Concentric cylinders shear strain

In this apparatus the polymer melt is sheared between concentric cylinders. The torque required to rotate the inner cylinder over a range of speeds is recorded so that viscosity and strain rates may be calculated. 

There are two well-accepted models for stress transfer. In the Cox model [94] the composite is considered as a pair of concentric cylinders (Fig. 19). The central cylinder represents the fiber and the outer region as the matrix. The ratio of diameters r/R) is adjusted to the required Vf. Both fiber and matrix are assumed to be elastic and the cylindrical bond between them is considered to be perfect. It is also assumed that there is no stress transfer across the ends of the fiber. If the fiber is much stiffer than the matrix, an axial load applied to the system will tend to induce more strain in the matrix than in the fiber and leads to the development of shear stresses along the cylindrical interface. Cox used the following expression for the tensile stress in the fiber (cT/ ) and shear stress at the interface (t) ... 

The first theoretical considerations concerning n (p) and G (p) of concentrated 3-D emulsions and foams were based on perfectly ordered crystals of droplets [4,5,15-18]. In such models, at a given volume fraction and applied shear strain, all droplets are assumed to be equally compressed, that is, to deform affinely under the applied shear thus all of them should have the same shape. Princen [15,16] initially analyzed an ordered monodisperse 2-D array of deformable cylinders and concluded that G = Qiox(p < (/), and that G jumps to nearly the 2-D Laplace pressure of the cylinders at the approach of ( > = 100%, following a ( — dependence. 

Experimentally, the dynamic shear moduli are usually measured by applying sinusoidal oscillatory shear in constant stress or constant strain rheometers. This can be in parallel plate, cone-and-plate or concentric cylinder (Couette) geometries. An excellent monograph on rheology, including its application to polymers, is provided by Macosko (1994). 

All the major manufacturers of viscometers and rheometers have Internet sites that illustrate and describe their products. In addition, many of the manufecturers are offering seminars on rheometers and rheology. Earlier lists of available models of rheometers and their manufacturers were given by Whorlow (1980), Mitchell (1984), and Ma and Barbosa-Canovas (1995). It is very important to focus on the proper design of a measurement geometry (e.g., cone-plate, concentric cylinder), precision in measurement of strain and/or shear rate, inertia of a measuring system and correction for it, as well as to verify that the assumptions made in deriving the applicable equations of shear rate have been satisfied and to ensure that the results provided by the manufecturer are indeed correct. 

However, the use of the parallel plate geometry is not recommended for viscosity measurements, because the shear strain rate variation along the gap between the plates is larger than that experienced in concentric cylinder systems.However, there might be advantages when using the geometry in oscillatory studies. ... 

Rheology is a powerful method for the characterization of HA properties. In particular, rotational rheometers are particularly suitable in studying the rheological properties of HA. In such rheometers, different geometries (cone/plate, plate/plate, and concentric cylinders) are applied to concentrated, semi-diluted, and diluted solutions. A typical rheometric test performed on a HA solution is the so-called "flow curve". In such a test, the dynamic viscosity (q) is measured as a function of the shear rate (7) at constant strain (shear rate or stress sweep). From the flow curve, the Newtonian dynamic viscosity (qo), first plateau, and the critical shear rate ( 7 c), onset of non-Newtonian flow, could be determined. 

The effect of temperature on the mechanical properties of a liquid can be investigated using a special type of dynamic mechanical analyser called an oscillatory rheometer. In this instrument the sample is contained as a thin film between two parallel plates. One of the plates is fixed while the other rotates back and forth so as to subject the liquid to a shearing motion. It is possible to calculate the shear modulus from the amplitude of the rotation and the resistance of the sample to deformation. Because the test is performed in oscillation, it is possible to separate the shear modulus (G) into storage (G ) and loss modulus (G") by measuring the phase lag between the applied strain and measured stress. Other geometries such as concentric cylinders or cone and plate are often used depending on the viscosity of the sample. 

Shear rate (shear-strain rate, velocity gradient) n. The rate of change of shear strain with time. In concentric-cylinder flow where the gap between the cylinders is much smaller than the cylinder radii, shear rate is almost uniform throughout the fluid and is given by 7r(i i + R2)N/ R2 — Ri), where Ri and are the radii of the cylinders, one rotating, the other stationary, and N is the rotational speed in revolutions per second. The universally used unit of shear rate is s . In tube flow, the shear rate varies from zero at the center to its maximum at the tube wall where, for a Newtonian liquid, it... 

In order to accoimt for a change in bed state, some investigators have adopted empirical stress train models based wholly on rheometric tests, because simple mechanical analogs caimot accoimt for such a change. An illustrative case is the model of Isobe et which was developed for a mud tested in a concentric-cylinder rheometer. A sample stress-shear (strain) rate relationship is shown in Fig. 27.8. The model (Fig. 27.9) attempts to mimic the characteristic hysteresis loop arising out of a phase lag between the applied stress and the resulting shear rate. Based... 

Thus, to predict sedimentation, one has to measure the viscosity at very low stresses (or shear rates). These measurements can be carried out using a constant stress rheometer (Carrimed, Bohlin, Rheometrics or Physica). A constant stress very small torques and using an air bearing system to reduce the frictional torque) is applied on the system (which may be placed in the gap between two concentric cylinders or a cone-plate geometry) and the deformation [strain y or compliance J = (y/cr) Pa ] is followed as a function of time [39-41]. 

Measurements on the solutions are performed using the Contraves Low-Shear-30-slnus rheometer. The fluid is contained between concentric cylinders that are set In oscillatory motion so that each fluid element encounters a periodically varying strain, with angular frequency u). Since the resulting stress In the fluid due to elastic effects Is In phase and the viscous response 90° out of phase with the strain, the two responses, G and G" respectively, can readily be separated and given as a function of frequency (G zero for a purely viscous fluid G" Is zero for a purely elastic material). 

Several different geometries can be used. In simple shear, a material is placed between two parallel plates and one or both are moved along the parallel direction. Shear can induce crystallization in monodisperse particles [172-175]. Stress versus strain-rate measurements have been made in 2D [176-180] and 3D [181-187]. Couette shear is an approximation to simple shear using material between two concentric cylinders with one or both of the cylinders rotating [170,171,178,188-192]. In pure shear, a material is placed between two parallel plates and one or both are moved along the perpendicular direction in such a way that the volume of the system is conserved [193]. In pure compression, the plates are moved perpendicularly, but the volume is decreasing [194,195]. 

Partially Plastic Thick-Walled Cylinders. As the internal pressure is increased above the yield pressure, P, plastic deformation penetrates the wad of the cylinder so that the inner layers are stressed plasticady while the outer ones remain elastic. A rigorous analysis of the stresses and strains in a partiady plastic thick-waded cylinder made of a material which work hardens is very compHcated. However, if it is assumed that the material yields at a constant value of the yield shear stress (Fig. 4a), that the elastic—plastic boundary is cylindrical and concentric with the bore of the cylinder (Fig. 4b), and that the axial stress is the mean of the tangential and radial stresses, then it may be shown (10) that the internal pressure, needed to take the boundary to any radius r such that is given by... 

It is interesting to see what we can learn from the reptation model. The best way to see it is to look at a simple experiment. Take a polymer melt or a concentrated solution and place it between two plates, similar to the geometry shown in Figure 12.1 (in practice it can also be a gap between two co-axial cylinders). At time t = 0, apply a constant shear stress a, and measure the relative deformation, or strain 7, as it develops in time after the stress is switched on at f = 0. If a is small, the deformation will be proportional to the stress ... 

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