Pergamon Press Rice University, Houston, Texas , U. AlWract-This work concerns the formulation of a thermomechanical theory of a mixture where each constituent has its own temperature field.
The theory also contains the effects of nonlinear elasticity, non- linear heat conduction, nonlinear viscosity and diffusion. A linearized version of the general theory is also presented.
The thermo- mechanical balance equations for a mixture of general materials were first formulated in a major work by Truesdell [l]. Truesdell proposed equations of balance of mass, momentum and energy in a form which is generally accepted today. Kelly [2] followed a suggestion made later by Truesdell and Toupin [3, sect.
In a sequel to their first paper, Eringen and Ingram[6] investigated the restrictions on the constitutive equations for a mixture of reacting gases resulting from an entropy inequality for each constituent. They allowed each constituent to have its own temperature. Green and Naghdi[5] also proposed an entropy inequality for each constituent which allowed for multiple temperature. Dunwoody and Miiller[7] were the first to investigate a multiple temperature mixture theory in which only an entropy inequality for the mixture was proposed.
They con- sidered a mixture of two chemically reacting ideal gases. Steel[9] applied the formulation developed by Green and Naghdi to a mixture of two elastic solids with distinct temperatures.
However, they considered constitutive equations for a mixture of elastic materials with but one temperature. In reference [ 1 l] we adopted the entropy inequality proposed by Bowen and Wiese and investigated theories of heat conduction in a mixture of rigid bodies where each body has its own temperature field. The effects of chemical reactions have been omitted here.
These effects have been investigated for special mixtures in two recent papers by Bowen [ 12, I In section 3 constitutive equations which define the mixture are stated.
In section 4 we deduce the restrictions imposed on the constitutive equations by the entropy inequality. In section 5 the restrictions imposed on the constitutive equations by the axiom of material frame-indifference and symmetry considerations are considered. In section 6 linearized constitutive equations for a mixture of fluids and isotropic solids are presented. The identity linear transformation will be denoted by I.
Component indices will refer to a fixed time-independent rectangular Cartesian coordinate system with basis e,, q, e3. Spatial coordinate indices will be denoted by Latin minuscules. Material coordinate indices will be denoted by Latin majuscules. The gradient with respect to spatial co- ordinates will be denoted by grad, and the gradient with respect to material coordinates by V. The divergence with respect to spatial coordinates will be denoted by div.
A quantity corresponding to a particular constituent of the mixture will be identified by placing a Latin minuscule directly under the symbol for the quantity: F, r, The summation convention will apply only to summations over coordinate indices. Summa- tions over constituents will always be indicated by the summation symbol. The com- plete contraction operator will be denoted by the symbol C[lO, section The symbol Y,, where p is a positive integer greater than or equal to 2, will denote the set of tensors of order p symmetrical in the last p - 1 indices.
It is for this reason that we have been intentionally careless about confusing functions and their values. Unless some possibility for confusion exists, we have repeatedly used the same symbol for a function and its value and, in some cases, for several functions and their common value. Each body 99 is considered to be a set with a structure prkcribed by Noll[ An element of 9I is deioted by X.
A motion of i is a one parameter family of Jonfigurations g where t is the time. The position of the particle 5 at the time t is given by 2. The equations of balance of mass, linear momentum, moment of momentum and energy for the mixture are postulated to be 2. The entropy density of the mixture is defined by 2. As we mentioned in the Introduction, we are interested in a theory of a mixture of bodies capable of nonlinear thermoelastic effects, nonlinear heat conduction, non- linear viscous effects, diffusion and the effects of multiple temperatures.
In this section the constitutive equations for this mixture are proposed. Following the nomenclature used by Coleman and Noll[lS], a? It follows that 2. We shall assume, for simplicity, that the densities p:O do not depend on the positions X and that, therefore, they are known constants. Also the component functions associated with f will be assigned the same symbol as the value in question.
For example, we shall write from 3. The choice of independent variables in 3. See, for example, Bowen and Wiese[lO, sect. It was Miiller [8] who stressed for the first time the importance of a dependence of the constitutive functions on gradients of the deformation. Without this dependence, expressions for the partial stresses are obtained which are known to be too special. In this context see Bowen and Wiese[lO, sect. An admissible thermodynamic process for the mixture defined by 3. By an obvious generalization of the theorem by Coleman and No11 [ 15, sect.
Further generalizing the logic of Coleman and Noll[15, sect. We can easily include as part of the defining constitutive assumptions the restrictions imposed by 2. The interesting restrictions will arise from the inequality 2. We shall obtain this restriction by using a standard technique originally formulated by Coleman and Noll[15, sect. Substituting 3. Equation 4. By an argument similar to that Lf Coleman and Noll[15,sect.
Equations 4. As a result of 4. Further conditions on A may be obtained from equations 4. We shah see in the next section that the restrictions imposed by the axiom of material frame-indifference imply that no loss in generality is incurred by adopting the definition 4. It follows then from 4. How- ever, in section 6 we shall illustrate the results implied by 4. Preliminary Definitions A. Lin V is the set of all endomorphisms of V,, and we call elements of Lin V tensors.
The set of all symmetric tensors will be denoted by Sym V, the set of all skew tensors by Sk V. The inner product of V induces an inner product on Lin V, which will also be denoted by "-", given by S. Let f be a smooth, i. Mixtures with Diffusion Given A c 8, we denote the boundary, the interior and the closure of A by dA, int A, and cl A, respectively.
If A,. The structure we require on Y is given by the following axioms. We shall reserve the term surface for the relative closure of an oriented class C 1 two-dimensional differentiable manifold or the countable union of such closed manifolds. The boundary of a standard region is taken to be oriented in the positive sense with respect to that region, i.
A surface contained in the sense of set inclusion in another surface is a positive segment of that surface if it has the same orientation; if it has the opposite orientation it is called a negative segment. We define in an obvious manner the positive and negative normal vector to a surface.
Miscellaneous We use the expressions a. The end of a p r o o f will be denoted by the symbol II. The Power Function. Mechanical Theory 1. In our theory P, A,A,. Another appropriate name might be infinitesimal displace- ments. Mixtures with Diffusion We now list all the assumptions we make on the power function in this chapter.
Action and reaction law. Invariance of the total power for a constituent. Contact action of the power. Physically this assumption seems reasonable. It essentially says that only the motion of component i matters, and that between components there is no interaction at a distance. The next theorem will give us a representation for the power. By assumption A l a Pu A, c g ,. This is sufficient for most of the applications of mixture theory.
A Theorems 2 and 3 suggest the definitions for force and moment. MO A, cg has a similar interpretation. Now we are going to state and prove some well known results about forces and moments in this new framework. We do not express explicitly the obvious quantifiers. We present a proof only for Mi; the other case is handled similarly.
It is an immediate consequence of the invariance of the total power for a con- stituent. II III. The Balance Equations In the preceding section we obtained some very general results about balance of forces and moments using very weak assumptions on the power function. However, what we really want is to reduce the theory to an initial-boundary value problem which could be used to test the theory.
The appropriateness of these assumptions, in our case, must be verified a posteriori. Our first set of assumptions- R 1 to R 3 - allows us to represent the power as an integral of a density over an area or a volume. The densities appear to depend on the subbodies, but we show that in fact they do not; that they are the same for all subbodies in the case of volume densities, and that the area densities reduce to functions of surfaces.
However, the densities we get are only integrable functions and they are defined uniquely only almost everywhere with respect to the appropriate measure - a r e a or volume.
To establish existence of a partial stress tensor, we need some more assump- tions- C 1 and C2. These finally give us the local form of the balance equations. Also, we establish boundary value conditions appropriate to these equations. To conclude the section, we show how the equations of balance for the constituents of the mixture combine to give a global balance equation in a very natural way. Mixtures with Diffusion ij R2.
Let A be a subbody. The following proposition, which gives a representation for the power is an immediate consequence of our assumptions R 1-R 3. A similar remark holds for the other densities.
The assumptions A 1-A 5 allow us to simplify this. We shall show that the volume densities b i. However, we first need a technical lemma. If A and J be two subbodies such that.
Taking a sequence of open sets G, decreasing monotonically to W, we get blj -big 'fa. Gn-W This inequality and the monotone convergence theorem imply that I bl;'- big. II However, the last proposition tells us more. The next corollary is an immediate consequence of the proof of the last proposition. By the same technique of proof and using the additivity of P -, v , we have the following three propositions.
The next result is a relation between the surface tractions a u and zu' Let A be a subbody containing 5p in its interior and let A c A be another body having 5f as a part of its boundary, i.
Then equation 1 is valid. Choosing judiciously the velocity field, we get the result. This result would be very undesirable, as a simple construction shows us. The physical interpretation of this equation is that the force exerted by the part of component j in d on the part of component i in A is equal in magnitude and in opposite direction to the force exerted by the part or component j in A on the part of component i in cg. Obviously this is not a result to be expected; one even supposes it is wrong.
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