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Abstract A rheological equation for viscoelastic fluids is derived as an expansion of Noll's functional for sample fluids. The present expansion is developed in terms of the deformation rate referred to a co-rotational material frame. The resulting series of approximations for stress differs significantly from those proposed previously by Green and Rivlin and by Coleman and Noll. Applications to both simple-extensional and viscometric flow are considered. For viscometric flows the present model exhibits non-Newtonian effects similar to those observed in real fluids, with relatively few terms in the functional expansion. The first term in the expansion yields a "first-order" fluid similar to that proposed by Pao. Expressions are also given for the viscometric material functions of the present "third-order" fluid. As with higher-order fluids, these functions are Fourier integrals of certain hereditary or memory functions for the fluid. Introduction In a previous paper it was shown how one could construct an inverse for the co-rotational or Jaumann derivative. This is operation, termed "Jaumann integration" was then used in a somewhat intuitive manner to derive a rheological model for viscoelastic fluids, by means of a Boltzmann superposition integral. As discussed in the above paper, the resulting fluid model is essentially a generalization of the "Quasilinear" model proposed earlier by Oldroyd. Moreover, the rheological equation bears some resemblance to certain others which have been derived from molecular considerations. However, while the model was shown to exhibit some of the features of real-fluid behavior, including shear-dependent viscosity and normal-stress functions, both related to the linear-viscoelastic "memory" function, it was concluded that the model was in general not sufficient to describe nonlinear viscoelastic effects in their entirety. The present work is first of all an attempt to provide the above model with a somewhat more rigorous continuum-mechanical basis, by deriving it from Noll's general theory of materials with a "memory". More importantly, the intent of the present paper is to arrive systematically at certain improvements on the "quasi-linear" model previously proposed and, then, to relate the resulting fluid models to the "higher-order" fluids proposed by Green and Rivlin and by Coleman and Noll. Finally, it is hoped that the present work might suggest other types of functional expansions and perhaps stimulate investigation of the convergence of such expansions, as it relates to the physical properties of fluids. FORMULATION OF THE RHEOLOGICAL EQUATION The Material Matrizant For the present work it will be convenient to recall the notion of a material matrizant, which was introduced in an earlier paper. Thus, for a given second-order tensor field(x,t) and a material velocity field the material matrizant of A relative to v is a tensor field Mtt [A] which is to satisfy ........................................(2.1) where I is the unit [metric] tensor or idem factor and D/Dt is the material derivative: ........................................(2.2)

knowing the effective elastic and damping properties of the materials over a sufficiently large frequency range the speed and attenuation of the pulses can be predicted using the linear theory of viscoelasticity. Material dispersion appears to be relatively insignificant in glass-epoxy and boron-epoxy samples below 0.25 mHz, although further research in this area is needed. Ultrasonic methods have been shown to provide a useful way of studying dispersion in composite materials. In this paper attention has been restricted to disturbances propagating in the direction of the reinforcing filaments. A complete characterization of the dynamic behavior of unidirectional fiber-reinforced materials requires a knowledge of the remaining four dynamic moduli (assuming transverse isotropy). Hence additional testing is needed.

Many experimental investigations have shown that plastics in general, as well as elastomers and their solutions, do not behave like Newtonian liquids. They both have viscosity and elasticity, the combination of which, according to the Maxwell postulate, causes their properties to be dependent on time, and leads us to call them viscoelastic bodies. This dependence does not exist, either for the Newtonian liquid whose mechanical properties are determined by its viscosity, or for the elastic body whose properties are determined by its elasticity. The time element introduces a peculiar difficulty in measuring the mechanical properties of these substances. Whereas the well-known Maxwell body describes this time dependence in simple mathematical expressions, it does not apply to the experimental results. The properties of viscoelastic bodies are not described by a Maxwellian body, but rather by a distribution of an infinite number of Maxwellian bodies, each having a so-called relaxation time. The existence of this infinite distribution calls imperatively for an investigation of the properties of these viscoelastic bodies in as wide a range of frequencies or times as possible in order to define them. Most investigators have used one method in a range of frequencies no wider than about 1 to 100. This range can be extended by using the observation of Ferry that the properties of viscoelastic bodies investigated at different temperatures, in the case of dynamic investigations with sinusoidal vibrations, seem to be described by shifting one standard curve on the frequency axis. One can superimpose measurements at different temperatures and extend the range of frequencies to about 1 to 1,000,000. But using Ferry's method of reduced variables necessitates some assumptions that, though very probable, have as yet not been proven exactly by experiments. It therefore seemed important to have a method for investigating the mechanical properties of viscoelastic bodies in a frequency range of about 1 to 1,000,000 under isothermal conditions. This paper describes a basic method and the results obtained for several elastomers.

Based upon the phenomenological theory of non-linear responses of many variable systems (K. Okano and O. Nakada, J. Phys. Soc. Japan, 16, 2071 (1961)) the non-linear constitutive equations for isotropic viscoelastic materials are presented.Let σ(t) denote the stress tensor at an instant of time t, and D(t) be the displacement gradient ∇S (S being the displacement vector) or velocity gradient ∇V (V being the velocity vector) according as the material concerned is a viscoelastic solid or a viscoelastic liquid, then we have (K. Okono, Japanese J. Appl. Phys., 1, 302 (1961)) the following non-linear constitutive equation (up to the second order terms in D) for an isotropic viscoelastic material:σ(t)=∫t-∞[a(1)(t-τ)D(0)(τ)+b(1)(t-τ)D(2)(τ)]dτ+∫t-∞∫t-∞[a(2)(t-τ1, t-τ2)D(0)(τ1): D(0)(τ2)I+b(2)(t-τ1, t-τ2)D(2)(τ1): D(2)(τ2)I+c(2)(t-τ1, t-τ2)D(0)(τ1)·D(2)(τ2)+d(2)(t-τ1, t-τ2){1/2D(2)(τ1)·D(2)(τ2)+1/2D(2)(τ2)·D(2)(τ1)-1/3D(2)(τ1): D(2)(τ2)I}+e(2)(t-τ1, t-τ2){1/2D(1)(τ1)·D(2)(τ2)-1/2D(2)(τ2)·D(1)(τ1)}]dτ1dτ2+higher order terms.In the above equation I is the unit tensor andD(0)≡1/3∇·SI or 1/3∇·VID(1)≡1/2(D-D)D(2)≡1/2(D+D)-D(0)and a(1)(t), b(1)(t), a(2)(t1, t2), ect. are the scalar material functions characterizing the viscoelastic response of the system concerned. The third order terms in D are given in the text. (eq. 2.5).If the material concerned is incompressible the terms on the right hand of the above constitutive equation which are connected with a dilatational deformation (the terms containing a(1), a(2), b(2), c(2)) should be replaced by an indeterminate hydrostatic pressure: -pI