The Simple Fluid Concept

In [1], Green and Rivlin discussed constitutive equations for materials with memory on the basis of a constitutive assumption that the Cauchy stress \( \mathop \sigma \limits_\sim (t) \) at time t is a matrix-valued functional of the history of the deformation gradient matrix relative to the undeformed configuration of the material. In [2] Noll, restricting his discussion to the special case when the material is of the hereditary type, made the constitutive assumption that \( \mathop \sigma \limits_\sim (t) \) is a matrix-valued functional \( \mathop F\limits_\sim \left\{ {\mathop g\limits_\sim (s)} \right\} \) of the history of the deformation gradient matrix \({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{F}}\left\{ {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{g}}\left( s \right)} \right\}\) at lapsed time s relative to an arbitrarily chosen fixed reference configuration. He called materials for which such a constitutive assumption is valid simple materials (see §3). He sought to distinguish between simple solids and simple fluids by defining the latter as materials for which each element of the constitutive functional \( \mathop F\limits_\sim \) is invariant under all unimodular transformations \( \mathop H\limits_\sim \) of the reference configuration: $$ \user1{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{F}} \{ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{g}} (s)\} = {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{F}}\{ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{g}}(s){\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{H}}\} }\user1{.} $$ (1.1)