Cauchy Elastic Material - Mathematical Definition

Mathematical Definition

Formally, a material is said to be Cauchy-elastic if the Cauchy stress tensor is a function of the strain tensor (deformation gradient) alone:

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 depends on the choice of reference configuration. Typically, the reference configuration is taken as the relaxed (zero-stress) configuration, but need not be.

Frame indifference requires that the constitutive relation should not change when the location of the observer changes. Therefore the constitutive equation for another arbitrary observer can be written . Knowing that the Cauchy stress tensor and the deformation gradient are objective quantities, one can write:

\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}

where is a proper orthogonal tensor.

The above is a condition that the constitutive law 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 laws relating the deformation gradient to the first or second Piola-Kirchhoff stress tensor.

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