Dataset for 3D Numerical Modeling of Compressible and Cohesive Granular Flow Impact on Narrow Obstacles

Supplementary Material for: 3D Numerical Modeling of Compressible and Cohesive Granular Flow Impact on Narrow Obstacles Michael J. Kohler, Johan Gaume, Christophe Ancey, Betty Sovilla A. Supplementary Material A.1. Volume-Averaging Procedure A MP property x is volume-averaged within a domain D by weighting the individual property of each MP i with its Jacobian Ji through ⟨x⟩D=Pn i=1xiJi Pn i=1Ji .(A.1) Where n is the number of MPs in the averaging domain. Here we use the angle brackets to denote volume-averaged quantities. For impact pressure, which we define as the flow-wise component σxx of the Cauchy stress tensor, this relation simplifies to ⟨σxx⟩D=Pn i=1τxx,i Pn i=1Ji ,(A.2) where we utilized Equation (7). Here we perform this procedure in the so called impact domain. This fixed volume extends almost over the entire width of the obstacle (1 . 5 m) and its height matches the initial flow height of 1 . 75 m. The upstream extent reaches from x = − 0 . 5 dx to x =2 . 5 dx and is selected to (i) cover the compact support of the quadratic B-spline interpolation functions amounting 3 dx as visualized in Figure A.11 and (ii) capture the particles which slightly penetrate the obstacle, which is commonly observed in MPM. For a volume-averaged compaction measure, it proved more robust to track the total mass of all particles within the impact region, excluding the portion that extends into the obstacle. We then divide this mass by the volume of the region and compare it to the original (reference) density. Figure A.11: Illustration of the impact domain, showing the quadratic B-spline basis function and their 1D extent in x -direction, covering the length of the impact domain A1 A.2. Sensitivity Analysis Below, we systematically assess the sensitivity of the total impact force to key numerical and material parameters. We vary the grid size, the obstacle friction, and the materials Young’s modulus and preconsolidation pressure. Analogous to the main part, we focus on the exemplary material a and bacross a range of compressibility and Froude numbers to evaluate the robustness of the observed impact behavior. Furthermore, we assess the influence of the rigid obstacle assumption by explicitly simulating flexible obstacles with varying stiffnesses. A.2.1. Grid Size Changing the grid size from dx =0 . 25 m to dx =0 . 125 m leads to a consistent increase in the total impact force by approximately 5 % to 10 % for material a and 5 % to 15 % for material b, with only minor dependence on compressibility and impact velocity. Although this indicates a grid-size effect, the overall trends are robust and remain consistent across grid resolutions. At Fr =20 and ξ=0.05, however, spurious oscillations are observed in the force signal, as addressed in the main part of the paper. material a) 0 20 40 60 0 1 2 3 Fx (kN) 1e1 =5 0 20 40 60 0 1 2 3 1e1 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 0.1 0 2 4 0 2 4 6 Fx (kN) 1e2 =5 024 0 2 4 1e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 5 0.0 0.5 1.0 1.5 time t (s) 0 2 4 6 Fx (kN) 1e3 =5 0.0 0.5 1.0 1.5 time t (s) 0 2 4 6 1e3 =0.05 dx =0.25 m dx =0.125 m rel. var. 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 rel. variation Fr = 20 Figure A.12: material a ( M =0 . 4, β =0 . 15) in a nearly incompressible ( ξ =5) and a highly compressible configuration ( ξ =0 . 05) - effect of changing the grid size from dx =0 . 25 m to dx =0 . 125 m on impact force for Fr =0 . 1, 5, and 20. material b) 0 50 100 0.00 0.25 0.50 0.75 1.00 Fx (kN) 1e3 =5 0 20 40 60 0.0 0.5 1.0 1.5 2.0 1e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 0.1 0 2 4 0.0 0.5 1.0 Fx (kN) 1e3 =5 0 2 4 0 2 4 6 1e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 5 0.0 0.5 1.0 1.5 time t (s) 0.0 2.5 5.0 7.5 Fx (kN) 1e3 =5 0.0 0.5 1.0 1.5 time t (s) 0 2 4 6 1e3 =0.05 dx =0.25 m dx =0.125 m rel. var. 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 rel. variation Fr = 20 Figure A.13: material b ( M =1 . 2, β =0 . 45) in a nearly incompressible ( ξ =5) and a highly compressible configuration ( ξ =0 . 05) - effect of changing the grid size from dx =0 . 25 m to dx =0 . 125 m on impact force for Fr =0 . 1, 5, and 20. A2 A.2.2. Obstacle Friction Changing the obstacle friction coefficient from µ =0 . 25 to µ =2 . 5 leads to an increase in the total impact force, by less than 5 % for material a and between 5 % to 20 % for material b, with only minor dependence on compressibility and impact velocity. Thus, the influence of friction remains minor, and the overall trends are robust and consistent across friction conditions. At Fr =20 and ξ=0.05, spurious oscillations are again observed in the force signal, as discussed before and in the main part of the paper. material a) 0 20 40 60 0 1 2 3 Fx (kN) 1e1 =5 0 20 40 60 0 1 2 3 1e1 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 0.1 0 2 4 0 2 4 6 Fx (kN) 1e2 =5 024 0 1 2 3 4 1e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 5 0.0 0.5 1.0 1.5 time t (s) 0 2 4 6 Fx (kN) 1e3 =5 0.0 0.5 1.0 1.5 time t (s) 0 2 4 61e3 =0.05 =0.25 =2.5 rel. var. 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 rel. variation Fr = 20 Figure A.14: material a ( M =0 . 4, β =0 . 15) in a nearly incompressible ( ξ =5) and a highly compressible configuration ( ξ =0 . 05) - effect of changing the obstacle friction from µ=0.25 to µ=2.5 on impact force for Fr =0.1, 5, and 20. material b) 0 50 100 0.0 0.5 1.0 Fx (kN) 1e3 =5 0 20 40 60 0.0 0.5 1.0 1.5 2.0 1e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 0.1 0 2 4 0.0 0.5 1.0 Fx (kN) 1e3 =5 0 2 4 0 2 4 6 1e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 5 0.0 0.5 1.0 1.5 time t (s) 0 5 Fx (kN) 1e3 =5 0.0 0.5 1.0 1.5 time t (s) 0.0 2.5 5.0 7.5 1e3 =0.05 =0.25 =2.5 rel. var. 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 rel. variation Fr = 20 Figure A.15: material b ( M =1 . 2, β =0 . 45) in a nearly incompressible ( ξ =5) and a highly compressible configuration ( ξ =0 . 05) - effect of changing the obstacle friction from µ=0.25 to µ=2.5 on impact force for Fr =0.1, 5, and 20. A3 A.2.3. Young’s Modulus Changing the Young’s modulus from E =1000 kPa to E =2000 kPa results in less than a 10 % change in the total impact force for material a, but a significantly larger positive influence for material b. This stronger sensitivity in material b arises from its higher shear strength, leading to a larger proportion of elastic deformation during impact. Nevertheless, the overall trends remain robust and consistent across stiffness variations. material a) 0 20 40 60 0 1 2 3 Fx (kN) 1e1 =5 0 20 40 60 0 1 2 3 1e1 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 0.1 0 2 4 0.0 0.5 1.0 Fx (kN) 1e3 =5 0 2 4 0 2 4 1e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 5 0.0 0.5 1.0 1.5 time t (s) 0.0 2.5 5.0 7.5 Fx (kN) 1e3 =5 0.0 0.5 1.0 1.5 time t (s) 0 2 4 6 1e3 =0.05 E =1000 kPa E =2000 kPa rel. var. 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 rel. variation Fr = 20 Figure A.16: material a ( M =0 . 4, β =0 . 15) in a nearly incompressible ( ξ =5) and a highly compressible configuration ( ξ =0 . 05) - effect of changing the Young’s modulus from E =1000 kPa to E =2000 kPa on impact force for Fr =0.1, 5, and 20. material b) 0 50 100 0.0 0.5 1.0 Fx (kN) 1e3 =5 0 20 40 60 0 1 2 3 1e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 0.1 0 2 4 0 1 2 3 Fx (kN) 1e3 =5 024 0 2 4 6 81e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 5 0.0 0.5 1.0 1.5 time t (s) 0.0 0.5 1.0 Fx (kN) 1e4 =5 0.0 0.5 1.0 1.5 time t (s) 0 5 1e3 =0.05 E =1000 kPa E =2000 kPa rel. var. 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 rel. variation Fr = 20 Figure A.17: material b ( M =1 . 2, β =0 . 45) in a nearly incompressible ( ξ =5) and a highly compressible configuration ( ξ =0 . 05) - effect of changing the Young’s modulus from E =1000 kPa to E =2000 kPa on impact force for Fr =0.1, 5, and 20. A4 A.2.4. Preconsolidation Pressure Changing the preconsolidation pressure from pc,0 =10 kPa to pc,0 =5 . 5 kPa has only a minor influence on the total impact force, remaining mostly below 5 % for both material a and material b. An exception is observed for Fr =0 . 1 and ξ =0 . 05, where the impact force decreases by approximately 20 % to 30 %. At Fr =20 and ξ =0 . 05, we again observe spurious oscillations in the force signal, as addressed in the main part of the paper. material a) 0 20 40 60 0 1 2 3 Fx (kN) 1e1 =5 0 20 40 60 0 1 2 3 1e1 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 0.1 0 2 4 0 2 4 6 Fx (kN) 1e2 =5 024 0 1 2 3 4 1e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 5 0.0 0.5 1.0 1.5 time t (s) 0 2 4 6 Fx (kN) 1e3 =5 0.0 0.5 1.0 1.5 time t (s) 0 2 4 61e3 =0.05 pc ,0 = 10 kPa pc ,0 = 5.5 kPa rel. var. 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 rel. variation Fr = 20 Figure A.18: material a ( M =0 . 4, β =0 . 15) in a nearly incompressible ( ξ =5) and a highly compressible configuration ( ξ =0 . 05) - effect of changing the preconsolidation pressure from pc,0 =10 kPa to pc,0 =5 . 5 kPa on impact force for Fr =0.1, 5, and 20. material b) 0 50 100 0.00 0.25 0.50 0.75 1.00 Fx (kN) 1e3 =5 0 20 40 60 0.0 0.5 1.0 1.5 2.0 1e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 0.1 0 2 4 0.0 0.5 1.0 Fx (kN) 1e3 =5 0 2 4 0 2 4 61e2 =0.05 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 rel. variation Fr = 5 0.0 0.5 1.0 1.5 time t (s) 0.0 2.5 5.0 7.5 Fx (kN) 1e3 =5 0.0 0.5 1.0 1.5 time t (s) 0 2 4 6 1e3 =0.05 pc ,0 = 10 kPa pc ,0 = 5.5 kPa rel. var. 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 rel. variation Fr = 20 Figure A.19: material b ( M =1 . 2, β =0 . 45) in a nearly incompressible ( ξ =5) and a highly compressible configuration ( ξ =0 . 05) - effect of changing the preconsolidation pressure from pc,0 =10 kPa to pc,0 =5 . 5 kPa on impact force for Fr =0.1, 5, and 20. A5 A.2.5. Obstacle Stiffness Below, we present a sensitivity analysis to assess the effect of obstacle stiffness on the impact pressure and flow morphology. The reference configuration is identical to that described in Section A.3. The flexible obstacle is modeled using elastic Material Points governed by a St. Venant–Kirchhoffconstitutive law. The obstacle’s Young’s modulus, Eobs , is varied over the range Eobs ∈ 1 × { 10 0, 10 1, 10 2, 10 3, 10 4}MPa , where Eobs =1 × 10 1MPa corresponds to the Young’s modulus of the avalanching material. The Poisson’s ratio is kept constant at ν =0 . 3. The grid spacing and Material Point density are fixed at dx =0 . 25 m and # MP =8, respectively. Figure A.20 compares the resulting flow morphologies and velocity fields for different obstacle stiffnesses and for the rigidobstacle reference case at a representative time. For Eobs > 1 × 10 1MPa , only minor variations are observed, and the overall flow pattern and velocity field remain practically indistinguishable. Figure A.21 presents the corresponding impact pressure time histories and the time-averaged steady-state impact pressures, ¯τxx , which exhibit good convergence as the obstacle stiffness increases. Notably, the pressure response exhibits negligible differences once the obstacle stiffness approaches that of typical construction materials such as concrete. Figure A.20: Flow morphology and velocity field from the top view (lower panels) and longitudinal section (upper panels) for all studied Young’s moduli and for the rigid-obstacle reference case (Eobs =inf .). 0 2 4 6 8 10 time (s) 0 5 10 15 20 25 30 xx (kPa) 100101102103104 E (MPa) 0 5 10 15 20 25 30 xx (kPa) Young's modulus of concrete 100 101 102 103 104 E (MPa) Figure A.21: Left: Time history of the volume-averaged flow-wise stress component for different obstacle stuffinesses. Right: Time-averaged steady-state impact pressure as a function of obstacle stiffness. A6 A.3. Convergence and Mesh-Sensitivity To verify the numerical stability and resolution independence of the results, we perform a convergence study using a reference configuration representative of a mid-range flow regime. The grid size and material point (MP) density are systematically varied to assess their influence on the computed impact forces. The reference avalanche configuration is defined by the following parameters: Fr =1,v=4.143 m/s,H=1.75 m,W=15.75 m,L=50 m. The avalanche’s material parameters are: ρ0=300 kg/m3;E=1000 kPa, ν =0.1; pc,0=10 kPa,M=0.8, β =0.3,and ξ=0.5. A.3.1. Grid-Size The total impact force converges toward a stable value with increasing grid resolution (decreasing dx ), as shown in the right plot below. Small jumps in the force curve reflect discretization effects, as the obstacle geometry may or may not align with integer multiples of the grid spacing. Figure A.22: Left: Total impact force time history for different grid resolutions. Right: Time-averaged steady-state impact force versus grid resolution. The grid size used in the main simulations (dx =0.25 m) is indicated by the dotted line. A.3.2. Material Points per Cell The influence of MP density on the total impact force is small, and the results converge toward a consistent value with increasing MP density, as shown in the right plot below. This confirms that the chosen MP density (6-8) provides sufficient accuracy for the analyses presented in the main text. Figure A.23: Left: Total impact force time history for different grid resolutions. Right: Time-averaged steady-state impact force versus grid resolution. The grid size used in the main simulations (dx =0.25 m) is indicated by the dotted line. A7 A.4. Influence of M′on σxx for different compressibilities for the entire Froude range Below we show the dependence of the impact pressure ⟨σxx⟩H on the effective internal friction M′ for a the entire Fr compressibility space. The accompanying table lists the fitted parameters and relative standard errors of the model function used to capture these trends: 0 15 30 45 ′ ( ) M ′ (-) 0 100 200 xx H (kPa) Fr = 0.1 0 15 30 45 ′ ( ) M ′ (-) 0 100 200 Fr = 0.3 0 15 30 45 ′ ( ) M ′ (-) 0 100 200 Fr = 1 0 15 30 45 ′ ( ) M ′ (-) 0 100 200 Fr = 2 M ′ (-) 0 100 200 Fr = 3 M ′ (-) 0 100 200 Fr = 5 M ′ (-) 0 100 200 300 Fr = 7 M ′ (-) 0 200 400 600 Fr = 10 0 0.6 1.2 1.8 2.4 M ′ (-) 0 500 1000 Fr = 14 0 0.6 1.2 1.8 2.4 M ′ (-) 0 1000 2000 3000 Fr = 20 eq. (17) - fit =5 =0.5 =0.05 Figure A.24: Impact pressure ⟨σxx⟩H over effective internal friction M′ for Fr =0 . 1 to 20, and ξ = { 5 , 0 . 5 , 0 . 05 } , which corresponds to χP,25 = {100.7%,107.5%,205.3%} Fr ξa RS Eab RS Ebc RS Ec 0.1 0.05 63.7972 1.2890 689.1868 0.1427 1.1772 0.1627 0.5 125.8475 0.3533 727.1908 0.0775 1.8392 0.0716 5 239.7963 0.1866 553.2184 0.0935 2.9575 0.0527 0.3 0.05 5.2954 2.4313 76.4574 0.1907 1.2851 0.1987 0.5 13.1396 0.7200 80.7280 0.1408 1.9678 0.1136 5 24.3932 0.2097 65.1346 0.0886 2.9520 0.0497 1 0.05 0.3452 4.0553 5.8514 0.2822 1.4873 0.2459 0.5 0.8065 1.3331 6.9705 0.1847 2.2366 0.1234 5 2.5046 0.1975 5.8721 0.0817 3.2039 0.0362 2 0.05 0.0699 6.9532 1.7058 0.2993 0.9332 0.2677 0.5 0.3565 1.2773 1.7365 0.2879 1.6937 0.2194 5 0.8904 0.2148 1.4157 0.1329 3.0796 0.0654 3 0.05 0.5687 0.2855 0.7069 0.2564 0.9430 0.2489 0.5 0.4007 0.5845 1.0520 0.2500 1.4056 0.2146 5 0.4445 0.1080 1.1491 0.0857 2.3145 0.0587 5 0.05 1.3544 0.0593 0.1291 0.7052 1.9871 0.4042 0.5 0.7453 0.0736 0.5221 0.1401 1.1258 0.1562 5 0.7388 0.0772 0.5784 0.1280 1.9765 0.0874 7 0.05 1.2358 0.1078 0.4569 0.3241 0.4180 0.5351 0.5 0.9972 0.0814 0.4609 0.2009 0.9190 0.2063 5 0.8528 0.0469 0.5489 0.0957 1.3395 0.0872 10 0.05 1.3195 0.0216 0.1873 0.3235 1.8210 0.2612 0.5 1.4918 0.0398 0.1548 0.4849 2.0463 0.3521 5 1.1065 0.0237 0.4279 0.0827 1.2234 0.0900 14 0.05 1.2544 0.0361 0.2210 0.2495 0.7036 0.3639 0.5 1.6974 0.0338 0.2346 0.2707 0.0000 nan 5 1.3150 0.0329 0.4394 0.1199 1.0648 0.1489 20 0.05 1.1470 0.0274 0.1949 0.2295 1.0424 0.3082 0.5 1.4807 0.1013 0.5079 0.3255 0.9865 0.3119 5 1.9151 0.0219 0.0409 0.9565 3.6989 0.3502 Table A.5: Fitting parameters and relative standard error (RSE) of the model function Equation (17) in Figure 6 A8 A.5. Overview of the Entire Parametric Space by Means of Raster Plots of Force and Stress Below we provide an overview of the entire parametric space by plotting the mean impact pressure σxx and mean total impact force Fxas functions of cohesion βand internal friction Mfor each studied Froude number Fr: 0.0 0.4 0.8 1.2 M (-) Fr=0.1 Fr=0.3 Fr=1 Fr=2 Fr=3 0.0 0.15 0.3 0.45 (-) 0.0 0.4 0.8 1.2 M (-) Fr=5 0.0 0.15 0.3 0.45 (-) Fr=7 0.0 0.15 0.3 0.45 (-) Fr=10 0.0 0.15 0.3 0.45 (-) Fr=14 0.0 0.15 0.3 0.45 (-) Fr=20 101 102 103 xx H (kPa) plastic compressibility: P ,25 = 107.5% 0.0 0.4 0.8 1.2 M (-) Fr=0.1 Fr=0.3 Fr=1 Fr=2 Fr=3 0.0 0.15 0.3 0.45 (-) 0.0 0.4 0.8 1.2 M (-) Fr=5 0.0 0.15 0.3 0.45 (-) Fr=7 0.0 0.15 0.3 0.45 (-) Fr=10 0.0 0.15 0.3 0.45 (-) Fr=14 0.0 0.15 0.3 0.45 (-) Fr=20 101 102 103 xx H (kPa) plastic compressibility: P ,25 = 205.3% 0.0 0.4 0.8 1.2 M (-) Fr=0.1 Fr=0.3 Fr=1 Fr=2 Fr=3 0.0 0.15 0.3 0.45 (-) 0.0 0.4 0.8 1.2 M (-) Fr=5 0.0 0.15 0.3 0.45 (-) Fr=7 0.0 0.15 0.3 0.45 (-) Fr=10 0.0 0.15 0.3 0.45 (-) Fr=14 0.0 0.15 0.3 0.45 (-) Fr=20 101 102 103 xx H (kPa) plastic compressibility: P ,25 = 100.7% Figure A.25: Raster plots of mean impact pressure σxx over the entire range of cohesion βand internal friction Mfor each studied Froude number Fr 0.0 0.4 0.8 1.2 M (-) Fr=0.1 Fr=0.3 Fr=1 Fr=2 Fr=3 0.0 0.15 0.3 0.45 (-) 0.0 0.4 0.8 1.2 M (-) Fr=5 0.0 0.15 0.3 0.45 (-) Fr=7 0.0 0.15 0.3 0.45 (-) Fr=10 0.0 0.15 0.3 0.45 (-) Fr=14 0.0 0.15 0.3 0.45 (-) Fr=20 101 102 103 104 Fx (kN) plastic compressibility: P ,25 = 205.3% 0.0 0.4 0.8 1.2 M (-) Fr=0.1 Fr=0.3 Fr=1 Fr=2 Fr=3 0.0 0.15 0.3 0.45 (-) 0.0 0.4 0.8 1.2 M (-) Fr=5 0.0 0.15 0.3 0.45 (-) Fr=7 0.0 0.15 0.3 0.45 (-) Fr=10 0.0 0.15 0.3 0.45 (-) Fr=14 0.0 0.15 0.3 0.45 (-) Fr=20 101 102 103 104 Fx (kN) plastic compressibility: P ,25 = 107.5% 0.0 0.4 0.8 1.2 M (-) Fr=0.1 Fr=0.3 Fr=1 Fr=2 Fr=3 0.0 0.15 0.3 0.45 (-) 0.0 0.4 0.8 1.2 M (-) Fr=5 0.0 0.15 0.3 0.45 (-) Fr=7 0.0 0.15 0.3 0.45 (-) Fr=10 0.0 0.15 0.3 0.45 (-) Fr=14 0.0 0.15 0.3 0.45 (-) Fr=20 101 102 103 104 Fx (kN) plastic compressibility: P ,25 = 100.7% Figure A.26: Raster plots of mean total impact force Fx over the entire range of cohesion βand internal friction Mfor each studied Froude number Fr A.6. Sensitivity of Compaction Effects on Impact Pressure to the Chosen Reference Compressibility In the main analysis, we used χP,25 =205 . 3% as the reference compressibility for computing δχ . Here, we instead use χP,25 =107.5% to demonstrate that, although the absolute magnitudes are affected, the overall patterns remain robust: 0.0 0.4 0.8 1.2 M (-) 0.00 0.02 0.06 -0.01 0.02 0.01 -0.04 -0.12 0.00 -0.00 -0.08 -0.24 -0.02 -0.08 -0.23 -0.43 Fr=0.1 0.02 0.00 -0.00 0.01 0.01 -0.02 -0.04 -0.09 0.03 0.02 -0.07 -0.28 0.01 -0.04 -0.20 -0.46 Fr=0.3 0.01 0.00 0.03 0.00 -0.03 -0.03 -0.10 -0.18 0.00 -0.09 -0.16 -0.33 -0.06 -0.11 -0.21 -0.48 Fr=1 -0.04 -0.04 -0.04 -0.04 -0.03 -0.00 -0.10 -0.18 -0.03 -0.11 -0.18 -0.34 -0.05 -0.15 -0.27 -0.53 Fr=2 0.0 0.15 0.3 0.45 (-) 0.0 0.4 0.8 1.2 M (-) 0.04 -0.00 -0.01 0.00 0.01 -0.00 -0.02 -0.13 -0.01 -0.08 -0.16 -0.30 -0.09 -0.17 -0.32 -0.52 Fr=3 0.0 0.15 0.3 0.45 (-) 0.09 0.08 0.10 0.09 0.08 0.06 -0.00 -0.06 -0.02 -0.03 -0.10 -0.18 -0.04 -0.12 -0.23 -0.37 Fr=5 0.0 0.15 0.3 0.45 (-) 0.13 0.16 0.15 0.15 0.16 0.12 0.07 0.05 0.09 0.05 0.02 -0.06 0.04 -0.02 -0.10 -0.21 Fr=7 0.0 0.15 0.3 0.45 (-) 0.35 0.35 0.34 0.35 0.24 0.20 0.16 0.13 0.02 -0.03 -0.06 -0.04 0.05 0.07 -0.01 -0.16 Fr=10 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 (-) Figure A.27: δχ , referenced to χP,25 =107 . 5%, over cohesion β and internal friction M for different Froude numbers Fr . The framed fields highlight material a(lower left) and material b (upper right). Blue colors represent a decrease in impact pressure, while green colors indicate an increase in pressure. 0.1 0.3 1 2 3 5 7 101420 Fr (-) 1.0 0.5 0.0 0.5 1.0 (-) a) subcritical transitional supercritical 0.01 0.1 1 Ma (-) (-) b) subsonic transonic supersonic 0.4 0.8 1.2 1.6 2.0 2.4 M ′ (-) Figure A.28: δχ , referenced to χP,25 =107 . 5%, over Froude Fr and Mach number Ma for various values of effective internal friction M′ . The Mach number is computed based on the bulk density of the compressible configuration with χP,25 =107.5. A9