  In this short note we develop a constitutive relation that is linear in both
the Cauchy stress and the linearized strain, by linearizing implicit
constitutive relations between the stress and the deformation gradient that
have been put into place to describe the response of elastic bodies (see
\cite{rajagopal2003implicit}), by assuming that the displacement gradient is
small. These implicit equations include the classical linearized elastic
constitutive approximation as well as constitutive relations that imply
limiting strain, as special subclasses.
