymbol{F}</math> and <math>\boldsymbol{\tau}</math> is the push forward of <math>\boldsymbol{S}</math>. |
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=== Summary of conversion formula === |
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⚫ | |||
⚫ | Key: <math display="block"> 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},</math> <math display="block">\boldsymbol{P}=J\boldsymbol{\sigma}\boldsymbol{F}^{-T},\quad\boldsymbol{\tau}=J\boldsymbol{\sigma},\quad |
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⚫ | |||
⚫ | |||
\boldsymbol{M}=\boldsymbol{C}\boldsymbol{S}</math> |
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{|class="wikitable" style="text-align: center" |
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== Summary of relations between stress measures == |
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{{Navbox |
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|name = Stress Tensor |
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|title = [[Stress tensor (disambiguation)|Stress Tensor]] |
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|listclass = hlist |
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|list1 = |
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*[[First Piola-Kirchhoff stress tensor]] (<math>\boldsymbol{P}</math>) |
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*[[Second Piola-Kirchhoff stress tensor]] (<math>\boldsymbol{S}</math>) |
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*[[Kirchhoff stress tensor]] (<math>\boldsymbol{\tau}</math>) |
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*[[Biot stress tensor]] (<math>\boldsymbol{T}</math>) |
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*[[Mandel stress tensor]] (<math>\boldsymbol{M}</math>) |
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|state = show |
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⚫ | |||
{|class="wikitable collapsible" width="100%" style="font-size:smaller; background:white" align=center |
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⚫ | |||
|- |
|- |
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! scope="col" | Equation for |
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⚫ | |||
| |
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! scope="col" | <math>\boldsymbol{\tau}</math> |
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! scope="col" | <math>\boldsymbol{P}</math> |
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! scope="col" | <math>\boldsymbol{S}</math> |
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! scope="col" | <math>\boldsymbol{T}</math> |
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! scope="col" | <math>\boldsymbol{M}</math> |
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|align=center | <math>\boldsymbol{M}</math> |
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|- |
|- |
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| <math>\boldsymbol{\sigma}=\,</math> |
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| <math>\boldsymbol{\sigma}</math> |
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| <math>J^{-1}\boldsymbol{\tau}</math> |
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| <math>J^{-1}\boldsymbol{P}\boldsymbol{F}^{T}</math> |
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| <math>J^{-1}\boldsymbol{F}\boldsymbol{S}\boldsymbol{F}^{T}</math> |
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| <math>J^{-1}\boldsymbol{R}\boldsymbol{T}\boldsymbol{F}^{T}</math> |
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| <math>J^{-1}\boldsymbol{F}^{-T}\boldsymbol{M}\boldsymbol{F}^{T}</math> (non isotropy) |
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|- |
|- |
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| <math>\boldsymbol{\tau}=\,</math> |
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| <math>J\boldsymbol{\sigma}</math> |
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| <math>\boldsymbol{\tau}</math> |
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| <math>\boldsymbol{P}\boldsymbol{F}^{T}</math> |
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| <math>\boldsymbol{F}\boldsymbol{S}\boldsymbol{F}^{T}</math> |
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| <math>\boldsymbol{R}\boldsymbol{T}\boldsymbol{F}^{T}</math> |
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| <math>\boldsymbol{F}^{-T}\boldsymbol{M}\boldsymbol{F}^{T}</math> (non isotropy) |
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|- |
|- |
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| <math>\boldsymbol{P}=\,</math> |
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| <math>J\boldsymbol{\sigma}\boldsymbol{F}^{-T}</math> |
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| <math>\boldsymbol{\tau}\boldsymbol{F}^{-T}</math> |
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| <math>\boldsymbol{P}</math> |
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| <math>\boldsymbol{F}\boldsymbol{S}</math> |
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| <math>\boldsymbol{R}\boldsymbol{T}</math> |
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| <math>\boldsymbol{F}^{-T}\boldsymbol{M}</math> |
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|- |
|- |
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| <math>\boldsymbol{S}=\,</math> |
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| <math>J\boldsymbol{F}^{-1}\boldsymbol{\sigma}\boldsymbol{F}^{-T}</math> |
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| <math>\boldsymbol{F}^{-1}\boldsymbol{\tau}\boldsymbol{F}^{-T}</math> |
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| <math>\boldsymbol{F}^{-1}\boldsymbol{P}</math> |
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| <math>\boldsymbol{S}</math> |
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| <math>\boldsymbol{U}^{-1}\boldsymbol{T}</math> |
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| <math>\boldsymbol{C}^{-1}\boldsymbol{M}</math> |
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|- |
|- |
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| <math>\boldsymbol{T}=\,</math> |
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| <math>J\boldsymbol{R}^{T}\boldsymbol{\sigma}\boldsymbol{F}^{-T}</math> |
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| <math>\boldsymbol{R}^{T}\boldsymbol{\tau}\boldsymbol{F}^{-T}</math> |
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| <math>\boldsymbol{R}^{T}\boldsymbol{P}</math> |
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| <math>\boldsymbol{U}\boldsymbol{S}</math> |
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| <math>\boldsymbol{T}</math> |
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| <math>\boldsymbol{U}^{-1}\boldsymbol{M}</math> |
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|- |
|- |
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| <math>\boldsymbol{M}=\,</math> |
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| <math>J\boldsymbol{F}^{T}\boldsymbol{\sigma}\boldsymbol{F}^{-T}</math> (non isotropy) |
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| <math>\boldsymbol{F}^{T}\boldsymbol{\tau}\boldsymbol{F}^{-T}</math> (non isotropy) |
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| <math>\boldsymbol{F}^{T}\boldsymbol{P}</math> |
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| <math>\boldsymbol{C}\boldsymbol{S}</math> |
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| <math>\boldsymbol{U}\boldsymbol{T}</math> |
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| <math>\boldsymbol{M}</math> |
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⚫ | |||
⚫ | |||
|- |
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⚫ | |||
⚫ | |||
⚫ | |||
|- |
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!colspan=7 |<math>\boldsymbol{P}=J\boldsymbol{\sigma}\boldsymbol{F}^{-T},\quad\boldsymbol{\tau}=J\boldsymbol{\sigma},\quad |
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⚫ | |||
⚫ | |||
⚫ | |||
⚫ | |||
|}<noinclude> |
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</noinclude> |
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== References == |
== References == |
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[[Category:Continuum mechanics]] |
[[Category:Continuum mechanics]] |
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[[Category:Gustav Kirchhoff]] |
[[Category:Gustav Kirchhoff]] |
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[[Category:Tensor physical quantities]] |
Latest revision as of 02:07, 27 August 2023
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 alternative measures of stress can be defined:[1][2][3]
- The Kirchhoff stress ().
- The nominal stress ().
- The Piola–Kirchhoff stress tensors
- The first Piola–Kirchhoff stress (). This stress tensor is the transpose of the nominal stress ().
- The second Piola–Kirchhoff stress or PK2 stress ().
- The Biot stress ()
Definitions
Consider the situation shown in the following figure. The following definitions use the notations shown in the figure.
In the reference configuration , the outward normal to a surface element is and the traction acting on that surface (assuming it deforms like a generic vector belonging to the deformation) is leading to a force vector . In the deformed configuration , the surface element changes to with outward normal and traction vector leading to a force . Note that this surface can either be a hypothetical cut inside the body or an actual surface. The quantity is the deformation gradient tensor, is its determinant.
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
or
where is the traction and is the normal to the surface on which the traction acts.
Kirchhoff stress
The quantity,
is called the Kirchhoff stress tensor, with the determinant of . 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.
Piola–Kirchhoff stress
Nominal stress/First Piola–Kirchhoff stress
The nominal stress is the transpose of the first Piola–Kirchhoff stress (PK1 stress, also called engineering stress) and is defined via
or
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]
Second Piola–Kirchhoff stress
If we pull back to the reference configuration we obtain the traction acting on that surface before the deformation assuming it behaves like a generic vector belonging to the deformation. In particular we have
or,
The PK2 stress () is symmetric and is defined via the relation
Therefore,
Biot stress
The Biot stress is useful because it is energy conjugate to the right stretch tensor . The Biot stress is defined as the symmetric part of the tensor where is the rotation tensor obtained from a polar decomposition of the deformation gradient. Therefore, the Biot stress tensor is defined as
The Biot stress is also called the Jaumann stress.
The quantity does not have any physical interpretation. However, the unsymmetrized Biot stress has the interpretation
Relations
Relations between Cauchy stress and nominal stress
From Nanson's formula relating areas in the reference and deformed configurations:
Now,
Hence,
or,
or,
In index notation,
Therefore,
Note that and are (generally) not symmetric because is (generally) not symmetric.
Relations between nominal stress and second P–K stress
Recall that
and
Therefore,
or (using the symmetry of ),
In index notation,
Alternatively, we can write
Relations between Cauchy stress and second P–K stress
Recall that
In terms of the 2nd PK stress, we have
Therefore,
In index notation,
Since the Cauchy stress (and hence the Kirchhoff stress) is symmetric, the 2nd PK stress is also symmetric.
Alternatively, we can write
or,
Clearly, from definition of the push-forward and pull-back operations, we have
and
Therefore, is the pull back of by and is the push forward of .
Summary of conversion formula
Key:
Equation for | ||||||
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(non isotropy) | ||||||
(non isotropy) | ||||||
(non isotropy) | (non isotropy) |
See also
- Stress (physics)
- Finite strain theory
- Continuum mechanics
- Hyperelastic material
- Cauchy elastic material
- Critical plane analysis
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
- ^ Three-Dimensional Elasticity. Elsevier. 1 April 1988. ISBN 978-0-08-087541-5.