Strain function

The crystal formed by any given strain of crystalliferous bacterium is a constant and a strain function. Constancy of insect susceptibility spectra for given bacillus strains tells us this, and differences between strains indicate uniqueness. Krywienczyk and Angus (22) demonstrated that dissolved endotoxins of strains of B. thurin-... 

The strain is given by the relative length or elongation a = L/L where L and are the lengths of the sample in the deformed and undeformed states, respectively. Dividing the stress f/A by the strain function (a - a- ) indicated in the simplest molecular theories of rubberlike elasticity (38,39) then gives the elastic modulus or reduced stress (38-42)... 

Factorizability has also been found to apply to polymer solutions and melts in that both constant rate of shear and dynamic shear results can be analyzed in terms of the linear viscoelastic response and a strain function. The latter has been called a damping function (67,68). 

Thus semi-intuitively we can write down our strain function ... 

Note 3 For elastomers, which are assumed incompressible, the modulus is often evaluated in uniaxial tensile or compressive deformation using X - as the strain function (where X is the uniaxial deformation ratio). In the limit of zero deformation the shear modulus is evaluated as... 

The choice of a suitable initiator represents an important step in creating a well-defined polymerization system in terms of initiation efficiency and confrol over propagation. The entire system can only be designed on a stoichiometric base when a quantitative and fast initiation occurs. This is of enormous importance, because the composition of the entire polymerization mixture needs to be varied within small increments in order to control the microstructure. The catalyst needs to be carefully selected from both chemical and practical points of view. Schrock [5,10,12,109,110] and Grubbs systems [6], both highly active in the ROMP of strained functionalized olefins, can offen be used. Since fhe preparafion and in particu-... 

In Eq. (1), a is the equilibrium stress (Nm 2) supported by the swollen specimen a is the stretched specimen length divided by the unstretched length (extension ratio) v2 is the volume fraction of dry protein and p is the density of dry protein. In the common case of tetrafunctional crosslinks, the concentration of network chains n (mol network chains/g polymer) is exactly one-half the concentration of crosslinks, so that n = 2c. The hypothesis that a specimen behaves as if it were an ideal rubber can be confirmed by observing a linear relation with zero intercept between a and the strain function (a — 1/a2) and by establishing a direct proportionality between a and the absolute temperature at constant value of the extension ratio, as stipulated by Eq. (1). 

Theory also predicts the reduced stress or modulus [/ ], defined as the ratio of the nominal stress to the strain function (a - or2), to be independent of the elongation a. Experimentally, however, the modulus is found to change with a, generally decreasing linearly with decreasing reciprocal elongation. 

Equation 15.6 shows that the splitting is indeed proportional to the strain function (A -Ar1). Thus, the induced orientational order parameter (see Equation 15.1) may be characterised by the slope P of the doublet splitting A versus (A -Ar1). Experimentally, in PDMS model networks, u amounts to a few 10"1 [23, 25]. 

In these systems, the 2H20 splitting is not linear in the strain function (A,2-A, 1) (Figure 15.16). This indicates that the microscopic deformation ratio Xt (at the chain... 

The TICA specimen preparation procedure has been described elsewhere (4). The mechanical measurements were made with the Rheometrics Mechanical Spectrometer (RMS) which measures the in-phase and out-of-phase stress response (a and b component respectively) of a specimen being subjected to a sinusoidal shear strain. The instrumental set-up was reported by Lee (6). The frequency of the strain function was kept constant at 1.6 Hz (10 rad/sec). All temperature scan experiments were scanned at 2 C/min rate. The temperature was scanned down at the same rate when the maximum temperature was reached. 

It is worth mentioning that the strain function is not temperature dependent and that the influence of temperature is only applied on the memory function or relaxation modulus through the shortening of the relaxation times with increasing temperatures. 

In terms of network analogy, the damping function may be viewed as the expression of the retraction of the strands as compared to the continuum. The Lodge model thus corresponds to no retraction (affine deformation, a=0 in equation (30) ), the Doi-Edwards equation corresponds to complete retraction (a=0.2), whereas incomplete retraction makes the damping function more softly decreasing (0 < a < 0.2). In the later cases, the deformation is non-affine since there is a difference between that of the continuum and that of the network strands. Wagner [33] showed that the Doi Edwards strain function... 

A comparison of Equations (15) and (16) suggests two distinct interpretations. In retaining the rubber strain function, it may be assumed that N increases with X., according to... 

The validity of Equation (16) being established, the craze initiation data may be analyzed according to Equation (19). The plot. In X, versus X,i, presented in Figure 15, demonstrates that Equation (19) gives a satisfactory description of intrinsic craze initiation. In view of the assumptions made above, the value of k = 0.66 calculated from the slope of the straight line is in sufficient agreement with the value of X = 0.50 determined from Equation (16). A possible reason for the fact that both values are not identical may be that a modified rubber strain function should be introduced. If this function causes AOj to increase more rapidly withX,j as compared to the normal type of rubber strain function, the fit of Equation (16) to the experimental data will yield a higher value of k. The rubber strain function of Treloar, for example, is of that type. 

Let us consider now the response of the solid standard model located on the right-hand side of Figure 10.6 to a step strain function (Fig. 10.7). In this case,... 

An improved swelling/shrinking strain function combined with an increased thermal expansion of the bentonite giving a good match of the mechanical (stress, strain) behavior of the buffer by the KTH/SKI team. 

We have used the generalized phenomenological Maxwell model or Boltzmann s superposition principle to obtain the basic equation (Eq. (4.22) or (4.23)) for describing linear viscoelastic behavior. For the kind of polymeric liquid studied in this book, this basic equation has been well tested by experimental measurements of viscoelastic responses to different rate-of-strain histories in the linear region. There are several types of rate-of-strain functions A(t) which have often been used to evaluate the viscoelastic properties of the polymer. These different viscoelastic quantities, obtained from different kinds of measurements, are related through the relaxation modulus G t). In the following sections, we shall show how these different viscoelastic quantities are expressed in terms of G(t) by using Eq. (4.22). 

Fig. 8.20 The elastomeric stress-strain curve of PET at 353 K, re-plotted against the Gaussian strain function g X), showing near-ideal rubbery behavior. The slope suggests an entanglement molecular weight of Me = 2342 g/mole.

Note that the initial condition for y depends on the straining function Xj, which is still unknown at this stage. This will be determined during the course of analysis, and the Lighthill s requirement is that the subsequent term not be more singular than the preceeding term. 

To simplify the discussion of rubber elasticity, only uniaxial deformation will be considered in this chapter. More complicated strain functions will be considered in the chapter on gels. Consider a uniaxial deformation in the x-direction. It is convenient to define a deformation ratio ... 

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