Working equations uniaxial extension

It can be shown using Eq. (1-20) that the upper-convected Maxwell equation is equivalent to the Lodge integral equation, Eq. (3-24), with a single relaxation time. This is shown for the case of start-up of uniaxial extension in Worked Example 3.2. Thus, the simplest temporary network model with one relaxation time leads to the same constitutive equation for the polymer contribution to the stress as does the elastic dumbbell model. 

In order to avoid the necessity of measuring the stress-strain curves of the adhesive as a function of rate and temperature, we would like to have a constitutive equation for the adhesive, the parameters of which can be determined from relatively simple rheological measurements. Authors have taken multiple approaches to this problem. In some of the work, the adhesive stress-strain curves in uniaxial extension were measured which allows a direct comparison between the constitutive model and the data. In other cases, the model was used as a tool to... 

We can write this equation in different cocndinate systems using only the stress terms in Table 1.7.1. In the uniaxial extension and simple shear examples, which we worked for the neo-Hookean solid, the stress was homogeneous, so all its derivatives are zero and eq. 1.7.17 is satisfied. However, with more complex sluq)es, such as twisting of a cylinder shown in Figure 1.7.3, the stresses do vary across the sample and die stress balance is required to solve for the tractions on die surface. 

TABLE 7.2.1 / Working Equations for Uniaxial Extension Tensile strain... 

As discussed in the preceding section, = en and 22 = 33, and since T22 = t33, there is only one viscosity function in uniaxial extension. Table 7.2.1 summarizes the working equations for uniaxial extensional iheometry. 

While these functions have been adjusted to describe shear and uniaxial extensional flows, they seem to work poorly for planar extension of LDPE (Samurkas et al. 1989). Planar extensional flow represents a particularly difficult test for K-BKZ-type constitutive equations, since fits to shear data fix all the model parameters required for planar extension, and there is therefore no wiggle room left to obtain a fit to the latter. (This is because I = I2 in both shear and planar extension.) A recent non-K-BKZ molecular constitutive equation derived from reptation-related ideas shows improved qualitative agreement with planar extensional data (McLeish and Larson 1998). 

Rivlin and Saunders carried out experiments in simple tension, torsion, pure shear and pure shear superposed on simple extension to extend the range of combinations of principal extension ratio. This showed behaviour generally consistent with Equation (3.61). Their work illustrates the importance of exploring a wide range of combinations of extension ratio to establish the function U. The traditional form of materials test - the uniaxial stretch -involves a very specific mode of deformation. Finding materials parameters by least squares... 

Usually, sealants and adhesive materials for construction applications are evaluated by looking at the engineering side, butnotthe chemistry of the material. As a result, only tests that measure the mechanical properties are used. Most of the studies on the viscoelastic properties use traditional tests such as tensile testing to obtain data, which can be used in complicated mathematical equations to obtain information on the viscoelastic properties of a material. For example, Tock and co-workers studied the viscoelastic properties of stmctural silicone rubber sealants. According to the author, the behavior of silicone mbber materials subjected to uniaxial stress fields carmotbe predicted by classical mechanical theory which is based on linear stress-strain relationship. Nor do theories based on ideal elastomers concepts work well when extensions exceed... 

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