Biaxial extension/stretching

Blatz and Ko19) also devised an apparatus which allows strip-biaxial extension testing to be performed (Fig. 6). Here one of the stretch ratios, say X2, is held at unity while the sample sheet is extended in the direction of the stretch ratio Xj. 

Another apparatus for general biaxial extension testing, illustrated in Fig. 8, was built at our laboratory27). A square sheet of rubber, 11.5 cm long and 1 mm thick, is clamped at the edges by several pieces of movable chucks and stretched biaxially by means of the two bars that can be displaced separately bade and forth by a servo-meachnism. Tensile forces acting on the sheet in the stretching directions are transmitted to the bars and measured by two transducers mounted at the ends of each bar. After many modifications and improvements on the mechanical and electric parts,... 

There are currently no ISO standard methods for biaxial extension and such measurements are rarely made in industrial laboratories. However, biaxial stressing is of value in the consideration of the theory of elasticity and is preferred by many for producing data for input to finite element programmes, as well as being involved in certain practical applications of rubber. The British standard for finite element analysis on rubber19 outlines the two approaches, equibiaxial stretching of a flat sheet and inflation of a flat sheet. The principles of these are illustrated in Figure 8.14. 

Compression of a weakly structured food between parallel plates may achieve squeezing flow (Steffe, 1996). When lubricated parallel plates are used, the result is a form of biaxial extension. Biaxial extension may be used to measure biaxial viscosity, which is a reflection of resistance to radial stretching flow in a plane. Lubricated squeezing flow of a semi-solid... 

An elongational or extensional viscosity (%) develops as a result of a conformational transition when disperse systems are forced through constrictions, or compressed or stretched (Kulicke and Haas, 1984 Rinaudo, 1988 Barnes et al., 1989 Odell et al., 1989 Clark, 1992). The intuitive logic is that the random coils resist the initial distortion. % is believed to elicit the human sensation of stringiness (Clark, 1995). If shear viscosity is denoted iq, rheologists define a Trouton ratio as %/ti, wherein % > T) by a factor approximating 3 for uniaxial extension and 6 for biaxial extension. Alternatively stated, the Newtonian ly calculates to one-third to one-sixth % (Steffe, 1992). 

Another axisymmetric extensional flow is biaxial extension. In this deformation, there is a compression along the axis of symmetry that stretches in the radial direction, as shown in Fig. 9. The principal strain rate is defined as... 

Only crystallization induced by a tensile type deformation has been discussed here. Other types of deformation such as biaxial extension, shear and torsion should also be considered. Such deformations have been studied and analyzed for amorphous networks. However, there is a paucity of experimental data, as well as analysis, of the equilibrium aspects of crystallization induced by these deformations. In one available report the observed melting temperature of natural rubber networks increased substantially when subject to biaxial deformation.(41) An increase in melting temperature of about 50 °C was found for a biaxial stretching ratio of three. This increase is much larger than that observed for natural rubber when crystallized in simple extension. 

Uniaxial extension is not the only kind of stretching flow that one can visualize. If Cj = 2 = Cfi, 3 = 2 b, and 0, one has equal biaxial extension physically this can be realized in a sheet-stretching experiment. If one dimension of the sheet is constrained not to change so that 2, say, is zero, then 3 = i. This is called planar extension or pure shear. 

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... 

These remarks about reaching a steady state apply not only to uniaxial extensional flows, data for which appear in Figure 4.2.5, but for other extensional flows as well. Besides uniaxial extension, the two most important extensional flows are equal biaxial extension and planar extension. Kinematic tensors for these extensional flows were to have been found in Exercise 2.8.1. In uniaxial extension the material is stretched in one direction and compressed equally in the other two in equal biaxial extension the material is stretched equally in two directions and compressed in the third and in planar extension the material is stretched in one direction, held... 

Figure 14.20 Fracture behavior map for SPS films (fracture type by film annealing temperature [heat set] and biaxial extension ratios, 1-mm thick SPS homopolymer film biaxially stretched at 110°C at lOmm/s [19]).

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 ... 

Biaxial extension is another example of a nonviscometric deformation. It can be achieved, for example, by simultaneously stretching a sheet of material in its length and width directions or by blowing up a balloon. It plays a prominent role in the blow-molding process (Section 19.4). Show the components of the deviatoric stress and the rate-of-strain tensors for the biaxial extension of an incompressible material. What material function(s) would be needed to describe e equilibrium biaxial extension of a homogeneous fluid ... 

Flow is generally classified as shear flow and extensional flow [2]. Simple shear flow is further divided into two categories Steady and unsteady shear flow. Extensional flow also could be steady and unsteady however, it is very difficult to measure steady extensional flow. Unsteady flow conditions are quite often measured. Extensional flow differs from both steady and unsteady simple shear flows in that it is a shear free flow. In extensional flow, the volume of a fluid element must remain constant. Extensional flow can be visualized as occurring when a material is longitudinally stretched as, for example, in fibre spinning. When extension occurs in a single direction, the related flow is termed uniaxial extensional flow. Extension of polymers or fibers can occur in two directions simultaneously, and hence the flow is referred as biaxial extensional or planar extensional flow. 

The extensional flow is a deformation that involves stretching along streamlines. According to the resulting deformation, it can be classified as uniaxial, biaxial, or planar extension. 

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