A Study on the Failure Behavior of Sand Grain Contacts with Hertz Modeling, Image Processing, and Statistical Analysis

Abstract

The crushing behavior of particles is encountered in a large number of natural and engineering systems, and it is important for it to be examined in problems related to hydraulic fracturing, where proppant-proppant and proppant-rock interactions are essential to be modeled as well as geotechnical engineering problems, where grains may crush because the transmitted stresses at their contacts exceed their tensile strength. Despite the interest in the study of the crushing behavior of natural particles, most previous experimental works have examined the single-grain or multiple-grain crushing configurations, and less attention has been given in the laboratory investigation of the interactions of two grains in contact up to their failure as well as on the assessment of the methodology adopted to analyze the data. In the present study, a quartz sand of 1.18-2.36 mm in size was examined, performing a total of 244 grain-to-grain crushing tests at two different speeds, 0.01 and 1 mm/min. In order to calculate stresses from the measured forces, Hertz modeling was implemented to calculate an approximate contact area between the particles based on their local radii (i.e., the radius of the grains in the vicinity of their contact). Based on the results, three different modes of failure were distinguished as conservative, fragmentary, and destructive, corresponding to micro-scale, meso-scale, and macro-scale breakage, respectively. From the data, four different classes of curves could be identified. Class-A and class-B corresponded to an initially Hertzian behavior followed by a brittle failure with a distinctive (single) peak point. The occurrence of hardening prior to the failure point distinguished class-B from class-A. Two additional classes (termed as class-C and class-D) were observed having two or multiple peaks, and much larger displacements were necessary to mobilize the failure point. Hertz fitting, Weibull statistics, and clustering were further implemented to estimate the influence of local radius and elastic modulus values. One of the important observations was that the method of analysis adopted to estimate the local radius of the grains, based on manual assessment (i.e., eyeball fitting) or robust Matlab-based image processing, was a key factor influencing the resultant strength distribution and m-modulus, which are grain crushing strength characteristics. The results from the study were further compared with previously reported data on single- and multiple-grain crushing tests.