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.

Read more about this topic:  Cauchy Elastic Material

Famous quotes containing the words mathematical and/or definition:

    “What he loved so much in the plant morphological structure of the tree was that given a fixed mathematical basis, the final evolution was so incalculable.”
    —D.H. (David Herbert)

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)