Mechanical waveform memory in an athermal random medium

Abstract

Using numerical simulations it is shown that a random, athermal pack of soft frictional grains will store an arbitrary waveform that is applied as a small time-dependent shear while the system is slowly compressed. When the system is decompressed at a later time, an approximation of the input waveform is recalled in time-reversed order as shear stresses on the system boundaries. It is shown that this effect depends on friction between the grains, and is independent of some aspects of the friction model. By systematically increasing the complexity of the stored waveform, it is found that a pack of $10^4$ grains can recall any one of 128 different waveforms with 100% classification accuracy and 512 different waveforms with over 90% classification accuracy, as measured by a neural net trained only on the inputs. This type of waveform memory might be observable in other types of athermal random media that form internal contacts when compressed such as crumpled sheets and nest-like fiber assemblies.

Figures (15)

• Figure 1: Scheme for storage and recall of waveform data by a complex medium (blue cube) using compression as the reference input. (a) The waveform data to be stored are applied as a small, time-dependent shear strain while the medium is progressively compressed. Shear is used for data input as it is distinguished by spatial symmetry from the reference input (compression). For a memory to be formed, a microscopic nonlinear interaction between compression and shear should form an imprint within the medium. To retain the information the medium is held compressed for an arbitrary period. (b) To read out the memory the medium is progressively decompressed, which effectively reapplies the reference input in reverse order. If the medium has suitable properties, the waveform data are recalled in reversed order as shear stress at the system boundaries.
• Figure 2: Examples of a single grain used in the simulations, which is either spherical (grain on left) or has the exterior shape of four partially overlapping spherical grainlets with their centers at the vertices of a rigid tetrahedron (remaining grains). For tetrahedral grains the grainlet overlap $\mathcal{O}$ sets how non-spherical the grains are. Most of the simulations reported here used tetrahedral grains with $\mathcal{O}=0.6$.
• Figure 3: Random packs of tetrahedral grains with $\mathcal{O}=0.6$ and various numbers of grains $N$, formed using the protocol described in the text. The grains are confined by frictionless walls that are not shown. For visibility the grains are rendered in four different colors but all grains have identical properties. The initial condition for a 325-grain simulation is also shown, with the grains positioned on a regular lattice but with random velocities in a tall, rectangular box. After evolving as a granular gas to randomize positions and orientations the samples are compressed in the $z$-direction to form the dense packs shown here.
• Figure 4: Various quantities versus time $t$ for a memory-experiment simulation. The sample preparation period with zero friction ($t<0.64$ s) is not shown; conversely friction is active for the entire period shown here. After a final preparatory compression cycle there is an encoding period (left dashed box) during which the input signal $\gamma_0 I_5(u)$ (Fig. \ref{['fig-inputs']}(a)) is applied as an $(x-y)$ pure shear strain while the system is compressed in the $x$-$y$ plane. The system is held compressed during a storage period ($t=1.0$ - 1.2 s), then decompressed (right dashed box) to read out the memory response as shear stress $p_x-p_y$ on the system boundaries. (a) Sample filling factor $\phi(t)$ and average coordination number $Z(t)$. Note tetrahedral particles typically pack more densely than spheres hajiakbari09. (b) Measured pressures on the $x$, $y$, and $z$ walls. The responses of $p_x,p_y$ to the input signal are barely visible during the encoding period. (c) Difference between $p_x$ and $p_y$ on an expanded vertical scale, shown both when input $I_5(u)$ is applied and when the zero-shear input $I_0(u)$ is applied. On this expanded scale the response to $I_5(u)$ during the encoding period is readily visible, while differences during the readout period between the responses to $I_5(u)$ and $I_0(u)$ are barely visible. These differences, shown expanded in Fig. \ref{['fig-memeffect']}, constitute the waveform memory effect.
• Figure 5: Waveforms used during compression as inputs for memory experiments. In these figures the solid curves show $I_s(u)$ giving the applied shear strain as a function of the dimensionless compression $u$ for the $s^\textit{th}$ input waveform. In all cases $I_s(u)$ is the sum of one or more half-cycle cosine bumps (Appendix \ref{['app-inputsigs']}). It was found (Fig. \ref{['fig-memeffect']}) that the imprinted and recalled memory was proportional not to $I_s(u)$ but rather to its integral $J_s(u) = \int_0^u I_s(v)dv$ which is shown as the dashed curves. (a) Set of $S=6$ input signals used in Sec. \ref{['demonstration']} for a demonstration of the memory effect and to test its dependence on friction and other simulation parameters. (b) Example for $L=3$ of the set of $S=2^L$ binary-word input signals used in Sec. \ref{['limits']} to test the limits of memory complexity. For these signals $J_s(u)$ follows the bit pattern of the $L$-bit binary representation of $s-1$ (Appendix \ref{['app-inputsigs']}). For example, $J_2(u)$ (dashed curve in upper right of (b)) is $-,-,+$ to follow the 3-bit binary representation 001 of $s-1=1$.