Constitutive Equations of State

Consider that at time t=0, a flexible, linear polymer is suddenly deformed homogeneously and the deformation is kept constant thereafter. If we assume that each slip-link point changes its position affinely, Eq. (4.137) can be rewritten (Doi and Edwards 1978b) as  

we show, with an example of shear deformation, the difference in the prediction of stress relaxation between the use of Q(E) and Q (E). 

Let us consider shear flow, for which the deformation tensor E is given by 

It can be shown from Eq. (4.150) that for small values of y, becomes 

Using lAA, Doi and Edwards (1978b) derived the following constitutive equation  

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To investigate the thermodynamic properties of ideal gases, one may insert the constitutional equation of state into... 

We begin by inserting the constitutional equation of state in the caloric equation of state, Eq. (1.13.16) this leads to the important finding that (dE/dV)T = 0, regarded as a second criterion to be imposed on ideal gases. Thus, the energy of an ideal gas depends solely on temperature. As a result we now write out the differential energy in the abbreviated form dE = dE/dT)y dT, whence... 

There are a number of models which can be used to predict the effect that absorbed diluents, in particular water, have on a cured polymeric resin. One of the most powerful of these is Group Interaction Modelling (GIM), a continuum-type model with a set of versatile input parameters based on the number and type of chemical functional groups present in the network. This allows the complex chemistty of amine-cured epoxy resins to be catered for whilst retaining the speed afforded by using a set of linked constitutive equations of state for property prediction. 

In the next chapter, we discuss the use of various forms of time, derivatives in the formulation of constitutive equations of state. There are no clear guidelines that can be applied to determine what form of time derivative might be the best or most appropriate. Therefore, one should determine the usefulness of a given relation based on its ability to predict the experimentally observed rheological behavior of a given material. 

The jump conditions must be satisfied by a steady compression wave, but cannot be used by themselves to predict the behavior of a specific material under shock loading. For that, another equation is needed to independently relate pressure (more generally, the normal stress) to the density (or strain). This equation is a property of the material itself, and every material has its own unique description. When the material behind the shock wave is a uniform, equilibrium state, the equation that is used is the material s thermodynamic equation of state. A more general expression, which can include time-dependent and nonequilibrium behavior, is called the constitutive equation. 

Constitutive relation An equation that relates the initial state to the final state of a material undergoing shock compression. This equation is a property of the material and distinguishes one material from another. In general it can be rate-dependent. It is combined with the jump conditions to yield the Hugoniot curve which is also material-dependent. The equation of state of a material is a constitutive equation for which the initial and final states are in thermodynamic equilibrium, and there are no rate-dependent variables. 

Static and dynamic property The uses of these foams or porous solids are used in a variety of applications such as energy absorbers in addition to buoyant products. Properties of these materials such as a compressive constitutive law or equation of state is needed in the calculation of the dynamic response of the material to suddenly applied loads. Static testing to provide such data is appealing because of its simplicity, however, the importance of rate effects cannot be determined by this one method alone. Therefore, additional but numerically limited elevated strain-rate tests must be run for this purpose. 

Even if satisfactory equations of state and constitutive equations can be developed for complex fluids, large-scale computation will still be required to predict flow fields and stress distributions in complex fluids in vessels with complicated geometries. A major obstacle is that even simple equations of state that have been proposed for fluids do not always converge to a solution. It is not known whether this difficulty stems from the oversimplified nature of the equatiorrs, from problems with ntrmerical mathematics, or from the absence of a lamirrar steady-state solution to the eqrratiorrs. 

The intrinsic constitutive laws (equations of state) are those of each phase. The external constitutive laws are four transfer laws at the walls (friction and mass transfer for each phase) and three interfacial transfer laws (mass, momentum, energy). The set of six conservation equations in the complete model can be written in equivalent form ... 

The mathematical model describing the two-phase dynamic system consists of modeling of the flow and description of its boundary conditions. The description of the flow is based on the conservation equations as well as constitutive laws. The latter define the properties of the system with a certain degree of idealization, simplification, or empiricism, such as equation of state, steam table, friction, and heat transfer correlations (see Sec. 3.4). A typical set of six conservation equations is discussed by Boure (1975), together with the number and nature of the necessary constitutive laws. With only a few general assumptions, these equations can be written, for a one-dimensional (z) flow of constant cross section, without injection or suction at the wall, as follows. 

The constitutive equation or equation of state for our Newtonian fluid is... 

Since it is relatively easy to fit experimental adsorption data to a theoretical equation, there is some controversy as to what constitutes a satisfactory description of adsorption. From a practical point of view, any theory that permits the amount of material adsorbed to be related to the specific surface area of the adsorbent and that correctly predicts how this adsorption varies with temperature may be regarded as a success. From a theoretical point of view, what is desired is to describe adsorption in terms of molecular properties, particularly in terms of an equation of state for the adsorbed material, where the latter is regarded as a two-dimensional state of matter. 

By applying the fundamental physical properties of conservation of mass, energy and momentum across the shockwave, together with the equation of state for the explosive composition (which describes the way its pressure, temperature, volume and composition affect one another) it can be shown that the velocity of detonation is determined by the material constituting the explosive and the material s velocity. 

A means to find or estimate required constitutive properties that appear in the conservation equations. These can include equations of state, thermodynamic and transport properties, and chemical reaction rates. 

The constitutive relations along with the conservation equations give the basic equations of fluid mechanics, which are a set of five nonlinear partial differential equations involving the seven variables, p, g,e, P, and T. Because five equations [Eqs. (1), (2), (3), (5), and (6)] cannot determine seven quantities, the equations are closed by expressing any two variables of the set (p,e,P,T) in terms of the other two remaining variables. This is done by using the assumption of local equilibrium and thermodynamic equations of state. 

In combination with the flux equations and appropriate equations of state, (5) and (6) constitute the full set of model equations accounting for chemical osmosis in groundwater flow and solute transport. 

These mixing rules, when joined with the Redlich-Kwong equation of state, will constitute the Redlich-Kwong equation of state for mixtures that is consistent with the statistical mechanical basis of the van der Waals mixing rules. 

This equation constitutes the central equation of state-of-the-art modeling of pattern formation in electrochemistry. Its integration requires the knowledge of the electric field component normal to the interface, which is obtained from the solution of Laplace s equation. 

The fundamental rheological characterization of a material requires the experimental determination of a constitutive equation (a rheological equation of state) that mathematically relates stress and strain, or stress and strain rate. The constants in the constitutive equation are the rheological properties of the material. 

Closure of such differential equations requires the definitions of both constitutive relations for hydrodynamical functions and also kinetic relations for the chemistry. These functions are specified by recourse both to theoretical considerations and to rheological measurements of fluidization. We introduce the ideal gas approximation to specify the gas phase pressure and a caloric equation-of-state to relate the gas phase internal energy to both the temperature and the gas phase composition. It is assumed that the gas and solid phases are in local thermodynamic equilibrium so that they have the same local temperature. 

In Eqs. (6) and (7) e represents the internal energy per unit mas, q the heat flux vector due to molecular transport, Sh the volumetric heat production rate, ta, the mass fraction of species i, Ji the mass flux vector of species i due to molecular transport, and 5, the net production rate of species i per unit volume. In many chemical engineering applications the viscous dissipation term (—t Vm) appearing in Eq. (6) can safely be neglected. For closure of the above set of equations, an equation of state for the density p and constitutive equations for the viscous stress tensor r, the heat flux vector q, and the mass flux vector 7, are required. In the absence of detailed knowledge on the true rheology of the fluid, Newtonian behavior is often assumed. Thus, for t the following expression is used ... 

The rheological constitutive equation of the Rouse model is that of an upper-convected Maxwell model, with the consequence that steady-state elongational flow only exists for strain rates lower than l/(2A,i). The steady-state elongational wscosity depends then on strain rate ... 

The relationships between stress and strain, and the influence of time on them are generally described by constitutive equations or rheological equations of state (Ferry, 1980). When the strains are relatively small, that is, in the linear range, the constitutive... 

In a number of areas of modeling phase equilibria, the cubic equation of state (EOS) provided equal or even better results than the traditional approach based on the activity coefficient concept. In fact, for certain t5rpes of phase equilibria, the EOS is the only method that provided acceptable results. The solubility of solids in a supercritical fluid (SCF) constitutes such a case. For the solubility of a solid in a SCF [SCF (1) + solid solute (2)], one can write the well-known relation ... 

The constitutive equation of elasticity is represented by the Hookian spring (Fig. 16). Hook s law states that the stress is proportional to the strain... 

In order to use these general momentum conservation equations to calculate the velocity field, it is necessary to express viscous stress terms in terms of the velocity field. The equations which relate the stress tensor to the motion of the continuous fluid are called constitutive equations or rheological equations of state. Although the governing momentum conservation equations are valid for all fluids, the constitutive equations, in general, vary from one fluid material to another and possibly also from one type of flow to another. Fluids, which follow Newton s law of viscosity (although it is referred to as a law, it is just an empirical proposition) are called Newtonian fluids. For such fluids, the viscous stress at a point is linearly dependent on the rates of strain (deformation) of the fluid. With this assumption, a general deformation law which relates stress tensor and velocity components can be written ... 

In order to be able to solve the energy equation (3.71) or (3.75), some constitutive equations are required. We will now consider equation (3.71) for pure substances. This necessitates the introduction of the caloric equation of state u = u( t), v). By differentiation we obtain... 

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