Activated solids: Spontaneous deformations, non-affine fluctuations, softening, and failure

Abstract

Internal activity can fundamentally reshape the mechanical behavior of solids, yet its role in softening and failure remains incompletely understood. In this study, we investigate spontaneous deformations in activated solids via non-affine fluctuations that quantify local rearrangements relative to global strain. Using scaling analysis and numerical simulations, we show that non-affinity in crystalline solids grows quadratically with active speed, increases linearly with persistence time before saturating, and scales inversely with the distance to the melting density. Spatial correlations reveal an activity-dependent growing correlation length, while relaxation dynamics are governed by the active persistence time. With increasing activity, the distributions of local non-affinity broaden, become more skewed, and develop heavy tails, eventually forming a secondary maximum that signals coexisting small and large non-affinities; this heterogeneity precedes defect formation and two-step melting from solid to hexatic and ultimately to fluid. Finally, we demonstrate that spatially patterned activation provides a simple route to locally induce non-affinity and mechanical softening. Our predictions are experimentally testable and suggest a pathway to tunable mechanics in adaptive metamaterials, with implications for mechanical regulation in biological systems.

Figures (14)

• Figure 1: A schematic illustration of the activated solid. (a) An ideal triangular lattice that serves as the reference for the calculation of non-affinity $\chi$. (b) A representative configuration at $\tilde{D}_t = 1.0$ and $\Lambda = 8.0$, is shown color coded by local non-affine parameter $\chi$; see the color bar. (c) A single coarse-graining region $\Omega$ is defined as the volume enclosed by the first nearest neighbors, determined by mean inter-particle separation $a$ at a given denisty $\rho$. This region includes one central particle with reference position $\mathbf{R}_0$ (cyan) and actual location $\mathbf{r}_0$ (dark green). The reference (cyan, $\{\mathbf{R}_i\}$) and actual locations (dark green, $\{\mathbf{r}_i\}$) of its six nearest neighbors are shown as well.
• Figure 2: (a) Representative time series of the system-averaged non-affinity $X(t)$ at $\Lambda = 8.6$ (athermal, top) and $\Lambda = 4.0$ (thermal, bottom); other parameters are given in the legends. (b) Scaled excess non-affinity $\langle X^{(a)} \rangle / \Lambda^2$ as a function of the active persistence $\tilde{D}_r^{-1}$. The data are well fit by $\langle X^{(a)} \rangle / \Lambda^2 = a/(D_r + c)$, with $a = 9.21 \times 10^{-3} \pm 4.57 \times 10^{-4}$ and $c = 397.96 \pm 20.57$. At large $\tilde{D}_r$ (low persistence), $\langle X^{(a)} \rangle$ grows approximately linearly with the persistence time $\tilde{D}_r^{-1}$ and saturates beyond $\tilde{D}_r^{-1} > c^{-1}$. (c) $\langle X^{(a)} \rangle / a_{\rho}$ versus $\Lambda$ for $N = 4096$ particles at fixed $\tilde{D}_r=3.0$ and different densities $\rho$, where $X^{(a)} = X(\Lambda) - X_{\mathrm{eq}}$. Filled (open) symbols correspond to athermal ($\tilde{D}_t = 0$) and thermal ($\tilde{D}_t = 1.0$) systems, respectively. Fits to $a_{\rho}\Lambda^{\nu}$ yield $\nu = 2.00 \pm 0.01$ (athermal) and $\nu = 2.11 \pm 0.06$ (thermal). Inset: density dependence of $a_{\rho} \sim (\rho - \rho_0)^{-1}$, with $\rho_0 \sigma^2 = 0.947 \pm 0.008$ (athermal) and $0.971 \pm 0.004$ (thermal).
• Figure 3: Probability distributions $P(\chi)$ of local non-affinity $\chi_i=\chi$ on log-log axes. (a) At fixed $\tilde{D}_r=3.0$, increasing active speed $\Lambda$ shifts and broadens the distributions toward larger $\chi$, with a secondary high-$\chi$ peak emerging at large $\Lambda$. (b) At fixed $\Lambda=6.0$, decreasing $\tilde{D}_r$ drives a crossover from unimodal to bimodal distributions, reflecting the onset of rare, large non-affine rearrangements.
• Figure 4: Normalized spatial correlations of the local non-affinity $\chi$ (a) at $\Lambda=5$ for varying $\tilde{D}_r$ and (b) corresponding correlation length $\xi$ across active speeds $\Lambda$ and rotational diffusivities $\tilde{D}_r$. $\xi$ increases with active speed and persistence in the persistent-active and intermediate regimes, while in the low-persistence limit, correlations are short-ranged and largely $\Lambda$-independent.
• Figure 5: Normalized time autocorrelation $C_X(\tau)$ at $\Lambda=5$ for several rotational diffusion constants $\tilde{D}_r$. Dashed lines indicate the reference exponential decay $\exp(-2 \tilde{D}_r \tau)$.