Equibiaxial extension

Figure 3. Modulus contributions from chemical cross-links (Cx, filled triangles) and from chain entangling (Gx, unfilled symbols) plotted against the extension ratio during cross-linking, A0, for 1,2-polybutadiene. Key O, GN, equibiaxial extension , G.v, pure shear A, Gx, simple extension Gx°, pseudo-equilibrium rubber plateau modulus for a polybutadiene with a similar microstructure. See Ref. 10.

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

Prediction of the second normal stress difference in shear and thermodynamic consistency obviously requires the use of a different strain measure including of the Cauchy strain tensor in the form of the K-BKZ model. With the ratio of second to first normal stress difference as a new parameter, Wagner and Demarmels [32] have shown that this is also necessary for accurate prediction of other flow situations such as equibiaxial extension, for example. 

Rg. 6.7 Equibiaxial extension. The stresses a, and original unit square of material of thickness t cause it to deform to the dimensions shown. 

Hi) Inflation of a Thin-Walled Spherical Balloon In this case, if the balloon radius expands by a factor k, equibiaxial extensions of ratio k will be set up in the balloon, with a shrinkage ratio 1 /k of the wall thickness to maintain the rubber volume constant. The circumferential stresses ti and t2 are equal and given by... 

Studies of bimodal networks have focused elongation because of the simplicity of this type of deformation. Examples of other deformations, equibiaxial extension (compression), shear, and torsion, are discussed later. 

There are numerous other deformations of interest, including compression, biaxial extension, shear and torsion. - Equibiaxial extension was obtained by inflating sheets of unimodal and bimodal networks of PDMS. Upturns in the modulus occur at high biaxial extensions, as expected. Pronounced maxima precede the upturns (figure 7.22), which is yet to be explained by molecular theories. 

Another relationship treats biaxial extension (50). If equibiaxial extension is assumed, such as in a spherical rubber balloon, then... 

TABLE 7J.1 / Working Equations for Compression (Equibiaxial Extension)... 

We note that although equal biaxial extension is just the reverse of uniaxial, the invariants of B are different. Therefore we would expect material functions measured in each deformation to be different in general. Another common approach to equibiaxial extension is to let Ob = and Cfr = 2i, basing length change on the sides rather than the thickness of the samples. 

Another stretching flow that has been used to characterize the nonlinear behavior of melts is equibiaxial extension, usually called simply biaxial extension. This flow can be generated by clamping a circular sample around its rim and stretching it radially, as demonstrated by Hachmann and Meissner [160]. The biaxial strain gg is given by ... 

For small or slow equibiaxial extension, the biaxial start-up function can be found using the Boltzmann superposition principle ... 

To generate equibiaxial extension, sheet inflation, lubricated squeeze flow, and the rotary clamp technique have all been used [9, p. 261]. The most rehable of these is the one based on the rotary clamp technique, and the latest version of this instrument has been described by Hachman and Meissner [160]. This impressive instrument is presently the only rheometer capable of generating homogeneous biaxial extension, but it is somewhat complex and difficult to use. Lubricated squeeze flow is much simpler, but there are important limitations on its capabilities due the difficulty of maintaining lubrication [220].Isakief a/. [161] used lubricated squeeze flow to carry out stress relaxation experiments for determination of the damping... 

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