PSI - Issue 13

4

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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Surface waves

Fig. 3. Numerically produced contours of normalized octahedral shear stress, i.e. stress / maximum pressure A induced by dynamic collision, in homogeneous, isotropic, linear elastic spheres. Collision occurs at the bottom of each sphere, and the normalized duration of contact c P T / D is (a) 0.8 and (b) 8. The snapshots are taken at normalized time c P t / D (a) 1.6 and (b) 3.2 after the collision. In (a), surface waves of shorter wavelengths propagating upwards along the curved free surface are visible while in (b), solely the quasi-static-like stress distribution can be found. Owing to the symmetric nature of the model, only contours in the right half parts are illustrated. Furthermore, for graphical simplicity, no fracture is incorporated in the calculations. 3. Fracture criteria for three-dimensional problems In the above preliminary numerical speculations using a plain yet efficient finite difference technique, the duration of collision or the length of the sphere-plate contact time seems to play a crucial role in comprehending the generation of two fracture patterns. In the calculations, however, the impact load is given at the bottom as a simple function of normal compressive stress only, which, in reality, may not be mechanically true. Therefore, more rigorous simulations based on the Discontinuous Galerkin (DG) method that include also the mechanics of dynamic contact between the sphere and plate are performed in order to verify whether the above finite difference speculations are, at least qualitatively, acceptable or not. The DG method used here is based on the second order numerical scheme proposed by Etienne et al. (2010), with the explicit leapfrog scheme in time and a centered flux choice for the inner element faces. This flux choice has very good non-dissipative properties (see Benjemaa et al., 2009; Delcourte et al., 2009). The unilateral contact of the sphere with the rigid plate is modeled by the normal compliance law introduced by Martins and Oden (Martins and Oden, 1983; Oden and Martins, 1985). Following this model, the normal stress is proportional to the “penetration depth” through a compliance coefficient. In the computations, the sphere is supposed to be axi-symmetric, and on the half disc (in the coordinates r and z ) a mesh with 2,032 vertexes, 3,863 triangles and 23,178 degrees of freedom (polynomial functions of degree two) is considered. The mesh is adapted such that it is much finer near the contact point (see Fig. 4(a)). The time step is chosen to be 1.49993  10  9 sec, while the compliance coefficient is set at 1.7  10 4 GPa/m. In Fig. 4, the close-up views of the impact zones are shown for two computations at 3  s after impact, with two different impact speeds of 4.67 (Fig. 4(b)) and 5.83 (Fig. 4(d)) m/s, respectively. In Fig. 4(b), the maximum of the shear stress (Tresca criterion) is plotted where an important concentration of the shear stress acting on the plane Orz in the compression zone is remarked near the point of impact. For the impact speed of 4.67 m/s, the fracture may occur in compression (acting on planes represented in Fig. 4(c)), giving the “top”-type fracture pattern. A different behavior exists for larger impact speeds. In Fig. 4(d), the maximum of the principal stress is plotted which are mostly in O  direction. It is noted that near the impact point a compression zone prevails, but on top of it a tension area is present. Since the strength in tension (Rankine criterion) of the ice is much smaller than that in compression, for the impact speed of 5.83 m/s, the fracture may occur in tension (acting on planes corresponding to one in Fig. 4(e)), resulting in the “orange segments”- type fracture pattern. Overall, the finite difference speculations are not contradictory to the DG observations, and a promising physical background has been pointed out for the experimentally found two fracture patterns. However, it seems still difficult to exactly simulate the dynamic fracture process, partly because our understanding of mechanics of actual three-dimensional dynamic fracture is rather inadequate. Numerical simulations involving fracture in brittle solids are often conducted in the three-dimensional framework, but typically with criteria derived from one- (or maximum two-) dimensional observations of pre- or post-failure behavior of solids. Besides the two distinct fracture patterns in ice spheres, our recent