True yield stress

Fig. 16. Correlation between microhardness at 0.1 min and true yield stress of PE M = 170000 (O) Mw = 2x 106 (A). The solid line is drawn according to Tabor s relation 28 )...

As discussed above, because materials often fail by the propagation of cracks, particularly brittle ones, it is difficult to measure the true yield stress, oy, of a material. Instead one measures a fracture stress, Op which is related to the size of the process zone and actual crack length by ... 

Equation (32) clearly illustrates the point that as rp becomes very small relative to c, i.e., for large cracks, the fracture stress may be orders of magnitude smaller than the true yield stress of the material. 

The extrapolated yield stress gives 0.06 Pa and a plastic viscosity of 3.88 mPas. We can use this to estimate the force between the particles, which gives 425kBT/a, in fair agreement with the value determined using pair potential curves. Here the Casson model has been used to partially linearise a pseudoplastic system rather than a system with a true yield stress. 

Much less is known about the settling of particles in fluids exhibiting a yield stress. Barnes (39) suggests that this is partly due to the fact that considerable confusion exists in the literature as to whether or not the fluids used in the experiments do have a true yield stress 39. Irrespective of this uncertainty, which usually arises from the inappropriateness of the rheological techniques used for their characterisation, many industrially important materials, notably particulate suspensions, have rheological properties closely approximating to viscoelastic behaviour. 

Figure 4 shows that the use of particular instruments depends upon whether or not the formulation exhibits yield stress—i.e., there is stress necessary before material starts to shear. Most propellant formulations will either exhibit a true yield stress of certain magnitude or will not exhibit any yield stress at all. However, certain formulations may have a low shear rate exponent (n-value regression coefficient) of 0.6 or less. In this case, the material will perform as if there were true yield stress present even though it may not be measurable by the techniques presented. It is termed a pseudo-yield stress. Whenever the low exponent occurs, the material must always be treated as having a true yield stress. 

Fig. 19 Correlation between the ratio of the stress intensity factor to crack initatiation (Kci) to stress intensity factor at crack arrest (lCca) and the true yield stress (ayt) of epoxy materials (from [73])...

Doubt has been cast on the existence of a true yield stress a long time ago, and it is usually Professor Reiner, who is quoted to have said everything flows if you wait long enough, some 60 years ago. Barnes and Walters found an apparently well-defined yield stress for PVA latex adhesives employing a shear strain... 

From the slopes of the lines, the viscosity r) can be obtained at each applied stress. A plot of versus tr is shown in Figure 20.10, which shows a limiting viscosity rj o), while below and above the viscosity shows a sharp decrease with further increase in cr. ti(o) is referred to as the residual or zero shear viscosity, which is an important parameter for predicting sedimentation. critical stress above which the structure breaks down this is sometimes referred to as the true yield stress. 

Basically, a constant stress cr is applied on the system and the compliance J(Pa ) is plotted as a function of time (see Chapter 20). These experiments are repeated several times, increasing the stress in small increments from the smallest possible value that can be applied by the instrament). A set of creep curves is produced at various applied stresses, and from the slope of the linear portion of the creep curve (when the system has reached steady state) the viscosity at each applied stress, //, can be calculated. A plot of versus cr allows the limiting (or zero shear) viscosity /(o) and the critical stress cr (which may be identified with the true yield stress of the system) to be obtained (see also Chapter 4). The values of //(o) and <7 may be used to assess the flocculation of the dispersion on storage. 

The flow stresses of the intermetallic Ti5Si3 and TiSi2 compounds were determined in compression tests in air at the strain rate of =10"2 s"1 in the temperature range from 700 to 1500 °C. Figure 8 presents the true yield stress vs. temperature curves. 

The ductile-to-quasi-brittle transition occurs because the true fracture stress, as determined by P, decreases with increasing PS more rapidly than the yield stress, determined by P (Figure 9). The critical PS content (Vf>s) at the ductile-to-quasi-brittle transition can be determined from the condition that the true yield stress is equal to the true fracture stress. From equations 8 and 13,... 

The second term on the right-hand side of Eq. (8.5) describes the increase in true yield stress with true strain rate. If the initial yield stress is measured in a tensile test, the low strain means there is no contribution from orientation hardening (see the next section), and there is insignificant heating. Consequently, the strain rate effect can be isolated. The initial yield stress was foimd, for HOPE at 20 °C, to vary with the true strain rate according to... 

Strictly speaking, it is virtually impossible to ascertain whether any real material has a true yield stress or not, but nevertheless the concept of a yield stress has proved to be convenient in practice because some materials closely approximate to this type of flow behaviour, e.g. see [Barnes and Walters, 1985 Astarita, 1990 Schurz, 1990 and Evans, 1992]. The answer to the question whether a fluid has a yield stress or not seems to be related to the choice of a time scale of observation. Common examples of viscoplastic... 

Often, the two model parameters, Tq and /rg, are treated as curve fltting constants irrespective of whether or not the fluid possesses a true yield stress. 

Notwithstanding the continuing debate over the very existence of a true yield stress, the concept of an apparent yield stress has been found to be an extremely useful empiricism in many areas of science and engineering [Hartnett and Hu, 1989] (see also Chapter 1). A recent comprehensive review [Barnes, 1999] has critically assessed the various issues raised in the definition, measurement and application of apparent yield stress behaviour. 

Any operational definition of apparent yield stress should take into account both the inevitable rheometrical limitations in its determination, and the characteristic time of the process to which it pertains. Such an operational definition has been proposed for a true yield stress in the context of the classical stress relaxation experiment [Spaans and Williams, 1995]. 

Below a critical stress, eta, the viscosity reaches a limiting value, namely the residual (or zero shear) viscosity ctci may be denoted as the true yield stress of the emulsion, i.e. the stress above which the "structure of the system is broken down. Above o er, decreases rapidly with further increase of the shear stress (the shear thinning regime). It reaches another Newtonian value rj, which is the high shear limiting viscosity. 

From the slope of the linear portion of the creep curve (after the system reaches a steady state), the viscosity at each applied stress, is calculated. A plot of % versus (T (Figure 7.40) allows one to obtain the limiting (or zero shear) viscosity and the critical stress eta (which may be identified with the true yield stress of the system). 

Rheological measurements are used to investigate the bulk properties of suspension concentrates (see Chapter 7 for details). Three types of measurements can be applied (1) Steady-state shear stress-shear rate measurements that allow one to obtain the viscosity of the suspensions and its yield value. (2) Constant stress or creep measurements, which allow one to determine the residual or zero shear viscosity (which can predict sedimentation) and the critical stress above which the structure starts to break-down (the true yield stress). (3) Dynamic or oscillatory measurements that allow one to obtain the complex modulus, the storage modulus (the elastic component) and the loss modulus (the viscous component) as a function of applied strain amplitude and frequency. From a knowledge of the storage modulus and the critical strain above which the structure starts to break-down , one can obtain the cohesive energy density of the structure. 

The start-up pressure for pumping a Bingham plastic is expressed in terms of the true yield stress ... 

Thus, one measures creep curves as a function of the applied stress (starting from a very small stress of the order of 0.01 Pa). This is illustrated in Fig. 3.45. The viscosity Pu (which is equal to the reciprocal of the slope of the straight portion of the creep curve) is plotted as a function of the applied stress. This is schematically shown in Fig. 3.46. Below a critical stress the viscosity reaches a limiting value, p(o) namely the residual (or zero shear) viscosity. Above a , p decreases rapidly with a further increase in the shear stress (the shear thinning regime). It reaches another Newtonian value Poo, which is the high shear limiting viscosity. 0, may be identified as the critical stress above which the structure of the suspension is broken down . Ucr is denoted as the true yield stress of the suspension. 

White [25], measures talc-, talc/calcite-, and caldte-fiUed thermoplastic compounds [25]. They found the true yield stresses generally give much smaller values than the apparent yield values [21,22,24].Kim and White [25] also found that yield values exist in elongational flow... 

For both the notched and the smooth specimens, the engineering and true yield stress and strain and ultimate stress and strain can be gathered (with respect to the average axial values across the cross-section). Additionally, four ratios, two that describe the notch effect on yield behavior and two that describe the notch effect on postyield behavior, can be calculated. For each of the specimens, a notch strengthening ratio with respect to stress and a notch strengthening ratio with respect to strain can be calculated ... 

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