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C.Z. data; and and analyzed E.B.H., manuscript; R.C.H. on C.Z., and research; C.Z. designed research; R.C.H. performed and E.B.H., C.Z., contributions: Author n tutrldsre.Ulk h rpgto fpaewaves contacts plane of of propagation nonlinearity the Unlike to disorder. ascribed structural in and were waves measurements coherent predictions between and for differences The dispersion rea- materials. provide granular wave to ordered able of were predictions 38) (17, sonable balance calcula- effects force Cosserat for Theoretical accounted included that 37). and crystals (16, granular on configuration through based tions signals packing measured responses the same when of the ensemble magnitude the an is in over signal averaged diminishes the are of that portion portion incoherent coordination The trailing and 36). measure- properties 25, elastic many (16, to of number the link average a is provides signal the and the represents ments of that portion portion coherent a coherent leading The through into parts. measured decomposed incoherent be signal and can acoustic medium The granular 25). velocities disordered (24, material’s different the size and at frequency) particle (or propagate wavelength their waves on depending dispersion: feature to performed. in been yet scaling not velocity has and materials velocity granular disordered wave a 3D con- to However, and heterogeneity 35). disorder force 33, structure tact 32, packing (12, linking quantitatively amplitudes study wave and and compressional con- as wave), (such nonspherical shear modes 29), wave 32), 16, (27, (12, geometries heterogeneity tact reorganization stress packing 33), 34), 28, asperities (27, (32, interparticle arrangements quali- chain of been force deformation (28), has the to transition attributed This tatively 30–33). 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PHYSICS in 1D chains and 2D and 3D granular crystals, the transmit- sion of a 3D disordered granular material in a unique loading ted ultrasound signals for 3D disorder granular systems arise frame (9, 10, 48). To evaluate the relative roles of packing from a superposition of the signals traveling through differ- structure and interparticle forces on wave behavior, we com- ent paths in this heterogeneous medium (23, 27, 39). Even pared measurements of velocity, dispersion, and attenuation small amounts of structural disorder can dramatically influence with simulations having contact networks controlled by mea- wave propagation, causing departures from theoretical force- sured interparticle forces and contact networks having uniform dependent velocity scaling, dispersion, and attenuation (12, 16, interparticle stiffness (equivalent to uniform forces). We further 40, 41). Quantifying the influence of both structural disorder and compared predictions with those made on isotropically loaded force heterogeneity on dispersion may enhance the usefulness granular crystals to isolate the roles of packing structure dis- of using dispersion as a nondestructive probe of the state of a order. Our results revealed that both packing structure and granular medium. interparticle forces play important roles in controlling coherent Both experiments and numerical simulations for ultrasound wave velocities, velocity scaling, and dispersion, while packing propagation have shown that coherent waves (including com- structure alone can explain the majority of the observed wave pressional, shear, and rotational waves) decay in amplitude attenuation. with propagation distance. Frictional contact loss, viscoelas- tic/viscous contact dissipation, wave scattering, and material Results and Discussion damping have been assumed to be responsible for wave atten- Wave Velocity. The acoustic velocities of the coherent portions uation (25, 35, 42, 43). Local contact damping has been incorpo- of ultrasound waves transmitted through a random packing of rated in discrete element simulations (18, 35, 44–46), enabling sapphire spheres under increasing normal compression were analysis of the energy dissipation during wave propagation. computed using cross-correlation analyses (Materials and Meth- Additionally, a stiffness matrix incorporating all particle con- ods and SI Appendix, section 1C) and are presented in Fig. 1A. tacts of a granular packing has been used to build a lattice- The measured acoustic velocity across load steps was found to based model to examine the wave attenuation and scattering follow a power-law dependence on the average vertical sample s α for 2D granular systems (20, 21, 47). However, limited experi- stress, v ∝ (σzz ) , with exponents α of approximately 0.22 and mental data are available to highlight the quantitative roles of 0.20 for input Gaussian bursts centered at 0.4 and 0.8 MHz, packing disorder and force heterogeneity on wave attenuation respectively. For the case of 0.4 MHz, the exponent was found to in 3D. be close to 0.24 for load steps 2 to 6 and 0.21 for load steps 7 to 11, The primary objective of this paper is to quantitatively inves- showing a decreasing trend as load increases. The exponents fell 1 1 tigate the relative influence of packing structure disorder and within the range of 6 and 4 , comparable with reported empiri- interparticle force heterogeneity on ultrasound wave velocity, cal and semiempirical relationships in previous experiments and dispersion, and attenuation in a disordered granular material. numerical simulations (8, 16, 24). To achieve this, we combined in situ X-ray computed tomog- To simulate wave propagation through the packing at each raphy (XRCT), three-dimensional X-ray diffraction (3DXRD), load step, systems of equations were constructed for a spring and ultrasound wave measurements during uniaxial compres- network representing the experimentally measured structural

A B C

D E

Fig. 1. Ultrasound velocities obtained by experimental measurements, simulations of network responses, and theoretical predictions. (A) Experimentally measured wave velocities (purple triangles for 0.40 MHz and green circles for 0.80 MHz) compared with simulations (blue stars and magenta inverted triangles). Error bars are the SD across the five measurements. Histograms of simulated velocities calculated for each particle contacting the bottom platen, as described in the text, for heterogeneous (B) and uniform (C) networks at load steps 4, 6, 8, and 10. In simulations, a single half-period of a sinusoidal with a fixed frequency of 0.4 MHz was used as the input signal at the top platen. (D) Experimentally measured wave velocities (same symbols as A) compared with averaging unit cell velocities with heterogeneous [M(heterogeneous)] and uniform [M(uniform)] forces as well as unit cells with actual velocities but only constituting the fastest theoretical chain (Chain) and percolating force network (Network). (E) Dependence of theoretically estimated wave velocity on the effective vertical contact force calculated in three distinct ways and compared with isotropically loaded granular crystals. Dependence of experimentally measured wave velocities of 0.4 MHz on the effective vertical contact force for the fastest chain is shown by a black triangle.

2 of 9 | www.pnas.org/cgi/doi/10.1073/pnas.2004356117 Zhai et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 n ubr fprilscnetdi oc eclto ewrsae37 3,20 n 3 tla tp ,6 n 0 respectively. 10, and 8 6, 4, steps load at al. 231 et of and Zhai 250, network 233, percolating 1.69 347, sample, the are are the of networks throughout networks Elements percolating forces percolation steps. for contact force values load average in with threshold all The connected colored force contact. for particles in (particles respectively; opacity of particles N, fight and numbers of 0.687 of width, and centers and length, time connecting 0.652, lines for shortest 0.664, are Particle factors the 0.504, chains scaling step. with are fastest linear load the chains same each and particle the forces at and interparticle with measurements red), More strong rendered XRCT material. (in are sapphire from forces black) used determined interparticle the (in were for in strong forces grains constants found of elastic 621 be networks anisotropic all can the percolating measurements of and black), 3DXRD 3DXRD positions from and The measured color. XRCT tensor regarding the strain details by per-particle on indicated based are calculated values were stresses Stress 10. and 8, 6, 2. Fig. of influence the veloci- signifies acoustic spread the This in paths. forces variations acoustic local heterogeneous and cause containing ties which network 1B), the spread (Fig. for broader sig- significantly observed input A 0.4-MHz is a respectively. 1 for networks, Fig. 10 in uniform and 8, provided the 6, are and 4, nal peak these steps of signal load of Distributions input for (25). height force velocities the particle–platen the between the in dividing time peak first the by by wave particle a sample contacting calculated the each We simulations. for in velocity platen bottom the tacting the predict precisely velocity. to acoustic needed are to heterogeneity dis- force scaling structural and velocity order both network that similar The suggesting a measurements, forces. in exhibits experimental aligned are forces (used vertically which heterogeneous forces stiffness), the with average aligned than thus, lower horizontally and typically force the average of an finding influence the reflect- network, heterogeneous ing the reduced with a compared features velocity network total the uniform through The be velocity simulations. wave to in in the material taken measure used The was to analyses used platen MHz. cross-correlation were bottom same experiments 0.4 The half- the of signal. on single frequency received forces the fixed a contact a all following with of particles oscillate sum sinusoidal all to a forcing of by platen period network top contact the the contacting into input put An “heteroge- “uniform”). was struc- (called measured signal stiffness (called with uniform network but network spring disorder separate tural a force for and interparticle neous”) and disorder efrhreaie h otc ocsbtenprilscon- particles between forces contact the examined further We h etclcmoeto niiulpril stress, particle individual of component vertical The (A) paths. wave ultrasound fastest the and networks, force stresses, Particle B A B and C o h eeoeeu and heterogeneous the for otc ocsa odses4 ,8 n 0fralcnat (in contacts all for 10 and 8, 6, 4, steps load at forces Contact (B) 1 . section Appendix, SI ahwsietfida h ecltn ewr fsrn inter- strong was network of percolating network the through percolating velocity corresponding 2B The the forces. the Fig. as particle and percolation, identified in for was 10) (provided found and path be threshold 8, 6, could force 4, path steps the connected load as a sample. which identified for the was level in force contacts last all The across platen magnitude top force +0.01hf interparticle the by incre- level we to force Finally, the platen mented contacts). bottom particle-boundary lateral the were these (ignoring connected through threshold path that that continuous a above contacts for and magnitude searched we specified, force Next, was a retained. threshold with force contacts a all First, follows. as identified through 3) “chain” and or path forces, sample. wave the interparticle theoretical “strong” fastest the of to identifying corresponding network cells percolating unit over a only forces averaging theoretical [labeled 2) contact cells’ forces M(uniform)], theo- contact measured unit uniform sample’s and all experimentally M(heterogeneous)] the [labeled averaging using estimated 1) velocities by we wave velocity velocity, wave this (z retical acoustic on vertical theoretical Based the cell. the in calculated velocity from wave We neigh- extracted its unit forces, measurements. of interparticle a positions X-ray corresponding the constructed the using and we particle bors individual packing, each granular for cell the through agating coherence, sections. phase following of the loss in described and as dispersion on heterogeneity force h ecltn ewr fsrn nepril ocswas forces interparticle strong of network percolating The prop- waves ultrasound of path the understand further To hf i s 1.72hf i, i s 1.81hf i, ieto hog ahunit each through direction ) i s i with i s n 1.84hf and i, NSLts Articles Latest PNAS hf i s i en h average the being σ zz p tla tp 4, steps load at , i s respectively; i, σ zz p .Contact ). | f9 of 3 hf for i s i,

PHYSICS computed by averaging the velocity in the z direction across all An exponent of about 0.18 was found for experimentally mea- unit cells whose central particles belong to this network (the sured wave velocities (at 0.4 MHz) plotted against vertical forces numbers of particles in networks for load steps 4, 6, 8, and 10 on the fastest chain across load steps 4 to 11. For comparison, the are provided in Fig. 2B). experimentally measured coherent wave velocities scaled with The fastest theoretical chain, called chain in Fig. 1, was the average vertical forces on the percolating force network with obtained by comparing all possible acoustic paths through the a power-law exponent of 0.39 and with the average forces on all sample from the top to bottom platen. One particle contacting contacts with a power-law exponent of 0.53 (neither is shown in the top platen was labeled as the first particle of this acoustic Fig. 1). This demonstrates that the fastest chain is the only path path. Any particles contacting this first particle with a z position of those considered that follows the expected Hertzian velocity lower than the first particle’s z position were treated as a viable scaling of 1/6. path for wave transmission. This process was repeated until a We note that the apparent Hertzian scaling of wave veloc- path from the top platen to the bottom platen was constructed. ity with the effective vertical contact force on the fastest chain, ¯p All such paths were evaluated, and the time of flight for each hfz i, suggests an alternative, unexplored explanation for the 1 PN chain i i i p i transition of power-law exponent from 1/4 to 1/6. Specifically, path was computed by 2 i=1 tz , where tz = hz /vz , with hz being the vertical distance between centroids of the two particles the coherent wave may follow Hertzian predictions of a 1/6 p ¯p in the chain above and below particle p, and vz being the longitu- power-law scaling microscopically (i.e., with hfz i) across a broad ¯p dinal wave velocity of the unit cell for the particle, p. The factor range of macroscopic loads. However, the value of hfz i may 1 i of 2 was used because the wave travel distance of hz was counted scale nonlinearly with macroscopic sample pressure, σzz , because twice in the calculation. Among all paths, the chain demonstrat- interparticle forces tend to become more homogeneous under ing the shortest time of flight was defined as the fastest chain. increasing macroscopic pressure (9, 10, 48, 49). The tendency The sample height at each load step was divided by the shortest of interparticle forces to become more homogeneous with load time of flight to determine the theoretical velocity on this fastest suggests that larger increments in macroscopic pressure may ¯p wave path. Force percolation networks and the fastest chain for be needed at higher loads to induce similar increments in hfz i load steps 4, 6, 8, and 10 are provided in Fig. 2B. on the fastest chain. This implies that the scaling exponent ¯p Velocity predictions by all three methods (averaging over of wave velocity with hfz i should decrease with sample com- unit cells, on percolating strong force networks, through fastest pression, which is reflected in the widely observed 1/4 to 1/6 chains) are shown in Fig. 1D, along with experimentally mea- scaling transition (28). This possibility will be addressed in future sured velocities. Velocities obtained for the fastest chain and research and is not studied further here. A number of parti- the force percolation network, respectively, served as approxi- cles fractured during the experiment (detailed in SI Appendix, mate upper and lower bounds for the experimentally measured section 1A). The resulting fragments potentially introduce dif- velocities. The fastest chain prediction closely matched measured ferent contact laws and thus, different force-dependent veloc- wave velocities, particularly at high loads. This finding indicates ity scaling. Other factors related to the contact law between that the actual force magnitudes on the fastest chain are needed particles, including rough contact (50), plastic deformation of to explain the wave speed of the coherent pulse. In contrast, particle contacts (8), and particle shape (27), can also con- the predicted velocities averaged over all unit cells for both tribute to the transition of the exponent value and were not M(uniform) and M(heterogeneous) fall far below the measured investigated here. wave velocities. The case of M(uniform) resulted in similar veloc- The wave velocities predicted with the three methods ities to those of M(heterogeneous) and demonstrated smaller described above were also compared in Fig. 1E with theoreti- deviations. This finding indicates that averaging over all particles’ cal acoustic velocities of granular crystals to further evaluate the structural disorder and force heterogeneity when making veloc- role of structural disorder. The granular crystals included sim- ity predictions is not appropriate, as such averaging significantly ple cubic (SC; equivalent to monatomic chain), body-centered reduces expected wave speeds. cubic (BCC), hexagonal close packed (HCP), and a 2D trian- The three velocity predictions employ contact networks with gular lattice (Tri). Experimentally measured wave velocities for different levels of average contact force and different levels of most load steps lie between those for SC and HCP structures structural disorder. As illustrated in Fig. 2B, the percolating and closely follow BCC predictions across load steps 4 to 11. force network and fastest paths tend to develop along contacts This suggests that a BCC structure can be used to predict the with higher than average contact forces and particles with higher scaling of ultrasound wave velocity with force at sufficient large than average vertical compressive stress magnitudes. To further force levels. However, the chain prediction more closely tracks emphasize the role of contact forces in furnishing wave veloci- the particular variations in the scaling of measured velocities ties, we evaluated the scaling of both the predicted and measured ¯p with force, indicating that the particular details of local struc- velocities with respect to the effective vertical contact force, fz . ture and force magnitudes on this chain can significantly improve The effective vertical contact force was calculated as the sum predictions. of the vertical forces on each particle, p, along the wave path ¯p (entire sample, percolating force network, or fastest chain). fz is Wave Dispersion. To study wave dispersion, we computed spec- calculated by trograms of received signals through the granular sample,

 p   received signals through pistons in contact, and received signals Nc through simulated lattice networks. Corresponding spectrograms ¯p X c c c fz =  kf kai aj ez · ez , [1] are shown in Fig. 3 A–D for the input signal of 0.8 MHz at c=1 load step 8. For the simulations (Fig. 3 C and D), experimental input signals (i.e., the Gaussian bursts shown in Fig. 7A) were where subscripts indicate tensor indices, ez is the unit vector scaled to limit the maximum particle displacement to <1 µm (0, 0, 1)T , ac is a unit vector parallel to the contact force, point- and injected into the heterogeneous and uniform networks. In ing to particle p, and fc is the force vector acting on particle both the experimental and numerical settings, wave reflection p. Note that the term in the inner parentheses is the force- on the boundaries (e.g., platen–particle boundary at the top of weighted fabric tensor. As is shown in Fig. 1E, the predicted the sample) cannot be avoided due to the sample size. There- and measured velocities with different methods generally follow fore, the spectrogram analysis is only performed for the first ¯p power-law relationships with hfz i, showing different exponents. 5 µs (including three to five wave cycles) after the arrival of the

4 of 9 | www.pnas.org/cgi/doi/10.1073/pnas.2004356117 Zhai et al. 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Fig. hie al. et Zhai CDGH EFAB h esrddseso aissgicnl,ntcoeyfol- closely not significantly, varies dispersion measured The the 3A), Fig. to (similar signal input the with Compared cell unit averaging from predictions dispersion Theoretical pcrga fte80kzrcie signal received 800-kHz the of Spectrogram (A) predictions. theoretical and simulations, experiments, from obtained dispersion wave of Comparison ω A–D. m ω and m fsetormcnor t2,4,ad6%of 60% and 40, 20, at contours spectrogram of , T C m E–H and costefiemaue uptsignals. output measured five the across eosrtdsgicn distortion significant demonstrated D) T sbu ie ihsur aes(the makers square with lines blue as Do ria ie)adbrown and times) arrival of SD ±1 m E–H fwv nlssaeas shown also are analysis wave of µs n orsodn characteristic corresponding and , iherrbr representing bars error with , ,6(F 6 ), ,ad1 (H 10 and (G), 8 ), i rirr nt)truhtegaua akn ota fthe of that to packing granular the signal through received units) the the arbitrary of as ratio amplitude (in transmission spectrogram the load maximum define and of We frequency 5A. ratio Fig. input in of shown function as a step, as ratio force transmission to the due Attenuation. modes Wave excited the for enhanced emphasizing coherence 4B, heterogeneity. phase Fig. of in shown loss network shown uniform is the 4A network heterogeneous Fig. the with in con- for networks initial dispersion an spring The inputting the dition. after averaging in forces particles uniform by and all predictions heterogeneous of dispersion spectrum power further the made we persion, to needed are wave, packing, fastest the dispersion. the entire predict from supporting the subset of predicted the details than that the rather that than suggests This frequency) chain. with fastest delayed time for more arrival than, packing (i.e., is the stronger curves) Specifically, entire is curve). blue dispersion the (brown measured or chain experimentally in fastest (green the cells of forces unit that uniform instance, of or average actual the with from that lows t1%o h aiu amplitude. maximum the of 10% at 4 Fig. oc n (B and force ohglgtteiprac ffrehtrgniyo dis- on heterogeneity force of importance the highlight To .Dseso eain o CadBCgaua rsasaesonfor shown are crystals granular BCC and SC for relations Dispersion ). nerdcontact inferred (A) on based 8 step load at relations Dispersion . .(E–H (D). networks uniform and ) oc-nfr otc arc e otdlnsidct contour indicate lines dotted Red fabric. contact force-uniform ) n eosrtsabodrsra hnta for that than spread broader a demonstrates and teuto fiptsgaswseautdby evaluated was signals input of Attenuation iprinrltosbsdon based relations Dispersion ) NSLts Articles Latest PNAS | f9 of 5

PHYSICS Fig. 5. Wave attenuation due to scattering and viscous contact damping. (A) Transmission ratio for experiments with four different input signals across load steps. Error bars represent SDs across five measurements. Damped oscillations with different initial conditions (SI Appendix, section 4A) for 2κhrii = 0.2π (B) and 2κhrii = 0.7π (C), corresponding to 0.8 and 1.6 MHz, respectively, at load step 8. (D) Comparison of transmission ratios for the contact network with inferred forces (C = 0), the contact network with uniform forces (C = 0), and the contact network with inferred forces (C6= 0). The colors and markers are the same as in A for the different input frequencies. (E and F) Attenuation coefficients due to wave scattering for case of inferred forces (E) and uniform forces (F). (G) Modal damping ratios for both cases of inferred forces (markers with white faces) and uniform forces (markers with black faces).

received signal through the pistons in contact with one another Modal damping factors indicate whether the system is undamped (this latter amplitude is nearly frequency- and compression-level (ζ = 0), damped (0 < ζ < 1), critically damped (ζ = 1), or independent). The transmission ratio is below 0.3 at nearly all overdamped (ζ > 1). load levels for input frequencies 1.2 and 1.6 MHz. The trans- To study the relative impact of interparticle dissipation and mission ratio rises above 0.5 and close to 1.0 at the highest load Rayleigh scattering on wave attenuation, we compared the mea- levels for input frequencies of 0.4 and 0.8 MHz. We suspect sured wave attenuation with three types of simulations: 1) net- the observed attenuation arises mainly from two mechanisms: work simulations with inferred forces and C = 0, 2) network Rayleigh scattering (20, 21) and viscoelastic damping at particle simulations with uniform forces and C = 0, and 3) network sim- contacts (44, 47, 51). ulations with inferred forces and C 6= 0. Sinusoidal pulses of a To evaluate attenuation due to Rayleigh wave scattering, we single half-period with frequencies of 0.4, 0.8, 1.2, and 1.6 MHz calculated the scattering attenuation coefficient (SI Appendix, were injected into the top platen, and the sum of contact forces section 4A), η, by fitting autocorrelation functions of the longi- on the top platen and the bottom platen over all contacting par- −η(κ)t tudinal velocity with Cl = e cos(ωd κt) for different initial ticles was recorded. The transmission ratio was calculated as the conditions controlled by wave number κ as in refs. 20 and 21. For ratio of the magnitude of the first peak in the sum of contact viscoelastic damping at particle contacts, we employed a propor- forces received at the bottom platen to the magnitude of the first tional (also called Rayleigh but unrelated to Rayleigh scattering) peak of input signal at the top platen. As compared in Fig. 5D, damping model (52) and defined a damping matrix C linearly we observed 1) increasing transmission ratios, thus decreasing proportional to the mass and stiffness matrices C = cm M + ch H attenuation, with increasing compression; 2) more attenuation where cm and ch are the mass and stiffness proportional damp- with increasing frequency; 3) comparable transmission ratios for ing coefficients (cm , ch > 0), accounting for the damping arising contact networks with heterogeneous and uniform forces; 4) from background motion resistance (e.g., drag) and particle con- a slight increase in wave attenuation due to interparticle dis- tacts, respectively (47). The background damping was ignored sipation; and 5) Rayleigh scattering as the main attenuation in this study. We adopted a viscoelastic Hertzian contact model mechanism, capable of explaining much of the experimentally (45, 46) for dissipation at each individual contact along both observed attenuation. normal and tangential directions, yielding a damping matrix C To further demonstrate these points, we present attenuation proportional to H. The network response with damping was then coefficients due to scattering in Fig. 5 E and F and the modal obtained by integrating Eq. 6 using the ODE45 solver in Matlab. damping ratio due to interparticle dissipation in Fig. 5G. The 1 T −1 Modal damping ratios, ζ, were computed by ζ = 2 [Φ CΦ]Ω attenuation coefficient due to scattering on a uniform network where Φ is the undamped modal matrix, Ω is a diagonal matrix (Fig. 5F) was slightly lower than that of the heterogeneous one containing undamped natural frequencies of the system, and ζ (Fig. 5E) for high frequencies, suggesting that disorder due to is a diagonal matrix containing a damping ratio for each mode. force heterogeneity only slightly enhances attenuation due to

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(Haydon Fig. custom-built encoder in with a actuator shown in linear and placed next described then frame was load was sample but The regions diameter, crystallized disordered. locally outer largely developed 3.00-mm sample diameter, The height. inner into 8-mm 1.5-mm particles and pouring of by cylinder prepared aluminum was an material granular The surface). to ticle 94 from ranging radii with 103 Precision) (Bird spheres sapphire single-crystal Setup. Experimental Methods and Materials manipula- wave testing. for nondestructive and for materials methods behaviors of and tion wave development the particular support with may associated features longitudinal scopic of dis- attenuation packing for that overall responsible waves. in conclude ultrasound primarily role we is negligible a Therefore, order played attenuation. but coherence wave enhanced phase heterogeneity was of Force attenuation scattering. loss wave wave to that due suggest primarily results simulation sponding role. for secondary responsible a plots. playing primarily heterogeneity is relation force sec- disorder with dispersion dispersion, packing a in that playing conclude discernible significantly We heterogeneity primarily disorder hetero- force role packing ondary and with that dispersion, uniform suggested increases with exper- forces were Simulations geneous that measured. those No matched dispersion. imentally closely significant predictions indicating dispersion frequency, on dependent media. granular disordered in inter- wave scaling and velocity ultrasound disorder and for velocities responsible packing are the interparti- heterogeneity contact both force with Hertzian particle that to scaled conclude according and We chain this theory. packing on velocity the measured fastest forces through cle theoretical chain the any by on predicted quantitatively was are study acoustic this follows. of velocity, as findings summarized The wave attenuation. controlling and dispersion, in interpar- paths, the and heterogeneity determine disorder force quantitatively structure ticle packing to response of simulated ultrasound importance The was relative network. network spring this linear of a The construct used stress. were to forces sample interparticle of inferred and levels configuration corresponding packing the at inter- forces and particle tensors, fabric, stress granular situ intraparticle contact average of In orientation, 3D measurement crystal stress. the disordered sample enabled 3DXRD of a and levels XRCT different through to sent compressed material were frequencies cen- numerically different ter with and waves ultrasound disor- experimentally burst Gaussian stressed were investigated. externally materials of granular responses dered ultrasound study, this In Conclusions first as attenuation, the in 5G. of suggesting Fig. damping coefficient in (53)], interparticle illustrated restitution sapphire of a for role with tabulated secondary 0.07 ranges MHz, than [within 3 smaller and 0.9 On 0 be attenuation. between to wave frequencies all found on for scattering hand, 5 wave other differ- Fig. the of with in role shown dominant damping) are the contact conditions (no calculations initial case correlation ent scattering-only velocity the of for Examples scattering. wave hie al. et Zhai h nig fti eerhpoieisgtit h micro- the into insight provide research this of findings The corre- and heterogeneity force measured the Additionally, 3) be to found were velocities measured Experimentally 2) wave coherent the of velocity measured experimentally The 1) n nfr ogns eo 5n frrsruheso h par- the of roughness rms (for nm 15 below roughness uniform and µm h rnlrmtra esuidwsasml f621 of sample a was studied we material granular The B and C oillustrate to ζ was sri aeo 10 of rate granu- (strain the 1 above approximately the located to in platen linear downward steel the material stainless microstepping lar by the 0.01, drove to which 0.005 actuator, from ranging increments, strain Loading. Mechanical 6D schematically. Fig. geometry 1-ID-E strain. experimental sample beamline of increments (APS) between Source performed X-ray rota- were Photon and the translation Advanced the outside the 6C on (Fig. at configuration mounted was present operational frame stage load its tion The 6B. in Fig. shown in PXIe-5105). hutch is MHz; (60 system oscilloscope arbitrary an an The and UMI-7764), (PXIe-5423), and (PXI-7342 generator controller waveform motion a Chassis), Express eteeoeisetdtercie inl,adw eetdapa nthe in peak a selected we and and frequencies, rise 1.2 signals, these received of sharp At the frequencies this attenuation. inspected center However, therefore significant with we demonstrated signal. signals that coherent some a MHz in the featured 1.6 captured of typically be MHz arrival not 0.8 the could and signals at 0.4 rise the of sharp of frequencies analysis center cross-correlation at cross-correlation resulting coefficients The and 1C). section signal Appendix, (SI received themselves input average received the the of and analysis signals cross-correlation combining by unless calculated frequencies angular are study this 7 in Fig. specified. in used otherwise provided frequencies are all signals domain) input that frequency of Note and Features were domain 5). which section time Appendix, N, the (SI 200-µm (in 0.50 MHz of of 3.27 chain be compression 1D to axial a calculated of under frequencies spheres cutoff sapphire frequencies the with center diameter below The (16) fall MHz. to 1.60 bursts” and selected a “Gaussian 1.20, were transducer: 0.80, yielding 0.40, piezoelectric of Gaussian, gener- the frequencies a was center to signal with voltage each Each convolved time-varying transmitted, times. sinusoid a were five sending piezoelectric received signals by and the of ated transmitted using types was received Four which was sample. trans- of signal the piezoelectric The the below sample. using receiver the sample the above through ducer transmitted was signal an by Transmission. Ultrasound followed performed, were measurements sweep. 3DXRD frequency ultrasound and XRCT ing, Isse noper-measurements. in system (C frame. NI hutch. ( (B) APS. X-ray components. the outside major showing configuration APS ational the at mounted 6. Fig. AB C aevlcte o h oeetprino rnmte ae were waves transmitted of portion coherent the for velocities Wave ω hogotteeprmn,adXC n DR measurements 3DXRD and XRCT and experiment, the throughout ) itr ftela frame load the of Picture (A) setup. experimental of Schematic orsod otertto fteetr odfaedrn X-ray during frame load entire the of rotation the to corresponds xeietlgoer,soigtesml n coordinate and sample the showing geometry, Experimental D) −3 / oesr httelaigrmie quasistatic remained loading the that ensure to µm/s h rnlrsml a oddmcaial nsmall in mechanically loaded was sample granular The o10 to fe ah3XDmaueet nultrasound an measurement, 3DXRD each After −4 .Atrec nrmn fmcaia load- mechanical of increment each After ). −z D ieto.Telaigrt a limited was rate loading The direction. NSLts Articles Latest PNAS odfaemutdat mounted frame Load ) lutae the illustrates A | and f9 of 7 B.

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Fig. 7. Features of input signals and calculation of wave velocities. Gaussian burst signals with center frequencies of 0.40, 0.80, 1.20, and 1.60 MHz in the time (A) and frequency (B) domains. (C) Time of flight through sample and pistons, tm (Left), and pistons in contact, tp (Right). (D) Transmitted ultrasound signals for input Gaussian bursts centered at 0.8 MHz at step 7. Input signal (black) and average received signal through sample and platens (red) correspond to the left y axis. Cross-correlation between the average of received signals and input (green circles), cross-correlation between received signals through sample (blue dashed line), and cross-correlation between the signal through platens in contact and input (magenta triangles) use the right y axis.

cross-correlation that corresponded to a cross-correlation below 0.75 but Dynamical Responses of Contact Network. With the experimentally mea- was consistent with arrival times at previous load levels. To compute the sured contact network and interparticle forces described above, we con- wave speed through the granular sample, we subtracted the arrival time structed a spring network to further study the system’s dynamical response of the coherent wave obtained from two pistons in contact, tp, from the at each level of applied sample stress in the experiment (20, 21). We arrival time through the sample, tm, and divided the sample height (identi- assumed that the contact fabric remained unchanged by wave propa- fied by the separation distance of the loading platens in XRCT images) by the gation. We further assumed that, during ultrasound wave propagation, result, ts = tm − tp. particles oscillated with respect to their equilibrium positions (identified experimentally). Finally, we calculated a linear contact stiffness at each Inference of Interparticle Forces. By combining XRCT and 3DXRD datasets, interparticle contact by linearizing a Hertzian contact law at the measured all interparticle and particle-boundary force vectors were calculated using a interparticle force. modified version of the minimization procedure described in refs. 9, 10, 54, The governing equations of wave propagation in the granular sample are and 55. The minimization procedure involved an effort to satisfy equilibrium given by (Eqs. 2 and 3) and stress–force relations (Eq. 4) for each particle: Mu¨(t) + Cu˙ (t) + Hu(t) = 0, [6] p Nc X where M, C, and H are the mass, damping, and Hessian matrices, each with f i = 0, [2] dimensions 6Ns × 6Ns ; u is 6Ns × 1 vector containing the displacement and i=1 p p p rotational configuration of all particles relative to their reference positions; p N c u(t) = [... , xp(t), yp(t), zp(t), θp(t), βp(t), γp(t), ...]; and t is time. Matrices of X i i x × f = 0, [3] M, C, and H are described in detail in SI Appendix, section 3 and were i=1 notably constructed using the experimentally measured packing structure p Nc and in some calculations, the experimentally measured interparticle forces. X Λpxi ⊗ f i = ΛpV pσp, [4] Eq. 6 was numerically integrated using the ordinary differential equation i=1 solver ODE45 in Matlab, for given initial conditions. We focused numerical simulations on studying longitudinal wave prop- p i where Nc is the total number of contacts for particle p, f is the force agation along the vertical (z) direction of the sample to mirror input and vector at contact i, and xi is the position of the contact i. A parame- output signals controllable and measurable in experiments. To investigate Λp = σp/hσs i σp = σp + σp + σp / hσs i = the relative influence of packing structure and interparticle force hetero- ter h h was calculated with h ( xx yy zz) 3 and h s s s h(σxx + σyy + σzz)/3i being the per-particle and sample-averaged (average geneity on observed wave behaviors, we simulated wave propagation in indicated with h·i) hydrostatic stress, respectively. This parameter weakened two ways. The first way used both the experimentally measured contact the contribution of particles with small measured stress values (within the networks and interparticle forces in the construction of mass, damping, and noise level of measurements) in the weighted objective function that was Hessian matrices. The second way used the experimentally measured contact minimized to find contacts forces: network but assumed a uniform contact stiffness, derived from the aver- age interparticle force in the packing, at all interparticle contacts. Although kKst f − bst k2 + λkKeqfk2. [5] the latter system would be out of equilibrium if its interparticle contact forces were derived from the corresponding interparticle stiffness values, In this objective function, k ...k2 is the 2-norm, Kst (corresponding to Eq. 4) it provided a convenient method of decoupling the influence of force het- s s and Keq (corresponding to Eqs. 2 and 3) are 6Np × 3Nc matrices containing erogeneity from that of structural disorder. Further theoretical calculations s positions of all grains and contacts in the packing, bst is a 6Np vector con- with granular crystals then provided a comparison of these results with s taining all per-particle stresses, f is a Nc vector containing all contact forces scenarios without structural disorder. of the packing, and λ is a trade-off parameter. This information is also avail- able in refs. 9, 10, 56, and 57, but it is summarized here and detailed in Data Availability. All data are described in SI Appendix, section 6. Particle SI Appendix, section 2. We denote the overall number of contacts in the data are provided as Datasets S1–S5, and all data, including particle and s P p system Nc = Nc . In the minimization procedure, a Coulomb friction con- waveform data, can be accessed through www.zenodo.org (60). i i straint on each interparticle contact force vector was also enforced: ft ≤ µfn, i i where ft is the tangential force component and fn is the normal force com- ACKNOWLEDGMENTS. C.Z. and R.C.H. acknowledge support from The ponent at contact i. The friction coefficient was assumed to be 0.2 for both Johns Hopkins University’s Whiting School of Engineering. E.B.H. and sapphire–sapphire and sapphire–boundary contacts (58). Future work will R.C.H. acknowledge support for the design and construction of the quantify this friction coefficient and its variability with separate small-scale load frame from Lawrence Livermore National Laboratory’s Laboratory Directed Research and Development Program Grant 17-LW-009. We thank sliding experiments in the same load and ambient humidity range present Dr. J. Almer, Dr. P. Kenesei, and Dr. J. S. Park for help in executing the experi- in the experiments, similar to those performed by others (59). Renderings ments. We acknowledge the APS for synchrotron beamtime under proposal of contact forces at select load steps are provided in Fig. 2B. These render- GUP-59127. Use of the APS, an Office of Science User Facility operated for ings employ linear scaling of force vector lengths, widths, and opacities with the US Department of Energy (DOE) Office of Science by Argonne National force vector magnitude. Laboratory, was supported by US DOE Contract DE-AC02-06CH11357.

8 of 9 | www.pnas.org/cgi/doi/10.1073/pnas.2004356117 Zhai et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 0 .Gln .Tnk,A Lema A. Tanaka, H. Gelin, S. 20. Wang E. gran- dry 19. in propagation wave Elastic Magnanimo, V. High- Saitoh, K. ordered beads: Luding, in S. of propagation Cheng, wave H. lattice for 18. model constrained micropolar a Enhanced Luding, in S. Merkel, propagation A. Sound 17. Gilles, lattice. B. a on Coste, constrained C. beads of 16. behavior Low-frequency Coste, C. Gilles, B. chains. granular 15. disordered in filtering Frequency Luding, linear S. Lawney, a P. in B. disordered propagation 14. Pulse in Daraio, C. propagation Wang, S. wave Nesterenko, V. Acoustic Kim, J. Boechler, Herbold, E. N. 13. Wallen, S. Hiraiwa, propagation. M. Information media: during 12. Granular dissipation Luding, energy S. and rotations Particle 11. Hurley, R. Hall, S. Herbold, E. Zhai, C. 10. 1 .Sio,R .Srvsaa .Ldn,Rttoa on ndsree granular disordered in sound Rotational Luding, S. Shrivastava, K. R. Saitoh, K. 21. 7 .T wn,K .Dnes on rpgto n oc hisi rnlrmaterials. granular in chains force and propagation Sound Daniels, E. acoustics. K. material Owens, granular T. for E. law contact 27. hertz of validity the On Gilles, B. Coste, C. 26. granular stressed externally in propagation Ultrasound Velicky, B. granular Caroli, of C. networks ordered Jia, in X. mitigation 24. Wave Daraio, C. Ponson, L. Leonard, A. grain for 23. model continuum based micromechanics Granular Misra, A. Poorsolhjouy, P. 22. 8 .Gdad olna lsiiyadpesr-eedn aesed ngranular in speeds wave pressure-dependent and elasticity Nonlinear Goddard, J. 28. media. granular stressed in broadening pulse Sound Jia, X. Langlois, V. 25. 0 .O adn .RcatJ. lsi aevlcte ngaua soils. granular in velocities wave Elastic Jr., Richart F. Hardin, O. B. elas- Nonlinear 30. packings: Granular Schwartz, L. Johnson, L. D. 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PHYSICS