Abstract
Mechanical metamaterials at the microscale exhibit exotic static properties owing to their engineered building blocks1,2,3,4, but their dynamic properties have remained substantially less explored. Their design principles can target frequency-dependent properties5,6,7 and resilience under high-strain-rate deformation8,9, making them versatile materials for applications in lightweight impact resistance10,11,12, acoustic waveguiding7,13 or vibration damping14,15. However, accessing dynamic properties at small scales has remained a challenge owing to low-throughput and destructive characterization8,16,17 or lack of existing testing protocols. Here we demonstrate a high-throughput, non-contact framework that uses MHz-wave-propagation signatures within a metamaterial to non-destructively extract dynamic linear properties, omnidirectional elastic information, damping properties and defect quantification. Using rod-like tessellations of microscopic metamaterials, we report up to 94% direction-dependent and rate-dependent dynamic stiffening at strain rates approaching 102 s−1, as well as damping properties three times higher than their constituent materials. We also show that frequency shifts in the vibrational response allow for characterization of invisible defects within the metamaterials and that selective probing allows for the construction of experimental elastic surfaces, which were previously only possible computationally. Our work provides a route for accelerated data-driven discovery of materials and microdevices for dynamic applications such as protective structures, medical ultrasound or vibration isolation.
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Data availability
The datasets generated and/or analysed during this study are available in the main text or the Supplementary Information.
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Acknowledgements
We acknowledge financial support from Kansas City National Security Campus (PDRD# 705288). C.M.P. acknowledges support from the National Science Foundation (NSF) CAREER Award (CMMI-2142460), partial support from DEVCOM Army Research Laboratory’s Army Research Office through the MIT Institute for Soldier Nanotechnologies (ISN) under cooperative agreement number W911NF-23-2-0121 and the MIT MechE MathWorks Seed Fund. This work was carried out in part through the use of MIT.nano’s facilities. We thank A. Maznev, D. W. Yee and K. A. Nelson for helpful discussions, T. Butruille for experimental support, R. Glaesener for imperfect-lattice design and B. Aymon for explicit dynamics simulation support. Honeywell Federal Manufacturing & Technologies, LLC manages and operates the Department of Energy’s Kansas City National Security Campus under contract DE-NA-0002839
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Y.K., T.P. and C.M.P. designed the research. Y.K. and C.M.P. designed the samples. Y.K., J.L. and T.P. conducted the LIRAS experiments. S.D. performed the elastic-surface calculations. R.S. and S.D. performed the nanomechanical experiments. R.S., Y.K. and C.M.P. performed the acoustic finite-element calculations. Y.K., T.P., W.D. and C.M.P. interpreted the results. Y.K., S.D., R.S. and C.M.P. wrote the manuscript, with input from all authors.
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Extended data figures and tables
Extended Data Fig. 1 LIRAS setup schematics.
The red beam indicates the picosecond laser serving as the pump laser and the green beams indicate the CW laser as the probe laser. The CMOS camera serves to monitor the samples. The interferometric signal corresponding to the waveforms of the sample-surface displacement are registered by an avalanche photodiode. The dashed boxes indicate variations of the LIRAS setup for the selective excitation of longitudinal and flexural (a) and torsional (b) modes.
Extended Data Fig. 2 Eigenfrequency studies to validate rod-wave approximations.
a, Eigenfrequency analysis of monolithic IP-Dip pillars (density 1,170 kg m−3, Young’s modulus 2.7 GPa and Poisson’s ratio 0.49) of different aspect ratios. b, Corresponding computed wave velocities. c, Computation of octet eigenfrequencies of different heights with a fixed cross-section of 50 μm × 50 μm (that is, a 5 × 5 tessellation). d, Corresponding computed wave velocities. Although no clear longitudinal mode could be identified for short samples with heights ≤20 μm, taller samples asymptotically approached the theoretical rod-wave-velocity value (dashed lines in b and d). e, Bloch-wave analyses on n × n × ∞ tessellations (∞ tessellation in the z direction stems from Bloch–Floquet periodic boundary conditions) yield dispersion relations as a function of wavevector kz for each of these n × n cross-sections. Longitudinal and transverse bands for tessellations ranging from n = 5 to 9 are in agreement within the long-wavelength limit, indicating that cross-sectional tessellations of n ≥ 5 enable the homogenization approximation.
Extended Data Fig. 3 Signal processing for LIRAS measurements.
For each pump scheme, photovoltage is plotted as a function of time, corresponding to the surface displacement as a function of time. The thermal background is removed by subtracting the raw data with a fifth-order polynomial fit function. Subsequently, the background-subtracted time-domain signal is converted into the frequency domain using FFT. Here the signal of a 5 × 5 × 22 octet in the [100] orientation is chosen for demonstration purposes.
Extended Data Fig. 4 Acoustic spectra and dispersion relations of tetrakaidecahedron samples.
a, SEM images of selected [100] samples for illustration. Scale bar, 50 μm. b, Acoustic spectra experimentally determined using the centre-pump scheme of LIRAS. c, Transverse-pump scheme representative data. d, Dispersion relations extracted from the spectra.
Extended Data Fig. 5 Experimental dispersion trend analysis for spectral-peak identification.
Long-wavelength-limit dispersion relations extracted from Fig. 2e for monolithic IP-Dip polymer (left) and octet [100] (right) samples for a transverse-pump scheme. The spectral peaks with a non-dispersive behaviour (linear relation between frequency and wavenumber) correspond to the torsional modes. Following elasticity theory, longitudinal and torsional waves are non-dispersive in the long-wavelength limit, whereas flexural waves have distinctive quadratic dispersion46. Furthermore, as the longitudinal mode is not activated in the transverse-pump scheme, the torsional mode can be assigned to the spectral peak with a linear f(k) relation. The expected third-order and fifth-order harmonics of the flexural mode are calculated by multiplying the measured fundamental flexural frequencies by factors of 3 and 5, respectively.
Extended Data Fig. 6 Illustration of vibrational modes corresponding to the measured acoustic spectra using finite-element-method simulation.
Shaded regions show eigenfrequency-simulation lower (quasistatic effective Young’s modulus) and upper (high-strain-rate effective stiffness) bounds. Sample centre-pump and off-centre-pump FFT (pictured with a DC offset for clarity) and their corresponding longitudinal and bending modes, respectively, are pictured for monolithic and octet [100] samples. Note that the torsional mode is determined in separate spectra.
Extended Data Fig. 7 Strain-rate analysis.
A 5 × 5 × 20 octet sample in the [100] direction is chosen, demonstrating calculation of an approximate time-dependent strain rate from LIRAS measurements. From the time–displacement measurements (second panel), the strain is approximated by dividing by the sample height, whereas the strain rate is obtained from the time derivative of the strain–time signals (see Supplementary Information Sections III and IV). Owing to the time-dependent magnitude of the strain rate, we obtain characteristic strain rates by computing the r.m.s. strain rate for every wavelength (shown as windowed (blue)) as well as the cumulative r.m.s. strain rate over several wavelengths up to a certain point in time (shown as cumulative (red)) in the third panel. The r.m.s. strain rate over the entire signal is extracted as the primary characteristic strain rate for each experiment (dashed line in the third panel). The r.m.s. strain rates calculated over the first and last wavelengths are considered as the higher and lower strain-rate limits, respectively. The corresponding strain-rate range is thus presented as shaded regions in the fourth panel. The average of these values over the tallest five pillars/lattices (five lowest-wavenumber samples in the first panel) is taken as the characteristic strain rate and its limits for that structure. We selected this single value of r.m.s. strain rate to be representative for LIRAS experiments on each structure.
Extended Data Fig. 8 Quasistatic and dynamic uniaxial compression experiments.
Quasistatic compression (\(\dot{\varepsilon }=1{0}^{-3}\,{{\rm{s}}}^{-1}\)) of 5 × 5 × 5 octet and tetrakaidecahedron metamaterials and monolithic pillar (left). Stress–strain plots for octet (blue) and monolithic pillar (purple) samples for strain rates ranging from 10−2 s−1 to 102 s−1 (right). The quasistatic response is overlaid in grey for the higher-strain-rate responses.
Extended Data Fig. 9 Dynamic explicit simulation of octet unit cell at 60 s−1 strain rate.
3D plot of the maximum absolute logarithmic strain (ε) is shown for a particular time instant when the applied engineering strain is 5%. Maximum absolute logarithmic strain and strain rate as a function of time are shown at the four representative material points labelled in the 3D plot (middle panel). The maximum strain rate is seen at point 2 for the given applied engineering strain rate. The volume-averaged maximum absolute logarithmic strain rate is approximated to be 48% of the applied engineering strain rate.
Extended Data Fig. 10 Bloch-wave velocity extraction and comparison of numerical and experimental effective stiffness and effective shear stiffness.
Longitudinal and shear velocities were extracted from the slope of the linear portion of the longitudinal and shear bands of the Γ–X orientation, respectively. Transverse velocities were extracted from the linear portion of the shear band of the Γ–M orientation, but are not pictured. The bar plot depicts the difference in effective stiffness and effective shear stiffness obtained from Bloch-wave analysis45 and eigenfrequency simulations compared with LIRAS experiments. The higher shear stiffness observed in eigenfrequency analysis might indicate that the chosen material properties are higher than those in reality for this torsional mode, as the chosen material properties were matched to the strain rates of the longitudinal mode. In other words, the material point strain rates in the torsional mode are probably lower than those for the longitudinal mode.
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Kai, Y., Dhulipala, S., Sun, R. et al. Dynamic diagnosis of metamaterials through laser-induced vibrational signatures. Nature 623, 514–521 (2023). https://doi.org/10.1038/s41586-023-06652-x
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DOI: https://doi.org/10.1038/s41586-023-06652-x
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