Stiff phenomenon

In contrast, the asymptotic approach puts minimal strain on the computer but demands more of the modeller. The convergence of the computed solutions is usually easy to test with respect to spatial and temporal resolution, but situations exist where reducing the timestep can make an asymptotic treatment of a "stiff" phenomenon less accurate rather than more accurate. This follows because the disparity of time scales between fast and slow phenomena is often exploited in the asymptotic approach rather than tolerated. Furthermore, the non-convergence of any particular solution is often easier to spot in timestep splitting with asymptotics because the manner of degradation is usually catastrophic. In kinetics calculations, lack of conservation of mass or atoms signals inaccuracy rather clearly. 

A large ratio between the maximum and the minimimi eigenvalue of the Jacobian indicates a stiff situation nevertheless, when a single well-conditioned differential equation is integrated on a large interval, it may also present the stiffness phenomenon if the adopted algorithm has a small stability region. 

The other striking feature of nanotubes is their extreme stiffness and mechanical strength. Such tubes can be bent to small radii and eventually buckled into extreme shapes which in any other material would be irreversible, but here are still in the elastic domain. This phenomenon has been both imaged by electron microscopy and simulated by molecular dynamics by lijima et al. (1996). Brittle and ductile behaviour of nanotubes in tension is examined by simulation (because of the impossibility of testing directly) by Nardelli et al. (1998). Hopes of exploiting the remarkable strength of nanotubes may be defeated by the difficulty of joining them to each other and to any other material. 

Prediction of the strength of fiber-reinforced composite materials has not achieved the near-esoteric levels of the stiffness predictions studied in the preceding sections. Nevertheless, there are many interesting physical models for the strength characteristics of a matrix reinforced by fibers. Most of the models represent a very high degree of integration of physical observation with the mechanical description of a phenomenon. 

Hypothyroid myopathy occurs in about 30% of patients with hypothyroidism irrespective of its cause. Muscle pain, cramps, and stiffness may be seen, and are often exacerbated by cold weather. Pseudomyotonic features of delayed muscle contraction and relaxation are common. Myoedema (the mounding phenomenon) is due to the painless, electrically silent contracture produced on direct percussion. Muscle biopsy often shows a predominance of type 1 (slow-twitch) fibers, again analogous to that seen in experimental hypothyroidism (Figure 22). Muscle hypertrophy with weakness and slowness of movement occurs in the Debre-Semelaigne syndrome seen in severely hypothyroid children, and Hoffman s syndrome is a similar condition seen in adults with hypothyroidism, but is also accompanied by painful spasms. 

However, not all properties are improved by filler. One notable feature of the mechanical behaviour of filled elastomers is the phenomenon of stresssoftening. This manifests itself as a loss of stiffness when the composite material is stretched and then unloaded. In a regime of repeated loading and unloading, it is found that part of the second stress-strain curve falls below the original curve (see Figure 7.13). This is the direct opposite of what happens to metals, and the underlying reasons for it are not yet fully understood. 

Stick-slip motion is another issue that has been explored using SFA. It is found that the occurrence of stick-slip depends on the sliding velocity and the stiffness of the system, and the mechanism of the phenomenon can be interpreted in terms of periodic transition between liquid and solid states of the conhned lubricant [40],... 

The results indicated that cross-link formation increased the bulk modulus of the system. As noted, cross-linking was a pressure-induced effect that was facilitated by a change in the coordination at zinc when the pressure reached 5 GPa. The observation that stiffening of the film is a pressure-induced phenomenon is consistent with the differences in the measured elastic properties of films found on the tops of the asperities and those found in the valleys between asperities as mentioned above. Basically, in real systems, pressures high enough to form stiff cross-linked films are achieved on top of the asperities, but they are not encountered between the asperities. 

Substances where the element of interest is directly bonded to hydrogen, or is part of a low-mass molecule, may not be as enriched in heavy isotopes as would be expected from rule (3). This phenomenon, usually of 2" order importance, is most pronounced for substances with stiff bonds and at low temperatures. 

Another significant helical amplification in optically active copolymers with preferential screw-sense helicity is known as the majority rule phenomenon [ 17,18]. In this case, the screw sense of a helical main chain bearing nonracemic chiral side groups is controlled by the ee only and a population of preferential screw-sense helicity and optical activity were nonlinearly amplified by ee of chiral side groups. Since Pino et al. first reported this phenomenon in poly-a-olefins made of vinyl co-monomers bearing nonracemic chiral moieties [21], this majority rule has already been established in stiff polyisocyanates bearing a nonracemic chiral side chain [17,18]. 

There is, however, a limit to the amount of stretch that can be given to a polymer because the phenomenon of necking can intervene and cause rupture of the fiber. In other words, in a polymeric fiber there is a limit to the modulus enhancement that can be obtained by subjecting it to an ever higher draw ratio. As we shall see in Chapter 4, this led to other means of obtaining high stiffness polymeric fibers such as aramid and polyethylene. 

At a given force, the elasticity of covalent bonds of the amino acid backbone gives rise to a length increase. But thermal fluctuations act on the backbone, which on an average pulls the cantilever closer to the membrane, a phenomenon referred to as entropic elasticity of linear polymers. The wormlike chain model [50] describes the polymer as an elastic rod with bending stiffness submitted to thermal fluctuations that decrease the end-to-end distance of the rod. Alternatively, the freely jointed chain model calculates the... 

---