Force elastic

A polymer chain can be approximated by a set of balls connected by springs. The springs account for the elastic behaviour of the chain and the beads are subject to viscous forces. In the Rouse model [35], the elastic force due to a spring connecting two beads is f= bAr, where Ar is the extension of the spring and the spring constant is ii = rtRis the root-mean-square distance of two successive beads. The viscous force that acts on a bead is... 

Polymers owe much of their attractiveness to their ease of processing. In many important teclmiques, such as injection moulding, fibre spinning and film fonnation, polymers are processed in the melt, so that their flow behaviour is of paramount importance. Because of the viscoelastic properties of polymers, their flow behaviour is much more complex than that of Newtonian liquids for which the viscosity is the only essential parameter. In polymer melts, the recoverable shear compliance, which relates to the elastic forces, is used in addition to the viscosity in the description of flow [48]. 

The phenomena under discussion are viscoelastic we have only considered the elastic forces. Next we must incorporate viscous forces. As indicated above, we use Eq. (2.47) to express the proportionality between the viscous resistance to displacement and the velocity of the bead, dZj/dt ... 

Our strategy in proceeding, therefore, is to write separate expressions for the forces cited in items (1) and (2), and then set them equal to each other as required by item (3). Since we have discussed osmotic effects in Chap. 8 and elastic forces in Chap. 3, we shall invoke certain concepts and relationships from these chapters in this discussion. In this derivation we continue to omit numerical coefficients and some of the less pertinent parameters (although we retain Vj for the sake of Problem 5 at the end of the chapter), and focus attention on the relationship between a, M, and the interaction parameter x-... 

As noted above, this force is counterbalanced by an entropy-based, elastic force which prevents the molecule from uncoiling. Entropy elasticity was discussed in Sec. 3.4, where the elastic (subscript el) force between crosslinks is given by Eq. (3.19) to be... 

Feder-kiel, m. quill. -kleid, n. plumage -klemme, /. spring clip, - aft, /. elastic force elasticity, federkraftig, a. elastic, springy. 

Spring-kolben, m. Bologna fiask. -kraft, /. springiness, power of recoil, elastic force, elasticity. 

A stone dropped in a pond pushes the water downward, which is countered by elastic forces in the water that tend to restore the water to its initial condition. The movement of the water is up and down, but the crest of the wai c produced moves along the surface of the water. This type of wave is said to be transverse because the displacement of the water is perpendicular to the direction the wave moves. When the oscillations of the wave die out, there has been no net movement of water the pond is just as it was before the stone was dropped. Yet the wave has energy associated with it. A person has only to get in the path of a water wave crashing onto a beach to know that energy is involved. The stadium wave is a transverse wave, as is a wave in a guitar string. 

Elastic forces and couples are calculated from the elastic energy W ... 

Elastic force associated with a given deformation—... 

Here 5(t) is the distance of the mass from the origin and F the total force acting on the elementary mass. Note that at the point of an equilibrium, z = 0, the elastic force of the spring is not equal to zero, and it compensates the weight. If the mass m is taken away from this point an additional elastic force arises and the resultant force E is a superposition of the following forces ... 

The elastic force Feq — —k(l — /q) which together with weight provides equilibrium. The sign means that this force is directed upward, that is, opposite the z-axis. 

The elastic force Fe is due to an additional deformation caused by a motion of the spring, and in accordance with Hooke s law F — —ks. Here 5 is the displacement of the mass from an origin, and the sign indicates that the direction of this force and that of the movement are opposite to each other. In fact, when the mass moves down the spring is expanded and therefore an elastic force tends to move this mass upward. We observe the same in the case of spring compression. 

Here A is the amplitude, cp the initial phase, and coo the frequency of free vibrations. Thus, in the absence of attenuation free vibrations are sinusoidal functions and this result can be easily predicted since mass is subjected to the action of the elastic force only. In other words, the sum of the kinetic and potential energy of the system remains the same at all times and the mass performs a periodic motion with respect to the origin that is accompanied by periodic expansion and compression of the spring. As follows from Equation (3.105) the period of free vibrations is... 

The vertical spring and mass is an example of a stable system and by definition this means that an arbitrary small external force does not cause the mass to depart far from the position of equilibrium. Correspondingly, the mass vibrates at small distances from the position of equilibrium. Stability of this system directly follows from Equation (3.102) as long as the mechanical sensitivity has a finite value, and it holds for any position of the mass. First, suppose that at the initial moment a small impulse of force is applied, delta function, then small vibrations arise and the mass returns to its original position due to attenuation. If the external force is small and constant then the mass after small oscillations occupies a new position of equilibrium, which only differs slightly from the original one. In both cases the elastic force of the spring is directed toward the equilibrium and this provides stability. Later we will discuss this subject in some detail. 

At equilibrium the moment tg is balanced by the moment of the elastic force of the thread... 

Figure 4. Schematic description of the swelling process. The molecules of the swelling liquid start to penetrate inside the polymer framework from its surface (a) and to solvate the polymer chains. The polymer chain start to stretch out and to move away from one another the apparent volume of the polymer increases and the first nanopores are formed (b). Swelling stops when increasing elastic forces set up by the unfolding of the polymer chains counterbalance the forces which drive the molecules of the swelling agent into the polymer framework (c).

Similar instability is caused by the electrostatic attraction due to the applied voltage [56]. Subsequently the hydrodynamic approach was extended to viscoelastic films apparently designed to imitate membranes (see Refs. 58-60, and references therein). A number of studies [58, 61-64] concluded that the SQM could be unstable in such models at small voltages with low associated thinning, consistent with the experimental results. However, as has been shown [60, 65-67], the viscoelastic models leading to instability of the SQM did not account for the elastic force normal to the membrane plane which opposes thickness... 

In uniaxial deformation, the energetic contribution to the total elastic force [4,5,16,80-82] is given by the thermodynamically exact relation... 

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