Advanced elasticity/Cauchy elastic materials

A Cauchy-elastic material is one in which the Cauchy stress at each material point is determined only by the current state of w:deformation (with respect to an arbitrary reference configuration).[1] 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 w: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 w:constitutive equation for the Cauchy stress tensor can be formulated based on the w:deformation gradient:

${\displaystyle \ {\boldsymbol {\sigma }}={\mathcal {G}}({\boldsymbol {F}})}$

where ${\displaystyle {\boldsymbol {\sigma }}}$ is the Cauchy stress and ${\displaystyle {\boldsymbol {F}}}$ is the deformation gradient. Note that the function ${\displaystyle {\mathcal {G}}}$ depends on the choice of reference configuration.

The condition of material objectivity requires that the constitutive relation ${\displaystyle {\mathcal {G}}}$ should not change when the location of the observer changes. Therefore the w:constitutive equation for another arbitrary observer can be written ${\displaystyle {\boldsymbol {\sigma }}^{*}={\mathcal {G}}({\boldsymbol {F}}^{*})}$. Knowing that the Cauchy stress tensor ${\displaystyle \sigma }$ and the w:deformation gradient ${\displaystyle F}$ are objective quantities, one can write:

{\displaystyle {\begin{aligned}&{\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{aligned}}}

where ${\displaystyle {\boldsymbol {R}}}$ is a proper orthogonal tensor.

The above is a condition that the constitutive law ${\displaystyle {\mathcal {G}}}$ 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 w:deformation gradient to the first or second w:Piola-Kirchhoff stress tensor.

Isotropic Cauchy-elastic materials

For an isotropic material the Cauchy stress tensor ${\displaystyle {\boldsymbol {\sigma }}}$ can be expressed as a function of the left Cauchy-Green tensor ${\displaystyle {\boldsymbol {B}}={\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T}}$. The w:constitutive equation may then be written:

${\displaystyle \ {\boldsymbol {\sigma }}={\mathcal {H}}({\boldsymbol {B}}).}$

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

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

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

References

1. R. W. Ogden, 1984, Non-linear Elastic Deformations, Dover, pp. 175-204.