Incompressible networks

This is the most commonly used model for natural gas nets, and most algorithms for incompressible networks may be used for gas networks as well, simply by replacing eqn (3) by eqn (4) in the library of pressure drop correlations. There are several commercially available computer programs for gas networks, among which the ones from the British Gas Corporation (6) and Intercomp (7) are found. 

Consider an affine model of an incompressible network with polydisperse strands between crosslinks. Prove that the stress a in this network due to... 

Demonstrate that the true stress in uniaxially deformed incompressible networks is the derivative of the free energy per unit volume FjV with respect of the logarithm of deformation A ... 

Demonstrate that for uniaxial deformation of an incompressible network, the stress is given by Eq. (7.64) where the crosslink and entanglement moduli Gx and Ge, respectively, are defined in Eqs (7.43) and (7.47). 

The second term on the right-hand side vanishes for incompressible networks. For deformations caused by swelling, the deformation is isotropic and thus... 

A knowledge of v can give an indication of the transit time of a plug of chemical or an ensemble of cells through a microfluidic channel network and thus to assess whether there is enough time for complete mixing or chemical reaction. Both Eq. (11) and Eq. (12) are strictly only valid under idealized conditions (i.e. incompressible and non-viscous fluids and steady flow), but can still be helpful for overall estimation and assessment. 

The relaxing Gaussian network of Green and Tobolsky (4) is the earliest version of this model. Lodge (12) and Yamamoto (J5) independently derived constitutive equations for similar systems, based on a stress-free state for each newly created strand and a distribution of junction lifetimes which is independent of flow history. For Gaussian strands in an incompressible system ... 

As can be expected, the angular scattering intensity distribution has become anisotropic. From the data obtained, two values of the average correlation distance between neighboring nodules can be calculated, parallel and perpendicular to the deformation direction respectively. Although the accuracy of the data is not outstanding, it can be seen that d increases with the macroscopic extension ratio, Ax, whereas dx decreases only slightly (Fig. 10). It is not possible to ascertain experimentally whether dL is proportional to AJ1/2 as it should be if the deformation proceeded at constant volume L e., if the network could be considered incompressible. 

The boundary conditions are defined in the same way as with the flow analysis network. The nodes whose control volumes are empty or partially filled are assigned a zero pressure, and the gate nodes are either assigned an injection pressure or an injection volume flow rate. Just as is the case with flow analysis network, a mass balance about each nodal control volume will lead to a linear set of algebraic equations, identical to the set finite element formulation of Poisson s or Laplace s equation. The mass balance (volume balance for incompressible fluids) is given by... 

This model is useful strictly for incompressible fluids only, but for small pressure changes it may be used for gas networks as well. The equations for general network simulation are based on eqn (1) and eqn (2) and a variety of models analogous to eqn (3). It goes beyond the scope of this paper to elaborate on networks with loops, this is dealt with in details by for example Mach (1), Gay and Preece (2,3) and Carnahan and Wilkes (4). A variety of... 

Two types of stress are important in drying. The first is the total stress, which corresponds to the force per unit area acting on both the liquid emd the particle network. When the pores are filled with liquid, the stress is spread evenly over the whole green body, because the essentially incompressible liquid distributes the stress evenly in all directions. The second t3q>e of stress is the network stress, which is the force per unit area acting only on the particle network. When we consider the warping and cracking of the particle network, the stress on the particle network is important not the total stress. 

We now consider an extensional deformation of an incompressible rubber network (Fig. 3-7), where the stretch axes are oriented along the coordinate axes apd where the stretch ratios X, k2, and I3 are in directions 1, 2, and 3, respectively. For the example in Fig. 3-7, the deformation is a uniaxial extension that increases the length of the cylinder by a factor of X1 over its initial length. By volume conservation, the radius of the cylinder then shrinks to times the original radius. If the cross-link points are convected with... 

An example of the success of the temporary network model for a practical application is shown in Fig. 3-11. Here, the predictions of Eq. (3-24) are compared to experimental force-deflection data for impact tests in which a heavy flat-bottomed object is dropped onto a flat circular pad of dissipative Sorbothane rubber at various velocities and two different temperatures. Since the material is nearly incompressible under these conditions, the impact... 

Dry networks are typically incompressible, which means that their volume does not change appreciably when they are deformed ... 

The stress-strain relationships of elastomeric (rubbery) networks at low extension (or draw) ratios (k, which is the length of the deformed specimen divided by the length of the initial undeformed specimen) can be described in terms of Equation 11.37 [29], which is the simplest possible constitutive equation for the deformation of an isotropic incompressible medium. 

The Hagen-Poiseuille law is mathematically analogous to the Ohm s Law. In addition, the conservation of mass (or flow for incompressible fluid) of fluid is analogous to the law of conservation of charge and current in electrical systems. This analogy allows for the use of Kirchoffs equations for calculation of the distribution of the volumetric flow of liquid between channels in a microfluidic network once we know the resistances of all the channels in the network and the pressures at the inlet and outlet, we can calculate the speed of flow in any part of the network (Fig. 1). 

The equilibrium small-strain elastic behavior of an "incompressible" rubbery network polymer can be specified by a single number—either the shear modulus or the Young s modulus (which for an incompressible elastomer is equal to 3. This modulus being known, the stress-strain behavior in uniaxial tension, biaxial tension, shear, or compression can be calculated in a simple manner. (If compressibility is taken into account, two moduli are required and the bulk modulus. ) The relation between elastic properties and molecular architecture becomes a simple relation between two numbers the shear modulus and the cross-link density (or the... 

The most modern picture of membrane deformation recognizes that the membrane is a composite of two layers with distinct mechanical behavior. The membrane bilayer, composed of phospholipids and integral membrane proteins, exhibits a large elastic resistance to area dilation but is fluid in surface shear. The membrane skeleton, composed of a network of structural proteins at the cytoplasmic surface of the bilayer, is locally compressible and exhibits an elastic resistance to surface shear. The assumption that the membrane skeleton is locally incompressible is no longer applied. This assumption had been challenged over the years on the basis of theoretical considerations, but only very recently has experimental evidence emerged that shows definitively that the membrane skeleton is compressible. This has led to a new constitutive model for membrane behavior [Mohandas and Evans, 1994]. The principal stress resultants in the membrane skeleton are related to the membrane deformation by ... 

The apparent discrepancy between the Flory theory and the entanglement concept of Dossin and Graessley has been addressed by Gottlieb and Macosco [55]. They pointed out that the two parameters h and k, both measuring the severity of constraints are related. For the case of a perfect, incompressible, unswollen network the analytical relationship is given by... 

Thus, polymer networks as well as LMW compounds arc cltcssified with practically incompressible matter, and the variations of their volume on ten.sion and other kinds of deformation can be neglected (Treloar, 1970). 

An alternative model which also describes stress-strain data for larger deformation is presented by the Mooney-Rivlin equation [40, 41], The equation describes the rubber elasticity of a polymer network on the basis that the elastomeric sample is incompressible and isotropic in its unstrained state and that the sample behaves as Hookean solid in simple shear. In a Mooney-Rivlin plot of a uniaxial deformation, the experimental measured stress cr, divided by a factor derived from classical models, is plotted as function of the reciprocal deformation 1/A ... 

And at last, the third and the most fundamental factor is the ehange of nanocomposite structure at the introduction of particulate filler in high-elastieity polymeric matrix. As Balankin showed [9], classical theory of entropic high-elastieity has a number of principal deficiencies due to non-fulfilment for real rubbers of two main postulates of this theory, namely, essentially non-Gaussian statisties of real polymeric networks and lack of coordination of postulates about Gaussian statistics and incompressibility of elastic materials. Last postulate means, that Poisson s ratio v of these materials must be equal to 0.5. As it is known [10], Gaussian statistics of macromolecular coil is correct only in case of its dimension Dj=2.0, i.e., for coil in 0-solvent. Since between value Df and fractal dimension... 

This familiar equation is more usually represented as a consequence of the molecular theories of a rubber network. Here we see that it follows from purely phenomenological considerations as a simple constitutive equation for the finite deformation of an isotropic, incompressible solid. Materials that obey this relationship are sometimes called neo-Hookean. 

Between major branch points, an epicardial artery is modelled as an elastic tube. The flow of blood in each such tube of the network is assummed to be that of an incompressible Newtonian fluid whose motion can be adequately described as... 

---