Effects of weak disorder on stress-wave anisotropy in centered square nonlinear granular crystals

The present study describes wave propagation characteristics in a weakly disordered two-dimensional granular media composed of a square array of spheres accommodating interstitial cylindrical intruders. Previous investigations, performed experimentally as well as numerically, emphasized that wave-front shapes in similar systems are tunable via choice of material combinations. Here, we investigate the effects of statistical variation in the particle diameters and compare the effects of the resulting disorder in experiments and numerical simulations, finding good agreement.

Figure 1

Schematic diagrams showing the sensor locations in experiments. (a) Sensor configuration 1 had sensors 1A, 1B, 1C, and 1D located at $X_8{Y}_{10}$, $X_8{Y}_7$, $X_8{Y}_4$, and $X_8{Y}_{14}$ of the 20-by-20 sphere array for all material combinations. (b) Sensor configuration 2 had sensors 2A, 2B, 2C, and 2D located at $X_8{Y}_{10}$, $X_{12}{Y}_{10}$, $X_{16}{Y}_{10}$, and $X_8{Y}_{11}$ of the 20-by-20 sphere array for the ${\text{steel}}^{\mathrm{s}}$- ${\text{ptfe}}^{\mathrm{c}}$, ${\text{delrin}}^{\mathrm{s}}$-${\text{ptfe}}^{\mathrm{c}}$, and ${\text{steel}}^{\mathrm{s}}$-${\text{steel}}^{\mathrm{c}}$ crystals. Sensors 2A, 2B, 2C, and 2D were moved to locations $X_4{Y}_{10}$, $X_8{Y}_{10}$, $X_{12}{Y}_{10}$, and $X_4{Y}_{11}$ of the 20-by-20 sphere array (4 spheres closer to the impact), due to small signal amplitudes for the ${\text{delrin}}^{\mathrm{s}}$-${\text{steel}}^{\mathrm{c}}$, crystals at locations far from the striker impact. (c) Sensor configuration 3 had sensors 3A, 3B, 3C, and 3D located at $X_4{Y}_8$, $X_6{Y}_6$, $X_8{Y}_4$, and $X_8{Y}_{10}$ of the 20-by-20 sphere array for all material combinations. For all experiments, the striker sphere impacted the system between the $10\text{th}$ and $11\text{th}$ edge spheres with initial velocity $V_x$.

Figure 2

Schematic of extended granular crystal analyzed in the present study. Larger sized granules represent spheres (radius, $R$) while smaller ones are cylinders (radius, $r$). Wave propagation studies are performed on an extended set of this packing by giving a symmetric disturbance, setting initial velocities on four spherical granules as shown.

Figure 3

(Center) Different wave propagation regimes mapped on mass ($m_2/m_1$) and stiffness ($E_2/E_1$) ratio space for the cylinder-sphere (2-1) system. (Top) Diagrams depicting examples of different wave fronts achievable in the systems at selected mass and stiffness ratios, indicated with circles in the central figure. (Bottom) Four insets depicting the propagating wave-front shapes for the four material systems tested in experiments, indicated with diamond-shaped symbols in the central figure.

Figure 4

Peak acceleration (m/s${}^2$) in the wave front as a function of X1 (in units of spheres) for each sphere-cylinder material combination: (a) ${\text{steel}}^{\text{s}}$-${\text{ptfe}}^{\text{c}}$, (b) ${\text{delrin}}^{\text{s}}$-${\text{ptfe}}^{\text{c}}$, (c) ${\text{steel}}^{\text{s}}$-${\text{steel}}^{\text{c}}$, and (d) ${\text{delrin}}^{\text{s}}$-${\text{steel}}^{\text{c}}$. The results from numerical simulations for the ideal granular crystal are shown by the solid (red) lines. The mean and standard deviation are represented by a dashed lines and dark (blue) shaded regions for the MR-SI (15 realizations each with a single impact) and by dot-dashed lines and light (gray) shaded for the SR-MI (a single realization with 15 impacts).The experimental data are represented by black dots with error bars for the three realizations tested with 15 impacts each.

Figure 5

Peak acceleration (m/s${}^2$) in the wave front as a function of X1 (in units of spheres) for the ${\text{steel}}^{\text{s}}$-${\text{steel}}^{\text{c}}$ crystal. The results from numerical simulations for the ideal granular crystal are shown by the solid (red) lines. The mean and standard deviation for three single realizations with 15 multiple impacts (SR-MI) are represented by dot-dashed lines and shaded light gray regions. The experimental data are represented by black dots with error bars for the three realizations tested with 15 impacts each.

Figure 6

Relative wave-front arrival times along X1 (spheres) with respect to sensor location 1A, for the ${\text{delrin}}^{\text{s}}$-${\text{ptfe}}^{\text{c}}$ crystal. The results from numerical simulations for the ideal granular crystal are shown by the solid (red) lines. The mean and standard deviation are represented by a dashed line and dark (blue) shaded region for the MR-SI and by a dot-dashed line and light (gray) shaded region for the SR-MI. For each sensor location along X1, three sets of black dots and error bars represent the mean and standard deviation for the three single experimental realizations with multiple impacts.

Figure 7

Experimental and numerical simulation results for the X- and Y-acceleration profiles for sensor configuration 1 (see Fig. 1) of the ${\text{delrin}}^{\text{s}}$-${\text{ptfe}}^{\text{c}}$ granular crystal. (a) Simulation results for the ideal granular crystal. (b) Experimental results for a single packing (SR) impacted 15 times. (c) Numerical simulation results for 15 different realizations (MR), each impacted once. (d) Numerical simulation results for a single realization (SR), impacted 15 times. For visual clarity, the acceleration data at sensor locations 1B, 1D, and 1C are shifted by 1000, 2000, and 3000 m/s${}^2$, respectively, from sensor location 1A. The zero time in the simulations denotes the moment of impact, while the zero time in experiments is arbitrary since the recorded data are based off the signal arrival time of sensor 1A.

Figure 8

Peak acceleration (m/s${}^2$) in the wave front versus X2 (in units of spheres) for each sphere-cylinder material combination: (a) ${\text{steel}}^{\text{s}}$-${\text{ptfe}}^{\text{c}}$, (b) ${\text{delrin}}^{\text{s}}$-${\text{ptfe}}^{\text{c}}$ (c) ${\text{steel}}^{\text{s}}$-${\text{steel}}^{\text{c}}$, and (d) ${\text{delrin}}^{\text{s}}$-${\text{steel}}^{\text{c}}$. The results from numerical simulations for the ideal granular crystal are shown by the solid (red) lines. The mean and standard deviation are represented by dashed lines and dark (blue) shaded regions for the MR-SI and by dot-dashed lines andlight (gray) shaded regions for the SR-MI. For each sensor location along X2, a set of black dots and error bars represent the mean and standard deviation for the single experimental realization with multiple impacts.

Figure 9

Peak acceleration (m/s${}^2$) in the wave front as a function of X3 (in units of spheres) for each sphere-cylinder material combination: (a) ${\text{steel}}^{\text{s}}$-${\text{ptfe}}^{\text{c}}$, (b) ${\text{delrin}}^{\text{s}}$-${\text{ptfe}}^{\text{c}}$ (c) ${\text{steel}}^{\text{s}}$-${\text{steel}}^{\text{c}}$, and (d) ${\text{delrin}}^{\text{s}}$-${\text{steel}}^{\text{c}}$. The results from numerical simulations for the ideal granular crystal are shown by the solid (red) lines. The mean and standard deviation are represented by dashed lines and dark (blue) shaded regions for the MR-SI and by dot-dashed lines and light (gray) shaded regions for the SR-MI. For each sensor location along X2, a set of black dots and error bars represent the mean and standard deviation for the single experimental realization with multiple impacts.