Rate Dependent Yielding

For mild steel the tensile stress strain behavior in a constant strain-rate test is often approximated by two straight lines as shown earlier in Fig. 2.9(b) and here in Fig. 11.7. Here the permanent strain, Ep, after unloading is indicated and the total strain, Etot. at some arbitrary point is given by. 

It is well known that many materials have yield points that vary with strain rate. Notably mild steel has a significant variation in the yield stress with strain rate at high temperature, as do other metals. As a result, various rate dependent plasticity theories have been developed for metals and some of these have been extended to polymers. Early approaches used idealized stress-strain response such as that shown in Fig. 11.8 and Fig. 11.10. In Fig. 11.8 the material is assumed to be rigid but with a rate dependent 

A mechanical model to represent a rigid-plastic material is shown in Fig. 11.9(a). The model is simply a friction element that moves or slides only when the frictional resistance is overcome. Thus the constitutive equation for the friction element is 

Note that when the stress applied to the friction element reaches Oy, the strain increase is not defined and thus this element must be used in conjunction with other elements to define the strain (rate) changes. 

A rigid-elastic element is shown in Fig. 11.9(b) that moves linear elastically after friction is overcome. Stress-strain diagrams of the two materials are also shown in Fig. 11.9. 

---

One way to obtain long-term information is through the use of the time-temperature-superposition principle detailed in Chapter 7. Indeed, J. Lohr, (1965) (the California wine maker) while at the NASA Ames Research Center conducted constant strain rate tests from 0.003 to 300 min and from 15° C above the glass transition temperature to 100° C below the glass transition temperature to produce yield stress master curves for poly(methyl methacrylate), polystyrene, polyvinyl chloride, and polyethylene terephthalate. It should not be surprising that time or rate dependent yield (rupture) stress master curves can be developed as yield (rupture) is a single point on a correctly determined isochronous stress-strain curve. Whether linear or nonlinear, the stress is related to the strain through a modulus function at the yield point (mpture) location. As a result, a time dependent master curve for yield, rupture, or other failure parameters should be possible in the same way that a master curve of modulus is possible as demonstrated in Chapter 7 and 10. 

What is the purpose of the mathematical models rate dependent yield behavior. 

From stochastic molecnlar dynamics calcnlations on the same system, in the viscosity regime covered by the experiment, it appears that intra- and intennolecnlar energy flow occur on comparable time scales, which leads to the conclnsion that cyclohexane isomerization in liquid CS2 is an activated process [99]. Classical molecnlar dynamics calcnlations [104] also reprodnce the observed non-monotonic viscosity dependence of ic. Furthennore, they also yield a solvent contribntion to the free energy of activation for tlie isomerization reaction which in liquid CS, increases by abont 0.4 kJ moC when the solvent density is increased from 1.3 to 1.5 g cm T Tims the molecnlar dynamics calcnlations support the conclnsion that the high-pressure limit of this unimolecular reaction is not attained in liquid solntion at ambient pressure. It has to be remembered, though, that the analysis of the measnred isomerization rates depends critically on the estimated valne of... 

These reversible reactions are cataly2ed by bases or acids, such as 2iac chloride and aluminum isopropoxide, or by anion-exchange resias. Ultrasonic vibrations improve the reaction rate and yield. Reaction of aromatic aldehydes or ketones with nitroparaffins yields either the nitro alcohol or the nitro olefin, depending on the catalyst. Conjugated unsaturated aldehydes or ketones and nitroparaffins (Michael addition) yield nitro-substituted carbonyl compounds rather than nitro alcohols. Condensation with keto esters gives the substituted nitro alcohols (37) keto aldehydes react preferentially at the aldehyde function. 

Economics vary, depending onlocation and country. In Japan, for example, the quinone of choice is the disodium salt of l,4-dihydto-9,10-dihydtoxyanthtacene (DDA). Unlike AQ, DDA is direcdy soluble in white Hquor and thus readily penetrates the chip. AQ works well with both hardwoods and softwoods. Addition rates as low as 0.025% by weight give satisfactory increases (ca 1—2%) in pulping rate and yield, especially for hardwoods. In the United States in 1987, the FDA put a limitation to the amount of AQ that can be added to most pulp to 0.11%. This typically is the amount added to softwoods lesser amounts ate employed with hardwoods. 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

Constitutive relation An equation that relates the initial state to the final state of a material undergoing shock compression. This equation is a property of the material and distinguishes one material from another. In general it can be rate-dependent. It is combined with the jump conditions to yield the Hugoniot curve which is also material-dependent. The equation of state of a material is a constitutive equation for which the initial and final states are in thermodynamic equilibrium, and there are no rate-dependent variables. 

The data plotted in the figure clearly support the predicted positive dependence of crystal size on agitation rate. Precipitation in the crystal film both enhances mass transfer and depletes bulk solute concentration. Thus, in the clear film model plotted by broken lines, bulk crystal sizes are initially slightly smaller than those predicted by the crystal film model but quickly become much larger due to increased yield. Taken together, these data imply that while the initial mean crystal growth rate and mixing rate dependence of size are... 

From the weak dependence of ef on the surrounding medium viscosity, it was proposed that the activation energy for bond scission proceeds from the intramolecular friction between polymer segments rather than from the polymer-solvent interactions. Instead of the bulk viscosity, the rate of chain scission is now related to the internal viscosity of the molecular coil which is strain rate dependent and could reach a much higher value than r s during a fast transient deformation (Eqs. 17 and 18). This representation is similar to the large loops internal viscosity model proposed by de Gennes [38]. It fails, however, to predict the independence of the scission yield on solvent quality (if this proves to be correct). 

Kinetic examination of the methane yield shows behavior quite similar to that of methyl radical a pressure dependent yield of 0.406 molecule/100 e.v., a pressure independent yield of 0.126 molecule/100 e.v., and a rate constant ratio of kq/kf = 1.5 X 106 mole-1 cc. for the competing steps. 

The original patent uses platinum as the catalyst and calls for temperatures of 100-300° and pressures of 45-115 psi(47). We found that such rigorous conditions are not required for the hydrosilation reaction with most commercial sources of platinum on carbon. Usually vigorous stirring at slightly elevated temperatures, 40-80°, at 15 psi will give moderate yields of the product. The rates and yields are usually highly dependent on the method of preparation of the catalyst. However, ultrasound permits the reaction to occur at a useful rate at 30° at atmospheric pressure(48) ... 

Diphenylmethane reacts with dioxygen in the presence of potassium 1,1-dimethylethoxide in various solvents (dimethylformamide [DMF], hexamethylphosphoramide [HMPA], pyridine) to produce nearly 100% yields of benzophenone [284]. The adduct of benzophenone with dimethylsulfoxide (DMSO) [l,l-diphenyl-2-(methylsulfinyl)ethanol] is formed as the final product of the reaction. The stoichiometry of the reaction and the initial rate depends on the solvent (conditions 300 K, [Ph2CH2] = 0.1mol L [Me3COK] = 0.2mol L 1,p02 = 97kPa). 

Thus, the time traces can be fitted by an exponential equation and the linear concentration dependence of the observed rate constant yields the slope kon and the ordinate koff. 

Energy transfer [3,12] between molecules has also been used in the design of optical sensors. Here, an excited molecule (donor) can transfer its electronic energy to another species (acceptor). This process occurs without the appearance of a photon and results from dipole-dipole interaction between the donor-acceptor molecules. The rate depends on the fluorescence quantum yield of the donor, the overlap of the emission spectra of the donor with the absorption of the spectrum of the acceptor, and their relative orientation and distance. It is the overlap of... 

The photoablation behaviour of a number of polymers has been described with the aid of the moving interface model. The kinetics of ablation is characterized by the rate constant k and a laser beam attenuation by the desorbing products is quantified by the screening coefficient 6. The polymer structure strongly influences the ablation parameters and some general trends are inferred. The deposition rates and yields of the ablation products can also be precisely measured with the quartz crystal microbalance. The yields usually depend on fluence, wavelength, polymer structure and background pressure. 

However, in contrast to the VE system discussed in the previous section, we have no information on c for this system, but since the yield of ions from a given dose-rate depends mainly on the dielectric constant, it can be assumed that in the mixtures of aromatic hydrocarbons with which we are dealing here, the c and the rate constants are independent of m, so that we can use (5.10) and (5.14), treating c as constant. 

Mislow and co-workers (258) and Hammond (259) have shown that optically active diaryl sulfoxides, which are configurationally stable in the dark at 200°C, lose their optical activity after 1 hr at room temperature on irradiation with ultraviolet light. Similarly, an easy conversion of the trans isomer of thianthrene-5,10-oxide 206a into the thermodynamically more stable cis isomer takes place upon irradiation in dioxane for 2 hr. However, the behavior of a-naphthylethyl p-tolyl sulfoxide under comparable irradiation conditions is different, namely, it is completely decomposed after 4 min. These differences are not surprising because the photochemical racemization of diaryl sulfoxides occurs by way of the pyramidal inversion mechanism whereas decomposition of the latter sulfoxide occurs via a radical mechanism with the cleavage of the sulfur-carbon bond. It is interesting to note that photoracemization may be a zero-order process in which the rate depends only on the intensity of the radiation and on the quantum yield. 

The reaction path of thiamine-dependent catalysis is essentially unchanged in the presence of an apoenzyme, except that the enzyme active site residues increase reaction rates and yields and influence the substrate and product specificity. The X-ray crystal structures of TDP-dependent enzymes have clarified this view and permitted an understanding of the roles of the individual amino acids of the active site in activating and controlling the thiazolium reactivity [36-40]. 

The reaction between CH3CI and hydroxide ion to yield methanol and chloride ion follows a second order kinetics, i.e., the rate depends upon the concentration of both the reactants. 

---