Deformation simple extension

A full discussion of stress-strain relations at equilibrium for large deformations in rubbery cross-linked polymers is beyond the scope of this chapter there have been many investigations of uniaxial deformations (simple extension, compression,) torsion, and biaxial deformations, which have been critically reviewed elsewhere. A few comments will introduce the discussion of nonlinear viscoelastic behavior. 

The photoelastic measurements were carried out in simple extension using strip specimens. In addition to the force/ also the optical retardation S (hence also the birefringence An <5) could be determined and the modulus G, the deformational-optical function A and the stress-optical coefficient C = A/G were calculated using the equations [31]... 

We see that the relative change of entropy and internal energy at constant pressure is independent of the degree of twisting. This conclusion differs from that obtained for simple extension or compression. The entropic and energetic components in torsion are identical with the result for simple deformation. Equations (60) and (61) lead to the conclusion that there can be no thermomechanical inversions of heat and internal energy in torsion. 

There are two prominent elementary modes of deformation, viz. simple shear and simple extension. 

When the strain rate, s, is maintained constant, the deformation obtained is called steady simple extension or steady uniaxial extension (Dealy, 1982) and the extensional viscosity, is related to the normal stress difference ... 

The analysis of the balance of forces on the lateral surface of the beam carried out on the basis of the preliminary hypothesis indicates that the only nonzero component of the stress tensor is simple extension. In fact, some parts of the transverse section are under tension, and others are under compression, and the two effects combined produce the flexion. According to Figure 17.1, the component jxx of the strain is given by... 

For a long strip of rubber with a small cut of length c, deformed in simple extension, the tearing energy (Eq. 28) is (207)... 

If an elastomer sample in the form of a unit cube is deformed by pure shear, then the three principal extension ratios are A, = X, = 1, X = MX. [Compare with the case of simple extension where X = X = 1 NX.] Following the arguments of Section B, derive an expression relating aE and A, where [Pg.208]

The treatment of mechanical deformation in elastomers is simplified when it is realized that the Poisson ratio is almost 0.5. This means that the volume of an elastomer remains constant when deformed, and if one also assumes that it is essentially incompressible (XjXjXj = 1), the stress-strain relations can be derived for simple extension and compression using the stored energy fimction w. 

To make anisotropic networks, the linear chains have to be ordered prior to the crosslink reaction. This orientation can be achieved in the melt state under strain by simple extension. The crosslinking reaction is performed in a two-step process. In the first step, a well-defined weak network is synthesized, which is deformed with a constant load to induce the network anisotropy. The load has to exceed the threshold load which is necessary to obtain a uniform director orientation. In the second step, a second crosslinking reaction occurs and locks in the network anisotropy. This procedure is shown schematically in Figure 9.12. 

Simple Extension of Swollen Networks. The force required to deform an elastomeric sample, from state 2 to state 3 in Figure 13, is given by equation (57), in which a =L/L is the extension ratio for the isotropic swollen state relative to the deformed swollen state. The ratio VofV is the volume fraction of polymer in the swollen system, and Aq is the cross-sectional area of the dry sample, which is generally measured before the experiment. The area of the swollen sample is given by... 

DMA experiments are performed under conditions of very small strain so that the material response is in the linear viscoelastic range. This means that the magnitude of stress and strain are linearly related and the deformation behavior is completely described by the complex modulus function, which is a function of time only. The theory applies both for the case of a tensile deformation or simple extension and for shear. In the latter case the comparable modulus is with components G ico) and G" co). As a first-order approximation, E = 3G. The theory is developed assuming deformation under isothermal conditions, and temperature does not appear (nor is implicit) as a variable. 

Then, for example, in simple extension, the stress-deformation relationship becomes... 

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. 

Simple extension belongs to a different class of deformations. As depicted in the lower part of Fig. 3.1, hquid elements are stretched along the direction of flow without sliding relative to adjacent elements. Extensional flow, sometimes called elongational flow, is an important component in many polymer-processing operations, for example in fiber spinning and film formation. Extensional flows are difficult to generate and sustain in a controlled way, however. Most laboratory methods used to characterize polymer-flow properties involve simple shear flows. The effects of these two classes of deformation on chain conformations, caused by the respective relative motions, are sketched in Eig. 3.2. For simplicity and brevity only the response in shear flow will be discussed in this chapter. 

Leaderman. These tests have more often been performed for deformation in simple extension than in shear then equations 38 and 39 hold with D (defined in Section F below) substituted for J and appropriate changes in the stress and strain components. 

If an isotropic cubical element is elongated in one direction and the dimensional changes in the two mutually perpendicular directions are equal (Fig. 1-14), the experiment is termed simple extension if 7 u is positive, 722 and 733 are equal to each other and negative (or possibly zero). The stresses are tensile stress resulting from the applied force and Pa is the ambient pressure. This type of deformation results when a rod, strip, or fiber is subjected to a tensile force. Equation 42 for this case becomes ... 

The statement in connection with equations 56 and 57 that simple extension gives the same information as simple shear is limited not only to materials with n very near i but also to small deformations. With large deformations and/or large rates of deformation, the two types of strain show very different behavior. For example, in steady-state flow, the apparent shear viscosity (ratio of stress to rate of strain) commonly decreases with increasing rate of strain, whereas the apparent elonga-tional viscosity may remain constant or increase. Some examples will be shown in Chapters 13 and 17. 

Deformation Simple Shear Bulk Compression Simple Extension" Bulk Longitudinal... 

FIG. 2-1. Creep compliance plotted logarithmically for eight typical polymer systems viscoelastic liquids on left, viscoelastic solids on right, identified by numbers as described in the text. Deformation is shear, J t), except for curves VI and Vlll, which are for simple extension, D t). The dashed curves represent the compliance after subtraction of the flow contribution t/rjo- The solvent for the dilute solution, example I, is also shown as a dashed line. 

Viscoelastic behavior in simple extension or in bulk longitudinal deformation will in general combine the features of shear and bulk viscoelasticity, since the moduli E t) and M t) depend on both (7(0 and K t), as shown by equations 51 and 58 of Chapter 1 ( and analogous relations for E and M ). However, as already pointed out, shear effects predominate in E(t) and E. and bulk effects predominate... 

The linear viscoelastic phenomena described in the preceding chapter are all interrelated. From a single quite simple constitutive equation, equation 7 of Chapter 1, it is possible to derive exact relations for calculating any one of the viscoelastic functions in shear from any other provided the latter is known over a sufficiently wide range of time or frequency. The relations for other types of linear deformation (bulk, simple extension, etc.) are analogous. Procedures for such calculations are summarized in this chapter, together with a few remarks about relations among nonlinear phenomena. 

The equations in this chapter are formulated in terms of shear deformation, but analogous relations exist for bulk compression, simple extension, etc. 

For deformation in simple extension at constant rate of (practical) strain e, for which <7(r) is replaced by E(t), Smith has pointed out that it is convenient to deflne a constant-strain-rate modulus F(t) = This is related to the relaxation... 

At small deformations, viscoelastic information can in principle be obtained from stress-strain measurements at a constant strain rate, as shown for shear deformations in equations S6 to S9 of Chapter 3. Such experiments are often made in simple extension, but the deformations can become rather large so there are marked deviations from linear viscoelastic behavior. The most commonly used instrument is the Instron tester other carefully designed devices have been described. - The sample is usually a dumbbell or a ring. In the former case, the strain in the narrow section as checked by separations of several fiducial marks can be calculated from the separation between the clamps by a suitable multiplication factor. In... 

Frequently, a fiber is anisotropic, with different properties in different direc-tions. For practical purposes, the only types of deformation which can be readily measured are simple extension, torsion, and flexure. Extension and flexure should both measure Young s modulus E for elongation in the fiber direction, and should therefore yield the same result. Torsion measures the shear modulus G for a direction of slide perpendicular to the fiber direction, and in case of anisotropy these moduli E and G are not connected in any simple manner. Some examples of such behavior will be given in Chapter 16. 

Stress relaxation following sudden strain for large deformations in both shear and simple extension has been treated for the tube model by Doi and Edwards.22-74... 

The preceding sections have dealt almost solely with linear viscoelastic behavior in shear or in simple extension which for soft materials is essentially determined by shear deformation characteristics. However, the use of reduced variables in describing temperature and pressure dependence of viscoelastic properties has considerably wider applicability. 

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