Elastic force ideal energy

It is important to realize that this type of behavior is not just a simple addition of linear elastic and viscous responses. An ideal elastic solid would display an instantaneous elastic response to an applied (non-destructive) stress (top of Figure 13-74). The strain would then stay constant until the stress was removed. On the other hand, if we place a Newtonian viscous fluid between two plates and apply a shear stress, then the strain increases continuously and linearly with time (bottom of Figure 13-74). After the stress is removed the plates stay where they are, there is no elastic force to restore them to their original position, as all the energy imparted to the liquid has been dissipated in flow. 

In our statistical treatment of an ideal elastomer, we have assumed that the elastic force is entirely attributable to the conformational entropy of deformation, energy effects being neglected. That the theory reproduces the essential features of the elasticity of real elastomers attests to the basic soundness of this assumption. On the other hand, we know that in real elastomers such energy effects cannot be entirely absent, and deviations from the ideal elastomer model may be expected to occur. Let us now examine in greater detail the extent to which the neglect of energy effects is justified. We can rewrite equation (6-28) ... 

S.4.2.3 Efficient Energy Transduction Requires Coupling to Near-ideal Elastic Force Development... 

Elastic forces come into play as hydrophobic associations stretch interconnecting chain segments. Only if the elastic deformation is ideal does all of the energy of deformation become recovered on relaxation. To the extent that hysteresis occurs in the elastic deformation/ relaxation, energy is lost and the protein-based machine loses efficiency. Thus, the elastic consilient mechanism, whereby the force-extension curve can be found to overlay the force-relaxation curve becomes the efficient mechanical coupler within the vital force. The objective now becomes one of understanding the age-old problem of a reluctance to discard past idols. 

In an ideal or perfect elastomer the energy repeatedly invested in extension is repeatedly and completely recovered during relaxation. Ideality increases as the elastic force results from a decrease in entropy upon extension, because this occurs without stressing bonds to the breaking point. Elastin models and elastin itself in water provide examples of such entropic elastomers with about 90% of the elastic force being entropic, that is, the /e//ratio of Equation (4) is about 0.1. This is essential to human life expectancy, because the half-life of elastin in the mammalian elastic fiber is on the order of 70 years. This means that the elastic fibers of the aortic arch and thoracic aorta, where there is twice as much elastin as collagen, will have survived some billion demanding stretch-relaxation cycles by the start of the seventh decade of life. This represents an ultimate in ideal elasticity. 

When attempting to relate the adhesion force obtained with the SFA to surface energies measured by cleavage, several problems occur. First, in cleavage experiments the two split layers have a precisely defined orientation with respect to each other. In the SFA the orientation is arbitrary. Second, surface deformations become important. The reason is that the surfaces attract each other, deform, and adhere in order to reduce the total surface tension. This is opposed by the stiffness of the material. The net effect is always a finite contact area. Depending on the elasticity and geometry this effect can be described by the JKR 65 or the DMT 1661 model. Theoretically, the pull-off force F between two ideally elastic cylinders is related to the surface tension of the solid and the radius of curvature by... 

Suppose there are n moles of an ideal gas in a cubical container with sides each of length L in meters. Assume each gas particle has a mass m and that it is in rapid, random, straight-line motion colliding with the walls, as shown in Fig. 5.11. The collisions will be assumed to be elastic—no loss of kinetic energy occurs. We want to compute the force on the walls from the colliding gas particles and then, since pressure is force per unit area, to obtain an expression for the pressure of the gas. 

Under an applied stress ideally solid materials may deform, but they do not flow. When the force is released the deformation relcixes. The energy, supplied to the material in order to obtain the deformation, remcilns stored upon terminating the stress, this energy.is fully and immediately released. Such a system is called (ideally) elastic. 

In typical crystalline solids, such as metals, the energetic contribution dominates the force because the internal energy increases when the crystalline lattice spacings are distorted from their equilibrium positions. In rubbers, the entropic contribution to the force is more important than the energetic one. In ideal networks there is no energetic contribution to elasticity, so /e = 0. 

If a dwell time is required or desired in punch-and-die presses (Fig. 5.11, upper right), special drive systems must be used [B.48, B.97]. It is obvious from Fig. 5.11, lower left and right, that no such possibility exists in roller presses where a continuous rolling action densifies the material between approaching surfaces until, immediately after passing the point of closest approximation, the relative motion is reversed, the surfaces retract, and the pressing force drops, ideally to zero if no expansion due to compressed gas and/or stored elastic energy takes place. 

An ideal gas is one whose particles take up no space. Ideal gases experience no intermolecular attractive forces, nor are they attracted or repelled by the walls of their containers. The particles of an ideal gas are in constant, random motion, moving in straight lines until they collide with each other or with the walls of the container. Additionally, these collisions are perfectly elastic, which means that the kinetic energy of the system does not change. An ideal gas follows the gas laws under all conditions of temperature and pressure. 

An ideal-elastic or energy-elastic body deforms under the influence of a force by a definite amount which does not depend on the duration of the force. For comparison purposes, reference is made, not to the force, but to the force per unit surface area, i.e., the stress. The deformation may be a stretching, shearing, turning, compression, or bending (see Table 11-1). 

In contrast to ideal entropy-elastic bodies, real entropy-elastic bodies have an energy-elastic component. The force Fe resulting from this component is given for a uniaxial deformation by... 

This is identical with equation 3.26 if R = R. [However, Fiory, Hoeve and Ciferri [6] have pointed out that the inversion process between equation 3.35 and 3.36 is valid only at large n values. For n <10, the error is large.] The entropic origin of force was not assumed in the derivation of equation 3.37 so that this result is generally valid and not restricted to ideal rubbers. The energy contribution to rubber elasticity may be calculated from equation 3.16... 

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