Cauchy Elastic Material - Isotropic Cauchy-elastic Materials

Isotropic Cauchy-elastic Materials

For an isotropic material the Cauchy stress tensor can be expressed as a function of the left Cauchy-Green tensor . The constitutive equation may then be written:

In order to find the restriction on which will ensure the principle of material frame-indifference, one can write:

$\ \begin{array}{rrcl} & \boldsymbol{\sigma}^* &=& \mathcal{H}(\boldsymbol{B}^*) \\ \Rightarrow & \boldsymbol{R}\cdot \boldsymbol{\sigma}\cdot \boldsymbol{R}^T &=& \mathcal{H}(\boldsymbol{F}^*\cdot(\boldsymbol{F}^*)^T) \\ \Rightarrow & \boldsymbol{R}\cdot \mathcal{H}(\boldsymbol{B}) \cdot\boldsymbol{R}^T &=& \mathcal{H}(\boldsymbol{R}\cdot\boldsymbol{F}\cdot\boldsymbol{F}^T\cdot\boldsymbol{R}^T) \\ \Rightarrow & \boldsymbol{R}\cdot \mathcal{H}(\boldsymbol{B})\cdot \boldsymbol{R}^T &=& \mathcal{H}(\boldsymbol{R}\cdot\boldsymbol{B}\cdot\boldsymbol{R}^T). \end{array}$

A constitutive equation that respects the above condition is said to be isotropic.

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