Uniaxial extension worked examples

To solve Eq. (3-24) for a particular flow history, one must compute the tensor B and carry out the integration. An example of how this is done for start-up of steady uniaxial extension is given in Worked Example 3.1 at the end of this chapter. 

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. 

Problem 6.8 (Worked Example) Estimate the steady-state uniaxial viscosity of a suspension of 0.1% by volume of rod-like particles L = 6 yum long and d = 10 nm in diameter in a Newtonian oil of viscosity 100 P at an extension rate of 1 sec . ... 

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. 

This important result demonstrates the value of the tensor form of Newton s viscosity law. It is directly analogous to the result in Chapter 1, that the tensile modulus is three times the shear modulus, eq. 1.5.11. The three times rule for viscosity in steady uniaxial extension is often called the Trouton ratio. We see it holds true at low rates for the polymer melt in Figure 2.1.3. The following examples give applications of the Newtonian model to more complex deformations. Further examples appear at the end of the chapter. Bird, et al. (1987, Chapter 1) or any other good fluid mechanics book contains many worked Newtonian examples. 

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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