Figure 1

(a) Three curves for different values of $C_1$ are shown for the potential function, Eq. (5), with $n=1/2$. Curve (1): $C_1=-1/6$ and the value of $y_1$ corresponds to $(2/3{)}^{3/2}$. Curve (2): $C_1=-11/64$ and $y_1=0.517$. Curve (3): $C_1=-0.1869$ and $y_1=0.381$. (b) Solution of Eq. (4) for corresponding values of $C$. The width of the wave is larger for smaller values of $C$. Each curve starts at a different value of $y_1$ shown by filled circle in part (a). Curve (1): $C=-2/9$, $y_1=(2/3{)}^{3/2}$. Curve (2): $C=-11/48$, $y_1=0.517$. Curve (3): $C=-0.2492$, $y_1=0.381$. All three curves have been shifted horizontally for comparison.Reuse & Permissions

Figure 2

Development of a two solitary rarefaction strain waves in a chain of 10001 particles compressed with a static force of 4.37 N corresponding to a force-displacement relationship with $n=1/2$, $s=5\times {10}^{-5}\mathrm{m}$, $A=1\times {10}^5\mathrm{N}/{\mathrm{m}}^{1/2}$. The initial velocities for the first and last particles are $-0.775$ and $0.775\mathrm{m}/\mathrm{s}$. (a) The leading rarefaction pulse separates from the following oscillatory wave train (not shown) after travelling approximately 80 particles. (b) Two rarefaction pulses have crossed leaving an oscillatory tail behind. (c) Results from (b) are compared curve (1) with a right-moving wave in a chain where there is no collision with a left-moving wave curve (2). The minimum strain value of curve (1) is $4\times {10}^{-4}$ and the phase shift in time between curves (1) and (2) is 0.5 ms.Reuse & Permissions

Figure 3

Strain from discrete simulations with 1000 particles for $n=1/5$ (green open square), $1/2$ (blue open circle), and $4/5$ (red plusses) is compared to Eq. (9) (see corresponding lines). The relevant material parameters were $s=0$ and $A=1\times {10}^5$. Equation (9) slightly overestimates the width (FWHM) of the solitary rarefaction wave.Reuse & Permissions

Figure 4

Impact of a discrete chain of 500 particles at $3\mathrm{m}/\mathrm{s}$. The chain was compressed with a static force of 4.37 N with $n=1/2$, $s=5\times {10}^{-5}\mathrm{m}$, $A=1\times {10}^5\mathrm{N}/{\mathrm{m}}^{1/2}$. At time $t=0.03\mathrm{s}$ a compression pulse is followed closely by an oscillatory wave train (strain offset by 0.12). At time $t=0.16\mathrm{s}$ a rarefaction pulse is shown passing the compression pulse (strain offset by 0.06). At time $t=0.26\mathrm{s}$ the initial disturbance quickly disintegrates due to nonlinear dispersion into wave packets, periodic waves, and the leading rarefaction solitary wave after traveling approximately 475 particles. The strain is offset by 0.12 for visual clarity. The arrow (1) is pointing to a slowly moving wave packet (i.e., an envelope solitary wave).Reuse & Permissions