Tensile curve

Figure C-6 Tensile Curve for Boron-Epoxy (Adapted from [C-2])...

In the continuous chain model, a large part of the deformation during extension of the fibre consists of the shear deformation as a result of which the chain orientation distribution contracts. This leads to the concave shape of the tensile curve, often found for polymer fibres. Therefore, this description of the extension of the fibre implies a strain hardening process. 

The function N(0)=p(0)sin0 is proportional to the total number of domains in the fibre at an angle 0 with the fibre axis. The determination of the modulus using sonic frequencies will yield a higher value of g than the method by which the modulus is derived from the initial slope of the tensile curve. For random orientation of the chains in a fibre (sin20),=O.5 and the modulus of an isotropic fibre is given by Eiso 4g. 

The continuous chain model includes a description of the yielding phenomenon that occurs in the tensile curve of polymer fibres between a strain of 0.005 and 0.025 [ 1 ]. Up to the yield point the fibre extension is practically elastic. For larger strains, the extension is composed of an elastic, viscoelastic and plastic contribution. The yield of the tensile curve is explained by a simple yield mechanism based on Schmid s law for shear deformation of the domains. This law states that, for an anisotropic material, plastic deformation starts at a critical value of the resolved shear stress, ry =/g, along a slip plane. It has been... 

Indeed, it has been observed that the onset of yielding of isotropic polymers is approximately constant, 0.02< [<0.025, which implies that 0.04shear yield strain, the plastic shear deformation of the domain satisfies a plastic shear law. For temperatures below the glass transition temperature, the continuous chain model enables the calculation of the tensile curve of a polymer fibre up to about 10% strain [6]. Figure 7 shows the observed stress-strain curves of PpPTA fibres with different moduli compared to the calculated curves. 

Fig. 7 Comparison of the observed tensile curves of PpPTA fibres with three different moduli with the curves calculated with the continuous chain model [6]...

For fibres made from the same polymer but with different degrees of chain orientation the end points of the tensile curves, a5, are approximately located on a hyperbola. Typical examples of this fracture envelope are shown in Figs. 8... 

Fig. 8 Tensile curves of cellulose II fibres measured at an RH of 65% (1) Fibre B, (2) Cor-denka EHM yarn, (3) Cordenka 700 tyre yarn, (4) Cordenka 660 tyre yarn and (5) Enka viscose textile yarn [26]. The solid circles represent the strength corrected for the reduced cross section at fracture. The dotted curve is the hyperbola fitted to the end points of the tensile curves 1,3 and 5. The dashed curve is the fracture envelope calculated with Eqs. 9,23 and 24 using a critical shear stress rb=0.22 GPa... 

Fig. 9 The end points of the tensile curves of the various polyetherketone (POK) yarns spun by B.J. Lommerts [27-29]. The dashed curve has been calculated with Eqs. 9,23 and 24 using rb=0.25 GPa...

The relation between the end points of the tensile curve, ab and eh (= b), can be calculated with Eqs. 9,23 and 24. This relation is now by definition taken as the fracture envelope. Note that these equations only hold for elastic deformation. In order to account for some viscoelastic and plastic deformation, a value gv is used, which is somewhat smaller than the value for elastic deformation g. The dashed curves in Figs. 8 and 9 are the calculated fracture envelopes (neglecting the chain extension) for the cellulose II and the POK fibres, respectively. These figures show a good agreement between the observed and calculated fracture points. 

Fig. 11 Tensile curves of PET yarns made with different draw ratios... 

For a polymer fibre with a single orientation angle the modulus, E, or the slope at each point of the tensile curve, is a function of the tensile stress and given by... 

Fig. 74 Schematic representation of the tensile curve of a fibre during (1) first loading, (2) hypothetical elastic unloading, (3) unloading and (4)second loading. Note that the second part of curve 4 practically coincides with curve 1 [1,6]... 

The tensile curve of a polymer fibre is characterised by the yield strain and by the strain at fracture. Both correspond with particular values of the domain shear strain, viz. the shear yield strain j =fl2 with 0.04rotation angle of -0y=fl2 and the critical shear strain 0-0b=/iwith /f=0.1. For a more fundamental understanding of the tensile deformation of polymer fibres it will be highly interesting to learn more about the molecular phenomena associated with these shear strain values. 

Most conspicuous is the steep tensile curve for the S-SBR sample containing PTh-silica. The PTh-silica gives the best improvement in tensile properties in terms of tensile strength, modulus at 100%, and modulus at 300%, but elongation at break is lower. The PA- and PPy-silicas as well as the silane-treated silica result in only a slight... 

The tensile curves are characterized by a sharp decrease in tensile with loading at low loadings. For rayon, acrylic, and fiber glass the tensile was minimized at approximately 20 volumes. For nylon the minimum was reached at approximately 50 volumes. Beyond the minimum points in the curves the tensile increased but at a slower rate than the initial rate of decrease. Eventually the tensile reached that of the natural rubber matrix for all but the nylon fibers. Since the loading resulting in... 

In a subsequent investigation, with Roos and Kampschreur (1989), Northolt extended the modified series model to include viscoelasticity. For that an additional assumption was made, viz. that the relaxation process is confined solely to shear deformation of adjacent chains. The modified series model maybe applied to well-oriented fibres having a small plastic deformation (or set). In particular it explains the part of the tensile curve beyond the yield stress in which the orientation process of the fibrils takes place. The main factor governing this process is the modulus for shear, gd, between adjacent chains. At high deformation frequencies yd attains its maximum value, ydo at lower frequencies or longer times the viscoelasticity lowers the value of gd, and it becomes a function of time or frequency. Northolt s relations, that directly follow from his theoretical model for well-oriented fibres, are in perfect agreement with the experimental data if acceptable values for the elastic parameters are substituted. 

A combination of Eqs. (13.167) and (13.168) yields the typical concave shape of the elastic stress-strain curve of well-oriented fibres. In Fig. 13.98 the calculated stress-strain curve is compared with the experimental curve at decreasing stress of a Twaron 1000 fibre. It shows almost elastic behaviour. By including the simple theory the tensile curve with yield can be calculated as shown in Fig. 13.99. 

A simple explanation for the shape of the fracture envelope starts with the assumption that the tensile curve is linear with modulus E1. The work of fracture or the strain energy per unit volume up to the fracture point is given by... 

Figure 14.5. Tensile curve of the neat resin and SWNT reinforced nanocomposites (a) neat resin, (b) p-SWNT composite, (c) 0.5 wt% epoxy grafted-SWNT composite, (d) 1 wt% epoxy grafted-SWNT composite (11).

Although cavitation is believed to be the most important contribution to creep asymmetry, there are other contributing factors. In Fig. 4.7, for example, data from the lower portion of the tensile curve labeled KX01 were taken from specimens in which few cavities were observed. Therefore, cavitation cannot be used as the sole explanation for the difference in behavior in tensile and compressive creep. Other factors seem to be important in establishing creep... 

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