Modified Bingham model

The relation between stress and strain for the Modified Bingham model in Fig. 11.15(b) has three regions of behavior, linear elastic, linear viscoelastic and plastic flow. The relations between stress and strain for the first two regions in Fig. 11.16(d) can be described by. 

This result predicts a linear variation of strain with time and, therefore, does not well represent the case of creep for polycarbonate. Combining models b and d in Eig. 11.15 would give the desired form of creep response. (See HW problem 11.4.) The modified Bingham model is, however, compatible with the experimental data given in Eig. 11.14, in the sense that there appears to be a stress below which creep or relaxation will not occur. 

Using the modified Bingham model in the Nagdi-Murch analytical approach, the difference between viscoelastic and elastic strains in a creep test becomes. 

The yield stress (Luder s band formation) vs. creep to yield time from Eig. 11.12 is shown in Eig. 11.16 and compared to Eq. 11.35. The constants A, B and C in Eq. 11.35 were determined from the creep to yield data. Poisson s ratio was assumed to be 0.4 and all other parameters were determined for the modified Bingham model. A similar procedure was used to obtain the creep to yield behavior of a rubber-toughened adhesive (Brinson, et ah, (1975)). 

Develop the differential equation for modified Bingham model given by Eq. 11,29. 

Verify Eq. 11.35 for the creep to yield time assuming use of the modified Bingham model and the Crochet equation... 

The over-stress idea can be incorporated into the viscoelastic mechanical models by means of adding sliding elements to describe yield or termination points. With such an application, equations containing viscosity terms are obtained in a form similar to (24.114). For example, Brinson et al. (1975), Renieri et al. (1976), Sancaktar and Brinson (1979, 1980), Sancaktar (1981), Sancaktar and Padgilwar (1982), Sancaktar et al. (1984), and Sancaktar and Schenck (1985) used the modified Bingham model developed by Brinson (1974) to describe the shear and tensile material behavior of structural adhesives in the bulk and bonded forms ... 

The modified Bingham model and its comparison with the Maxwell model, tp = ju/E= relaxation time and to is the time to reach elastic limit... 

The first general parameterization to be reported by Dewar and co-workers was a third-generation modified INDO model (MINDO/3 Bingham, Dewar, and Lo, 1975). Some of the specific modifications to the INDO framework included the use of different t exponents in s and p type STOs on the same atom, the definition of pair parameters /Iab between two atoms A and B that were not averages of atomic parameters (actually, four such parameters... 

Butter, and other unctuous materials, may be qualitatively described by a modified Bingham body (Elliott and Ganz, 1971 Elliott and Green, 1972), which consists of viscous, plastic and elastic elements in series. The stress-strain behavior for the model proposed by Elliot and Ganz (1971) is shown in Figure 7.12B. Diener and Heldman (1968) proposed a more complex model to describe how butter behaves when a low level of strain is applied. The model consists of plastic and viscous elements in parallel, coupled in series with a viscous element in parallel with a combination of a viscous and an elastic element. Figure 7.12C shows the stress-strain curve for... 

II. The modified Bingham body - a useful rheological model. J. Text. Stud. 3, 194-205. Enjalbert, F., M. C., Nicot, C., Bayourthe, M., Vernay, Moncoulon, R. 1997. Effects of dietary calcium soaps of unsaturated fatty acids on digestion, milk composition, and physical properties of butter. J. Dairy Res. 64, 181-195. 

Rheology experiments also give information in the determination of wax appearance temperatures of crude oils. In this research, WATs of crude oils were determined by viscometry from the point where the experimental curve deviates from the extrapolated Arrhenius curve (Figure 4). It was observed that all crude oils, except highly asphaltenic samples, are Newtonian fluids above their wax appearance temperatures. The flow behaviour of crude oils is considerably modified by the crystallization of paraffins corresponding to the variation of the apparent viscosity with temperature. Below the WAT, flow becomes non-Newtonian and approaches that of the Bingham and Casson plastic model [17,18]. 

The use of the modified rheoviseometer, which enables handling of unstable mineral suspensions, has recently revealed that the Casson Equation fits the flow curve for the magnetite suspension better than the typically used Bingham plastic model [see the special issue of Coal Preparation entirely devoted to magnetite dense media [Coal Preparation ( 99Qi). 8(3 ).]... 

Despite the fact that equation (3.37) is applicable to all kinds of time-independent fluids, numerous workers have presented expressions for turbulent flow friction factors for specific fluid models. For instance, Tomita [1959] applied the concept of the Prandtl mixing length and put forward modified definitions of the friction factor and Reynolds number for the turbulent flow of Bingham Plastic fluids in smooth pipes so that the Nikuradse equation, i.e. equation (3.37) with n = 1, could be used. Though he tested the applicability of his method using his own data in the range 2000 < Reg(l — 4>f 3 — )< 10, the validity of this approach has not been established using independent experimental data. 

This is not simply because of the presence of rheology modifiers or other polymers a simple NaPA-stabilized GCC suspension will also show a plastic-like behaviour. Typically, this involves the suspension behaving like a solid at low shear rates and stresses, and only yielding to liquid-like behaviour above a certain yield stress [Pg.150]

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