In physics, a Cauchy-elastic material is one in which the stress at each point is determined only by the current state of deformation with respect to an arbitrary reference configuration.
The definition also implies that the constitutive equations are spatially local; that is, the stress is only affected by the state of deformation in an infinitesimal neighborhood of the point in question, without regard for the deformation or motion of the rest of the material.
However, many elastic materials of practical interest, such as steel, plastic, wood and concrete, can often be assumed to be Cauchy-elastic for the purposes of stress analysis.
Formally, a material is said to be Cauchy-elastic if the Cauchy stress tensor
alone: This definition assumes that the effect of temperature can be ignored, and the body is homogeneous.
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.
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: A constitutive equation that respects the above condition is said to be isotropic.
Therefore a Cauchy elastic material in general has a non-conservative structure, and the stress cannot necessarily be derived from a scalar "elastic potential" function.
Materials that are conservative in this sense are called hyperelastic or "Green-elastic".