The Rheology Handbook. Thomas Mezger

behavior begins. Then, visco­elastic or viscous flow is occurring.

      A sample’s structure is already deformed below the yield point, however, this deformation is only very small. The structure would completely reform after removing the load as long as the yield point has not been exceeded. In analysis, usually by use of a software program, the τ-value is specified as the one point just before the measuring curve deviates significantly from the straight fitting line. Before starting the analysis, the user has to define the bandwidth of the tolerated deviation (e. g. as 5 or 10 %).

      When using this method, the very small deflection of the measuring system in the range below the yield point can only be detected by very sensitive instruments showing a high resolution for the values of torque and deflection angle (or rotational speed, respectively). Therefore it is senseful to use here an air-bearing rheometer (see also Chapter 11.7.5b). This method should be selected for scientific experiments to be on the safe side to get the yield value still in the reversible elastic deformation range . Some examples of tests results obtained with ketchups and coatings are shown in [4.17].

      4.2.1.1.2b) Yield point by the “tangent crossover method”, using two fitting lines

      Also here, the measuring points are presented on a logarithmic scale (see Figure 4.4). The first line is fitted in the curve interval of the linear-elastic range, i. e., at low deformation values, as explained above. A second fitting line is adapted in the flow range, i. e., in the curve interval showing high values of τ and γ. This straight line in the flow range indicates clearly a higher slope value compared to the first one. The shear stress value at the crossover point of the two lines is taken as the yield point τ2. Sometimes, instead of the second (linear) straight line a non-linear fitting function is chosen to be fitted to the measuring points in the flow range (for example, in the form of a curved polynomial function).

      Please note: Using this method, the yield point value determined may occur already in the range of viscoelastic or viscous flow behavior (yield zone, see below under c). A certain part of the sample’s deformation would remain permanently if the up to then increasing shear stress were removed within that range, and the internal structure at rest would have been changed already irreversibly. If this should be avoided and in order to be for sure on the safe side, then should be preferred the method with a single fitting line in the linear-elastic range only, as explained above.

      4.2.1.1.3c) Yield zone (flow transition range)

      The transition between elastic deformation and viscous flow is often not occurring as a clear kink but merely as a gradual bend and change in the slope of the curve. In this case it is better to speak of a yield zone or a yield/flow transition range and not of a dot-like yield point value (for more information see Chapter 8.3.4.3).

      For “Mr. and Ms. Cleverly“

      4.2.1.1.4d) Yield point and flow point in the shear stress/deformation diagram

      If the transition from the linear-elastic (LE-) range to the flow range occurs quite gradually, the two yield point values τ1 (as limit of the LE-range) and τ2 (as tangent crossover point) may differ clearly when determined by use of the two methods described above (Figure 4.5). The determined two parameters correspond approximately to the two values which are obtained by an amplitude test (oscillatory test) as yield point τy and as flow point τf (see Chapter 8.3.4 and Figure 8.12). Here, this counts approximately on the one hand for the shear stress values of τ1 und τy and on the other hand for the values of τ2 und τf.

      However, if for brittle break behavior the transition from the linear-elastic (LE-) range occurs immediately, and therefore the measuring curve shows a clear kink, then τ1 and τ2 (or τy and τf respectively), both will meet together in one point (Figure 4.6). Correspondingly, also with an amplitude sweep, as yield point τy and flow point τf the same point would be obtained (see Chapter 8.3.4 and Figure 8.13).

       Figure 4.5: The logarithmic tau-gamma diagram shows that two different yield point values may result when applying the two different evaluation methods (limit of the LE-range, tangent crossover method)

       Figure 4.6: For brittle break behavior of the

      sample, the logarithmic tau-gamma diagram shows that for the two different yield point

      methods the result will be the same point

      4.2.1.1.5e) Yield point by the method of maximum deviation from the fitting line

      Here, a straight line is fitted through the entire range of measuring points of the logarithmic stress/deformation diagram with best fitting to the measuring points as well in the lower as well as in the upper shear stress range. Then the location is determined at which the distance between the measuring points and this straight line shows its maximum value. The corresponding shear stress value is then taken as the yield point. An advantage of this method is that also the whole deviation function can be presented versus the shear stress τ showing the distinctness of the bend of the logarithmic tau-gamma function. Another advantage is that several bends may be determined if occurring.

      Note: Yield point and flow point

      Beside the methods described above, there are existing further methods for yield point determination: in Chapter 3.3.4 using flow curves obtained by rotational tests, in Chapter 6.4 via creep curves, and – as already mentioned – in Chapter 8.3.4. The latter method is more meaningful in a scientific sense, since here are determined both the yield point and the flow point (oscillatory tests, amplitude sweeps). An overview on diverse methods for yield point determination is also given in Chapter 12.4.1a (guideline) and in the Index, as well as in [4.16].

      End of the Cleverly section

4.5References

[4.1]Dubbel, Taschenbuch für den Maschinenbau, editor K. H. Grote, Feldhusen, J., Springer Vieweg, Wiesbaden, 2014 (24th ed.)

[4.2]Meier-Westhues, U., Polyurethane – Lacke, Kleb- und Dichtstoffe; Vincentz, Hannover, 2007

[4.3]Baur, E. et al., Saechtling Kunststoff-Taschenbuch, Hanser, München, 2013 (31st ed.)

[4.4]Pahl, M., Gleissle, W., Laun, H.-M., Praktische Rheologie der Kunststoffe und Elastomere, VDI, Düsseldorf, 1995

[4.5]Schäffler, H., Bruy, E., Schelling, G., Baustoffkunde, Vogel, Würzburg, 1975 (Weber, S., Sch., B., Sch., 2012, 10th ed.)

[4.6]Young, T., A course of lectures on natural philosophy and the mechanical arts, 1807

[4.7]Poisson, S. D., Mémoire sur les equations generales de l’équilibre et du mouvement des corps solides élastiques et des fluides, J. École Polytechnique, Paris, 1829

[4.8]Domke, W., Werkstoffkunde und Werkstoffprüfung, Cornelsen Girardet, Essen, 2012 (10th ed.)

[4.9]Gordon, J. E., The science of structures and materials, Sci. Am. Library, New York, 1988 (in German: Strukturen unter Stress – mechanische Belastbarkeit in Natur und Technik, Spektrum, Heidelberg, 1989)

[4.10]Ilschner, B., Singer, R. F., Werkstoffwissenschaften und Fertigungstechnik, Springer, Berlin, 2010 (5th ed.)

[4.11]Magnus, K.; Müller-Slany, H. H., Grundlagen der Technischen