Dissipative Solitary Waves in Granular Crystals

We provide a quantitative characterization of dissipative effects in one-dimensional granular crystals. We use the propagation of highly nonlinear solitary waves as a diagnostic tool and develop optimization schemes that allow one to compute the relevant exponents and prefactors of the dissipative terms in the equations of motion. We thereby propose a quantitatively accurate extension of the Hertzian model that encompasses dissipative effects via a discrete Laplacian of the velocities. Experiments and computations with steel, brass, and polytetrafluoroethylene reveal a common dissipation exponent with a material-dependent prefactor.

Figure 1

(a) Schematic diagram of the experimental setup. (b) Solitary wave decay in a chain composed of 70 steel particles impacted by a steel bead with $v_{\mathrm{imp}}=v_1$. The (blue) solid curves correspond to the recordings for sensors placed in particles 9, 16, 24, 31, 40, 50, 56, and 63.

Figure 2

Optimization of the dissipation coefficients ($\alpha ,\gamma$) for a steel chain. (a) Difference $D(\alpha ,\gamma )$, as defined in Eq. (2), between the force maxima recorded in the experiment and our model. (b) Difference ${\Delta }_n(\alpha ,\gamma )$, as defined in Eq. (3), in wave forms between the experiment and our model for sensor $n=56$. The solid curves and dashed curves correspond to the minima obtained from (a) and (b), respectively. (c) Maximum force $F_m^{\mathrm{num}}(n)$ for experiments with $v_{\mathrm{imp}}=v_3$ (top curves) and $v_{\mathrm{imp}}=v_8$ (bottom curves, displaced by 5 units for clarity). The (red) circles correspond to the experiment, and the (green) thick curves give the numerical best fit with $(\alpha ,\gamma )=(1.81\pm 0.25,-5.58\pm 1.30)$. The dashed curves correspond to the extreme cases using the standard deviation found in the optimal parameters. (d) Velocity of traveling front versus the maximum force (in a log-log plot). The solid curve represents the best linear fit, which gives $v\propto F_m^{0.17}$; we also show a dashed line with slope $1/6$.

Figure 3

Force versus time for the steel chain with $v_{\mathrm{imp}}=v_2$ through sensors at positions (a) $n=16$ and (b) $n=56$. The (red) thick solid curve depicts the (smoothed; see text) experimental series, and the thin (blue) dashed curves, dotted curves, and solid curves, respectively, show the numerics with $(\alpha ,\gamma )=(1,-5.5)$, ($1.4,-6$), and ($1.81,-5.58$). The last case corresponds to the best fit (see text) for the dissipation parameters for the chains of steel beads.

Figure 4

Results for the Teflon (left column) and brass (right column) experiments. The top panels depict the same information as Fig. 2c for experiments with impact velocities (a) $v_3$ (top curves) and $v_8$ (bottom curves, displaced by 0.3 units for clarity) and (b) velocities $v_3$ (top curves) and $v_6$ (bottom curves, displaced by 7 units for clarity). The best fit for the dissipation parameters for Teflon and brass are, respectively, $(\alpha ,\gamma )=(1.68\pm 0.16,-1.56\pm 0.19)$ and $(\alpha ,\gamma )=(1.85\pm 0.13,-6.84\pm 0.66)$. The bottom panels show the same information as in Fig. 3. In (c), we depict the force versus time through the sensor at $n=38$ with $(\alpha ,\gamma )=(1,-1.56)$, ($1.4,-1.56$), and ($1.68,-1.56$). In (d), we show the same information for the sensor at $n=14$ with $(\alpha ,\gamma )=(1,-5.5)$, ($1.4,-6$), and ($1.85,-6.84$).