Nonlinear modulus

Then, equations 88 and 91 give the following expression for the nonlinear modulus ... 

The design of shape-memory devices is quite different from that of conventional alloys. These materials are nonlinear, have properties that are very temperature-dependent, including an elastic modulus that not only increases with increasing temperature, but can change by a large factor over a small temperature span. This difficulty in design has been addressed as a result of the demands made in the design of compHcated smart and adaptive stmctures. Informative references on all aspects of SMAs are available (7—9). 

Cord materials such as nylon, polyester, and steel wire conventionally used in tires are twisted and therefore exhibit a nonlinear stress—strain relationship. The cord is twisted to provide reduced bending stiffness and achieve high fatigue performance for cord—mbber composite stmcture. The detrimental effect of cord twist is reduced tensile strength. Analytical studies on the deformation of twisted cords and steel wire cables are available (22,56—59). The tensile modulus E of the twisted cord having diameter D and pitchp is expressed as follows (60) ... 

The bulk modulus of an ideal SWNT crystal in the plane perpendicular to the axis of the tubes can also be calculated as shown by Tersoff and Ruoff and is proportional to for tubes of less than 1.0 nm diameter[17]. For larger diameters, where tube deformation is important, the bulk modulus becomes independent of D and is quite low. Since modulus is independent of D, close-packed large D tubes will provide a very low density material without change of the bulk modulus. However, since the modulus is highly nonlinear, the modulus rapidly increases with increasing pressure. These quantities need to be measured in the near future. 

A typical behavior of amplitude dependence of the components of dynamic modulus is shown in Fig. 14. Obviously, even for very small amplitudes A it is difficult to speak firmly about a limiting (for A -> 0) value of G, the more so that the behavior of the G (A) dependence and, respectively, extrapolation method to A = 0 are unknown. Moreover, in a nonlinear region (i.e. when a dynamic modulus depends on deformation amplitude) the concept itself on a dynamic modulus becomes in general not very clear and definite. 

Under increasing strain the propint volume increases from the voids created around the unbonded solid particles. Nonlinearities in Young s modulus and Poisson s ratio then occur. Francis (Ref 50) shows this effect for a carboxy-terminated polybutadiene composite propellant with 14% binder as in Figure 12. He concludes that nonlinearities in low-temperature properties reduce the predicted stress and strain values upon cooling a solid motor, and therefore a structural analysis that neglects these effects will be conservative. However, when the predictions are extended to a pressurized fiberglas motor case, the nonlinearities in properties produce greater strains than those predicted with linear analysis... 

FIGURE 30.22 Mixing silica-filled compound Complex modulus versus strain amplitude as modeled with Equation 30.3 typical changes in nonlinear viscoelastic features along the mixing hne. 

If the Boltzmann superposition principle holds, the creep strain is directly proportional to the stress at any given time, f Similarly, the stress at any given lime is directly proportional to the strain in stress relaxation. That is. the creep compliance and the stress relaxation modulus arc independent of the stress and slrai . respectively. This is generally true for small stresses or strains, but the principle is not exact. If large loads are applied in creep experiments or large strains in stress relaxation, as can occur in practical structural applications, nonlinear effects come into play. One result is that the response (0 l,r relaxation times can also change, and so can ar... 

Using this factorizability of response into a time-dependent and a strain-dependent function. Landel et ai. (61,62) have proposed a theory that would express tensile stress relaxation in the nonlinear regime as the product of a time-dependent modulus and a function of the strain ... 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

The underlying nonlinearity function m (A). which is independent of the type of deformation, is very similar for different amorphous rubbers. For SBR, it is independent of the cross-link" density over moderate changes in cross-link density (62) and independent of the temperature down to —40°C, a temperature where the modulus has increased by a factor of 2 to 3 over the room-temperature value (61). The function A) is insensitive to the presence of moderate amounts of carbon black filler for strains up to about 100% (63). 

For elastomers, factorizability holds out to large strains (57,58). For glassy and crystalline polymers the data confirm what would be expected from stress relaxation—beyond the linear range the creep depends on the stress level. In some cases, factorizability holds over only limited ranges of stress or time scale. One way of describing this nonlinear behavior in uniaxial tensile creep, especially for high modulus/low creep polymers, is by a power... 

Orientation effects are strongly coupled to nonlinear behavior, discussed in Section V, and the stress-strain response discussed in Chapter 5, Orientation makes an initially isotropic polymer anisotropic so that five or nine modulus/compliance values arc required to describe the linear response instead of two, as discussed in Chapter 2. For an initially anisotropic polymer the various modulus/compliance components can be altered by the orientation. It may not be necessary to know all components for an... 

Cauchy-integral method, 219-220 cyclic wave functions, 224-228 modulus and phase, 214-215 modulus-phase relations, 217-218 near-adiabatic limit, 220-224 reciprocal relations, 215-217, 232-233 wave packets, 228-232 multidegenerate nonlinear coupling,... 

This approach is analytically correct for isothermal reactors and first-order rate laws, since concentration does not appear in the expression for the Thiele modulus. For other (nonlinear) rate laws, concentration changes along the reactor affect the Thiele modulus, and hence produce changes in the local effectiveness factor, even if the reaction is isothermal. Problem 21-15 uses an average effectiveness factor as an approximation. 

Aromatic poly(benzothiazole)s are thermally and thermooxidatively stable and have outstanding chemical resistance and third-order nonlinear optical susceptibility. Aromatic poly(benzothiazole)s can be spun into highly-oriented ultrahigh strength and ultrahigh modulus fibers. However, this type of polymer is insoluble in most organic solvents. Therefore, hexafluoroisopropylidene units are introduced in the polymer backbone to obtain soluble or processable aromatic poly(benzothiazole)s. 

Concret does not have well defined elastic and plastic regions due to its brittle nature. A maximum compressive stress value is reached at relatively low strains and is maintained for small deformations until crushing occurs. The stress-strain relationship for concrete is a nonlinear curve. Thus, the elastic modulus varies continuously with strain. The secant modulus at service load is normally used to define a single value for the modulus of elasticity. This procedure is given in most concrete texts. Masonry lias a stress-strain diagram similar to concrete but is typically of lower compressive strength and modulus of elasticity. 

The concentration and temperature Tg will, for example, be conditions of reactant concentration and temperature in the bulk gas at some point within a catalytic reactor. Because both c g and Tg will vary with position in a reactor in which there is significant conversion, eqns. (1) and (15) have to be coupled with equations describing the reactor environment (see Sect. 6) for the purpose of commerical reactor design. Because of the nonlinearity of the equations, the problem can only be solved in this form by numerical techniques [5, 6]. However, an approximation may be made which gives an asymptotically exact solution [7] or, alternatively, the exponential function of temperature may be expanded to give equations which can be solved analytically [8, 9]. A convenient solution to the problem may be presented in the form of families of curves for the effectiveness factor as a function of the Thiele modulus. Figure 3 shows these curves for the case of a first-order irreversible reaction occurring in spherical catalyst particles. Two additional independent dimensionless paramters are introduced into the problem and these are defined as... 

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