# Cauchy elastic material

A Cauchy-elastic material is one in which the Cauchy stress at each material point is determined only by the current state of

deformation (with respect to an arbitrary reference configuration). R. W. Ogden, 1984,*Non-linear Elastic Deformations*, Dover, pp. 175-204.] Therefore, the Cauchy stress in such a material does not depend on the path of deformation or the history of deformation. Neither does the stress depend on the time taken to achieve that deformation or the rate at which the state of deformation is reached. A Cauchy elastic material is also called a**simple elastic**material.However, unlike in a

hyperelastic material , the work done by the stresses does 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 potential function.A Cauchy elastic material must satisfy the requirements of material objectivity and the principle of local action, i.e., the constitutive equations are spatially local. This assumption excludes action at a distance from being present in a constitutive relation (no nonlocal materials allowed). Also it enforces the requirement that body forces, such as gravity, and inertial forces cannot affect the properties of the material.

Though a Cauchy elastic material is a mathematical idealization of elastic material behavior, this description applies to many of the purely mechanical constitutive relations for elastic materials found in nature.

**Constitutive equation**Neglecting the effect of temperature and assuming the body to be homogeneous, a

constitutive equation for the Cauchy stress tensor can be formulated based on thedeformation gradient ::$\backslash boldsymbol\{sigma\}\; =\; mathcal\{G\}(\backslash boldsymbol\{F\})$where $\backslash boldsymbol\{sigma\}$ is the Cauchy stress and $\backslash boldsymbol\{F\}$ is the deformation gradient. Note that the function $mathcal\{G\}$ depends on the choice of reference configuration.The condition of material objectivity requires that the constitutive relation $mathcal\{G\}$ should not change when the location of the observer changes. Therefore the

constitutive equation for another arbitrary observer can be written $\backslash boldsymbol\{sigma\}^*\; =\; mathcal\{G\}(\backslash boldsymbol\{F\}^*)$. Knowing that the Cauchy stress tensor $sigma$ and thedeformation gradient $F$ are**objective**quantities, one can write::$egin\{align\}\; \backslash boldsymbol\{sigma\}^*\; =\; mathcal\{G\}(\backslash boldsymbol\{F\}^*)\; \backslash \; Rightarrow\; \backslash boldsymbol\{R\}cdot\backslash boldsymbol\{sigma\}cdot\backslash boldsymbol\{R\}^T\; =\; mathcal\{G\}(\backslash boldsymbol\{R\}cdot\backslash boldsymbol\{F\})\; \backslash \; Rightarrow\; \backslash boldsymbol\{R\}cdotmathcal\{G\}(\backslash boldsymbol\{F\})cdot\backslash boldsymbol\{R\}^T\; =\; mathcal\{G\}(\backslash boldsymbol\{R\}cdot\backslash boldsymbol\{F\})end\{align\}$where $\backslash boldsymbol\{R\}$ is a proper orthogonal tensor.

The above is a condition that the constitutive law $mathcal\{G\}$ 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 secondPiola-Kirchhoff stress tensor .**Isotropic Cauchy-elastic materials**For an isotropic material the Cauchy stress tensor $\backslash boldsymbol\{sigma\}$ can be expressed as a function of the left Cauchy-Green tensor $\backslash boldsymbol\{B\}=\backslash boldsymbol\{F\}cdot\backslash boldsymbol\{F\}^T$. The

constitutive equation may then be written::$\backslash boldsymbol\{sigma\}\; =\; mathcal\{H\}(\backslash boldsymbol\{B\}).$

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

:$egin\{array\}\{rrcl\}\; \backslash boldsymbol\{sigma\}^*\; =\; mathcal\{H\}(\backslash boldsymbol\{B\}^*)\; \backslash Rightarrow\; \backslash boldsymbol\{R\}cdot\; \backslash boldsymbol\{sigma\}cdot\; \backslash boldsymbol\{R\}^T\; =\; mathcal\{H\}(\backslash boldsymbol\{F\}^*cdot(\backslash boldsymbol\{F\}^*)^T)\; \backslash Rightarrow\; \backslash boldsymbol\{R\}cdot\; mathcal\{H\}(\backslash boldsymbol\{B\})\; cdot\backslash boldsymbol\{R\}^T\; =\; mathcal\{H\}(\backslash boldsymbol\{R\}cdot\backslash boldsymbol\{F\}cdot\backslash boldsymbol\{F\}^Tcdot\backslash boldsymbol\{R\}^T)\; \backslash Rightarrow\; \backslash boldsymbol\{R\}cdot\; mathcal\{H\}(\backslash boldsymbol\{B\})cdot\; \backslash boldsymbol\{R\}^T\; =\; mathcal\{H\}(\backslash boldsymbol\{R\}cdot\backslash boldsymbol\{B\}cdot\backslash boldsymbol\{R\}^T).\; end\{array\}$

A

constitutive equation that respects the above condition is said to beisotropic .**References****See also***

Hyperelastic material

*Objectivity (frame invariance)

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