Non-linear spring

Internal viscosity (Section 4) provides another possible source of shear-rate dependence. For sufficiently rapid disturbances, a spring-bead model with internal viscosity acts like a rigid body for sufficiently slow disturbances it is flexible and indefinitely extensible. The analytical difficulties for coupled, non-linear spring-bead systems are equally severe in linear spring-bead systems with internal viscosity. Even the elastic dumbbell with internal viscosity has only been solved exactly in the limit of small e (559), where e is the ratio of internal friction coefficient to molecular (external) friction coefficient Co n. For this case, the viscosity decreases with shear rate. 

The Zimm model (Zimm, 1956) extends the spring model by considering intermolecular forces such as hydrodynamic forces (perturbations of the velocity field near beads by other beads), reduced excluded-volume effects (coil expansion and reduced contacts), non-linear spring forces (finitely extendable springs) and internal viscosities (coil sluggishness). One can obtain the following expression for the viscosity from the Zimm model ... 

Here, n is the normal unit vector and v>j is the relative velocity at the contact point. The elastic part of the contact force is represented by a non-linear spring, assumed proportional to the spring stiffness and to s- (sn displacement). Additionally, to account for viscoelastic material properties that cause energy dissipation, a damping factor Pn related to the coefficient of restitution is included in the model ... 

Palmer [9] examined the function of ankle joint at different gait speeds in sagital plane and divided moment-angle curve into three sub-phases Controlled Plantarflexion (CP), Controlled Dorsiflexion (CD) and Powered Plantarflexion (PP). He concluded that ankle function can be mimicked by linear spring in CP and by nonlinear spring in CD. In PP a torque actuator should be added to the non-linear spring. 

The interaction between metal particles is adopted as the non-linear spring potential [7], and the potential is listed below ... 

Because of the assumption that linear relations exist between shear stress and shear rate (equation 3.4) and between distortion and stress (equation 3.128), both of these models, namely the Maxwell and Voigt models, and all other such models involving combinations of springs and dashpots, are restricted to small strains and small strain rates. Accordingly, the equations describing these models are known as line viscoelastic equations. Several theoretical and semi-theoretical approaches are available to account for non-linear viscoelastic effects, and reference should be made to specialist works 14-16 for further details. 

It is likely that most biomaterials possess non-linear elastic properties. However, in the absence of detailed measurements of the relevant properties it is not necessary to resort to complicated non-linear theories of viscoelasticity. A simple dashpot-and-spring Maxwell model of viscoelasticity will provide a good basis to consider the main features of the behaviour of the soft-solid walls of most biomaterials in the flow field of a typical bioprocess equipment. 

A reasonable approximation for the force between two adjacent particles is given by the so-called FENE (finitely extendable non-linear elastic) spring force law (Bird et al. 1987a)... 

Comparison with experimental data demonstrates that the bead-spring model allows one to describe correctly linear viscoelastic behaviour of dilute polymer solutions in wide range of frequencies (see Section 6.2.2), if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. The validity of the theory for non-linear region is restricted by the terms of the second power with respect to velocity gradient for non-steady-state flow and by the terms of the third order for steady-state flow due to approximations taken in Chapter 2, when relaxation modes of macromolecule were being determined. 

Fig. 12. Equation of state (a) and phase diagram (b) of a bead-spring polymer model. Monomers interact via a truncated and shifted Lennard-Jones potential as in Fig. 6 and neighboring monomers along a molecule are bonded together via a finitely extensible non-linear elastic potential of the form iJpENE(r) = — 15e(iJo/

Thus, the ideal c4iain can be thought of as an entropic spring and obeys Hooke s law for elongations much smaller than the maximum elongation (. R < 7 max = bN). For stronger deformations, the Langevin function [Eq. (2.112)] for freely jointed chains or Eq. (2.119) for worm-like chains can be used to describe the non-linear relation between force and... 

A simple generic bead spring model of chains can be used to study universal polymer properties that do not depend on specific chemical details. Bonds between neighbouring Lennard-Jones particles in a chain can be represented by the finite extension non-linear elastic (FENE) potential. 

Figure 6.5. Concentrations of nitrate in small streams and lakes in forested catchments in northern New England in the spring (right) and summer (left) as a function of NOy deposition onto the landscape. Note the non-linear response, with nitrate concentrations tending to increase as deposition exceeds 6 to 8 kg N per hectare per year (600 to 800 kg N km yr ). The arrows indicate the average deposition rates for oxidized nitrogen compounds (NOy) estimated for the northeastern United States in Boyer et al. (2002) and Howarth et al. (1996). Modified from Aber et al. (2003)...

The seismic analysis of the core is performed with the two-dimensional special purpose computer codes CRUNCH-2D and MCOCO, which account for the non-linearities in the structural design. Both CRUNCH-2D and MCOCO are based on the use of lumped masses and inertia concepts. A core element, therefore, is created as a rigid body while the element flexibilities are input as discrete springs and dampers at the corners of the element. CRUNCH-2D models a horizontal layer of the core and the core barrel structures (Figure 3.7-7). The model is one element deep and can represent a section of the core at any elevation, MCOCO models a strip of columns in a vertical plane along a core diameter and includes column support posts and core barrel structures (Figure 3.7-8). The strip has a width equal to the width of a permanent reflector block. Both models extend out to the reactor vessel,... 

Non-linearity is, however, entering into play if the object is heavy and/or if the spring is compressed with the same force instead of elongated. 

Further models for polymer dynamics include the incorporation of stiffness parameters for both local and collective modes, and the approach of Bird and co-workers using the finitely extensible non-linear elastic (FENE) dumb-bell. The latter has been used to reproduce the non-Newtonian viscosity observed with polymer solutions even at the 6-temperature at high shear rates (frequencies), but not given by the simple (infinitely extensible) bead spring. 

Equations (25) - (29) determine the simplest approach to the dynamics of a macromolecule, even so, it appears to be rather complex if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. Due to these effects, all the beads in the chain ought to be considered to interact with each other in a non-linear way. To tackle with the problem, this set of coupled non-linear equations is usually simplified. There exist the different simpler approaches originating in works of Kirkwood and Riseman [46], Rouse [2], Zimm [5], Cerf [4], Peterlin [6] to the dynamics of a bead-spring chain in the flow of viscous liquid. The linearization is usually achieved by using preliminary-averaged forms of the matrix of hydrodynamic resistance (hydrodynamic interaction) [5] and the matrix of the internal viscosity [4]. In the last case, to ensure the proper covariance properties when the coil is rotated as a whole, Eq. (29) must be modified and written thus... 

---