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).[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 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.
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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 the deformation gradient:
where 
is the Cauchy stress and 
is the deformation gradient. Note that the function 
depends on the choice of reference configuration.
The condition of material objectivity
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 σ and the deformation gradient F are objective
quantities, one can write:
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.
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 h which will ensure the principle of material frame-indifference, one can write:
A constitutive equation that respects the above condition is said to be isotropic.
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