Photoelasticity images

Figure G.2 Photoelasticity image of a beam 3.5 x 20x400 mm in size under flexion (P, polarizer A, analyser).

Using a combination of linear and circular measurements, the direction and magnitude of the principle stress difference can be determined and the full 2D stress tensor can be reconstructed. However, in granular applications, the internal stress distribution of each particle is less important than the forces between particles. Section 2.4.3 explores techniques to extract forces directly from photoelastic images. 

Combining (2.57), (2.44), and (2.12) for the dark-field polariscope yields the full equation for a photoelastic image of a single disk ... 

Figure 7.3 Contrasting photoelastic images of an isotropically compressed quasi-2D system of photoelastic disks (a) and a shear-jammed system of the same photoelastic disks (b). Note the long quasi-linear force chains in the sheared system, vs. the rather short entangled force network of the isotropically compressed system. More details on the use of photoelastic materials to obtain quantitative force data are given later.

Figure 7.5 Photoelastic image of a heap formed from elliptical photoelastic particles via a protocol that is roughly similar to the seive method mentioned earlier [56]. Note that the force chains arriving at the base are roughly uniformly distributed in this, unlike what might occur when a force dip occurs.

Since the pressure is a reflection of the mean normal forces on a particle, hence the internal stress, it is possible to calibrate a measure of the fringe density against the pressure, P. One measure of the fringe density is the square magnitude of the photoelastic image intensity, or G. Thus, G, integrated over a particle, provides an empirical connection to the local P. 

Image analysis Modern visualization techniques produce amazing images. Section 2.4 explains several important techniques to extract quantitative information from images including particle tracking, photoelastic force measurement, and particle imaging velocimetry. 

Figure 2.24 Enlargement of small area (rectangle Figure 2.4) from 2D photoelastic disks under simple shear viewed through a dark-field circular polariscope. (a) Experiment. (Courtesy of J. Ren and R. P. Behringer, Duke University, Durham, NC.) (b) Calculated reconstruction used to extract the interparticle forces, (c) Least squares image used to determine the initial guesses for the contact points and final contact positions (circles).

Figure 7.20 Images showing the photoelastic response of disks that are subject to simple shear, after Ren et al. [63]. Top to bottom shows states that are, respectively, fragile, near shear jamming, and well above shear jamming. In these experiments, a layer of particles is sheared, using a special apparatus that allows for uniform shear strain across the whole layer. The initial state is one that is stress free and that has a parallelogram shape that is "Tilted down"" at the lower left corner. The final state is one where the tilt is reversed. Note that throughout the shearing process, the density of the system remains constant.

Significant progress has been made in 2D systems that allow all the particles to be visualized, for example, using photoelastic disks (cf. Chapter 7). To overcome the inherent difficulty of visualizing particle motion inside a 3D packing, various advanced techniques such as x-ray imaging and optical index matching have been developed [9,10]. At the same time, increased computation power... 

Figure 9.2 Force networks for systems of photoelastic disks that have been compressed isotropically (a) and sheared (b). In these images, brighter particles experience higher mean force. The sheared system on the right is in a regime of densities for which shear jamming occurs, and the resulting force networks, or force chains, are highly anisotropic.

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