Tension-elongation curve

Using the thermodynamic relation (1.19), we can find the average end vector R under a given tension f by the differentiation 

Because the vector R lies in parallel to the tension, we can write the result for the RF model in terms of its absolute value as 

The Langevin function r = L(t), described by the dimensionless elongation f=R/na, is measured relative to the total chain length na, and its inverse function can be expanded in the power series 

in the linear region, the tension is proportional to the elongation as 

Because the Langevin function and its inverse function are mathematically dilHcult to treat, we introduce here a simple nonlinear model chain whose tension is described by 

---

Fig. 1.5 (a) Tension-elongation curve of the Langevin chain (solid line) and its Gaussian approximation... 

Let us next find the tension-elongation curve. The average end-to-end distance R can be found by the fundamental relation... 

Figure 1.16(b) shows the tension-elongation curves at three different temperatures. At T = -0.5 in the transition region, there appears a wide plateau in R, and we notice the existence of the critical tension 3.0 for x = —0.5 at which chain segments start to be reeled out from the globules. For the balance between a globule of the size f and a hydrated coil of the same size, we And a scaling law... 

Fig. 4.7 Tension-elongation curve of a cross-linked rubber. Experimental data (circles), affine network theory by Gaussian chain (broken line), affine network theory (4.107) by Langevin chain (solid line).

This chapter is devoted to the molecular rheology of transient networks made up of associating polymers in which the network junctions break and recombine. After an introduction to theoretical description of the model networks, the linear response of the network to oscillatory deformations is studied in detail. The analysis is then developed to the nonlinear regime. Stationary nonhnear viscosity, and first and second normal stresses, are calculated and compared with the experiments. The criterion for thickening and thinning of the flows is presented in terms of the molecular parameters. Transient flows such as nonhnear relaxation, start-up flow, etc., are studied within the same theoretical framework. Macroscopic properties such as strain hardening and stress overshoot are related to the tension-elongation curve of the constituent network polymers. 

The mechanical properties of the polylmide film were measured in tension tests. To do this polylmide films were cast and cured on glass substrates. The films were removed and tensile test specimens were stamped from the films using a die. The specimens were then pulled to fracture at a cross-head speed of 10 mm/min. The mechanical properties obtained from the load-elongation curves are given in Table I. 

Fig. L(a) Three common working definitions of the yield point for metals. (1) Load maximum (2) tangent method, (3) firoofistress" or "strain-offset method. (The proof-strain is commonly taken to be 0 /%, but is quite arbitrary.) (b) Load elongation curves for polymers. (I) Brittle, (2) strain softening, (3) cold-drawing, (4) strain-hardening, (5) rubbery. Typical definitions of the yield point are marked by arrows on curves (2), (3) and (4). Any one polymer can show behaviour raiding from (1) to (5) depending on test conditions, e.g. temperature, strain-rate, tension...

From (20) one sees that the macroscopic modulus of elasticity E of rubber-like substances is proportional to the absolute temperature. This fact was proved by the experiments of K. H. Meyer, I. S. Ornstein, and their collaborators carried out for the first part of the elongation curve. The equation (20) further shows that there is proportionality between the tension S and the elongation Ai, a fact which also stands in fair agreement with many experimental investigations of the first part of the elongation curve of rubber having a low degree of vulcanization. 

A variety of specimen shapes and sizes can be used but the most common is a smooth-bar tensile coupon, as described in ASTM E 8, Test Method for Tension Testing of Metallic Materials." In smooth-bar tests, the changes are measured in terms of time to failure, ductility (elongation or reduction-in-area), maximum load achieved, and area bounded by a nominal stress-elongation curve or a true stress/ true strain curve, which are often supported by fractogra-phic examination. The specimen is exposed to the environment while it is stressed under a constant displacement rate. 

Figure 9.12 Typical stress/elongation curve for a rubber compound in tension.

Deformation (e) of PO vs time (/) at uniaxial strain in a static-mechanical field with tension a, i.e., the dependence s(t, a) is given in Table 1, column 8. Mechanical parameters of PO determined by specific points and regions of the load-elongation curve a(s, Fi) are given in paragraphs 1 and 2, or special methods used to determine their depen-... 

The apparent difference between the curves for tension and compression is due solely to the geometry of testing. If, instead of plotting load, we plot load divided by the actual area of the specimen, A, at any particular elongation or compression, the two curves become much more like one another. In other words, we simply plot true stress (see Chapter 3) as our vertical co-ordinate (Fig. 8.7). This method of plotting allows for the thinning of the material when pulled in tension, or the fattening of the material when compressed. 

One of the most informative properties of any material is their mechanical behavior specifically the determination of its stress-strain curve in tension (ASTM D 638). This is usually accomplished in a testing machine by measuring continuously the elongation (strain) in a test sample as it is stretched by an... 

The tensile strength with sole leather is usually 2-3 kilos per sq. mm. and with good belting leather should be at least 3 kilos in the latter case, the less the elongation on tension the higher the quality. The leather should not change or crack when bent to a curve with a diameter ten times the thickness. 

In practice, up to 90% of polyurethanes are used in compression, a few percent in torsion, and very little in tension. There is considerable data on the tensile stress against tensile strain (elongation) for polyurethanes. Most polyurethane specification sheets provide this data. Figure 7.3 and Figure 7.4 show typical stress-strain curves for both polyester and polyether polyurethanes. 

---