The recorded images were also used to obtain the position of the centroid and the locations of the particles contacts at each time instant (see Fig. 45.2). This was achieved by converting the recorded grayscale image to a binary image and the particle centroids and radius were then detected using a circular Hough transform with Matlab’s image processing toolbox. The resulting centroids and radii were then used to find contacting particles: if the distance between two centroids was less than two times the radii of the corresponding particles, the particles were taken to be contacting with appropriate normal and tangent vectors. Similarly, contact points between particles and the boundaries of the fixture were also detected. 45.3 Results and Discussion The Fig. 45.3 shows the representative exx, exyandeyy strain fields for polyurethane grains at some time instant during the impact experiment. In order to compute the average stress values from average strain values for each grain, linear elastic model is used for both materials. While the assumption is definitely valid for grains with small strains, it may no longer be valid for polyurethane grains with large strains. But hyperelasticity, plasticity and rate dependency based models for polyurethane are not considered for scope of this paper. The analysis of the velocity fields obtained from the DIC results reveals that although the wave velocity for the polyurethane is around 93 m/s, the wavefront moves at approximately 5 m/s through the granular media. The numerical optimization presented in [13] links particle positions, accelerations, contact points, and stresses to interparticle forces. Three particle-scale equations accomplish this connection: momentum balance, stress-force relations, and constraint equations. These equations are combined in a multiobjective optimization problem that can be solved to obtain inter-particle forces. In [13], the authors have ignored the effect of local accelerations and have thus ignored any effect of only the mean particle acceleration in their analysis. In the current work, we have studied the effect of local acceleration gradients within the particles and have included integral terms with moments of local accelerations about the particle centroid. In [13], the plane stress assumption has been considered in constitutive response but as seen in Fig. 45.4, the comparison of plane stress and plane strain conditions for same experiment at a particular instant reveals good agreement between the two cases in terms of inter-particle forces. The good agreement between the plane stress and plane strain conditions indicate that the contribution of elastic energy stored in the grains is limited. High inter-particle forces are observed due to collisions between grains indicating the Fig. 45.2 The particle edges (red circles) and contact points (blue circles) for a representative image as detected by the circular Hough transform based approach described above 45 Experimental Inference of Inter-Particle Forces in Granular Systems Using Digital Image Correlation 381
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