Constitutive equation linear regime

Watanabe et al (1996) describe the non-linear dynamic rheology of a concentrated spherical silica-particle-filled ethylene glycol/glycerol system. The strain dependence was well described by the BBCZ-type constitutive equation, until the shear-thickening regime. The shear thickening was qualitatively described in relation to the structure of the suspended filler particles. 

The equation for solvent transport consists of a diffusional term and a term due to osmotic pressure. The osmotic pressure term arises by using linear irreversible diermodynamics arguments (20). The osmotic pressure is relat to the viscoelastic properties of the polymer through a constitutive equation. In our analysis, the Maxwell element has been used as the constitutive model. Thus, the governing equations for solvent transport in the concentrated regime are... 

Equation (4.2) includes translational and collisional effects but not static effects. However, most engineering applications (e.g., chute flows) and other natural flow situations (mudflows, snow avalanches, debris flow, etc.) appear to fit into a regime for which the total stress must be represented by a linear combination of a rate-independent static component plus the rate-dependent viscous component just described (Savage, 1989). The flow patterns observed for material flow in rotary kilns appear to fit these descriptions and the constitutive equations for granular flow may apply within the relevant boundary conditions. 

The restriction of linear viscoelastic behavior—small deformations—is more serious for equations 56 to 63 than for most of the preceding treatments, because with constantly increasing strain or stress the nonlinear regime may soon be reached. Then formulation in terms of a more complicated constitutive equation is necessary. 

Transient Response Creep. The creep behavior of the polymeric fluid in the nonlinear viscoelastic regime has some different features from what were foimd with the linear response regime. First, there are no ready means of relating the creep compliance to the relaxation modulus as was done in the linear viscoelastic case. In fact, the relationship between the relaxation properties and the creep properties depends entirely on the exact constitutive relationship chosen for the response of the material, and numerical inversion of the specific constitutive law is ordinarily necessary to predict creep response from the relaxation behavior (or vice versa). For most cases, the material properties that appear in the constitutive equations are written in terms of the relaxation response. We discuss this subsequently in the context of the K-BKZ model. 

The established tools of nonlinear dynamics provide an elaborate and versatile mathematical framework to examine the dynamic properties of metabolic systems. In this context, the metabolic balance equation (Eq. 5) constitutes a deterministic nonlinear dynamic system, amenable to systematic formal analysis. We are interested in the asymptotic, the linear stability of metabolic states, and transitions between different dynamic regimes (bifurcations). For a more detailed account, see also the monographs of Strogatz [290], Kaplan and Glass [18], as well as several related works on the topic [291 293],... 

In the limit of vanishingly small Reynolds numbers, forces due to convective momentum flux are negligible relative to viscous, pressure, and gravity forces. Equation (12-4) is simplified considerably by neglecting the left-hand side in the creeping flow regime. For fluids with constant /r and p, the dimensionless constitutive relation between viscons stress and symmetric linear combinations of velocity gradients is... 

The deformation response of a material to a given loading regime is described by generalized equations known as constitutive relations. For uniaxial loading in the limit of small strains, the simplest of these is known as Hooke s Law and linearly relates the stress to the strain ... 

The above brief treatment was necessary for two reasons. First, it introduced several important concepts (e.g., linear constitutive relations) and quantities (e.g., pressure tensor, heat flow vector) of nonequilibrium fluids. The microscopic versions of the same quantities are used as phase variables in theoretical manipulations and NEMD simulations. Second, it showed the crucial importance of NEMD simulations because the applicability of the Navier-Stokes equations is limited to the close-to-equilibrium regime where the transport coefficients are independent of the generating field. 

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