Uniaxial stress-strain behaviour

The case v = —1/4, p = 1, obtained under assumptions valid for high crosslink densities, results in the uniaxial stress-strain behaviour... 

A typical uniaxial stress-strain behaviour of an HDPE geomembrane is also shown in Fig. 8.29. While maximum elongation is several hundred per cent, yield occurs at 10-15% strain. A safety factor of 1.5-2.5 is usually taken into account when employing geomembranes, which brings any allowable strain down to 5-6.5%. Geomembranes are viscoelastic materials and their stress-strain behaviour is time dependent, so over the long term, applied stresses could lead to their creep failure. 

Thus, this consideration shows that the thermoelasticity of the majority of the new models is considerably more complex than that of the phantom networks. However, the new models contain temperature-dependent parameters which are difficult to relate to molecular characteristics of a real rubber-elastic body. It is necessary to note that recent analysis by Gottlieb and Gaylord 63> has demonstrated that only the Gaylord tube model and the Flory constrained junction fluctuation model agree well with the experimental data on the uniaxial stress-strain response. On the other hand, their analysis has shown that all of the existing molecular theories cannot satisfactorily describe swelling behaviour with a physically reasonable set of parameters. The thermoelastic behaviour of the new models has not yet been analysed. 

Stress-strain measurements at uniaxial extension are the most frequently performed experiments on stress-strain behaviour, and the typical deviations from the phantom network behaviour, which can be observed in many experiments, provided the most important motivation for the development of theories of real networks. However, it has turned out that the stress-strain relations in uniaxial deformation are unable to distinguish between different models. This can be demonstrated by comparing Eqs. (49) and (54) with precise experimental data of Kawabata et al. on uniaxially stretched natural rubber crosslinked with sulphur. The corresponding stress-strain curves and the experimental points are shown in Fig. 4. The predictions of both... 

Fig. 4. Stress-strain behaviour in uniaxial extension O experimental values by Kawabata et al. ------------- theoretical results...

Oppermann, W. Rennar, N., Stress-Strain Behaviour of Model Networks in Uniaxial Tension and Compression. Progr. Coll. 8c Polym. Sci 1992, 75(1) 49-54. 

The tensile test provides an insight into the stress/strain behaviour of a material under uniaxial tensile loading and makes it possible to distinguish between brittle and ductile materials. It is a useful tool for quality control and general comparison of properties, but it cannot be considered representative for applications with load/time scales widely different from those of the standard test. 

The form of the stress-strain behaviour depends upon the loading geometry employed. The present analysis will be restricted to simple uniaxial tension or compression when specimens are deformed to an extension ratio A in the direction of the applied stress. It is possible to replace Ai by A but Equation (5.144) requires that A2A3 = 1/Ai and so A2 = A3 = 1/A. Equation (5.142) can therefore be written as... 

As shown in Fig. 3.4 stress-strain tests on uniaxially aligned fibre composites show that their behaviour lies somewhere between that of the fibres and that of the matrix. In regard to the strength of the composite, Ocu, the rule of mixtures has to be modified to relate to the matrix stress, o at the fracture strain of the fibres rather than the ultimate tensile strength, o u for the matrix. 

The important point to note from this Example is that in a non-symmetrical laminate the behaviour is very complex. It can be seen that the effect of a simple uniaxial stress, or, is to produce strains and curvatures in all directions. This has relevance in a number of polymer processing situations because unbalanced cooling (for example) can result in layers which have different properties, across a moulding wall thickness. This is effectively a composite laminate structure which is likely to be non-symmetrical and complex behaviour can be expected when loading is applied. 

The strength properties of solids are most simply illustrated by the stress-strain diagram, which describes the behaviour of homogeneous brittle and ductile specimens of uniform cross section subjected to uniaxial tension (see Fig. 13.60). Within the linear region the strain is proportional to the stress and the deformation is reversible. If the material fails and ruptures at a certain tension and a certain small elongation it is called brittle. If permanent or plastic deformation sets in after elastic deformation at some critical stress, the material is called ductile. 

Figure 5 demonstrates the different behaviour resulting from Eqs. (49) and (54) in the case of uniaxial compression. We also tested the elastic potential (Eq. (44)) in the two cases v = 1/2 and v = —1/4 by comparing the corresponding stress-strain relations with biaxial extension experiments which cover relatively small as well as large deformation regions for an isoprene rubber vulcanizate. In the rectan-... 

Three significant characteristic points of the stress-strain curve are distinguishable. The first kink describes the elastic limit, the second kink is the yield strength and the third point describes the breaking point of the foil for uniaxial short term exposure. Up to the elastic limit, ETFE-foU shows an almost linear elastic behaviour. Ftooke s law prevails (at least for shortterm loads). From the elastic limit point up to the yield strength, ETFE... 

In the field of rubber elasticity both experimentalists and theoreticians have mainly concentrated on the equilibrium stress-strain relation of these materials, i e on the stress as a function of strain at infinite time after the imposition of the strain > This approach is obviously impossible for polymer melts Another complication which has thwarted the comparison of stress-strain relations for networks and melts is that cross-linked networks can be stretched uniaxially more easily, because of their high elasticity, than polymer melts On the other hand, polymer melts can be subjected to large shear strains and networks cannot because of slippage at the shearing surface at relatively low strains These seem to be the main reasons why up to some time ago no experimental results were available to compare the nonlinear viscoelastic behaviour of these two types of material Yet, in the last decade, apparatuses have been built to measure the simple extension properties of polymer melts >. It has thus become possible to compare the stress-strain relation at large uniaxial extension of cross-linked rubbers and polymer melts ... 

Natural rubber (NR) is a well studied elastomer. Of particular interest is the ability of NR to crystallize, specifically the strain-induced crystallization that takes place whilst the material is stretched. Moreover, in many elastomer applications, network chain dynamics under external stress/strain are critical for determining ultimate performance. Thus, a study on how the strain-induced crystallization affects the dynamics of a rubbery material is of outmost importance. Lee et al [1] reported their initial findings on the role of uniaxial extension on the relaxation behavior of cross-linked polyisoprene by means of dielectric spectroscopy. Nonetheless, to our best knowledge no in-depth study of the effects of strain induced crystallization on the molecular dynamics of NR has been undertaken, analyzing the relaxation spectra and correlating the molecular motion of chains with its structure. Broadband dielectric spectroscopy (BDS) has been chosen in order to study the dynamic features of segmental dynamics, because it is a comparatively simple technique for the analysis of the relaxation behaviour over a suitable frequency interval. This study is important from a basic and practical point of view, since an elongated crosslinked polymer at equilibrium may be considered as a new anisotropic material whose distribution of relaxation times could be affected by the orientation of the chains. 

Tg especially wl en deformed under the influence of an overall hydrostatic compressive stress. This behaviour is illustrated in Fig. 5.37 where true stress-strain curves are given for an epoxy resin tested in uniaxial tension and compression at room temperature. The Tg of the resin is 100°C and such cross-linked polymers are found to be brittle when tested in tension at room temperature. In contrast they can show considerable ductility in compression and undergo shear yielding. Another important aspect of the deformation is that glassy polymers tend to show strain softening . The true stress drops after yield, not because of necking which cannot occur in compression, but because there is an inherent softening of the material. 

In Fig 3 the uniaxial fibre stress-strain relation is shown for a constant strain rate of 1.67 10" Hz. Equation (1) can also be used to determine cyclic stress strain response (an example is shown in Fig. 3) or even an analytical description of the relaxation behaviour of the fibres, however time dependent phenomena will not be discussed here. 

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