Equi-biaxial stretch

Here there are two equal stretches present, in the 1-2 plane and 

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Rajeev RS, Harkin-Jones E, Soon K, McNally T, Menary G, Armstrong CG, Martin PJ. Studies on the effect of equi-biaxial stretching on the exfoliation of nanoclays in polyethylene terephthalate. Eur Polym J 2009 45 332-340. 

Fig. 9 Engineering stress-strain curves for the stretched films (a)uniaxial, MD (b) uniaxial, TD (c)equi-biaxial stretching...

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.

Fig. 46 a Stress contributions of the strained filler clusters for the different pre-strains (upper part), obtained as in Fig. 45b. The solid lines are adapted with the integral term of Eq. (47) and the log-normal cluster size distribution Eq. (55), shown in die lower part. The obtained parameters of the filler clusters are Qe /d 3=26 MPa, =25, and b=0.8. b Uniaxial stress-strain data (symbols) as in Fig. 45c. The insert shows a magnification for the smaller strains, which also includes equi-biaxial data for the first stretching cycle. The lines are simulation curves with the log-normal cluster size distribution Eq. (55) and material parameters as specified in the insert of Fig. 45a and Table 4, sample type C40...

Here, as in all following sub-sections, except for equi-biaxial deformation, the principal direction of stretch is the 2-direction, and constancy of volume given by eq. (6.27) applies. 

Takei, A., Jin, L., Hutchinson, J.W. Ridge localizations and networks in thin films compressed by the incremental release of a large equi-biaxial pre-stretch in the substrate. Adv. Mater. 26,... 

Fig. 3.6. Comparison of Gaussian theory of elasticity with experimental data, (i) Compression and low extension— the solid line indicates the theoretical result (ii) two-dimensional equi-biaxial extension. In the upper curve, the origin is displaced and indicates hysteresis when the sample stretches to A2=3 (iii) simple extension to break (curve (a)). Curves (b) and (c) are hysteresis curves with displaced origins. (From Treloar, 1944.)...

Some measurements regarding biaxial extension have involved the direct stretching of a sheet of sample in two perpendicular directions within its plane, by two independently variable amounts. In the equi-biaxial case, the deformation is equivalent to compression. A good account of such experimental results [131] has been... 

Crystallinity of the stretched films were reached its maximum(40%) at the highest uniaxial stretching of 300%, and maximum(34%) at the equi-biaxal stretching of 200%. At above 200% for the biaxial stretching, it didn t show significant change. 

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