Rubber stretched, uniaxially

Using the equation given in Problem 2.22, find the orientation ctor / for a rubber stretched uniaxially to A = 3.0. 

The samples we used were vulcanizates of natural rubber (NR) and styrene-butadiene copolymer rubbers (SBR), carbon-filled and unfilled. Table 1 summarizes their preparative data. Incompressibility of these vulcanizates and some other vulcanizates were checked by dipping, stretching uniaxially, and weighing a specimen in water. 

In the field of rubber elasticity both experimentalists and theoreticians have mainly concentrated on the equilibrium stress-strain relation of these materials, i e on the stress as a function of strain at infinite time after the imposition of the strain > This approach is obviously impossible for polymer melts Another complication which has thwarted the comparison of stress-strain relations for networks and melts is that cross-linked networks can be stretched uniaxially more easily, because of their high elasticity, than polymer melts On the other hand, polymer melts can be subjected to large shear strains and networks cannot because of slippage at the shearing surface at relatively low strains These seem to be the main reasons why up to some time ago no experimental results were available to compare the nonlinear viscoelastic behaviour of these two types of material Yet, in the last decade, apparatuses have been built to measure the simple extension properties of polymer melts >. It has thus become possible to compare the stress-strain relation at large uniaxial extension of cross-linked rubbers and polymer melts ... 

Blatz and Ko19 added a special attachment to their uniaxial tensile testing equipment (Fig. 5). In this, a sheet specimen of rubber is stretched by chucks attached to its four edges, but, differing from the apparatus of Rivlin and Saunders, the chucks can be moved smoothly on the rigid tracks so that it is possible to strech... 

H and 2H NMR have been used in styrene-butadiene rubber (SBR) with and without carbon-black fillers to estimate the values of some network parameters, namely the average network chain length N. The values obtained from both approaches were checked to make sure that they were consistent with each other and with the results of other methods [71, 72, 73]. To this purpose, a series of samples with various filler contents and/or crosslink densities were swollen with deuterated benzene. The slopes P=A/ X2-X 1) obtained on deuterated benzene in uniaxially stretched samples were measured. The slopes increase significantly with the filler content, which suggests that filler particles act as effective junction points [72, 73]. 

For filler reinforced rubbers, both contributions of the free energy density Eq. (35) have to be considered and the strain amplification factor X, given by Eq. (39) differs from one. The nominal stress contributions of the cluster deformation are determined by oAtfJ=dWA/dzA, where the sum over all stretching directions, that differ for the up- and down cycle, have to be considered. For uniaxial deformations E =e, E2=Ej= +E) m- one obtains a positive contribution to the total nominal stress in stretching direction for the up-cycle if Eqs. (29)-(36) are used ... 

Figure 45a-c shows an adaptation of the developed model to uniaxial stress-strain data of a pre-conditioned S-SBR-sample filled with 40 phr N220. The fits are obtained for the third stretching cycles at various prestrains by referring to Eqs. (38), (44), and (47) with different but constant strain amplification factors X=Xmax for every pre-strain. For illustrating the fitting procedure, the adaptation is performed in three steps. Since the evaluation of the nominal stress contribution of the strained filler clusters by the integral in Eq. (47) requires the nominal stress aR>1 of the rubber matrix, this quantity is developed in the first step shown in Fig. 45a. It is obtained by demanding an intersection of the simulated curves according to Eqs. (38) and (44) with the measured ones at maximum strain of each strain cycle, where all fragile filler clusters are broken and hence the stress contribution of the strained filler clusters vanishes. The adapted polymer parameters are Gc=0.176 MPa and neITe= 100, independent of pre-strain. According to the considerations at the end of Sect. 5.2.2, the tube constraint modulus is kept fixed at the value Ge=0.2 MPa, which is determined by the plateau modulus Gn° 0.4 MPa [174, 175] of the uncross-linked S-SBR-melt (Ge=l/2GN°). The adapted amplification factors Xmax for the different pre-strains ( max=l> 1-5, 2, 2.5, 3) are listed in the insert of Fig. 45a.

Figure 46a shows that a reasonable adaptation of the stress contribution of the clusters can be obtained, if the distribution function Eq. (55) with mean cluster size =25 and distribution width b= 0.8 is used. Obviously, the form of the distribution function is roughly the same as the one in Fig. 45b. The simulation curve of the first uniaxial stretching cycle now fits much better to the experimental data than in Fig. 45c. Furthermore, a fair simulation is also obtained for the equi-biaxial measurement data, implying that the plausibility criterion , discussed at the end of Sect. 5.2.2, is also fulfilled for the present model of filler reinforced rubbers. Note that for the simulation of the equi-biaxial stress-strain curve, Eq. (47) is used together with Eqs. (45) and (38). The strain amplification factor X(E) is evaluated by referring to Eqs. (53) and (54) with i= 2= and 3=(l+ ) 2-l.

We now consider an extensional deformation of an incompressible rubber network (Fig. 3-7), where the stretch axes are oriented along the coordinate axes apd where the stretch ratios X, k2, and I3 are in directions 1, 2, and 3, respectively. For the example in Fig. 3-7, the deformation is a uniaxial extension that increases the length of the cylinder by a factor of X1 over its initial length. By volume conservation, the radius of the cylinder then shrinks to times the original radius. If the cross-link points are convected with... 

Figure 3.7 Cross-linked rubber network before (left) and after (right) a uniaxial extensional deformation. The stretching of the polymer strands connecting cross-link points is affine—that is, directly proportional to the macroscopic deformation. (From Larson 1988.)...

A rubber band with molar mass between crosslinks Afs = 3000 g mop is uniaxially stretched to three times its original length. After achieving equilibrium at 21 °C it is allowed to contract adiabatically back to the unstretched state. 

It has been shown that the life time in the creep process of rubbery polymers scatters largeley but obeys the specified statistical distribution which are introduced theoretically based on some assumptions. Two assumptions are made here that "one crack leads the body to failure" and "the v th crack leads the body to failure". The former assumption leads the exponential distribution of tg, and the latter the unimodal distribution when v>2. It has been explained from experiments that the distribution of tg for pure rubbers of vulcanized SBR and NR are the exponential, type and for filled systems the unimodal type. Theory introduced here can be applied not only to the creep failiire but also to the failure process varing stress level such as uniaxial extension with constant strain rate. It has been demonstrated that the distribution of Xg, the stretch ratio at breedc in the constant rate of extension, is well estimated by the theory substituting the parameters n and c which are obtained from creep failure experiment to eq(l9). 

Stress-strain measurements at uniaxial extension are the most frequently performed experiments on stress-strain behaviour, and the typical deviations from the phantom network behaviour, which can be observed in many experiments, provided the most important motivation for the development of theories of real networks. However, it has turned out that the stress-strain relations in uniaxial deformation are unable to distinguish between different models. This can be demonstrated by comparing Eqs. (49) and (54) with precise experimental data of Kawabata et al. on uniaxially stretched natural rubber crosslinked with sulphur. The corresponding stress-strain curves and the experimental points are shown in Fig. 4. The predictions of both... 

Such ideas formed the basis of the Kuhn and Grun model for the stress-optical behaviour of rubbers. The birefringence of a uniaxially stretched rubber is given by... 

The choice of fluorescent probe depends on a variety of factors. It has already been pointed out that what is determined directly is information which characterises the distribution of orientations of the fluorescent molecules. The ideal experiment would be one in which the polymer molecules themselves contained fluorescent groups. Stein has considered the theory of the fluorescence method specifically for a uniaxially oriented fluorescent rubber but no experiments to study orientation have been reported for such a system. Nishijima et al have, however, made some qualitative observations on the polarisation of the fluorescent light from polyvinylchloride films which had been first stretched and then irradiated with light of wavelength 185 nm to produce fluorescent polyene segments. 

Nominal stress versus den ion ratio, for uniaxial stretching of a sample of crosslinked natural rubber. Full line shows the Gaussian prediction (eqn 3.38). 

Uniaxial Extension. A rubber strip of original length Lo is stretched uni-axially to a length L, as illustrated in Figure 1. The stretch and elongation are AL = L — Lq and k = LILo, respectively. The strain e (also known as the relative deformation, linear dilation, or extension) and the elongation or extension ratio X are related by... 

A stretched rubber sample subjected to uniaxial load contracts reversibly on heating. 

The theory of Bernstein, Kearsley and Zapas [20] and developments of it (e.g. Zapas and Craft [21]) - so-called BKZ theories - are aimed in particular at large deformation behaviour. The Gaussian model of rubber elasticity tells us that in uniaxial stretching the true stress o is in the form... 

A stress-strain isotherm for the uniaxial deformation of natural rubber, at ambient temperature, that was cross-linked in the liquid state is shown in Fig. 8.1.(5) Here f is the nominal stress defined as the tensile force,/, in the stretching direction divided by the initial cross-section, and a is the extension ratio. Using the most rudimentary form of molecular rubber elasticity theory f can be expressed as (6-9)... 

THE NATURE OF THE MELTING TRANSITION IN A UNIAXIALLY STRETCHED RUBBER SAMPLE... 

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