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Today, even though the Clausius-Duhem inequality is widely considered to be of central importance in the subject of continuum thermomechanics, it is also believed to be a somewhat special interpretation of a more fundamental (second) law of thermodynamics. In this work, which is concerned with the relation between thermodynamics and stability for a class of non-Newtonian incompressible fluids of the differential type, we find it essential to introduce the additional thermodynamical restriction that the Helmholtz free energy density be at a minimum value when the fluid is locally at rest. As a background to our main considerations we begin by introducing the general class of Rivlin-Ericksen fluids of complexity n and obtain, for this class, a preliminary set of thermodynamical constitutive restrictions. We then give detailed attention to the special case of fluids of grade 3 and arrive at fundamental inequalities which restrict its (temperature dependent) material moduli. When the moduli are taken to be constant we find that these inequalities require that a body of such a fluid be stable in the sense that its total kinetic energy must tend to zero in time, no matter what its previous mechanical and thermal fields, provided it is both mechanically isolated and immersed in a thermally passive environment at constant temperature from some finite time onward. When the material constants of a fluid of grade 3 are such that the Clausius-Duhem inequality is satisfied but the free energy is not at a minimum in equilibrium, we show that for a broad class of reasonably posed problems the flows are necessarily asymptotically unbounded. Finally, we determine the stability character of non-trivial base flows for fluids of grade 3 with constant material moduli, and establish a uniqueness theorem for the initial-boundary value problem and a uniqueness theorem for problems involving sufficiently slow steady flows.

Constitutive modeling within peridynamic theory considers the collective deformation at each time of all the material within a δ-neighborhood of any point of a peridynamic body. The assignment of the parameter δ, called the horizon, is treated as a material property. The difference displacement quotient field in this neighborhood, rather than the extension scalar field, is used to generate a three-dimensional state-based linearly elastic peridynamic theory. This yields an enhanced interpretation of the kinematics between bonds that includes both length and relative angle changes. A free energy function for a linearly elastic isotropic peridynamic material that contains four material constants is proposed as a model, and it is used to obtain the force vector state and the associated modulus state for this material. These states are analogous to, respectively, the stress field and the fourth-order elasticity tensor in classical linear theory. In the limit of small horizon, we find that only three of the four peridynamic material constants are related to the classical elastic coefficients of an isotropic linear elastic material, with one of the three constants being arbitrary. The fourth peridynamic material constant, which accounts for the coupling effect of both bond length and relative angle change, has no effect on the limit, but remains a part of the peridynamic model. The determination of the two undetermined constants is the subject of future investigation. Peridynamic models proposed elsewhere in the literature depend on the deformation state through its dilatational and deviatoric parts and contain only two peridynamic material constants, in analogy to the classical linear elasticity theory. Observe from above that our model depends on both length and relative angle changes, as in classical linear theory, but, otherwise, is not limited to having only two material constants. In addition, our model corresponds to a nonordinary material, which represents a substantial break with classical models.

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