Hysteresis diagram

To determine the values of material characteristics, the stress and extension signals are plotted against each other. Damping (mechanical loss) in a material causes a phase shift between strain and stress. When the two signals are superimposed, a hysteresis diagram is obtained on which four different characteristic quantities can be defined stresses, strains, elastic moduli, and mechanical losses. 

If the material is linear-viscoelastic, there is a phase difference, 8, between the extension and the stress, and the hysteresis diagram takes the form of an ellipse. Equations 6-51, 6-52, and 6-53 can in this case be solved anal) ically ... 

The hysteresis loops to be found in the literature are of various shapes. The classification originally put forward by de Boer S in 1958 has proved useful, but subsequent experience has shown that his Types C and D hardly ever occur in practice. Moreover in Type B the closure of the loop is never characterized by the vertical branch at saturation pressure, shown in the de Boer diagrams. In the revised classification presented in Fig. 3.5, therefore. Types C and D have been omitted and Type B redrawn at the high-pressure end. The designation E is so well established in the literature that it is retained here, despite the interruption in the sequence of lettering. 

Fig. 4.25 Adsorption isotherms showing low-pressure hysteresis, (a) Carbon tetrachloride at 20°C on unactivated polyacrylonitrile carbon Curves A and B are the desorption branches of the isotherms of the sample after heat treatment at 900°C and 2700°C respectively Curve C is the common adsorption branch (b) water at 22°C on stannic oxide gel heated to SOO C (c) krypton at 77-4 K on exfoliated graphite (d) ethyl chloride at 6°C on porous glass. (Redrawn from the diagrams in the original papers, with omission of experimental points.)...

FIG. 16 Phase diagram of fluid vesicles as a function of pressure increment p and bending rigidity A. Solid lines denote first-order transitions, dotted lines compressibility maxima. The transition between the prolate vesicles and the stomatocytes shows strong hysteresis efifects, as indicated by the error bars. Dashed line (squares) indicates a transition from metastable prolate to metastable disk-shaped vesicles. (From Gompper and KroU 1995 [243]. Copyright 1995 APS.)... 

In Part III heterogeneous equilibria involving clathrates are discussed from the experimental point of view. In particular a method is presented for the reversible investigation of the equilibrium between clathrate and gas, circumventing the hysteresis effects. The phase diagrams of a number of binary and ternary systems are considered in some detail, since controversial statements have appeared in the literature on this subject. 

Figure 1. Schematic diagram of a hysteresis curve for a typical ferromagnet showing magnetization (a) as a function of the applied magnetic field (H) Saturation magnetization Is Indicated by Os- Inset shows the multidomain structure and subdomain superparamagnetlc clusters.

The experiments were performed at a constant inflow concentration of ascorbic acid ([H2A]) in the CSTR. Oscillations were found by changing the flow rate and the inflow concentration of the copper(II) ion systematically. At constant Cu(II) inflow concentration, the electrode potential measured on the Pt electrode showed hysteresis between two stable steady-states when first the flow-rate was increased, and then decreased to its original starting value. The results of the CSTR experiments were summarized in a phase diagram (Fig. 6). 

Figure 3.37. Pd-H phase diagram. T-Xprojection (onto a plane at constant pressure P = 100 Pa) obtained from the experimental P—x isotherms. Because of hysteresis the data obtained in absorption or desorption experiments are slightly different.

Figure 2, Room temperature phase diagram of the PLZT system illustrating phases present and typical hysteresis loops associated with each phase compositions 1, 2 and 3 are 9565, 7065 and 12040, respectively...

Since fluid shear rates vary enormously across the radius of a capillary tube, this type of instrument is perhaps not well suited to the quantitative study of thixotropy. For this purpose, rotational instruments with a very small clearance between the cup and bob are usually excellent. They enable the determination of hysteresis loops on a shear-stress-shear-rate diagram, the shapes of which may be taken as quantitative measures of the degree of thixotropy (G3). Since the applicability of such loops to equipment design has not yet been shown, and since even their theoretical value is disputed by other rheologists (L4), they are not discussed here. These factors tend to indicate that the experimental study of flow of thixotropic materials in pipes might constitute the most direct approach to this problem, since theoretical work on thixotropy appears to be reasonably far from application. Preliminary estimates of the experimental approach may be taken from the one paper available on flow of thixotropic fluids in pipes (A4). In addition, a recent contribution by Schultz-Grunow (S6) has presented an empirical procedure for correlation of unsteady state flow phenomena in rotational viscometers which can perhaps be extended to this problem in pipe lines. 

The stationary-state response curves, or bifurcation diagrams shown in Figs 1.13(b) and 1.12(f), represent two of the simplest possible patterns monotonic variation and a single hysteresis loop respectively. These are the only qualitatively different responses possible for the cubic autocatalytic step on its own. They are also found for a first-order exothermic reaction in an adiabatic flow reactor (see chapter 6). With only slightly more complex chemical mechanisms a whole array of extra exotic patterns can be found, such as those displayed in Fig. 1.14. The origins of these shapes will be determined in chapter 4. 

Fig. 1.14. Four more of the possible stationary-state bifurcation diagrams for chemical systems (see also Fig. 1.2) in flow reactors (a) isola (b) mushroom (c) isola + hysteresis loop ...

Fig. 7.5. The 0ad 1 parameter plane showing the hysteresis line and the isola cusp described by eqns (7.34)—(7.36). The plane is divided into five regions (see inset for details in vicinity of cusp point) corresponding to the qualitative forms in Fig. 7.4. The numerical values are appropriate to the exponential approximation, y = 0, but the qualitative form of the diagram holds for ally < J.

The conditions for the hysteresis and isola boundaries on the 0ad-tN diagram can be obtained parametrically, although in the case of the isola curve we must solve a cubic equation for 0ad. The expressions are for hysteresis... 

Examples of several of the diagrams found in this example will be presented below, but first they need to be put into perspective. The hysteresis variety in Fig. 13 is actually only a portion of the whole. There is another swallowtail point at much smaller values of w, related to the limit point that... 

---