Photoelastic Force

In (Section 2.3.2) photoelasticity was introduced as a technique to visualize stress in granular materials. In this section I will describe three techniques to directly 

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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. 

The photoelastic measurements were carried out in simple extension using strip specimens. In addition to the force/ also the optical retardation S (hence also the birefringence An <5) could be determined and the modulus G, the deformational-optical function A and the stress-optical coefficient C = A/G were calculated using the equations [31]... 

Fig. 4.17 (a) Force vectors obtained in a two-dimensional assembly of photoelastic disks under... 

Thus, by using the photoelasticity effect, we managed to ascertain for the first time that the hypercrossHnked polystyrene network can experience inner stresses of shrinkage on drying, stresses of extension on sweUing in good solvents, and also can acquire a fuUy relaxed state, totally free from any stresses, when wetted with methanol. Dusek and Prins [163] have postulated a pecuHar reference state when polymeric chains do not exert any forces on network junction points (see also Chapter 1, Section 2.2), but this state of... 

This being so the engineer who has a problem to solve may have to resort to a model merely to obtain a solution from the deductive calculus of the theory. The use of photoelasticity is an example. Consider a photoelastic specimen used to model the theory of simple beam behaviour. This is for illustration purposes only, of course, since the deductive calculus of a simple elastic beam is easily solved. Both model and theory assume at least the following Newtonian mechanics elastic behaviour of materials symmetrical bending and no resultant forces on the system. The theoretical derivation of the elementary equations of... 

Experiments and results. The viscoelastic material chosen was an optically smooth polyurethane, reconunended for dynamic studies in photoelasticity SM4 Vishay, with E = 3.6 MPa). The surfaces were wiped with an alcohol-soaked cloth, dried with warm air and left, sheltered from dust, for 30 min for the equilibrium with room temperature to be reached. Then, two strips of thickness /i=3.175 mm with various lengths L and widths b varying in the range 5-20 cm and S-20 mm, respectively, w e gently superimposed, and they adhered under only molecular attraction forces, without additional adhesive. In order to avoid the dwell time effect (/9), strips were coupled during the same contact duration 30 min, for all experiments. Moreover, temperature (23°C) and humidity (84%) were kept constant. In these conditions, the reproducibility is better than 3%. 

Stress-induced birefringence or photoelasticity (Figure 2.4) has been an important technique for studying granular materials since 1950s [176-178,193, 194,207-214]. It can be used to visualize and measure forces in 2D packings. 

Figure 2.4 2D photoelastic disks under simple shear viewed through a dark-field circular polariscope. The force is roughly proportional to the brightness of the particles (see text). An anisotropic force network is evident. (Courtesy of J. Ren and R. P. Behringer, Duke University, Durham, NC.)... 

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. 

T = 3.05 mm. The bowing of the surface causes the contact length to depend on force, (b) The photoelastic signal depends on the force per unit length P = F/L and the texture analyzer measures the force F. If the radius of the cylinder is independent of thickness then F is proportional to P. In some particles, the side is bowed (a) and F/P is not a constant. The plot shows F/P (dots) for a bowed particle as a function of strain displacement 6/D. The line is a calculation of L assuming the bow is circular with diameter 38 mm. For this particle, the photoelastic measurement overestimates the true force during the first 2.2% strain. 

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.

Figure 7.12 Response for increasing amounts of spatial disorder (a) regular hexagonal packing of monodisperse photoelastic disks (b) polydisperse disk packing (c) packing of pentagons [62]. In this and the next several figures, the response is shown in color for an ensemble of multiple experiments where a point force was applied to the upper boundary of the indicated stacking of photoelastic particles. More details about photoelastic techniques are given at the end of this chapter.

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