Flow stress diagram

The m values were determined from the log strain rate versus log flow stress diagram. 

The hydrogen effect on ductility and the flow stress will be considered first on the example of non-alloyed titanium. The Ti - H phase diagram is given in Fig. 1, and Fig. 2 shows the temperature dependence of ductility of Ti-a H alloys, A , for several X values. Tensile tests were run at a rate e 10" s . Ductility of the commercial... 

Melt fracture occurs when the rate of shear exceeds a critical value for the melt concerned at a particular temperature (that is, the critical shear rate ). There is a corresponding critical shear stress and the relevant point on the flow curve (or the shear rate-shear stress diagram) is known as the critical point. It is believed that it is reached in the die entry region (that is, where material is being funnelled from the die reservoir into the capillary of a capillary rheometer)—which, in an extruder, corresponds with the point at which melt moves into the die parallel portion of the die. Some further complicating effects may occur at the wall of the die. 

Fig. 21. Schematic diagram of process flow stress paths for elements of material undergoing extrusion through a conical die of small semi-angle, and a uniaxial tensile test...

Plastic flow can also set up residual stresses. The beam shown in Fig. I6-2(a) is supported at two points and loaded by two equal forces Fapplied near each end. At any point between the two supports the stress in the outside fibers is constant, tensile on the top of the beam and compressive on the bottom. These stresses are a maximum on the outside surfaces and decrease to zero at the neutral axis, as indicated by the stress diagram at the right of (a). This diagram shows how the longitudinal stress varies across the section AA, when all parts of the beam are... 

Much of the past and present research in nonnewtonian fluids has consisted of measuring their stress-rate-of-strain curves (such as Fig. 1.5) and trying to find mathematical descriptions of these curves. The study of the flow behavior of materials is called rheology (from Greek words meaning the study of flow ), and diagrams like Fig. 1.5 are often called rheograms. 

Fig. 12 Generalized flow phase diagram obtained for CPCl/NaSal system derived from a superimposition between flow curves at different concentrations and temperatures, using normalized coordinates ct/Gq and /Tg. No stress plateau is observed beyond the critical conditions Op/Go > 0.9 and y-Tg 3 0.5. From Berret et al. [137]...

Mechanical Behavior of Materials. Different kinds of materials respond differently when they undergo basic mechanical tests. This is illustrated in Eigure 15, which shows stress—strain diagrams for purely viscous and purely elastic materials. With the former, the stress is reheved by viscous flow and is independent of strain. With the latter, there is a direct dependence of stress on strain and the ratio of the two is the modulus E (or G). 

Fig. 4. Diagram illustrating the method for determining a yield stress by constructing a flow curve in the coordinates of the Casson equation. The content of the filler is more for curve 2 than for curve 1...

Fig. 3. Flow diagram illustrating the principal environmental compartments affected by, and the principal environmental stresses that can arise from, active and abandoned coal mine voids and coal mine waste deposits.

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 return of shear deformation after elimination of an external stress at a moment tj is also plotted in Fig. 23 (curve d). This diagram shows the flow processes before and after discharge using Eq. (36) as well as a pure elastic deformation = o/G0 and a irreversible flow according to Eq. (41). The elastical deformation yei disappears immediately after discharge. 

Fig. 414 Phase diagram for PE0 aPB0,7PE058 determined on the basis of tube inversion measurements, which detect the finite yield stress of immobile gel phases or the flow behaviour of fluid sols (Booth et al. 1997 Luo et al. 1992).

Fig, 19. (a) Sketch of the channel-die apparatus used for the deformation experiment. Dimensions are in millimetres. The compression stamp is moved along the deformation direction D. The flow of the sample is constrained by the rigid walls of the die in the direction C, and free flow is possible in the direction F. (b) Stress (cr)-strain(e) diagram resulting from channel-die extrusion of bisphenol-A polycarbonate at 300 K and a strain rate of e = 0.01 s l. (c, d) Dipolar DECODER spectra of 13C-labelled bisphenol-A polycarbonate before and after deformation. The spectra exhibit a characteristic star-like ridge pattern. Each of three types of corners (C, D, F) in the pattern corresponds to vectors oriented along a particular direction in the channel-die used for the experiment, (e, f) The anisotropy caused by the deformation becomes readily visible in the difference spectrum (deformed minus undeformed). For clarity, the negative (f) and positive contours (e) have been drawn separately. (Reproduced from Utz et al. with permission.)... 

Figure 8-5 Shearing Stress-Rate of Shear Diagrams. (A) Newtonian liquid, viscous flow, (B) dilatant flow, (C) pseudoplastic flow, (D) plastic flow.

The major types of fluid flow behavior can be described by means of basic shear diagram of shear rate versus shear stress, such as Figures 1-2 and 1-3. In Figure 1-2, the shear stresses are plotted against the shear rates (independent variable) which is the conventional method. However, some authors plot shear rates against the shear stresses (independent variable) as shown in Figure 1-3. With the introduction of controlled-stress rheometers, the use of shear stress as the independent variable is often desirable. 

Figure 1-2 Basic Shear Diagram of Shear Rate versus Shear Stress for Classification of Time-Independent Fiow Behavior of Fiuid Foods Newtonian, Shear-Thinning, and Shear-Thickening. Also, some foods have yield stress that must be exceeded for flow to occur Bingham and Herschel-Bulkley (H-B).

A flow model may be considered to be a mathematical equation that can describe rheological data, such as shear rate versus shear stress, in a basic shear diagram, and that provides a convenient and concise manner of describing the data. Occasionally, such as for the viscosity versus temperature data during starch gelatinization, more than one equation may be necessary to describe the rheological data. In addition to mathematical convenience, it is important to quantify how magnitudes of model parameters are affected by state variables, such as temperature, and the effect of structure/composition (e.g., concentration of solids) of foods and establish widely applicable relationships that may be called functional models. 

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