Material constitutive relation

Nonlinear effects are usually described in terms of polarization P through the material constitutive relation ... 

The polarization P is given in tenns of E by the constitutive relation of the material. For the present discussion, we assume that the polarization P r) depends only on the field E evaluated at the same position r. This is the so-called dipole approximation. In later discussions, however, we will consider, in some specific cases, the contribution of a polarization that has a non-local spatial dependence on the optical field. Once we have augmented the system of equation B 1.5.16. equation B 1.5.17. equation B 1.5.18. equation B 1.5.19 and equation B 1.5.20 with the constitutive relation for the dependence of Pon E, we may solve for the radiation fields. This relation is generally characterized tlirough the use of linear and nonlinear susceptibility tensors, the subject to which we now turn. 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

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. 

The contribution to the stress from electromechanical coupling is readily estimated from the constitutive relation [Eq. (4.2)]. Under conditions of uniaxial strain and field, and for an open circuit, we find that the elastic stiffness is increased by the multiplying factor (1 -i- K ) where the square of the electromechanical coupling factor for uniaxial strain, is a measure of the stiffening effect of the electric field. Values of for various materials are for x-cut quartz, 0.0008, for z-cut lithium niobate, 0.055 for y-cut lithium niobate, 0.074 for barium titanate ceramic, 0.5 and for PZT-5H ceramic, 0.75. These examples show that electromechanical coupling effects can be expected to vary from barely detectable to quite substantial. 

Up to this point we have restricted consideration to materials for which the dielectric function is a scalar. However, except for amorphous materials and crystals with cubic symmetry, the dielectric function is a tensor therefore, the constitutive relation connecting D and E is... 

Let us consider a sphere composed of a material described by the constitutive relation (5.46). We assume that the principal axes of the real and imaginary parts of the permittivity tensor coincide this condition is not necessarily satisfied except for crystals with at least orthorhombic symmetry (Born and Wolf, 1965, p. 708). If we take as coordinate axes the principal axes of the permittivity tensor, the constitutive relation (5.46) in the sphere is... 

I he field scattered by any spherically symmetrical particle composed of materials described by the constitutive relations (2.7)-(2.9) has the same form as that scattered by the homogeneous sphere considered in Chapter 4. However, the functional form of the coefficients an and bn depends on the radial variation of e and ju. In this section we consider the problem of scattering by a homogeneous sphere coated with a homogeneous layer of uniform thickness, the solution to which was first obtained by Aden and Kerker (1951). This is one of the simplest examples of a particle with a spatially variable refractive index, and it can readily be generalized to a multilayered sphere. 

Scattering problems in which the particle is composed of an anisotropic material are generally intractable. One of the few exceptions to this generalization is a normally illuminated cylinder composed of a uniaxial material, where the cylinder axis coincides with the optic axis. That is, if the constitutive relation connecting D and E is... 

The creep rate when boundary sliding is rate-limiting has been treated and discussed by Beere [13, 14]. If a viscous constitutive relation is used for grainboundary sliding (i.e., the sliding rate is proportional to the shear stress across the boundary), the macroscopic creep rate is proportional to the applied stress, and the bulk polycrystalline specimen behaves as a viscous material. An analysis of the sliding-controlled creep rate of the idealized model in Fig. 16.4 is taken up in Exercise 16.2. 

In expressing F, G, and H in terms of the dependent variables, we use certain properties of the materials involved (e.g., that the reaction is first order, so G = Vkc). These are sometimes called constitutive relations because they invoke the constitutions of the various components. They are not principles applying to everything, as are natural laws, but apply only to the materials in question. 

Of course, we have to express f, g, and h in terms of a common variable, u for example, by means of a constitutive relation for the material under study often u = h. 

In the broadest sense, I found the analogy with fluid mechanics to be very helpful. Just as kinematics provides the geometrical framework of fluid mechanics by exploring the motions that are possible, so also stoicheiometry defines the possible reactions and the restrictions on them without saying whether or at what rate they may take place. When dynamic laws are imposed on kinematic principles, we arrive at equations of motion so, also, when chemical kinetics is added to stoicheiometry, we can speak about reaction rates. In fluid mechanics different materials are distinguished by their constitutive relations and allow equations for the density and velocity to be formulated thence, various flow situations are examined by adding appropriate boundary conditions. Similarly, the chemical kinetics of the reaction system allow the rates of reaction to be expressed in terms of concentrations, and the reactor is brought into the picture as these rates are incorporated into appropriate equations and their boundary conditions. 

When solving flow and heat transfer problems in polymer processing we must satisfy conservation of mass, forces or momentum and energy. Momentum and energy balances, in combination with material properties through constitutive relations, sometimes result in... 

As with any theory of material behavior, we have to make constitutive assumptions in order to define the peculiar mixture of a poroelastic material and a compressible bi-component fluid. Among other quantities, we must state constitutive relations for the mass supply a and the momentum supply m, which give rise to adsorption/desorption and to diffusion, respectively. 

To close the problem, constitutive relations of powders must be introduced for the internal connections of components of the stress tensor of solids and the linkage between the stresses and velocities of solids. It is assumed that the bulk solid material behaves as a Coulomb powder so that the isotropy condition and the Mohr-Coulomb yield condition may be used. In addition, og has to be formulated with respect to the other stress components. 

The Standard Linear Solid Model combines the Maxwell Model and a like Hook spring in parallel. A viscous material is modeled as a spring and a dashpot in series with each other, both of which other, both of which are in parallel with a lone spring. For this model, the governing constitutive relation is ... 

The generic equations of balance are statements of truth, which is a priori self-evident and which must apply to all continuum materials regardless of their individual characteristics. Constitutive relations relate diffusive flux vectors to concentration gradients through phenomenological parameters called transport coefficients. They describe the detailed response characteristics of specific materials. There are seven generic principles (1) conservation of mass, (2) balance of linear momentum, (3) balance of ro-... 

Instead of velocity gradients, displacement gradients can be used in relation (8.38). In this form, relations of the kind (8.38) are established on the basis of the phenomenological theory of so-called simple materials (Coleman and Nolle 1961). To put the theory into practice, function (8.38) should be, for example, represented by an expansion into a series of repeated integrals, so that, in the simplest case, one has the first-order constitutive relation (8.37). Let us note that the first person who used functional relations of form (8.38) for the description of the behaviour of viscoelastic materials was Boltzmann (see Ferry 1980). 

One can see that there are several forms for the representation of the constitutive relation of a viscoelastic liquid. Of course, we ought to say that all the types of constitutive relation we discussed in this section are equivalent. We can use any of them to describe the flow of viscoelastic liquids. However, the description of the flow of a liquid in terms of the internal variables allows one to use additional information, if it is available, about microstructure of the material, and, in fact, appears to be the simplest one for derivation and calculation. We believe that the form, which includes the internal variables, reflects a deeper penetration into the mechanisms of the viscoelastic behaviour of materials. From this point of view, all the representations of deformed material can be unified and classified. 

Although the microscopic theory remains to be the real foundation of the theory of relaxation phenomena in polymer systems, the mesoscopic approach has and will not lose its value. It will help to understand the laws of diffusion and relaxation of polymers of various architecture. The information about the microstructure and microdynamics of the material can be incorporated in the form of constitutive relation, thus, allowing to relate different linear and non-linear effects of viscoelasticity to the composition and chemical structure of polymer liquid. 

Assumption 6.5. In view of Remark 6.2, it is assumed that all the material flow rates associated with lo1 are determined by appropriate functions of the process state variables (e.g., via feedback control laws, constitutive relations or pressure-flow correlations). 

For further progress it is necessary to specify how E varies with D, or how P depends on Ea. For this purpose, we introduce the constitutive relations D - e(T,V)E or P - ot0(T,V)F0, where e is the dielectric constant and a0 is a modified polarizability. (Conventionally, the polarizability is defined through the relation P - oE, but no confusion is likely to arise through the introduction of this variant.) Note several restrictions inherent in the use of these constitutive relations. First, the material under study is assumed to be isotropic. If this is not the case, e and c 0 become tensors. Second, the material medium must not contain any permanent dipole moments in the preceding constitutive relations P or E vanishes when E0 or D does. Third, we restrict our consideration to so-called linear materials wherein e or a0 do not depend on the electric field phenomena such as ferroelectric or hysteresis effects are thus excluded from further consideration. These three simplifications obviously are not fundamental restrictions but render subsequent manipulations more tractable. Finally, in accord with experimental information available on a wide variety of materials, e and aQ are considered to be functions of temperature and density assuming constant composition, these quantities vary with T and V. 

It has been experimentally observed that for small deformations, the strain in a body is linearly proportional to the applied stress. In one dimension this is known as Hooke s law, relating the elongation of a spring or elastic material to the tensile force. A principle such as this, which relates stress to strain, is known as a constitutive relation, and can be generalized to three-dimensional, non-piezoelectric solids [1] ... 

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