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In [[physics]], a '''Cauchy-elastic material''' is one in which the [[stress (mechanics)|stress]] at each point is determined only by the current state of [[deformation (engineering)|deformation]] with respect to an arbitrary reference configuration.<ref name=Ogden/>  A Cauchy elastic material is also called a '''simple elastic''' material.
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It follows from this definition that the stress in a Cauchy-elastic material does not depend on the path of deformation or the history of deformation, or on the time taken to achieve that deformation or the rate at which the state of deformation is reached.  The definition also implies that the [[constitutive equation]]s are spatially local; that is, the stress is only affected by the state of deformation in an infinitesimal neighborhood of the point in question, without regard for the deformation or motion of the rest of the material. It also implies that body forces (such as gravity), and inertial forces cannot affect the properties of the material. Finally, a Cauchy elastic material must satisfy the requirements of [[objectivity (frame invariance)|material objectivity]].
 
Cauchy-elastic materials are mathematical abstractions, and no real material fits this definition perfectly. However, many elastic materials of practical interest, such as steel, plastic, wood and concrete, can often be assumed to be Cauchy-elastic for the purposes of [[stress analysis]].
 
== Mathematical definition ==
Formally, a material is said to be Cauchy-elastic if  the [[Cauchy stress tensor]] <math> \boldsymbol{\sigma} </math> is a function of the [[strain tensor]] ([[deformation gradient]]) <math>\boldsymbol{F}</math> alone:
:<math>\ \boldsymbol{\sigma} = \mathcal{G}(\boldsymbol{F}) </math>
This definition assumes that the effect of temperature can be ignored, and the body is homogeneous. This is the [[constitutive equation]] for a Cauchy-elastic material.
Note that the function <math>\mathcal{G}</math> depends on the choice of reference configuration. Typically, the reference configuration is taken as the relaxed (zero-stress) configuration, but need not be.
 
[[objectivity (frame invariance)|Frame indifference]] requires that the constitutive relation <math>\mathcal{G}</math> should not change when the location of the observer changes. Therefore the [[constitutive equation]] for another arbitrary observer can be written <math> \boldsymbol{\sigma}^* = \mathcal{G}(\boldsymbol{F}^*) </math>. Knowing that the [[Stress (physics)|Cauchy stress tensor]] <math> \sigma </math> and the [[deformation gradient]] <math> F </math> are '''objective''' quantities, one can write:
 
:<math>
  \begin{align}
            & \boldsymbol{\sigma}^* &=& \mathcal{G}(\boldsymbol{F}^*) \\
    \Rightarrow & \boldsymbol{R}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{R}^T &=& \mathcal{G}(\boldsymbol{R}\cdot\boldsymbol{F}) \\
    \Rightarrow & \boldsymbol{R}\cdot\mathcal{G}(\boldsymbol{F})\cdot\boldsymbol{R}^T &=& \mathcal{G}(\boldsymbol{R}\cdot\boldsymbol{F})
\end{align}
</math>
where <math>\boldsymbol{R}</math> is a proper orthogonal tensor.
 
The above is a condition that the [[constitutive equation|constitutive law]] <math> \mathcal{G} </math> has to respect to make sure that the response of the material will be independent of the observer. Similar conditions can be derived for [[constitutive equation|constitutive laws]] relating the [[deformation gradient]] to the first or second [[Piola-Kirchhoff stress tensor]].
 
== Isotropic Cauchy-elastic materials ==
For an isotropic material the [[Stress (physics)|Cauchy stress tensor]] <math> \boldsymbol{\sigma} </math> can be expressed as a function of the [[Finite strain theory#The Left Cauchy-Green deformation tensor|left Cauchy-Green tensor]] <math> \boldsymbol{B}=\boldsymbol{F}\cdot\boldsymbol{F}^T </math>. The [[constitutive equation]] may then be written:
 
:<math>\ \boldsymbol{\sigma} = \mathcal{H}(\boldsymbol{B}). </math>
 
In order to find the restriction on <math> h </math> which will ensure the principle of material frame-indifference, one can write:
 
:<math>\ \begin{array}{rrcl} & \boldsymbol{\sigma}^* &=& \mathcal{H}(\boldsymbol{B}^*) \\
\Rightarrow & \boldsymbol{R}\cdot \boldsymbol{\sigma}\cdot \boldsymbol{R}^T &=& \mathcal{H}(\boldsymbol{F}^*\cdot(\boldsymbol{F}^*)^T) \\
\Rightarrow & \boldsymbol{R}\cdot \mathcal{H}(\boldsymbol{B}) \cdot\boldsymbol{R}^T &=& \mathcal{H}(\boldsymbol{R}\cdot\boldsymbol{F}\cdot\boldsymbol{F}^T\cdot\boldsymbol{R}^T) \\
\Rightarrow & \boldsymbol{R}\cdot \mathcal{H}(\boldsymbol{B})\cdot \boldsymbol{R}^T &=& \mathcal{H}(\boldsymbol{R}\cdot\boldsymbol{B}\cdot\boldsymbol{R}^T). \end{array}</math>
 
A [[constitutive equation]] that respects the above condition is said to be [[isotropic]].
 
==Non-conservative==
Even though the stress in a Cauchy-elastic material depends only on the state of deformation, the work done by stresses may depend on the path of deformation.  Therefore a Cauchy elastic material has a non-conservative structure, and the stress cannot be derived from a scalar "elastic potential" function.  Materials that are conservative in this sense are called [[hyperelastic material|hyperelastic]] or "Green-elastic".
== References ==
<references>
 
<ref name=Ogden>
  R. W. Ogden, 1984, ''Non-linear Elastic Deformations'', Dover, pp. 175-204.
</ref>
 
</references>
 
== See also ==
* [[Hyperelastic material]]
* [[Objectivity (frame invariance)]]
 
[[Category:Continuum mechanics]]
[[Category:Elasticity (physics)]]

Latest revision as of 21:44, 12 November 2014

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