An axiomatic theory of mixtures with diffusion

Abstract

All previous axiomatic treatments of mixture theory neglected diffusion. In particular, GURTIN & DE LA PENI-tA 1 and OLIVER 2 considered rigid mixtures while WILLIAMS 3 allowed mechanical effects, but without diffusion. The main purpose of this paper is to derive dynamical local equations for the mechanical behavior of a mixture with diffusion; the equations of energy and entropy will be treated in a future paper. 4 With this in mind, I present an axiomatic treatment based on the notion of the power as a linear functional on the space of virtual velocity fields. The norm we use in ~ rules out multipolar interactions, but it is a simple exercise to change the norm in ~ to include multipolar interactions, e.g., to include the velocity gradient. I remark, however, that in this instance one must introduce a normalization condition in order to obtain uniqueness for the force measures. The approach we use, which introduces the concept of force in a very natural manner, also applies without change to the case of a single continuum, regarded as a mixture with only one constituent. I now give a summary of this paper. In Part I I give some preliminary definitions. In Part II the power is introduced. The power will be assumed to be a biadditive, bounded linear functional on 5. With this assumption we are able to represent the power as an integral of the velocity field with respect to a certain measure, which we define to be the force. The total power for a constituent will then be assumed to have a certain invariance property which is a consequence of the principle of objectivity. This assumption will give us the balance of force and moment for a constituent. In Part I I I I introduce some additional assumptions on the force measures #. This, with some smoothness conditions, will give us the existence of a stress tensor for each component of the mixture and the existence of an interaction stress for each pair of distinct components of the mixture. I also derive local balance equations and discuss relevant boundary value problems for the mixture. The equations we get differ from the classical ones, as presented, e.g., by BOWEN. 5