Trainable amorphous matter: tuning yielding by mechanical annealing

TL;DR

The paper demonstrates that training disordered solids with cyclic shear can encode memories that finely tune the yield point over a wide range, unlike conventional thermal annealing. By combining experiments on PNIPAM colloidal glasses with MD simulations of bi-disperse soft spheres, the authors show that the yield strain $\gamma_{\mathrm{Y}}$ tracks the training amplitude $\gamma_{\mathrm{T}}$, while the material can simultaneously soften and develop enhanced non-affine dynamics and shear bands beyond $\gamma_{\mathrm{T}}$. Importantly, the internal energy alone does not determine the mechanical response; MA and TA imprint distinct structural memories, and hybrid thermo-mechanical protocols access material states unattainable by either route alone. These findings establish MA as a powerful design principle for programmable, trainable amorphous matter with implications for soft robotics and materials engineering. The work highlights a mechanism by which memory effects and localized rearrangements govern macroscopic yielding, enabling targeted control over stiffness, brittleness, and flow through controlled preparation history.

Abstract

Living organisms can demonstrate highly adaptable and sophisticated responses using memory resulting from repeated exposure to external conditions or training. However, realizing similar adaptability in mechanical responses in inanimate, physical materials presents an outstanding challenge in several fields, including soft matter, materials science, and in the domain of soft robotics, to name a few. Our study focuses on disordered solids, which are model systems that resemble granular matter, foam and other disordered, soft solids. Here, combining bulk rheology, in-situ optical imaging, and numerical simulations, we demonstrate how training via cyclic shear can encode memories that tune the yield point in a unique way and over unprecedented ranges. Our study reveals that such tunability is intricately linked to the plasticity, non-affine deformations, and formation of shear bands. Remarkably, our numerical simulations illustrate that systems with identical internal energies, prepared via different protocols (mechanical or thermal), can display markedly different rheological responses, indicating that energy alone does not determine mechanical behavior. Moreover, while the yield strain increases with training amplitude, the material simultaneously softens, contrasting with the thermal case where both quantities increase monotonically with increasing annealing. Our results open up possibilities for memory-induced tuning of mechanical response in trainable amorphous matter, independently or in combination with thermal annealing, far beyond the material--feature space achievable via the latter alone.

Figures (4)

• Figure 1: Training induced tunability of the yield point in model glasses. (a) Schematic of the shear strain perturbation protocol applied to an amorphous solid during training and read-out: black lines represent the training cycles where strain is varied in an asymmetric cycle: $0 \rightarrow \gamma_{\mathrm{T}}\rightarrow 0$. After $N$ training cycles, a uniform strain is applied (shown in red) for read-out, which exceeds $\gamma_\mathrm{T}$. The corresponding 3D representations of the un-sheared and sheared configurations from our particle based simulations are shown in (b) and (c) respectively. Stress-strain response of untrained and trained systems during readout in experiments (d) and simulations (e). Here we have used x-y component of the stress tensor and subtracted out any pre-stress in the system. In both cases, untrained systems show ductile yielding, while trained systems show more brittle-like yielding, occurring just beyond the training amplitude $\gamma_\mathrm{T}$. This tunability of the yield point can be observed more clearly in the normalized differential shear modulus ($G/G_0$; see text for the definition) during readout in experiments (f) and simulations (g). (h) The yield strain $\gamma_\mathrm{Y}$ (see the SI section S1 for the method of identifying the yield point) shows a one-to-one correlation with the applied $\gamma_\mathrm{T}$ over a wide range, both in experiments and simulations. For the experimental data the mean and error bars are computed from three independent samples.
• Figure 2: Emergence of non-affine motion during readout and shear banding. (a) Relative non-affinity $\tilde{\Delta}$ is plotted as a color map during the readout phase for various training amplitudes $\gamma_{\mathrm{T}}$. For all values of $\gamma_{\mathrm{T}}$ the increase in relative non-affinity begins predominantly when the applied strain $\gamma$ goes beyond the training amplitude $\gamma_{\mathrm{T}}$ (the red dashed line indicates $\gamma=\gamma_{\mathrm{T}}$). (b)--(d) Experimental non-affinity maps for a system trained at $\gamma_{\mathrm{T}} = 0.10$ are shown at different points along the readout:(b) at $\gamma= 0.05$ (below $\gamma_{\mathrm{T}}$, marked by a diamond), (c) at $\gamma= 0.11$ (just beyond $\gamma_{\mathrm{T}}$, marked by a square), and (d) at $\gamma= 0.16$ (well beyond $\gamma_{\mathrm{T}}$, marked by a circle). These maps visually capture the growth in non-affinity when the system is strained beyond the training amplitude. (e) Relative non-affinity $\tilde{\Delta}$ during readout from simulations for a system trained at $\gamma_{\mathrm{T}}= 0.10$, showing a similar onset of increase beyond the training amplitude. (f)--(h) Non-affinity maps from particle based simulations, corresponding to the strain values $\gamma = 0.05, \gamma=0.11,$ and $\gamma= 0.16$, provide a scenario similar to the one observed experimentally.
• Figure 3: Comparison of thermal and mechanical annealing. (a) Schematic of a rugged energy landscape, where the arrows represent annealing protocols achieved via mechanical and thermal means. (b) Total potential energy of the inherent structures ($E_\mathrm{IS}$) of thermally annealed systems as a function of inverse temperature (left, blue shaded region) and mechanically annealed systems as a function of shear amplitude (right, red shaded region); here both $T$ and $\gamma_\mathrm{T}$ controls the degree of annealing in respective cases. For TA, $E_\mathrm{IS}$ changes monotonically whereas, for MA we see a non-monotonic evolution of $E_\mathrm{IS}$. (c) Readout response (blue, light red and dark red represents the thermally annealed system, mechanically annealed system with $\gamma_{\mathrm{T}}= 0.04$ and $\gamma_{\mathrm{T}}= 0.22$, respectively) under uniform shear for the three equi-energy (marked by a horizontal dashed grey dashed line in (b) sub-figure) configurations. Inset shows differential modulus corresponding to the stress–strain curves, highlighting distinct yielding behavior among these systems. For (d)--(f) blue diamond symbols represent TA data (from the Ref. Misaki_PNAS_2018 extracted using PlotDigitizer software), red squares represent data from our experiments, and circles correspond to data from our MD simulations for MA. (d) Yield strain $\gamma_{\mathrm{Y}}$, shows for TA only a mild increase in $\gamma_{\mathrm{Y}}$ while MA offers a broad tunability of the yield point across a wide range of $\gamma_{\mathrm{T}}$. (e) Normalized shear modulus $\Tilde{G}$ (see the SI section S1 for details) shows an increasing trend with thermal annealing whereas mechanical annealing seem to make the material softer. (f) Normalized brittleness (see the SI section S1 for details) exhibits peak brittleness at intermediate training amplitudes for MA, whereas TA shows a monotonic increase in brittleness. In all panels, red and blue dashed lines are guides to the eye for mechanical and thermal annealing trends, respectively.
• Figure 4: Material property tuning through thermo-mechanical annealing. (a) Protocol for thermo-mechanical annealing, where sequences of thermal annealing (TA) and mechanical annealing (MA) were applied before a uniform shear readout. (b) Initial shear modulus $G_0$ from simulations for TA (blue diamonds) and MA (red circles) shows stark difference in trends. Combined TA–MA sequences (green squares) reach states inaccessible to either TA or MA alone. (c) Yield stress $\sigma_{\mathrm{Y}}$ versus Young’s modulus $E$ for various disordered materials are digitized from the Ref. Cubuk_science_2017 (open symbols), and the slope of the black line indicates the proposed universal yield strain 2.9%. Our experimental results (solid red squares) and simulation results (solid red circles for MA and solid blue diamonds for TA) are overlaid. (d) Zoomed in view of (c), highlighting that mechanically annealed data points deviate from the dashed line along the vertical direction, indicating that mechanical annealing can defy universal mechanical response of known disordered materials. Data for the untrained systems are shown as solid black squares (experiment) and solid black circles (simulation).