Constitutive equations relations

For the analysis of experimental force-deformation data, it is necessary to use a suitable constitutive equation for the material under test. The constitutive equation relates the stresses and strains that are generated in the wall during compression, and therefore relates the tensions and stretch ratios. For example, Liu et al. (1996) used a Mooney-Rivlin constitutive equation to investigate the compression of polyurethane microcapsules and the functions f, /2 and fa are produced in... 

In a typical macroscopic assumption of proportionality between polarization and applied electric field, P = e0(c — 1 )E, where e is the dielectric constant, and eq3 reduces to the traditional Poisson—Boltzmann equation (the concentrations cH and c0h being in general much smaller than ce). However, if the correlations between neighboring dipoles are taken into account, the following constitutive equation relating the polarization to the macroscopic electric field is obtained7... 

Having closed the g equations, one must relate the Reynolds stresses to g in order to have a closed system. Again two approaches have been hotly debated. The more common approach uses the constitutive equation Eq. (26), together with an additional constitutive equation relating the turbulent viscosity to the turbulent kinetic energy. 

Constitutive equations relate stress tensor to various kinematic tensors and can be used to solve flow and other engineering problems. The development of constitutive equations to describe the state of stress continues to be an active area of research. 

Olir discussion on diffusion will be restricted primarily to binary systems containing only species A and B. We now wish to determine how the molar diffusive flux of a species (i.e., Ja) is related to its concentration gradient. As an aid in the discussion of the transport law that is ordinarily used to describe diffusion, lesll similar laws ftom other trans K)it processes. For example, in conductive heat transfer the constitutive equation relating the heat flux q and the temperature gradient is Fourier s law ... 

It is valid for each continuum independent of the individual material properties and is therefore one of the fundamental equations in fluid mechanics and subsequently also in heat and mass transfer. The movement of a particular substance can only be described by introducing a so-called constitutive equation which links the stress tensor with the movement of a substance. Generally speaking, constitutive equations relate stresses, heat fluxes and diffusion velocities to macroscopic variables such as density, velocity and temperature. These equations also depend on the properties of the substances under consideration. For example, Fourier s law of heat conduction is invoked to relate the heat flux to the temperature gradient using the thermal conductivity. An understanding of the strain tensor is useful for the derivation of the consitutive law for the shear stress. This strain tensor is introduced in the next section. 

Empirical constitutive equations relating the local permeability K, specific resistance, and solidosity to the effective stress are... 

As usual, we have invoked the summation convention and in addition have assumed that the material properties are homogeneous. For an isotropic linear elastic solid, the constitutive equation relating the stresses and strains is given by... 

Relationship of superficial liquid flow rate q to liquid pressure gradient, Darcy s law. Empirical constitutive equations relating permeability K, specific resisfance a, and soli-dosify (vol fraction of solids) e, to the effective pressure. 

The relationship between the rate of a chemical reaction and the species concentrations cannot be predicted and -must be determined from experiment. Detailed discussions of the analysis of experimental reaction rate data to get the constitutive equation relating the reaction rate to species concentrations are given in Kinetics and Mechanism, 3rd ed., by J. W. Moore and R. G. Pearson, John Wiley Sons, New York (1981), Chaps. 2 and 3 Chemical Reaction Engineering, 2nd ed., by O. Levenspiel, John Wiley Sons, New York (1972), Chap. 3 and Introduction to Chemical Engineering Analysis, by T. W. F. Russell and M. M. Denn, John Wiley Sons, New York (1972), Chap. 5., ... 

In order to complete the analysis we require constitutive equations relating both the pressure gradient and the torque density to the local velocity field q. Assuming these relations to be linear, the least restrictive assumption we can make is to suppose that these vectors possess a Taylor series expansion. Thus... 

These three equations are the general equations for melt spinning in the absence of radial variation of velocity and temperature. Ideally, if a constitutive equation relating stress to deformation and temperature were available, the stress and diameter profiles could be calculated from the equations. Further, if a relation could be found between stress and... 

The mechanical state of a drying body is described by the coupled system of differential equations composed of the equilibrium of internal forces and constitutive equations relating stresses with themechano-thermal-hydro strains. The internal equilibrium of forces is expressed by stresses as follows ... 

This equation is not independent from eq. (7.8-4). It can be proved by adding the two equations and making use of the Graham s law of diffusion. This means that eq. (7.8-4) is the only independent constitutive equation relating fluxes and concentration gradients. 

The neccessary constitutive equation relating the fluxes and mole fraction gradients is the following so-called modified Stefan-Maxwell equation (from eq. 8.6-6) ... 

The well-known dual-phase-lag heat conduction model introduces time delays to account for the responses among the heat flux vector, the temperature gradient and the energy transport. The dual-phase-lag heat conduction model has been used to interpret the non-Fourier heat conduction phenomena. The onedimensional dual-phase-lag constitutive equation relating heat flux to temperature gradient is expressed as (Xu, 2011 Zhou et al., 2009)... 

As mentioned in Sections 1 and 3, a rheological constitutive equation relates the components of the three-dimensional extra-stress (matrix) to the components of the strain (matrix) or the rate of strain (matrix) in any given flow field. Extensional viscosity data of the kind shown in Figure 5 should therefore be explainable on the basis of a proper constitutive equation. Unfortunately, there is no simple constitutive equation that accurately predicts the behavior of a polymer melt in all the commonly encountered flow situations. Some equations do a remarkably good job... 

As an aid in visualizing viscoelastic response, we introduce two linear mechanical models to represent the extremes of the mechanical response spectrum. The spring in Fig, 18.1a represents a linear elastic or Hookean solid whose constitutive equation (relation of stress to strain and time) is simply x = Gy, where G is... 

The constitutive equation relating generalised strain and stress is given by the diagonal matrix. 

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