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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.<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> Some such stress measures that are widely used in continuum mechanics, particularly in the computational context, are:
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| #The Kirchhoff stress (<math>\boldsymbol{\tau}</math>).
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| #The Nominal stress (<math>\boldsymbol{N}</math>).
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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 Piola-Kirchhoff stress or PK2 stress (<math>\boldsymbol{S}</math>).
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| #The Biot stress (<math>\boldsymbol{T}</math>)
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|
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| == Definitions of stress measures ==
| | It depends on the quality of the Wordpress theme but even if it's not a professional one you will be able to average 50-60$ EACH link. Thus, it is important to keep pace with this highly advanced age and have a regular interaction with your audience to keep a strong hold in the business market. A pinch of tablet centric strategy can get your Word - Press site miles ahead of your competitors, so here are few strategies that will give your Wordpress websites and blogs an edge over your competitors:. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. If you are happy with your new look then click "Activate 'New Theme'" in the top right corner. <br><br>As you know today Word - Press has turn out to be a tremendously popular open source publishing and blogging display place. The higher your blog ranks on search engines, the more likely people will find your online marketing site. We also help to integrate various plug-ins to expand the functionalities of the web application. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>Usually, Wordpress owners selling the ad space on monthly basis and this means a residual income source. As of now, Pin2Press is getting ready to hit the market. Use this section to change many formatting elements. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. After that the developer adds the unordered list for navigations. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. When you have virtually any issues with regards to wherever as well as tips on how to employ [http://iz.sa/wordpressbackupplugin533248 wordpress backup plugin], you are able to email us with our site. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. Wordpress template is loaded with lots of prototype that unite graphic features and content area. <br><br>Website security has become a major concern among individuals all over the world. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. If this is not possible you still have the choice of the default theme that is Word - Press 3. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins. |
| Consider the situation shown the following figure. The following definitions use the notations shown in the figure.
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| {|align="center"
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| |[[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.
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| === Cauchy stress ===
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| 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
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| :<math>
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| d\mathbf{f} = \mathbf{t}~d\Gamma = \boldsymbol{\sigma}^T\cdot\mathbf{n}~d\Gamma
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| </math> | |
| or
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| :<math>
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| \mathbf{t} = \boldsymbol{\sigma}^T\cdot\mathbf{n}
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| </math>
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| where <math>\mathbf{t}</math> is the traction and <math>\mathbf{n}</math> is the normal to the surface on which the traction acts.
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| | |
| === Kirchhoff stress ===
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| The quantity <math>\boldsymbol{\tau} = J~\boldsymbol{\sigma}</math> is called the '''Kirchhoff stress tensor''' and is used widely in numerical algorithms in metal plasticity (where there
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| is no change in volume during plastic deformation).
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| === 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 Piola-Kirchhoff stress (PK1 stress) <math>\boldsymbol{P}</math> and is defined via
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| :<math>
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| 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
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| </math>
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| or
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| :<math>
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| \mathbf{t}_0 = \boldsymbol{N}^T\cdot\mathbf{n}_0 = \boldsymbol{P}\cdot\mathbf{n}_0
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| </math>
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| 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.''
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| === Second Piola-Kirchhoff stress ===
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| If we pull back <math>d\mathbf{f}</math> to the reference configuration, we have
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| :<math>
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| d\mathbf{f}_0 = \boldsymbol{F}^{-1}\cdot d\mathbf{f}
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| </math>
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| or,
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| :<math>
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| d\mathbf{f}_0 = \boldsymbol{F}^{-1}\cdot \boldsymbol{N}^T\cdot\mathbf{n}_0~d\Gamma_0
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| = \boldsymbol{F}^{-1}\cdot \mathbf{t}_0~d\Gamma_0
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| </math>
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| The PK2 stress (<math>\boldsymbol{S}</math>) is symmetric and is defined via the relation
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| :<math>
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| d\mathbf{f}_0 = \boldsymbol{S}^T\cdot\mathbf{n}_0~d\Gamma_0 = \boldsymbol{F}^{-1}\cdot \mathbf{t}_0~d\Gamma_0
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| </math> | |
| Therefore,
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| :<math>
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| \boldsymbol{S}^T\cdot\mathbf{n}_0 = \boldsymbol{F}^{-1}\cdot\mathbf{t}_0
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| </math>
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| | |
| === Biot stress ===
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| The Biot stress is useful because it is [[energy conjugacy|energy conjugate]] to the [[Finite strain theory|right stretch tensor]] <math>\boldsymbol{U}</math>. The Biot stress is defined as the symmetric part of the tensor <math>\boldsymbol{P}^T\cdot\boldsymbol{R}</math> where <math>\boldsymbol{R}</math> is the rotation tensor obtained from a [[polar decomposition]] of the deformation gradient. Therefore the Biot stress tensor is defined as
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| :<math>
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| \boldsymbol{T} = \tfrac{1}{2}(\boldsymbol{R}^T\cdot\boldsymbol{P} + \boldsymbol{P}^T\cdot\boldsymbol{R}) ~.
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| </math> | |
| The Biot stress is also called the Jaumann stress.
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| The quantity <math>\boldsymbol{T}</math> does not have any physical interpretation. However, the unsymmetrized Biot stress has the interpretation
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| :<math>
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| \boldsymbol{R}^T~d\mathbf{f} = (\boldsymbol{P}^T\cdot\boldsymbol{R})^T\cdot\mathbf{n}_0~d\Gamma_0
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| </math>
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| == Relations between stress measures ==
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| ===Relations between Cauchy stress and nominal stress===
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| From [[Nanson formula|Nanson's formula]] relating areas in the reference and deformed configurations:
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| :<math>
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| \mathbf{n}~d\Gamma = J~\boldsymbol{F}^{-T}\cdot\mathbf{n}_0~d\Gamma_0
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| </math>
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| Now,
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| :<math>
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| \boldsymbol{\sigma}^T\cdot\mathbf{n}~d\Gamma = d\mathbf{f} = \boldsymbol{N}^T\cdot\mathbf{n}_0~d\Gamma_0
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| </math>
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| Hence,
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| :<math>
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| \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
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| </math>
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| or,
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| :<math>
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| \boldsymbol{N}^T = J~(\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma})^T = J~\boldsymbol{\sigma}^T\cdot\boldsymbol{F}^{-T}
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| </math>
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| or,
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| :<math>
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| \boldsymbol{N} = J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma} \qquad \text{and} \qquad
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| \boldsymbol{N}^T = \boldsymbol{P} = J~\boldsymbol{\sigma}^T\cdot\boldsymbol{F}^{-T}
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| </math>
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| In index notation,
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| :<math>
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| N_{Ij} = J~F_{Ik}^{-1}~\sigma_{kj} \qquad \text{and} \qquad
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| P_{iJ} = J~\sigma_{ki}~F^{-1}_{Jk}
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| </math>
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| Therefore,
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| :<math>
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| J~\boldsymbol{\sigma} = \boldsymbol{F}\cdot\boldsymbol{N} = \boldsymbol{P}\cdot\boldsymbol{F}^T~.
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| </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.
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| | |
| ===Relations between nominal stress and second P-K stress===
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| Recall that
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| :<math>
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| \boldsymbol{N}^T\cdot\mathbf{n}_0~d\Gamma_0 = d\mathbf{f}
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| </math> | |
| and
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| :<math>
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| d\mathbf{f} = \boldsymbol{F}\cdot d\mathbf{f}_0 = \boldsymbol{F} \cdot (\boldsymbol{S}^T \cdot \mathbf{n}_0~d\Gamma_0)
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| </math>
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| Therefore,
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| :<math>
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| \boldsymbol{N}^T\cdot\mathbf{n}_0 = \boldsymbol{F}\cdot\boldsymbol{S}^T\cdot\mathbf{n}_0
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| </math>
| |
| or (using the symmetry of <math>\boldsymbol{S}</math>),
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| :<math>
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| \boldsymbol{N} = \boldsymbol{S}\cdot\boldsymbol{F}^T \qquad \text{and} \qquad
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| \boldsymbol{P} = \boldsymbol{F}\cdot\boldsymbol{S}
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| </math>
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| In index notation,
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| :<math>
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| N_{Ij} = S_{IK}~F_{jK} \qquad \text{and} \qquad P_{iJ} = F_{iK}~S_{KJ}
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| </math>
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| Alternatively, we can write
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| :<math>
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| \boldsymbol{S} = \boldsymbol{N}\cdot\boldsymbol{F}^{-T} \qquad \text{and} \qquad
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| \boldsymbol{S} = \boldsymbol{F}^{-1}\cdot\boldsymbol{P}
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| </math>
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| | |
| ===Relations between Cauchy stress and second P-K stress===
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| Recall that
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| :<math>
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| \boldsymbol{N} = J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}
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| </math>
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| In terms of the 2nd PK stress, we have
| |
| :<math>
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| \boldsymbol{S}\cdot\boldsymbol{F}^T = J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}
| |
| </math>
| |
| Therefore,
| |
| :<math>
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| \boldsymbol{S} = J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T} = \boldsymbol{F}^{-1}\cdot\boldsymbol{\tau}\cdot\boldsymbol{F}^{-T}
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| </math>
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| In index notation,
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| :<math>
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| S_{IJ} = F_{Ik}^{-1}~\tau_{kl}~F_{Jl}^{-1}
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| </math>
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| Since the Cauchy stress (and hence the Kirchhoff stress) is symmetric, the 2n PK stress is also symmetric.
| |
| | |
| Alternatively, we can write
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| :<math>
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| \boldsymbol{\sigma} = J^{-1}~\boldsymbol{F}\cdot\boldsymbol{S}\cdot\boldsymbol{F}^T
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| </math>
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| or,
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| :<math>
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| \boldsymbol{\tau} = \boldsymbol{F}\cdot\boldsymbol{S}\cdot\boldsymbol{F}^T ~.
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| </math>
| |
| | |
| Clearly, from definition of the [[push-forward]] and [[pull-back]] operations, we have
| |
| :<math>
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| \boldsymbol{S} = \varphi^{*}[\boldsymbol{\tau}] = \boldsymbol{F}^{-1}\cdot\boldsymbol{\tau}\cdot\boldsymbol{F}^{-T}
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| </math>
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| and | |
| :<math>
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| \boldsymbol{\tau} = \varphi_{*}[\boldsymbol{S}] = \boldsymbol{F}\cdot\boldsymbol{S}\cdot\boldsymbol{F}^T~.
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| </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>.
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| | |
| == See also ==
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| * [[Stress (physics)]]
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| * [[Finite strain theory]]
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| * [[Continuum mechanics]]
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| * [[Hyperelastic material]]
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| * [[Cauchy elastic material]]
| |
| | |
| == References ==
| |
| <references />
| |
| | |
| [[Category:Solid mechanics]]
| |
| [[Category:Continuum mechanics]]
| |
It depends on the quality of the Wordpress theme but even if it's not a professional one you will be able to average 50-60$ EACH link. Thus, it is important to keep pace with this highly advanced age and have a regular interaction with your audience to keep a strong hold in the business market. A pinch of tablet centric strategy can get your Word - Press site miles ahead of your competitors, so here are few strategies that will give your Wordpress websites and blogs an edge over your competitors:. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. If you are happy with your new look then click "Activate 'New Theme'" in the top right corner.
As you know today Word - Press has turn out to be a tremendously popular open source publishing and blogging display place. The higher your blog ranks on search engines, the more likely people will find your online marketing site. We also help to integrate various plug-ins to expand the functionalities of the web application. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself.
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