General Constitutive Relations

The three-dimensional rate of strain at the point is expressed by a rate-of-strain tensor analogous to (17.1). Given a general constitutive relation between the deviatoric stress and rate-of-strain tensors for a particular material, the mechanical response of that material can be described completely (in principle) under 

---

Test results provides the hypothesis that syntactic foam is rate insensitive and that the static uniaxial strain stress-strain curve actually represents the general constitutive relation. Disagreement between the experimental data and the predicted behavior is greatest at low stresses (1 kbar) where experimental stresses are about double those predicted analytically. The discrepancy decreases at the higher stress levels and virtually disappears at and beyond 7 kbar. This range... 

The general theory is applied to thermohydraulic modelling of bentonite buffer with an assumption of a rigid skeleton. We get the constitution from the general constitutive relations with appropriate choices of the free energies and the dissipation function. The chosen specific free energies of the components are the following... 

Table 5.2 lists the expressions used for the transport properties in the model. Both the liquid- and vapor-equilibrated properties are given along with some general constitutive relations. The discussion below about the expressions focuses only on how they relate to the physical model for an in-depth discussion and derivation of the expressions, the reader is referred to our paper on the subject [39]. 

A general constitutive relation proposed by Han [73] for LCM processes is expressed as ... 

This equation stems from the general constitutive relation of a thin arbitrarily oriented lamina given by Elq. (4.31). Stresses and electric flux density of the individual laminae may be smmnarized in in-plane resultants N and out-of-plane resultants M by integration over the laminate thickness, in the latter case, in consideration of the distance to the middle surface ... 

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. 

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. 

Piezoelectric solids are characterized by constitutive relations among the stress t, strain rj, entropy s, electric field E, and electric displacement D. When uncoupled solutions are sought, it is convenient to express t and D as functions of t], E, and s. The formulation of nonlinear piezoelectric constitutive relations has been considered by numerous authors (see the list cited in [77G06]), but there is no generally accepted form or notation. With some modification in notation, we adopt the definitions of thermodynamic potentials developed by Thurston [74T01]. This leads to the following constitutive relations ... 

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... 

In the preceding sections the optical response of matter has been described by a scalar dielectric function e, which relates the electric field E to the displacement D. More generally, D and E are connected by the tensor constitutive relation (5.46), which we write compactly as D = e0e E. The dielectric tensor is often symmetric, so that a coordinate system can be found in which it is diagonal ... 

Constitutive Relations. A more general representation of the nonlinear polarization is that of a power series expansion in the electric field. For molecules this expansion is given by... 

To close the set of governing equations, additional constitutive relations must be introduced. Following the general effective stress principle, one finds... 

The relations (8.30) and (8.31) make up a general form for a non-linear single-mode constitutive relation. To specify the constitutive equation for a given system, one ought to determine the unknown function in (8.31) relying on experimental evidence. A particular form of relation (8.30) and (8.31), called canonical form (Leonov 1992), embraces many empirical constitutive equations (Kwon and Leonov 1995). One can obtain the canonical form of constitutive relation (Leonov 1992), if one neglects the viscosity term in the stress tensor (8.30), which is quite reasonable for polymer melts, and put an additional assumption on matrix M... 

Equations (10) are generally valid for both liquid and gas phases if reactions take place there. They represent nothing but a differential mass balance for the film region with the account of the source term due to the reaction. To link this balance to the process variables like component concentrations, some additional relationships - often called constitutive relations (see Ref. [16]) - are necessary. For the component fluxes Ni, these constitutive relations result from the multicomponent diffusion description (Eqs. (1), (2)) for the source terms, from the reaction kinetics description. The latter strongly depends on the specific reaction mechanism [27]. The reaction rate expressions lli usually represent nonlinear dependencies on the mixture composition and temperature of the corresponding phase. 

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] ... 

Chemistry generally constitutes the first step in hierarchical process design. This relates to process synthesis, specifically the selection of reaction routes and separation agents (Chen et al., 2003). Anastas and Allen (2002) proposed a set of strategies based on the Green Chemistry principles discussed above, which include ... 

The complexity of the Maxwell-Stefan equations and the generalized Fick s law have lead many investigators to use simpler constitutive relations that avoid the mathematical complexities (specifically, the use of matrix algebra in applications). In this chapter we examine these effective diffusivity or pseudobinary approaches. 

Let us now turn our attention to multicomponent systems. An exact analytical solution of the multicomponent penetration model for ideal gas mixtures has been presented by Olivera-Fuentes and Pasquel-Guerra (1987). Their analysis, which, in many ways, is similar to the film model analysis of Section 8.3.5, is generalized below to any system described by constitutive relations of the form of Eqs. 2.2.9 or 3.2.5. 

The analysis of turbulent eddy transport in binary systems given above is generalized here for multicomponent systems. The constitutive relation for j y in multicomponent mixtures taking account of the molecular diffusion and turbulent eddy contributions, is given by the matrix generalization of Eq. 10.3.1... 

---