Constitutive equation differential form

Boggs combined, in a mathematically elegant approach, a constitutive equation, including normal stress terms with equations of motion to form differential equations similar to the Navier-Stokes equations. He found that viscoelasticity had a destabiliz-... 

There are many ways of writing equations that represent transport of mass, heat, and fluids trough a system, and the constitutive equations that model the behavior of the material under consideration. Within this book, tensor notation, Einstein notation, and the expanded differential form are considered. In the literature, many authors use their own variation of writing these equations. The notation commonly used in the polymer processing literature is used throughout this textbook. To familiarize the reader with the various notations, some common operations are presented in the following section. 

Equations (9.6) and (9.7) make up the simplest set of constitutive equations for dilute polymer solutions, which, after excluding the internal variables j, can be written in the form of a differential equation that has the form of the two-constant contra-variant equation investigated by Oldroyd (1950) (Section 8.6). 

One may use the linear viscoelastic data as a pure rheological characterization, and relate the viscoelastic parameters to some processing or final properties of the material inder study. Furthermore, linear viscoelasticity and nonlinear viscoelasticity are not different fields that would be disconnected in most cases, a linear viscoelastic function (relaxation fimction, memory function or distribution of relaxation times) is used as the kernel of non linear constitutive equations, either of the differential or integral form. That means that if we could define a general nonlinear constitutive equation that would work for all flexible chains, the knowledge of a single linear viscoelastic function would lead to all rheological properties. 

These constitutive equations differ in their mathematical form the Wagner equation is an integral equation whereas the Phan Thien Tanner model is a differential one. 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

CONSTITUTIVE EQUATIONS IN DIFFERENTIAL FORM FOR MULTIAXIAL TENSION STATES... 

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... 

The above equations represent differential forms properly adapted to describe electromagnetic phenomena in terms of applied fields. In what follows we introduce constitutive relations, assuming that V and A4 are collinear with q and T-Lq respectively, namely... 

When the temperature is not constant, the bulk heat transfer equation complements the system and involves Equations 5.240, 5.241, and 5.276. The heat transfer equation is a special case of the energy balance equation. It should be noted that more than 20 various forms of the overall differential energy balance for multicomponent systems are available in the literature." " The corresponding boundary condition can be obtained as an interfacial energy balance." - Based on the derivation of the buUc and interfaciaT entropy inequalities (using the Onsager theory), various constitutive equations for the thermodynamic mass, heat, and stress fluxes have been obtained. 

It is shown that the development of the equations governing THM processes in elastic media with double porosity can be approached in a systematic manner where all the constitutive equations governing deformability, fluid flow and heat transfer are combined with the relevant conservation laws. The double porosity nature of the medium requires the introduction of dependent variables applicable to the deformable solid, and the fluid phases in the two void spaces. The governing partial differential equations are linear in view of the linearized forms of the constitutive assumptions invoked in the formulations. The linearity of these governing equations makes them amenable to solution through conventional mathematical techniques applicable to the study of initial boundary value problems in mathematical physics (Selvadurai, 2000). Such solutions should serve as benchmarks for appropriate computational developments. 

Summary. A procedure really specific for the rational thermodynamics is introduced in this section in the form of several principles put forward to derive the thermodynamically consistent constitutive equations. In their most general form, the constitutive equations were proposed as functions (3.118) on the basis of the principles of determinism, local action, differential memory, and equipresence. They were further reduced to the form (3.121) considering the same material throughout the body and applying the principle of objectivity. Because of our interest in fluids only, the constitutive equations were further modified to this material type by means of the principle of material symmetry giving the final form (3.127). Two special types of fluid were defined by (3.129) and (3.130). 

Because fields (4.121) are controlled from the outside (of the mixture), constitutive equations are relations between (4.119), (4.120) according to the constitutive principle of determinism their independent variables form the thermokinetic process (4.119) giving their values as responses (4.120). For simplicity, we restrict to recent past and nearest surroundings of the considered response by constitutive principles of differential memory and local action. Constitutive equations for responses (4.120) are then functions of the following values of thermokinetic process (4.119) and their (time and space) derivatives taken in a considered instant and place of response (in referential description introduced in Sect.4.1 similarly as in Sect. 3.1, i.e. as (4.1), Py = PyOi-y, t), T = T(Ky, t)), namely... 

Equation 1 is the continuum equilibrium condition, Eq. 2 are the constitutive equations, which specify the material behavior, and Eq. 3 are the kinematics, the displacement-deformation conditions. The forming machine defines the boundary conditions for this set of partial differential equations. It can be seen without going into details that those equations are essentially grade two in the displacements, which means that boundary conditions are in the displacements or in the first derivative of displacements. Eirst derivatives of displacements are in all constitutive equations connected to stress at least due to elasticity, which is common for all materials. Erom these two types of press machines can be derived, namely, path-driven machines, where boundary conditions for the displacements are prescribed, and force-driven machines, where stress boundary conditions are prescribed, which are integrated to the press force. Erom this it follows also that despite the possibilities of servo presses to operate under different modes of the drive, path, force, or energy, nothing really new is added, because the drive can only introduce either boundary condition at a time. 

The dependence of the stress on the strain-deformation history of macro-molecular liquids can be incorporated in two ways. The stress constitutive equation can be formulated as a differential equation, in which the extra stress r is the solution of an equation that is typically of the general form... 

The expression (3.3) is known as the Gibbs Jundamental form of the system (it represents a special Pfaffian form, as it is called in differential calculus). Every process that can be performed by the system must satisfy this differential form. On the left-hand side is the total differential of energy (because energy is a state function ) the energy forms located on the right-hand side of the equation generally do not constitute total differentials, although they can often be summed up into total differentials (see later). 

There are numerous other constitutive equations of both differential and integral type for polymer melts, and some do a better job of matching data from a variety of experiments than does the PTT equation. The overall structure of the differential equations is usually of the form employed here The total stress is a sum of individual stress modes, each associated with one term in the linear viscoelastic spectrum, and there is an invariant derivative similar in structure to the one in the PTT equation, but with different quadratic nonlinearities in t and Vv. The Giesekus model, for example, which is also widely used, has the following form ... 

The integral in (1.2.5) is really a special case (where e t) is differentiable) of a Stieltjes integral. Gurtin and Sternberg (1962) base their rigorous formulation of Linear Viscoelasticity on constitutive equations which have this Stieltjes form. We adopt a convenient notation of theirs, and write (1.2.5) as... 

The traditional discussion of mechanical (spring and dashpot) models and the related topic of differential forms of the constitutive equations will not be included here, but are treated extensively in several older references, Gross (1953), Ferry (1970), Bland (1960) for example. See also Nowacki (1965), Flugge (1967) and Lockett (1972). A consistent development of the theory is possible without these concepts. However, they do provide insights into the nature of viscoelastic behaviour and physically motivate exponential decay models. 

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