Dynamic Avalanches: Rate-Controlled Switching and Race Conditions

Abstract

Avalanches are rapid cascades of rearrangements driven by cooperative flipping of hysteretic local elements. Here we show that flipping dynamics and race conditions -- where multiple elements become unstable simultaneously -- give rise to dynamic avalanches that cannot be captured by static models of interacting elements. We realize dynamic avalanches in metamaterials with controlled flipping times and demonstrate how this allows us to modify, promote, and direct avalanche behavior. Our work elucidates the crucial role of internal dynamics in complex materials and introduces dynamic design principles for materializing targeted pathways and sequential functionalities.

Figures (5)

• Figure 1: (a) Force-extension response of a hysteretic element; insets show typical configurations for $s=0$ and $s=1$. (b) Close-up of the $0\rightarrow 1$ transition dynamics in air (light) and in silicon oils for viscosities $\eta=350, 500, 1000$ cSt (increasingly dark). (c) Sample 1 is a metamaterial containing two hysteretic elements ($B_1, B_2$) coupled in series with an (elliptical) spring (scale bar 1 cm). (d) Numerical model of dynamic hysteretic elements, with spring constant $k_s$, masses $m_1$ and $m_2$, and viscous dampers $\eta_1$ and $\eta_2$.
• Figure 2: (a,b) Force-extension curves for sample 1 submersed in oil (a) and air (b). The switching thresholds $U_2^+(00)$ and $U_1^-(10)$ for the up and down transitions of states $(00)$ and $(10)$ are indicated (as well as $U_2^-(01)$ in (b)). Both exhibit a negative $G$. Note the overshoot of the force during the $(10)\rightarrow (01)$ avalanche at $U=19.75$ mm. (c,d) Corresponding t-graphs, with the associated switching thresholds indicated. (e,f) Orbits of the numerical model with $v:=\dot{U}=2\cdot10^{-4}$ mm/s, $G=-0.4$ mm, $m_i=(20,10)$ g and $\eta_i=(2.0,2.0)$ Ns/m (e) and $\eta_i=(1.8,2.2)$ Ns/m (f), where $\bar{u}_2=u_2+u_k$, and $u_1+\bar{u}_2=U$. Zoom ins: In (e), dashed lines represent transitions $(10)\rightarrow(00)$, $(01)\rightarrow(00)$, and $(00)\rightarrow(01)$; in (f), the transition $(10)\rightarrow(00)$ is replaced by $(10)\rightarrow(01)$. (g) Avalanche vs no-avalanche parameter regime for designs with the same masses, different pairs of $\eta_i$ and range of $G$ and $v$ (star indicates $G$ and $v$ for panels (e,f)). Dynamic avalanches occur on the right side of each curve, including in regimes where the hysteron model predicts the absence of avalanches (white).
• Figure 3: Race conditions. Sample 2 features three serially coupled hysteretic elements ($B_1-B_3$), a spring between $B_1$ and $B_2$, and the third element can be damped (green). (a-b) Avalanche transition of sample 2 in air, and corresponding t-graph (for switching thresholds, see Supplemental Material, Fig. S13). The relevant gap for the $(011)\rightarrow(110)$ avalanche is positive, whereas the gap for $(100)\rightarrow(001)$ avalanche is negative, so that the latter is a dynamic avalanche which violates condition (i) i.e., the intermediate state 000 is stable (dashed). (c-d) Avalanche transition of sample 2, where element three is damped by submersing rigidly attached wings ($D$, green) in oil, and the corresponding t-graph. The $(011)\rightarrow(101)$ dynamics avalanche violates condition (ii) as it does not follow the scaffold (dotted). Scale bars, 1 cm.
• Figure 4: Three-bit counter. (a) T-graph of a three-bit counter. (b) Target curve $F(U)$. (c,d) High-speed images during the avalanche transition in air and in silicone oil. Scale bars, 1 cm.
• Figure A1: Design of three-bit counter. (a) The transitions between subsequent states must occur at one of the six critical switching forces. (b) Target curve $F(U)$. (c) Experimentally observed $F(U)$ curve for sample 3, with critical forces and displacements highlighted.