Energy-Elasticity

Acoustic emission is a naturally occurring phenomenon within materials, and the term Acoustic Emission is used to define the spontaneous elastic energy released within material or by a process, in the form of transient elastic waves. (2)... 

Elastic energy release due to subcritical crack growth is one recognized source of structure-related AE within its acknowledged lunitations, AEBIL provides a viable means of early on-line deteetion and localization of stable crack propagation. 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

Melt Viscosity. The study of the viscosity of polymer melts (43—55) is important for the manufacturer who must supply suitable materials and for the fabrication engineer who must select polymers and fabrication methods. Thus melt viscosity as a function of temperature, pressure, rate of flow, and polymer molecular weight and stmcture is of considerable practical importance. Polymer melts exhibit elastic as well as viscous properties. This is evident in the swell of the polymer melt upon emergence from an extmsion die, a behavior that results from the recovery of stored elastic energy plus normal stress effects. 

Elastic energy released when specimen breaks... 

The argument, at its simplest, is as follows. The primary function of a spring is that of storing elastic energy and - when required - releasing it again. The elastic energy stored per unit volume in a block of material stressed uniformly to a stress a is ... 

If you blow up a balloon, energy is stored in it. There is the energy of the compressed gas in the balloon, and there is the elastic energy stored in the rubber membrane itself. As you increase the pressure, the total amount of elastic energy in the system increases. 

To make the flaw grow, say by 1 mm, we have to tear the rubber to create 1 mm of new crack surface, and this consumes energy the tear energy of the rubber per unit area X the area of surface torn. If the work done by the gas pressure inside the balloon, plus the release of elastic energy from the membrane itself, is less than this energy the tearing simply cannot take place - it would infringe the laws of thermodynamics. 

From what we have said already, we can write down an energy balance which must be met if the crack is to advance, and fast fracture is to occur. Suppose a crack of length fl in a material of thickness t advances by 8a, then we require that work done by loads > change of elastic energy + energy absorbed at the crack tip, i.e. 

Now, as the crack grows into the plate, it allows the material of the plate near the crack to relax, so that it becomes less highly stressed, and loses elastic energy. 8Lf is thus negative, so that is positive, as it must be since G,- is defined positive. We can... 

Fig. 4. The Griffith criterion for the fixed grip condition, (a) cracked plate with fixed ends (b) schematic of elastic energy...

Fig. 4. Schematic of the JKR treatment of contact mechanics calculations. The point (, a, P) corresponds to the actual state under the action of interfacial forces and applied load P. P is the equivalent Hertzian load corresponding to contact radius a between the two surfaces, ( o, P) and (S], a, P ) are the Hertzian contact points. The net stored elastic energy and displacement S are calculated as the difference of steps 1 and 2.

The net stored elastic energy is given by stress distribution is given by... 

The JKR theory is essentially an equilibrium balance of energy released due to interfacial bond formation and the stored elastic energy. For simple elastic solids the deformation as a function of load, according to the JKR theory is given by... 

The most recent model of de Gennes and coworkers [100,101], the threshold toughness is related to the extra surface and elastic energy of the individual chains as they are pulled out. The threshold toughness and the critical crack propagation speed are given by... 

To evaluate the influx solution experimentally for an A/B cantilever beam configuration as shown in Fig. 1, we apply Griffith s theory at the critical moment of fracture, such that the incremental change in stored elastic energy U. with change in crack length a, is Just sufficient to overcome the fracture surface energy S... 

In which we summarize the terms as CTq = cohesive fracture strength of A = modulus of A, h = thickness of A layer L = influx length to a maximum length Lc, I — influx coverage of the A-B interface (/ < 1) = fraction of elastic energy dissipated in the A layer ( < 1). 

An alternative energy approach to the fracture of polymers has also been developed on the basis of non-linear elasticity. This assumes that a material without any cracks will have a uniform strain energy density (strain energy per unit volume). Let this be IIq. When there is a crack in the material this strain energy density will reduce to zero over an area as shown shaded in Fig. 2.65. This area will be given by ka where )k is a proportionality constant. Thus the loss of elastic energy due to the presence of the crack is given by... 

Figure 6 Variation of stored elastic energy (W) with the percent NBR content in NBR-CSPE blend.

Earlier studies [14,15] clearly reveal that there is a reaction between two polymers and that the extent of reaction depends on the blend ratio. As 50 50 ratio has been found to the optimum (from rheological and infrared studies) ratio for interchain crosslinking, the higher heat of reaction for the NBR-rich blend may be attributed to the cyclization of NBR at higher temperatures. There is an inflection point at 50 50 ratio where maximum interchain crosslinking is expected. Higher viscosity, relaxation time, and stored elastic energy are observed in the preheated blends. A maximum 50-60% of Hypalon in NBR is supposed to be an optimum ratio so far as processibility is concerned. 

Stored elastic energy (Fig. 12) also increases with shear rate both for preblends and preheated blends. Here again, we see that the W values increase sharply with NBR, attain a maximum at 50 50 level, and beyond 50% NBR the stored elastic energy decreases. 

Rheological parameters, such as relaxation time, shear modulus, and stored elastic energy, are determined from the extrudate swell and stress-strain data as previously described. Representative examples of the variation of these parameters with blend ratios for both blends are shown in Figs. 16-18. Figure 16 shows that relaxation time for both preblends without heating and... 

Stored elastic energy also increases up to 45-50% of Thiokol rubber and then decreases gradually until the end in the preheated blends (Fig. 18). A similar phenomenon is also noticed in the preblends without heating where the inflection point shifts toward the slightly higher level of Thiokol rubber. 

The plot of the rheological parameters (relaxation time, shear modulus, and stored elastic energy) are shown in Figs. 22-24. The relaxation time increases as the ACM content is increased to attain a maximum at 60 40 = ACM XNBR blend ratio for the preblends. For lower shear rate the rise is sharp and after 60 40 blend ratio, // remains almost constant, whereas for the higher shear rate region the rise is not sharp and after 60 40 blend ratio ty decreases as ACM percent increased in the blend. In the case of the preheated blends the /y increases up to 50 50 blend ratio and then decreases with the addition of ACM in the blend. The preheating increases the ty in both shear rate regions. 

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