Hookean spring

We shall follow the same approach as the last section, starting with an examination of the predicted behavior of a Voigt model in a creep experiment. We should not be surprised to discover that the model oversimplifies the behavior of actual polymeric materials. We shall continue to use a shear experiment as the basis for discussion, although a creep experiment could be carried out in either a tension or shear mode. Again we begin by assuming that the Hookean spring in the model is characterized by a modulus G, and the Newtonian dash-pot by a viscosity 77. ... 

The mass of the subchain is pictured as concentrated in a bead, connected to adjacent beads by Hookean springs which, individually, obey Eq. (3.45). 

Ea = Arrhenius activation energy Es = excess stress energy AEr = potential barrier for bond rotation Eel = molecular elastic energy F = mean force potential f = average force on the chain fb = bond breaking force H0 = Hookean spring constant kB = Boltzmann constant... 

Substituting Eq. (12) into Eq. (11) permits us to derive the Hookean spring force law, well-known in the classical theory of rubber elasticity ... 

The inclusion of internal viscosity raises considerably the free-energy storage capacity of a rapidly deforming macromolecule as compared to the idealized Hookean spring model and could play a decisive role in mechanochemical reactivity in transient elongational flow. 

Zimm [34] extended the bead-spring model by additionally taking hydrodynamic interactions into account. These interactions lead to changes in the medium velocity in the surroundings of each bead, by beads of the same chain. It is worth noting that neither the Rouse nor the Zimm model predicts a shear rate dependency of rj. Moreover, it is assumed that the beads are jointed by an ideally Hookean spring, i.e. they obey a strictly linear force law. 

A Hookean spring is one that obeys Hooke s law, i.e. the force is... 

The results were compared to MD-simulations [317]. Whereas the scattering function of pure PEO could be well described, the dynamics of the salt-loaded samples deviates from the predictions obtained with various electrostatic interaction models. The best but still not perfect and - at least for longer times -unphysical model assumes Hookean springs between chains to simulate the Na-ion mediated transient cross-links [317]. 

In general, the Hookean spring represents bond flexing while the Newtonian dashpot represents chain and local segmental movement. It is customary to attempt to relate stress-strain behavior to combinations of dashpots and springs as indicators of the relative importance of bond flexing and segmental movement. 

FIGURE 14.1 Stress-strain plots for a Hookean spring (a) where E (Equation 14.1) is the slope, and a Newtonian dashpot (b) where is a constant (Equation 14.3). 

In this model, derived originally for star-shaped branched molecules, polymer molecules are represented by beads connected by identical Hookean springs, and the decrease in viscosity with branching is expressed by the g1/2 rule. 

PROBLEM PR.14 Show how a Hookean spring works against an inverse-power van der Waals interaction in a force balance of a sphere and a flat surface. 

The typical creep curve for a plastic fat is shown in Figure 7.8 and demonstrates the effect of working on the structure of butter. The softening that occurs in plastic fats is dependent on both the amount of force or deformation applied and also on the testing time (deMan and Beers, 1987). When a force is applied (i.e., when the sample is compressed), there is an initial elastic response (A), which can be represented by a Hookean spring (deMan et al., 1985 deMan and Beers, 1987). If the yield stress is exceeded,... 

FIGURE 7-5 Schematic diagram of the periodic forced oscillations of a simple Hookean spring. 

Application of the PRISM theory to Gaussian chains is the simplest case. Each site is bonded to its neighbors by simple entropic Hookean springs. Nonbonded interactions are neglected. In this case, one obtains... 

Figure 14.19 Stress-strain plots for (a) a Hookean spring where E is the slope (6) a Newtonian dash pot where s is constant, (c) stresstime plot stress for relaxation in the Maxwell model, and (d) stresstime plot stress for a Voigt-Kelvin model.

F = IksTfi R, is that of a linear, or Hookean, spring, given in Eq. (3-10). It is an appropriate expression when the molecule is stretched to no more than about a third of its maximum extension. 

At small relative extensions R) worm-like chains behave as Hookean springs ... 

Theories for polymer dynamics of dilute polymer solutions include the elastic (Hookean) spring model (Kuhn, 1934) which considers that the system is mechanically equivalent to a set of beads attached with a spring. The properties are then based on a spring constant between beads and the friction of beads through solvent. The viscosity of a Hookean system is then described by... 

H) Hookean dumbbells The beads are joined by a Hookean spring which has zero length if there are no forces acting on the beads the tension in the connector F(c) = HR, where H is a Hookean spring constant. 

One notices that olcrm is just the instantaneous response of the Hookean spring, and from equation (3-1) it is seen to be equal to the reciprocal of E. From the results of Chapter 2, Section A, we have... 

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