Uncovering avalanche sources via acceleration measurements

Avalanches are impulsive, discrete deformation events occurring across many materials and processes (e.g., plasticity, ferroic switching). Traditional experimental studies rely on acoustic emission (AE) signals, analyzing features such as amplitude and duration under the assumption that they reflect the underlying avalanche source properties. However, AE signals are shaped by an acoustic transfer function dependent on propagation paths, interference, and reflections, which can vary with source location, thereby distorting the source characteristics and hindering direct inference. Extracting avalanche sources through deconvolution is a double-blind problem because both the source and transfer function are unknown. The research question addressed here is how to experimentally access and quantify intrinsic features of avalanche sources (amplitude and characteristic time) independent of the transfer function. The paper proposes an acceleration-based measurement approach and associated analysis that enforce a constant transfer function across events in the same test and enable elimination of its effect, thereby uncovering the true temporal evolution and magnitude of the underlying avalanche sources. The study focuses on deformation twinning in magnesium single crystals—a process where nucleation and rapid forward growth of needle twins produce avalanches—and aims to clarify source timescales, distributions, and relationships among source features compared with those seen in AE signals.

Classical avalanche studies emphasized universal behaviors and power-law statistics across disparate systems, while more recent work probes the physical mechanisms and process-specific characteristics of avalanches (e.g., distinguishing dislocation slip, twinning, and martensitic transformations). Prior statistical analyses of AE features have inferred physical constraints (e.g., twin boundary velocity bounds), but are confounded by transfer-function effects. AE signals are commonly modeled as a convolution of an unknown source with an unknown transfer function dependent on source-sensor geometry. Past efforts assumed specific transfer function models or spatial invariance within an experiment, which can be invalid. Limited attempts to recover sources via deconvolution exist but rely on assumptions and yield few robust sources. In twinning, nucleation and forward growth can be very fast, even exceeding shear wave speeds, suggesting microsecond-scale sources. The present work builds upon this context by designing an experiment where the transfer function is constant across events and by using frequency-domain ratios of measured signals to eliminate it, allowing direct probing of source statistics.

• Material and process: Magnesium single-crystal samples (99.99% purity) cut to ~1.5 mm × 0.6 mm × 8 mm with basal {0001} faces on the large sides and prismatic {10-10} on ends. Uniaxial compression along the 8 mm axis promotes {10-12}/{10-11} extension twinning. Six loading experiments were conducted at various deformation stages, with in-situ visualization confirming twinning-dominated deformation via successive twin nucleation.
• Acceleration-based experimental concept: To enable measurable accelerations, one sample end is attached to a small mass (aluminum stage) carrying an accelerometer; an elastic rod connects this stage to the test machine. The combined sample–rod–mass system has natural frequency f_sys ∈ [3, 5] kHz. The accelerometer (Kistler 8640A50) has resonance frequency f_acc ∈ [29, 32] kHz and cutoff frequency f_0 ≈ 60 kHz. Frequencies up to f_0 correspond to microsecond-scale excitations typical of avalanches. Setup parameters are chosen so that wavelengths λ at measured frequencies satisfy λ > c_l/f_e = 96 mm, much larger than the sample length L = 8 mm, rendering strain and stress effectively uniform and making acceleration proportional to macroscopic plastic strain change Δε_p(t) (hence to the microscopic source volume change ΔV_i(t) via Δε_i = ΔV_i/V_s).
• Signal acquisition: AE and acceleration signals are simultaneously recorded at 10 MSa/s/channel (Vallen AMSY-6). Trigger thresholds: 25.6 dB (AE) and 67.2 dB (acceleration). During six experiments, 1648 AE signals and 743 acceleration signals were detected.
• Source model: Each avalanche source s_i(t) is modeled as a monotonic sigmoid capturing abrupt plastic strain increase: s_i(t) = A_i/2 · tanh(t/τ_i + b_i) + c_i/2, where A_i is source amplitude (proportional to Δε_i and ΔV_i in arbitrary units), τ_i is the characteristic time, and b_i, c_i account for shifts.
• Transfer-function elimination: In frequency domain, the measured acceleration spectrum X_i(ω) = S_i(ω) H(ω), where H(ω) is the (constant-within-test) transfer function. For any two signals i and j, S_i(ω)/S_j(ω) = X_i(ω)/X_j(ω). Choose a reference measured acceleration signal X_R(ω) from the same test with large amplitude and high high/low-frequency intensity ratio ρ, indicating a rapid source. Model a rapid reference source S_R(t) as S_R(t) = 0.5 + 0.5 tanh(t/τ_R) with τ_R = 1 µs, A_R = 1 AU, b_R = 1, c_R = 0. Then compute sources via Ŝ_i(ω) = X_i(ω) S_R(ω)/X_R(ω), restrict f to [0, 60] kHz, zero-pad signals to a common length (L = 150,000 samples) to allow division, and inverse transform to ŝ_i(t).
• Parameter extraction: For each computed source, define t = 0 at its maximal slope and fit Eq. (1) within windows w ∈ {40, 100, 200, 300, 400, 600} µs centered at t = 0, choosing the fit with highest adjusted R^2 subject to w > 6 τ_i to ensure sufficient coverage of the rise. Extract A_i and τ_i. Robustness checks examine different X_R and τ_R values (τ_R tested at 0.5, 1, 2, 5, 10, 100 µs).
• Dynamics and resolution: Time resolution ≈ 2.9 µs (set by f_0) implies cascades of events separated by less than round-trip acoustic travel time Δt_waves ≈ 2.8 µs across the 8 mm sample coalesce into a single source in measurements. Most avalanches are widely separated (seconds), enabling distinct capture.
• Additional setup details: Compression at constant crosshead speed 0.005 mm/min (strain rate ~1e-5 s^-1). AE sensor: Fujicera 1045S with bandpass 95–850 kHz (100–1500 kHz sensor band), so system and accelerometer resonances do not affect AE. The moving mass m ≈ 14 g; elastic-rod stiffness chosen to avoid spurious avalanches and allow decay of acceleration within milliseconds for event separation.

• Detection summary: 1648 AE signals and 743 acceleration signals were recorded; 659 acceleration-based sources achieved adjusted R^2 > 0.9 (including 214 from an individual test). Signals failing the fit were predominantly near-threshold (≈80% had peaks < 2× threshold).
• Successful transfer-function elimination: Recovered sources ŝ_i(t) are smooth sigmoid-like without the oscillatory content of measured signals, validating that the method effectively removes the (constant-within-test) transfer function.
• Characteristic times bounded: Extracted τ_i lie on the microsecond scale and do not exceed ≈37 µs across six experiments. This upper bound is not imposed by instrumentation and indicates an intrinsic limit tied to the twinning process. Lower bounds are limited by time resolution (~2.9 µs).
• Distributions: Across six experiments (659 sources), A_i follows a power-law over ~2 decades with exponent α ≈ 1.71 (maximum-likelihood estimate). The ratio v_i = A_i/τ_i (proportional to average twinned-volume growth rate dV_i/dt and to average plastic strain rate) also follows a power-law over ~2 decades with exponent α ≈ 1.83. In contrast, τ_i spans a narrow range and is bounded above (~37 µs).
• Amplitude–duration relationships: Measured AE and acceleration signal features show strong amplitude–duration power-law relations: AE A vs D with α ≈ 1.34 (for 50 ≤ D_AE ≤ 2000 µs) and r ≈ 86%; acceleration A_acc vs D_acc with α ≈ 2.06 (for 500 ≤ D_acc ≤ 6000 µs) and r ≈ 80%. However, source-level A_i vs τ_i shows weak correlation (r ≈ 8%) with A_i broadly scattered over decades for a given τ_i, indicating no clear physical law ties these source features.
• AE–source linkage: For avalanches producing both AE and acceleration signals (561 pairs), A_i scales approximately as A_i ∝ (A_AE)^0.94 over 0.08–4.5 mV with modest correlation (r ≈ 28%), indicating large AE tends to imply large sources on average but is not predictive for individual events.
• Robustness to reference choice: Changing the reference signal X_R with high ρ yields nearly identical CCDFs of τ_i and A_i; >90% of events showed τ_i variation < 6 µs across references. τ_R choices ≤ 2 µs (0.5, 1, 2 µs) produced virtually identical τ_i distributions and maximal fit counts; τ_R = 5–10 µs narrowed distributions and altered maxima, while τ_R = 100 µs degraded fits (−35% events with R^2_adj > 0.9) and produced physically inconsistent τ_i inferences.
• AE durations do not reflect source durations: τ_i values are orders of magnitude smaller than many measured signal durations, demonstrating that AE/acceleration signal durations primarily reflect the transfer function rather than source dynamics.

The study demonstrates that acceleration measurements configured to ensure uniform sample strain/stress and a constant transfer function across events, combined with a frequency-domain ratio approach, can effectively remove transfer-function effects and recover intrinsic avalanche source features. This addresses the core challenge in AE-based avalanche research: disentangling source physics from propagation and sensor artifacts. The uncovered bounded microsecond-scale τ_i indicates an inherent timescale limit for twinning-related avalanches, plausibly tied to the kinetics of twin nucleation/forward growth and microstructural constraints. The power-law behavior of A_i and v_i (with exponents 1.71 and 1.83) suggests scale-free aspects consistent with dynamic criticality, yet the deviation from mean-field predictions indicates material- and mechanism-specific behavior in deformation twinning. Crucially, the strong amplitude–duration correlations observed in measured AE/acceleration signals originate from the transfer function, not from the sources themselves; hence, commonly inferred relations between AE amplitude and duration cannot be straightforwardly mapped to source amplitude and timescale. The weak A_i–τ_i correlation underscores that avalanche size and duration at the source are not governed by a single simple law. Together, these results refine interpretation of AE data and provide a pathway to directly quantify source-level dynamics in twinning.

This work introduces and validates a new experimental–analytical framework that uses acceleration measurements to eliminate transfer-function effects and expose avalanche source functions in real time. Applied to deformation twinning in Mg single crystals, the method reveals (i) microsecond-scale characteristic times with an intrinsic upper bound (~37 µs), (ii) scale-free distributions for source amplitude and average growth rate (A_i and v_i) with exponents 1.71 and 1.83, and (iii) the absence of a clear physical law linking source amplitude and timescale, in contrast to strong amplitude–duration correlations seen in measured signals that arise from the transfer function. The approach identifies a physically meaningful source feature (v_i = A_i/τ_i) directly tied to twinned-volume growth dynamics. Future work should: extend the method to other materials and avalanche mechanisms; improve temporal resolution to resolve closely spaced cascades; establish absolute calibration to convert A_i from arbitrary to physical units; investigate microstructural origins of the τ_i upper bound; and further quantify and model the transfer function’s role in conventional AE analyses.

• Temporal resolution (~2.9 µs) limits separation of cascades with inter-event times comparable to the sample’s acoustic round-trip (~2.8 µs), causing such cascades to appear as single events.
• Source amplitudes are reported in arbitrary units due to unknown proportionality constants, preventing absolute quantification without calibration.
• The assumption of a constant transfer function holds within a single test but may vary across tests with changes in alignment, contacts, or boundary conditions.
• Small-amplitude signals near the detection threshold often fail the sigmoid fit (lower adjusted R^2), potentially biasing statistics toward larger events.
• Results are obtained for specific Mg single-crystal geometry and loading; generalization to other materials/geometries requires validation.
• The method relies on selecting a suitable reference signal and on modeling the source as a sigmoid, which, while justified by monotonic plastic strain increase, is still a modeling assumption.