Eccentric rotating geometries

The other eccentric geometries described in Sections 5.7.1-5.7.3 all accomplish an oscillatory motion in a similar way by rotating the sample through a deformation fixed in laboratory space. These oscillations are not identical to sinusoidal oscillations, but in the region of linear viscoelastic response, the dynamic moduli can readily be obtained from the solution for an ideal elastic or ideal viscous material (Abbot et al., 1971 Pipkin, 1972). 

For small strains in the linear viscoelastic range, the measured force components can be converted to E and E by using the static cantilever equations and substituting for the elastic modulus E. 

Notice that in eqs. 5.7.1 and 5.7.2 the sample dimensions appear to high powers thus accurate measurement and sample uniformity are essential. Diameter variations or stresses molded into the sample can cause large oscillations in the measured force components. 

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Other eccentric rotating geometries (a) tilted rotating hemispheres (also called balance rheometer), (b) eccentric rotating cylinders, and (c) tilted rotating cone and plate. 

As stated, we begin with the special problem of flow between two rotating cylinders whose axes are parallel but offset to produce the eccentric cylinder geometry shown in Fig. 5 1. In the concentric limit, this is the famous Couette flow problem, which was analyzed in Chap. 3. 

Figure 5.7.2a illustrates the eccentric rotating disk (ERD) geometry (recall Exercise 1.10.7 and Example 2.3.1). A sample is placed between two disks that rotate at the same angular velocity but about offset or eccentric axes. Surface tension holds the sample between the disks. The flow between these eccentric rotating disks results in a shearing motion, with material elements moving in circular paths with respect to each other. A coordinate system r, j, z that rotates with the lower disk (Figure 5.7.2b) can describe the relative motion between particles (Figure 5.7.2c). The deformation is seen to be of constant magnitude, but continually changing direction.

Eccentric rotating disks, (a) Cross section of the experimental geometry, (b) Top view showing particle paths, (c) Relative displacements of particles in a coordinate system rotating with the lower disk. 

Eccentric rotating cylinders can give accurate and G data but only at rather small strains (Broadbent and Walters, 1971). At larger deformation, cavitation or extrusion of the sample from the gap can occur. This also seems to be the problem for such other eccentric geometries as the tilted rotating disks or cone and plate (Davis and Macosko, 1973 Walters, 1975). 

Note that in the literature this geometry js called the Maxwell orthogonal rheometer or eccentric rotating disks, ERD (Macosko and Davis, 1974 Bird, et al., 1987, also see Chapter 5). Usually, the coordinates X2 = > and X3 = z are used. 

The prosthesis for total knee joint replacement consists of femoral, tibial, and patellar components. Compared to the hip joint, the knee joint has a more complicated geometry and movement biomechanics, and it is not intrinsically stable. In a normal knee, the center of movement is controlled by the geometry of the ligaments. As the knee moves, the ligaments rotate on their bony attachments and the center of movement also moves. The eccentric movement of the knee helps distribute the load throughout the entire joint surface [Burstein and Wright, 1993]. 

Several other eccentric geometries have been described. Three of them are shown in Figure 5.7.5. Kepes (1968) and Kaelble (1969) developed tilted rotating hemispheres also known as the Kepes balance rheometer. Figure 5.7,3 shows that this rheometer can measure li and G data accurately. ... 

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