A theory for elastic fluids is developed using a tensor analogous to the Cauchy deformation measure to characterize the microstructure. The resulting mechanical theory is equivalent to that formally proposed by Hand. A free energy function is introduced and the constitutive functions are required to satisfy a dissipation inequality in any possible motion. Sufficient conditions for satisfying this inequality are obtained for constitutive functions that are linear in the rate of deformation. Several molecular theories of suspensions that are of this form appear to be inconsistent with some of these conditions. A particular theory that satisfies the dissipation inequality is analyzed in steady shear and elongational flows. This five‐constant model describes a shear‐thinning fluid with a nonvanishing extinction angle at high shear rates. Yield phenomena exhibited by some suspensions can be incorporated easily here through the material parameters.