PSI - Issue 13

Koji Uenishi et al. / Procedia Structural Integrity 13 (2018) 670–675 Uenishi et al. / Structural Integrity Procedia 00 (2018) 000–000

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Keywords: Ice; Mechanics of dynamic contact; Wave interaction

1. Introduction Icy bodies can be found at small scales like snow crystals or ice cubes for drinks as well as at larger scales such as glaciers on the Earth and in the rings of Saturn (e.g. Bridges et al., 1984) in space. In a ring system, energy loss during collisions of ice particles may govern the mean free path between collisions, the physical properties of the ring, the cooling rate in the system, etc., and quantitative information about the coefficient of restitution is definitely needed for more precise estimation of such energy loss (Dilley and Crawford, 1996). The coefficient of restitution depends considerably on the relative impact velocity, but because of the inadequate amount of data sets associated with this velocity dependency, the coefficient is often presumed to be constant regardless of the impact velocity in the analyses of inelastic collisions, energy loss and cooling in the system. Hence, in our earlier investigation (Uenishi et al., 2013), the mechanical properties of ice as a solid / granular material and the scale effect of impact velocity have been studied by observing free fall of ice spheres (diameter D = 25 or 50 mm) impinging upon a plate of ice (thickness 60 mm) with a high-speed digital video camera (Photron FASTCAM SA5) at a frame rate of 7,000 frames per second (fps). Seventeen different falling distances between 40 and 450 mm have been set, without any rotation and fracture of the spheres, and from the experimentally recorded photographs before and after the collisions of the spheres, the relation between the coefficient of restitution and the relative velocity of the ice spheres as well as its fluctuations have been quantitatively evaluated. Then, using the obtained data sets and the numerical Event-Driven (ED) method, the cooling process related to collisions of 3,000 ice spheres has been simulated in a two-dimensional square. The results have shown, for example, that if not only the velocity-dependent restitution coefficient but also its fluctuations are considered, the final temperature in the system becomes about 4 % lower than that without taking the fluctuations into account (Uenishi et al., 2013, 2016b). Thus, the scale effect of the relative impact velocity on the collisional behavior of (a collection of) icy bodies as a granular material has been quantitatively demonstrated. 2. Dynamic fracture of ice spheres due to impact loading: Two typical patterns In the second series of our laboratory experiments, fracture of the impinging ice spheres is involved. Cracks may be easily generated in icy bodies if they are abruptly immersed in warm water (owing to thermal shock), but here, only fracture induced by mechanical impact has been treated, with the falling distances of ice spheres ( D = 50 mm) being between 2.15 and 2.7 m (7 different distances). Collisions of 200 ice spheres against a flat polycarbonate plate (thickness 10 mm) have been recorded by the same high-speed digital video camera, but at a frame rate of 150,000 fps. Figure 1 indicates the two representative fracture patterns observed. In Fig. 1(a), only the sections near the free surface in the lower hemisphere are fragmented upon collision and the relatively large top-shaped part remains unfractured (here called “top”- type fracture pattern). On the contrary, upon collision against the transparent polycarbonate plate, the sphere in Fig. 1(b), prepared under the same (temperature, etc.) conditions, is divided mainly into three or four large segments of similar size (like segments of oranges; named “orange segments”-type fracture pattern). Both fracture patterns have been recognized repeatedly during the experiments, and further experiments using new receptacles that produce more single crystal-like transparent ice spheres do show the presence of the two specific fracture patterns even without the pre-existence of apparent material inhomogeneities (e.g. air bubbles) in ice spheres (see Fig. 2; Now the diameter of ice spheres is D = 60 mm and photographs are taken by Photron FASTCAM-MAX 120K). The mechanical background behind the above experimental observations of the generation of the two distinct fracture patterns can be roughly described with simple finite difference simulations of wave propagation and interaction in the spheres. In the calculations with the second order spatiotemporal accuracy, a homogeneous, isotopic and linear elastic ice is assumed, with the density 920 kg/m 3 , Young’s modulus 9.5 GPa and Poisson’s ratio 0.34 by referring to Middleton and Wilcock (1994). This combination of material properties gives the longitudinal (shear) wave speed as c P  4,000 m/s ( c S  2,000 m/s), respectively. Constant grid spacing  x = 1.0 mm with the constant time step  x /(2 c P ) is employed in order to efficiently clarify the essential features of wave propagation in a sphere. For a representative impact loading due to collision, a simplified time history of pressure approximated by a sine-squared curve P ( t ) = A sin 2 (  t / T ) (for 0  t  T ;