Constitutive equation definition

The close fit of the experimental data and the values predicted by the constitutive modified Halpin-Tsai equations I and II (24) and (25), as seen in Fig. 43 (for NR) illustrates the appropriate definition of the IAF. Table 10 also confirms that newly devised equations (24) and (25) provide astounding results because their predictions conform to the experimental data. The introduction of IAF imparts a definitive change to the predicting ability of the constitutive equations for polymer/filler nanocomposites (Fig. 43 Table 10). 

The set of constitutive equations for the dilute polymer solution consists of the definition of the stress tensor (6.16), which is expressed in terms of the second-order moments of co-ordinates, and the set of relaxation equations (2.39) for the moments. The usage of a special notation for the ratio, namely... 

To complement the equations obtained from the application of the conservation principles, it is required to use some equations based on physical, chemical, or electrochemical laws, that model the primary mechanisms by which changes within the process are assumed to occur (rates of the processes, calculation of properties, etc.). These equations are called constitutive equations and include four main categories of equations definition of process variables in terms of physical properties, transport rate, chemical and electrochemical kinetics, and thermodynamic equations. 

Compatibility conditions address the fact that the various components of the strain tensor may not be stated independently since they are all related through the fact that they are derived as gradients of the displacement fields. Using the statement of compatibility given above, the constitutive equations for an isotropic linear elastic solid and the definition of the Airy stress function show that the compatibility condition given above, when written in terms of the Airy stress function, results in the biharmonic equation = 0. 

Newtonian constitutive equation, (2 80), that the normal component of the surface force or stress acting on a fluid element at a point will generally have different values depending on the orientation of the surface. Nevertheless, it is often useful to have available a scalar quantity for a moving fluid that is analogous to static pressure in the sense that it is a measure of the local intensity of squeezing of a fluid element at the point of interest. Thus it is common practice to introduce a mechanical definition of pressure in a moving fluid as... 

The definition (2 85) is a purely mechanical definition of pressure for a moving fluid, and nothing is implied directly of the connection for a moving fluid between p and the ordinary static or thermodynamic pressure p. Although the connection between p and p can always be stated once the constitutive equation for T is given, one would not necessarily expect the relationship to be simple for all fluids because thermodynamics refers to equilibrium conditions, whereas the elements of a fluid in motion are clearly not in thermodynamic equilibrium. Applying the definition (2-85) to the general Newtonian constitutive model, (2-80), we find... 

Specification of the processes included in the model. definition of constitutive equations... 

If we accept this as the definition of the laminar-flow Reynolds number, then for any constitutive equation which can be integrated twice to give the nonnewtonian equiyalent of the Poiseuille equation, Eq. 15.9 cian be used to define a working Reynolds number. For example, for power-law fluids (Eq. 15.7) this leads (Prob. 15.7) to... 

Primitives and definitions are used to formulate general postulates (e.g., the First and Second Laws, balances of mass, momentum, etc.) valid for all (in fact for a broad class of) material models. Real materials are expressed through special mathematical models in the form of constitutive equations which describe idealized materials expressing features important in assumed applications. Moreover, the same real material may be described by more models with various levels of description. The levels are motivated by the observer s time and space scales— typically the time and space intervals chosen (by the observer) for description of a real material having its own... 

The manner of the description of the state plays no role in this chapter the only thing which is important here is that we are able to say whether two states are the same or not. Therefore, the results are general and valid for all constitutive models or at least for those discussed in the following Chaps. 2 and 3. But we emphasize that phenomenological models expressed by constitutive equations, i.e., by a concrete choice of state, may be various and therefore, the concrete meaning of concepts discussed in this chapter (like work, definition of equilibrium, entropy values, etc.) may differ among such models, cf., e.g., [10,17,47,101]. This will be demonstrated in constitutive models discussed in the following Chaps. 2, 3 (see Sects. 2.1-2.3, 3.6-3.S). 

Concrete definition of equilibrium state must be performed for each constitutive model (characterized by the observer s scales of Sect. 1.1 and mainly Sect. 2.3) by time fixing of some quantities from those determining their states (see Rem. 6). Time persistency is usually difficult to achieve (because of molecular fluctuation) and therefore to describe real materials by such constitutive models we must add to constitutive equations (as their regularity) the conditions of stability by which the time permanence of equilibrium state S4 is assured. For details see Sects. 2.1-2.4, 3.8, 4.7 and Rems. 7,9, 11 in Chap. 2. Although one eqrulibrium state would suffice, typically there are more equilibrium states often forming the equilibrium process as their time sequence, see Rem. 12. 

We call the fields (3.114)-(3.116) fulfilling the balances of mass (3.63), (3.65), momentum (3.76), moment of momentum (3.93), and energy (3.107) a thermodynamic process, because only these are of practical interest. Then we denote the fields (3.114) as the thermokinetic process and the fields (3.115) as the responses (we limit to the models with symmetric T (3.93) in more general models we must introduce also the torque M into responses (3.115), cf. Rems. 17, 32). The fields (3.116) are controlled from the outside (at least in principle). Just constitutive equations, which express the difference among materials, represent the missing equations and are relations between (3.114) and (3.115) [6, 7, 9, 10, 23, 34, 38, 40, 41, 44, 45], Referring to Sect. 2.1 we briefiy recall that constitutive equations are definitions of ideal materials which approximate real materials in the circumstances studied (i.e at chosen time and space scales). Constitutive equations may be proposed in rational thermodynamics using the constitutive principles of determinism, local action, memory, equipresence, objectivity, symmetry, and admissibility. 

Recall that the mixture invariance described in Sect. 4.4 means that all balances (4.95)-(4.102) remain valid with primed quantities defined by transformations (4.108)-(4.113). But here we proceed further with functions (4.222), (4.223) instead of (4.103), the Eqs.(4.108)-(4.112) permit the formulation of linear constitutive equations with primed quantities by constitutive principles analogously as in Sect. 4.5. Remaining parts of Sects. 4.5 and 4.6 will be done with analogous (but primed) definitions keeping the rule that the definitions themselves are mixture invariant (cf. above (4.114)). Procedure and description will be similar as in Sect. 4.4. Quantities or expressions which do not change by using (4.108)-(4.113) we denote as mixture invariant, e.g. quantities (4.113). For simplicity, we use the primes for mixture invariant quantities rather exceptionally. 

Thus, the constitutive equations and field equations together, along with the jump conditions and boundary condition, should lead to a definite theory, predicting specific answers to particular problems. 

These three parameters (or other equivalent dimensionless groups) must appear in whatever formulation of this type of problem (the esterification reaction in PVRs), possibly together with other parameters which take into account other aspects such as additional phenomena (for example concentration polarization of the membrane), the presence of products in the initial mixture, the concentration of the catalyst and more complex constitutive equations. The dimensionless parameters have a more general validity than the individual dimensional parameters that appear grouped into them and characterize more univocally the behaviour of the system. The adoption of the parameter 5, the ratio of the characteristic rate of permeation to the characteristic rate of reaction, can be extended to any PVR and in general also to any membrane reactor. With this approach the comparison between different studies on PVRs is more direct and meaningful. On the other hand, the less acceptable, though often employed, dimensional parameter A/V, is comprised in the definition of 5. 

The Weissenberg number compares the elastic forces to the viscous effects. It is usually used in steady flows. One can have a flow with a small Wi number and a large De number, and vice versa. Sometimes the characteristic time of the flow in the deflnition of the Deborah number has been taken to be the reciprocal of a characteristic shear rate of the flow in these cases, the Deborah number and the Weissenberg number have the same definition. Pipkin s diagram (see Fig. 3.9 in Tanner 2000) classifies shearing flow behavior in terms of De and Wi, and provides a useful guide for the choice of constitutive equations. 

Stress and Strain Definitions. As remarked, the material functions relate the stress and strain responses of the material through a constitutive equation. For the elastic material, there is no time dependence and the relationships are relatively simple. In the case of linear viscoelasticity, equations that take into accoimt the time history of the stresses or strains are required. First, stress and strain are defined. 

Book content is otganized in seven chapters and one Appendix. Chapter 1 is devoted to the fnndamental principles of piezoelectricity and its application including related histoiy of phenomenon discoveiy. A brief description of crystallography and tensor analysis needed for the piezoelectricity forms the content of Chap. 2. Covariant and contravariant formulation of tensor analysis is omitted in the new edition with respect to the old one. Chapter 3 is focused on the definition and basic properties of linear elastic properties of solids. Necessary thermodynamic description of piezoelectricity, definition of coupled field material coefficients and linear constitutive equations are discussed in Chap. 4. Piezoelectricity and its properties, tensor coefficients and their difierent possibilities, ferroelectricity, ferroics and physical models of it are given in Chap. 5. Chapter 6. is substantially enlarged in this new edition and it is focused especially on non-linear phenomena in electroelasticity. Chapter 7. has been also enlarged due to mary new materials and their properties which appeared since the last book edition in 1980. This chapter includes lot of helpful tables with the material data for the most today s applied materials. Finally, Appendix contains material tensor tables for the electromechanical coefficients listed in matrix form for reader s easy use and convenience. 

Algebraic equations (14.3) correspond to constitutive equations, which are generally based on physical and chemical laws. They include basic definitions of mass, energy, and momentum in terms of physical properties, like density and temperature thermodynamic equations, through equations of state and chonical and phase equilibria transport rate equations, such as Pick s law for mass transfer, Fourier s law for heat conduction, and Newton s law of viscosity for momentum transfer chemical kinetic expressions and hydraulic equations. 

It is important to characterize the physical law in (3.18) i.e., the mechanical energy must be transferred smoothly. Then if we formulate a constitutive law, a condition is required for the constitutive equation such as positive definiteness... 

The constitutive equations (2.7) and (2.12) of derived adsorption-induced deformation model and deformable lattice model are regarded as dynamical systems. Its definition[178] is as ... 

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