Biomechanics of a moth scale at ultrasonic frequenciesZhiyuan Shena, Thomas R. Neila, Daniel Roberta, Bruce W. Drinkwaterb, and Marc W. Holderieda,1

aSchool of Biological Sciences, University of Bristol, BS81TQ Bristol, United Kingdom; and bDepartment of Mechanical Engineering, University of Bristol,BS81TR Bristol, United Kingdom

Edited by John B. Pendry, Imperial College London, London, United Kingdom, and approved October 9, 2018 (received for review June 13, 2018)

The wings of moths and butterflies are densely covered in scalesthat exhibit intricate shapes and sculptured nanostructures. Whilecertain butterfly scales create nanoscale photonic effects, mothscales show different nanostructures suggesting different func-tionality. Here we investigate moth-scale vibrodynamics to un-derstand their role in creating acoustic camouflage against batecholocation, where scales on wings provide ultrasound absorberfunctionality. For this, individual scales can be considered asbuilding blocks with adapted biomechanical properties at ultra-sonic frequencies. The 3D nanostructure of a full Bunaea alcinoemoth forewing scale was characterized using confocal microscopy.Structurally, this scale is double layered and endowed with differ-ent perforation rates on the upper and lower laminae, which areinterconnected by trabeculae pillars. From these observations aparameterized model of the scale’s nanostructure was formedand its effective elastic stiffness matrix extracted. Macroscale nu-merical modeling of scale vibrodynamics showed close qualitativeand quantitative agreement with scanning laser Doppler vibrom-etry measurement of this scale’s oscillations, suggesting that thegoverning biomechanics have been captured accurately. Impor-tantly, this scale of B. alcinoe exhibits its first three resonancesin the typical echolocation frequency range of bats, suggesting ithas evolved as a resonant absorber. Damping coefficients of themoth-scale resonator and ultrasonic absorption of a scaled wingwere estimated using numerical modeling. The calculated absorp-tion coefficient of 0.50 agrees with the published maximum acous-tic effect of wing scaling. Understanding scale vibroacousticbehavior helps create macroscopic structures with the capacityfor broadband acoustic camouflage.

moth scale | acoustics | ultrasonics | vibration | porous materials

The nocturnal acoustic arms race between moths and echo-locating bats has been ongoing for 65 My. To defend them-

selves against the bats’ biosonar (most relevant frequencies from20 to 150 kHz, with wavelengths from 16.6 to 2.3 mm), differentmoth species have evolved a wealth of active and passive defensestrategies. Several moth taxa have independently evolved earsthat can detect the ultrasonic frequencies of the biosonar calls ofan approaching bat (1), which allow them to respond with eva-sive flight behaviors (2). In addition, Arctiinae, Geometridae,and some other moths produce loud ultrasound clicks whenunder attack, which can startle bats, alert them to the moths’toxicity, or even jam the bats’ biosonar (1, 3, 4). Recent findingssuggest some other moth species mimic such aposematic ultra-sound clicks (5).* The many nontoxic moth species withouthearing capability, however, have to rely on passive acousticcamouflage to avoid capture by bats (6, 7).Like in most flying insects, moth and butterfly wings consist of

a solid thin chitinous membrane suspended between a networkof stiffer wing veins. In contrast to most other insects, however,both the upper and lower wing surface of moths and butterfliesare covered with arrays of overlapping scales, which has given theorder Lepidoptera its scientific name (Greek lepidos = scale;pteron = wing). The scales and wing membrane are part of theinsect exoskeleton consisting of a sclerotized biomaterial matrixof mainly chitin and protein (8). A typical moth scale is anchoredinto a socket in the wing membrane with a narrow pedicel and

widens into a flattened blade (9). Each scale itself is a highlysculptured porous structure, and scales show diverse morphol-ogies even on a single wing (10). The highly sculptured scalestructure implies sophisticated evolutionary adaptations, analo-gous to the highly organized nanoscale photonic structures forvisual signaling (11, 12). Across moths, scale morphologies arediverse and hence provide a large candidate pool for biophysicaladaptations. Previous studies have highlighted the role of moth-scale morphology in creating multiple functions of moth wingssuch as aerodynamics, thermal regulation, and wettability (13–15). Additionally, moth wings have been hypothesized asthe main organ bringing about acoustic camouflage. Micro-reverberation chamber testing revealed that scale-covered mothwings are more absorbent at frequencies from 40 to 60 kHz thanwings with scales removed (7). There is, however, no explanationfor how the moth wing, and more specifically its microstructure,creates this acoustic absorber functionality.Lepidoptera wing scales are usually less than 0.25 mm wide, and

thus always smaller than one-tenth of the smallest wavelength batsuse for echolocation. Even the cross-section thickness of the wingincluding upper- and lower-scale layers is always below the rele-vant wavelengths. Because wings are therefore ultrathin absorberswith subwavelength thickness, rigid porous absorption is in-efficient, leaving the alternative of resonant absorber functionality(16). In resonant absorbers a resonant mass and spring system willprovide maximum absorption at the frequency it is tuned to. Otherresonant systems show maximum sound transmission at resonance(17). Both high absorbance and transmittance are viable strategies

Significance

Ultrathin sound absorbers offer lightweight solutions frombuilding acoustics to sonar cloaking. The scales on moth wingshave evolved to reduce the echo returning to bats, and weinvestigate their resonant sound-absorber functionality. Reso-nant absorbers are most efficient at resonance, and laserDoppler vibrometry (LDV) revealed that an individual mothscale’s three resonance modes indeed span the biosonar fre-quencies of bats. The porous anisotropic nanostructure of suchscales was parameterized and its effective stiffness propertiescalculated. Modal analysis on a 3D model accurately predictsresonance modes and frequencies found by LDV, and confirmsabsorption performance-matching measurements. Our abilityto model the absorbers contributing to evolved biosonarcamouflage has implications for developing bioinspired thinand lightweight resonant sound absorbers.

Author contributions: Z.S., D.R., B.W.D., and M.W.H. designed research; Z.S. performedresearch; Z.S., D.R., B.W.D., and M.W.H. contributed new reagents/analytic tools; Z.S. andT.R.N. analyzed data; and Z.S., T.R.N., D.R., B.W.D., and M.W.H. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1To whom correspondence should be addressed. Email: marc.holderied@bristol.ac.uk.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1810025115/-/DCSupplemental.

Published online November 12, 2018.

*O’Reilly LJ, Neil T, Holderied MW, 16th International Meeting on Invertebrate Sound &Vibration, September 14–17, 2017, Rauischholzhausen, Germany.

12200–12205 | PNAS | November 27, 2018 | vol. 115 | no. 48 www.pnas.org/cgi/doi/10.1073/pnas.1810025115

to reduce backscatter and hence detectability to bat biosonar.Since scales are the basic elements of moth wings, their vibrationalresponse is essential in understanding the acoustic behavior ofentire moth wings and hence the acoustic camouflage effect inmoth acoustic ecology and behavior.Moth-wing scales show a hierarchical design, with a scale tiling

pattern at the large scale, scale shape at the next, and scale in-ternal structure of nanometer order. As a step to understandingthis sophisticated natural structure, this paper focuses on ex-ploring the vibrational behavior of a single freestanding scale.Our prediction is that moth scales are resonant systems, and thattheir resonances are at biologically relevant frequencies used bybats for echolocation. We further provide an accurate numericalmodel of the dynamic behavior of a scale, which captures thegoverning physical phenomena at work. Numerical modeling isused to show that the scale resonator can achieve high absorp-tion coefficients at resonance.Existing resonant absorbers are made of solid materials. In

some designs a layer of porous material is added to achievehigher absorption coefficients or a broadband performance (18).The moth scale represents a resonator design composed of res-onating microperforated scales. Elucidating the acoustic mech-anisms behind moth-wing sound absorbance aims at developingbioinspired sound-absorbing materials with a thickness belowtheir functional wavelength for applications in noise control,architectural acoustics, and bioinspired radar and sonar targetconcealment.

ResultsScale Shape and Structure. Scanning electron microscopy (SEM)of a Bunaea alcinoe wing (Fig. 1A, dorsal view) shows thatneighboring scales overlap both laterally and along their longi-tudinal axis. Scales have a leaf-like shape with the blade con-tinuously widening from the basal socket, ending in a wide apicalmargin deeply incised forming several elongated extensions.Each scale possesses a top lamina and a bottom lamina con-nected by a framework of trabeculae forming the intertrabecularsinus. Both laminae consist of parallel longitudinal ridges, andneighboring longitudinal ridges are connected via a series ofcross-ribs (Fig. 1 B and E–H). This intricate arrangement makesscales highly porous with a large proportion of air-filled space.One typical scale (Fig. 1D) was selected for detailed structural

characterization, vibrational analysis, and dynamic modeling. Weused confocal microscopy to obtain a high-resolution 3D imageof scale nanostructure (Fig. 1 D–F). As a typical example ofscales covering the dorsal aspect of the wing, this scale (Fig. 1D)is 295 μm long from its socket to the tip of the longest apicalextension. The blade is roughly triangular with the greatest widthat the base of the incisions being 189 μm, and a distance betweenthe tips of the two lateral extensions of 214 μm. The longitudinalribs (Fig. 1E) in the top lamina are 1.00 ± 0.13 μm wide (mean ±SD; n = 5) and their distance is 2.59 ± 0.11 μm. In the bottomlamina (Fig. 1F) they are 0.90 ± 0.14 μm wide and spaced by1.97 ± 0.11 μm. Cross-ribs in the top lamina are 0.29 ± 0.03 μmwide and separated by 1.01 ± 0.11 μm, and in the bottom lamina0.25 ± 0.03 μm wide and separated by 0.98 ± 0.11 μm. As a re-sult, both laminae are highly perforated with 31% of the top und30% of the lower lamina area consisting of voids. For the areashown in Fig. 1 E and F the scale is 3.18 ± 0.6 μm thick betweenthe peaks of the longitudinal ridges in upper and lower laminae.

The 3D Unit Cell: Structure. The nanoscale morphological analysisreveals the globally periodic structure of the scale, where an el-ementary unit, a structural cell comprising porous laminae andtrabecular pillars, repeats in the 2D plane of the scale surface.Based on a transverse cross-section (Fig. 2A) through the 3Disosurface shown in Fig. 1 G and H, a parameterized model ofsuch a unit cell was developed (Fig. 2B). Both the top lamina and

the bottom lamina are simplified as corrugated perforated platestructures. The corrugation is formed by juxtaposing a row ofelliptical shells, which are then truncated by a cutting plane (Fig.2A). Arrays of elliptical holes are then punctured in the troughsof the corrugation to mimic the perforation formed by the cross-ribs. The trabeculae that link the upper and lower laminae aresimplified as arrays of vertically positioned cylindrical pillars.Twenty independent parameters are needed to describe theparameterized moth-scale model (SI Appendix, Table S1). Be-cause the ridges are spaced more widely in the top comparedwith the bottom lamina, the unit cell includes two ridge periodsin the top and three in the bottom lamina (Fig. 3A and SI Ap-pendix, Table S1). For comparison with butterfly scales, which inall documented cases have the lower lamina unperforated, amore “butterfly-like” unit cell was created, where the lowerlamina was not perforated but that was otherwise identical, in-creasing the mass of this butterfly-like unit cell by 9.3% relativeto the moth unit cell.

The 3D Unit Cell: Effective Stiffness Matrix. Due to the periodicnature of the parameterized moth-scale model, the single unitcell acts as the representative element of the whole scale struc-ture (Fig. 3A). A set of boundary conditions (SI Appendix, TableS2) is applied to introduce either a pure axial or a pure shearstrain to the unit cell. Fig. 3 B and C show the six surfaces wherevarious displacement boundary conditions are applied. Thestress distribution under such strains was calculated using finiteelement modeling. The stiffness matrix [1] was calculated basedon the simulated stress distribution (Fig. 3 D–I) (for details seeMaterials and Methods).

Fig. 1. Scale arrangement and structure. (A–C) SEM images of B. alcinoescales: (A) Partly disrupted tiling of scales; (B) perforated top lamina of ascale; (C) cross-section of a fractured scale revealing the intertrabecular sinusbetween the two laminae. (D–F) Confocal microscopy of the scale: (D) Indi-vidual scale used for further analysis. (Magnification 20×.) White squareindicates observation area of (E) top lamina and (F) bottom lamina. (Mag-nification100×.) (G and H) Isosurface 3D visualizations of a midsection shownin the yellow square in D of the individual scale; (G) the top lamina and (H)bottom lamina with longitudinal ridges and cross-ribs. In H the lower laminafaces upward, oriented with the basal socket of the scale to the back and theapical ridge facing toward the front.

Shen et al. PNAS | November 27, 2018 | vol. 115 | no. 48 | 12201

BIOPH

YSICSAND

COMPU

TATIONALBIOLO

GY

Stiffnessmatrix �cij�

=

2666664

21.89 2.88 2.152.88 11.5 1.212.15 1.21 8.06

0 0 00 0 00 0 0

0 0 00 0 00 0 0

2.78 0 00 1.13 00 0 1.36

3777775×GPa. [1]

The ridges and the corrugated profile of laminae make the scalematerial anisotropic in scale plane, with the stiffness in the ridgedirection 1.75× the value of the stiffness transverse to the ridge(Fig. 2B). The in-plane anisotropic nature of the nanostructuremeans that the scale shows different bending moduli along theridge direction and the transverse direction. The perforatedstructure has a porosity of 57%. The perforated structure thushas an effective density of 43% of bulk chitin.

Scale Vibrations at Ultrasonic Frequencies: Modeling. The first threecalculated resonances are at 28.4, 65.2, and 153.1 kHz, respectively(Fig. 4D–F), ultrasonic frequencies overlapping with and spanningthe majority of the entire bat biosonar range. The first vibrationmode is a pitch vibration around the x axis. The second mode is atwisting vibration around the middle longitudinal y axis of thescale. The third mode is a yaw vibration of the scale constrainedwithin the flat scale plane, rotating around the z axis.In the more butterfly-like unit cell, which differed from the moth

scale only in that the perforation of the lower lamina was filled, thestiffness matrix changed to [2] and the resulting resonances shiftedupward substantially to 88.4, 150.9, and 406.0 kHz, respectively.

Stiffness matrix �cij�

=

2666664

23.71 4.61 2.144.61 13.78 1.412.14 1.41 6.59

0 0 00 0 00 0 0

0 0 00 0 00 0 0

1.04 0 00 1.09 00 0 1.39

3777775× 1010Pa. [2]

Scale Vibrations at Ultrasonic Frequencies: Measurements. Usinglaser Doppler vibrometry (Materials and Methods) the averagevibrational spectra were measured, and three resonances werefound at frequencies of 27.6, 90.8, and 152.3 kHz, respectively(Fig. 5). The deflection shapes of the resonances are shown inFig. 4 A–C. The average displacement amplitude at the reso-nance peaks is at best 2.5× the response at the nonresonancefrequencies, showing that the scale vibrator has a broadbanddisplacement response.

Scale Damping Coefficients and Scaled-Wing Absorption Coefficients.A finite element model was built to simulate the ultrasonic ab-sorption performance of a surface coated in an array of scales.Assuming Rayleigh damping, two damping parameters were usedto capture the various damping mechanisms. These were foundusing a parameter search until the calculated displacement spectrabest matched the measured spectra (Fig. 6A) (19). The resultingRayleigh damping coefficients were α = 1.51 × 104 and β = 8.30 ×10−8, which are equivalent to a modal damping ratio of 4.5%.With this experimentally extracted damping, the simulated

transmission, reflection, and absorption coefficients of a scaledwing were calculated from a finite element model (Fig. 6B).Modeling covers 20 to 80 kHz, including the first resonance andrepresenting the most relevant bat echolocation range. Asexpected, the absorption coefficient spectrum shows a peak atthe first resonance frequency of the scale, with a maximum of0.50 at 29 kHz. In contrast, the absorption peak disappears for amodel just consisting of the wing membrane. Changing the ma-terial mass or stiffness shifts the peak, moving it potentiallyoutside the pertinent echolocation frequencies.

Discussion and ConclusionThis paper represents an effort to numerically and experimen-tally characterize moth-scale biomechanics and vibrational be-havior. Resonant functionality of scales could be adaptive increating acoustic camouflage against bat echolocation. As pre-dicted, both laser Doppler vibrometry (LDV) and modelingconfirmed resonant behavior of the scale, and the first three

BA

Fig. 2. Schematic showing (A) how the 3D model of the scale was param-eterized and (B) the implemented 3D model containing 2 × 10 unit cells.

G H I

D E F

A B C

Fig. 3. The moth-scale model: (A) The parameterized single unit; B and Cshow the different boundary conditions. Note that each boundary includes allof the facets on the same plane. (D–I) Simulation results of stress distributionand deformation (solid line frame shows the original shape) in the single unitunder different boundary conditions (SI Appendix, Table S2). The unit cellundergoes a pure (D) «xx, (E) «yy, (F) «zz, (G) γxy, (H) γyz, and (I) γxz strain.

12202 | www.pnas.org/cgi/doi/10.1073/pnas.1810025115 Shen et al.

resonances occurred within and fully included the relevant batecholocation frequency range (20–150 kHz). In contrast, themodified butterfly-like scale with solid lower lamina had muchhigher calculated resonance frequencies well outside the fre-quency range relevant for bat echolocation. This supports thenotion that the moth’s double-perforated scale nanostructure hasevolved to create resonance frequencies in response to batecholocation. The three resonance modes we found in our ex-ample scale have the potential to reduce echo strengths (back-scatter) around the respective frequencies. Ultrasonic wavesreaching a moth wing at one of these resonant frequenciesshould be transmitted (17) and/or absorbed (assuming resonantabsorber; refs. 16 and 20) preferentially. This reduces thebackscattered target strength and hence detectability to batbiosonar, affording an evolutionary advantage by reducedpredation pressure.A parameterized nanostructural model of a moth scale has

been developed and used to extract effective material properties.A macroscopic dynamic finite element simulation of a moth scaleusing these effective scale properties has been able to replicatethe experimental LDV well. Mode shapes found in the simula-tion (Fig. 4 D–F) match those found by vibrational character-ization (Fig. 4 A–C), but note that the measured rotation axis ismoved somewhat to the right of the scale’s midline. Importantly,calculated first and third resonance frequencies differ by just0.8 kHz (2.9%) and 1.2 kHz (1.0%) from the measured values.This match strongly suggests that the chosen modeling approachcaptures the most relevant governing biophysical parameters.Only the second resonance frequency differs more substantially(28%). The remaining deviations from calculated resonancesmight exist because of the following: Firstly, the scale curvaturehas been simplified as spherical, while the true scale-curvature

profile is more complex. Secondly, the perforation rate was set asconstant while it changes over different parts of the scale.Thirdly, the incident sound wave for LDV was not normal to thescale surface (clear line of sight needed for laser), and the wavefront was not perfectly flat [speaker close enough to reach ade-quate sound-pressure level (SPL) exposure]. In addition, theorientation of the principal axes of the effective anisotropicmaterial stiffness matrix are fixed in the scale model. In the truescale, however, the material property primitive axes reorientatethemselves normal to the development of the scale curvature.The very good overall model–experiment agreement in terms ofthe mode shapes and resonance frequencies suggests, however,that these factors, which could be modeled in the future, playa secondary role.Ultrasonic absorption modeling shows that the observed res-

onance behavior of the single scale creates relevant ultrasonicabsorption functionality of the scaled wing (assuming an array ofscales). The absorption coefficient peak value of 0.50 is com-parable with the empirical report on moth-wing absorption (7,21). The wing model is simplified as one layer formed by arectangular grid of identical cover scales on a flat wing mem-brane. A real moth wing is composed of multiple layers of coverand base scales of different sizes, shapes, and degrees of overlap.The ultrasonic absorption coefficient spectra of a real moth wingcould therefore be higher than the numerical estimated valuehere. The different morphologies of scales suggest differingresonance frequencies. Such resonance variation of differentscales may have evolved to cumulatively achieve more broad-band acoustic absorption. Note that our current model does notinclude potential acoustic effects of the scale microperforation,which might further increase absorption.The resonant moth-scale absorber we investigated here has a

morphology that differs substantially from existing resonant ab-sorber designs. Moth-scale resonators have the advantage of asmall footprint, are easily assembled into densely overlappingarrays, and resonator properties can be tuned via multiple-parameter tuning. The parametric model of the moth-wingstructure paves the way to understanding and reconciling notonly the acoustic but also the aerodynamic and thermal func-tionality of moth scales. Our model sheds light on designs ofbiomimetic lightweight acoustic metamaterials that achieve

Fig. 5. Mechanical responses of a scale. The vibrational spectrum was cal-culated by averaging amplitude spectra over all scanning points. (Inset) Scaleshape and the scanning area.

Fig. 4. Modeled and measured resonances of the moth scale. (A–C) Scan-ning LDV results of the first three resonances of the scale. Resonance fre-quencies: (A) 27.6 kHz; (B) 90.8 kHz, and (C) 152.3 kHz. (D–F) Simulation ofmode shape of a single scale with curvature radius of 700 μm. The colorprofile shows the normalized z component (the out-of-scale plane dis-placement of the vibrating scale). (D) Rotational vibration around the x axis,pivoting at the clamped edge, at frequency 28.4 kHz; (E) twisting vibrationaround the y axis at 65.2 kHz; and (F) rotational vibration around the z axis,at 153.1 kHz. Gray outline of scale indicates rest position for comparison.Color bar indicates displacement amplitude.

Shen et al. PNAS | November 27, 2018 | vol. 115 | no. 48 | 12203

BIOPH

YSICSAND

COMPU

TATIONALBIOLO

GY

specific acoustic absorption and transmission for applications innoise mitigation.

Material and MethodsSpecimens. Live B. alcinoe (Cabbage tree emperor moth; Stoll, 1780) wereobtained from wwb.co.uk/ as pupae on July 28th 2016. The pupae werehoused in a temperature-controlled cabinet (Economic Deluxe, Snijders Sci-entific), where they were subject to a 12-h night/day cycle in which tem-perature varied between 25 °C and 30 °C while humidity was maintained ata constant level of 70%. We checked daily for successful eclosion, whichhappened during the first week of August 2016. Intact specimens were killedand pinned in a natural position with the wings oriented horizontally to thedorsal plane. After being dried at room temperature for 2 wk, scale speci-mens behind the bifurcation of the third vein on the front dorsal area of theright forewing were removed from the wing using a fine brush. Individualscales were then mounted in an upright position by clamping their stalk endusing microsurgery tweezers (B5SA; Bondline Electronics Ltd.) for LDV (Pol-ytec PSV-400; POLYTEC GmbH).

Scale Microstructure. Nanostructure of individual B. alcinoe moth scales wasobtained using SEM (Zeiss Evo15 with Lab6 emitter) and confocal microscopy(Leica TCS SP5). For SEM, sections of wing were mounted on adhesive carbontabs (EM Resolutions Ltd) and coated with 5 nm of gold (Quorum Q150R ES;Quorum Technologies Ltd.). Scales were imaged in both high-vacuum modeusing an SE1 detector and variable-pressure mode using a VPSE G3 detector.We used an electron high tension of 15–20 kV with 50–100-pA probe.(Magnification range 250× to 10K×.)

For confocal microscopy, a single target scale was immersed in mountingmediumglycerol. It was sealed between twomicroscopy slides with nail polish(ethyl acetate and butyl acetate as main ingredients) as a sealing material.Autofluorescence of the scale material was strong enough to obtain clearconfocal images without further labeling (22). The excitation and lightemission frequency sweep spectra of the scale were characterized. The op-timal confocal microscopic setting was determined by trial and error as:excitation wavelength = 488 nm, emission band = 495–720 nm, pinhole =60 nm, optical section thickness = 0.46 μm, z step = 80 nm. The 100× lensrealized a pixel size of 30 × 30 nm, hence creating 3D voxels of 30 ×30 × 80 nm.

Modeling of the Effective Material Property. The 3D data obtained fromconfocal microscopy were turned into a voxel space from which a 3D iso-surface model was created and saved in stereolithography (STL) format usingMATLAB (R2016a; The MathWorks). This 3D model of the moth scale in STLformat was then imported into a finite element methods software (COMSOL5.2a; COMSOL Inc) to identify and parameterize an idealized unit cell formodeling of the effective material properties. The effective material ex-traction process was greatly simplified due to the periodicity of the modelsince a single unit cell can be adopted as the representative element of thewhole structure (Fig. 3A). Expanding the unit cell in three dimensions resultsin a 3D material possessing the symmetry of point group 2mm (Hermann–Mauguin point-group notation), with a twofold symmetry axis in the z di-rection and two mirror planes in the x-z and y-z plane passing through thetwofold symmetry axis (Fig. 2B) (23). This 2mm symmetry leads to a 3Dmaterial with anisotropic elastic behavior, which can be represented by thefollowing constitutive equations:

½σ�= ½c�½e�, [3]

2666664

σxxσyyσzzτxyτyzτxz

3777775=

2666664

c11 c12 c13c12 c22 c23c13 c23 c33

0

0c44 0 00 c55 00 0 c66

3777775

2666664

«xx«yy«zzγxyγyzγxz

3777775, [4]

where ½c� is the stiffness matrix; σxx, σyy, and σzz are the normal stress com-ponents; τxy, τyz, and τxz are the shear stress components; «xx, «yy, and «zz arethe normal strain components; γxy, γyz, and γxz are the shear straincomponents.

The nonzero values in the stiffness matrix are extracted from the finiteelement parameterized single unit cell model static stress–strain simulation(Fig. 3A), assuming the nanostructure is made of homogeneous chitin. Sixboundary conditions (SI Appendix, Table S2) were set: u, v, and w representdisplacement variables in the x-, y-, and z directions. d = 0.01 μm representsan infinitesimal displacement to induce strain in the unit cell. Each boundarycondition introduces either a pure axial strain or a shear strain in the unitcell, while the other five strain elements remain zero. The strain vector ½e� in[4] thus has only one nonzero element. The stress distribution of the singleunit cell element under such a boundary condition is then calculated (Fig. 3D–I) and the effective stress elements in the stress vector ½σ� are calculated byaveraging the simulated stress distribution on the respective boundaries.Similar method had been adopted in refs. 24–26 to extract the effectiveelastic material property of a single-layer perforation. The chitin’s Youngmodulus is 65 GPa and the density is 1,300 kg/m3 (27).

Modeling Scale Vibrations at Ultrasonic Frequencies. To calculate the vibrationof a single scale, a macroscale finite element model as shown in Fig. 4 was builtin COMSOL. The polygonal outline of the scale is extracted from the confocalimage as shown in Fig. 1D using the software ImageJ (1.46r; National Institutesof Health). The polygon is then imported into COMSOL and the curved scalemodel is formed by penetrating the extruded outline polygon with a sphericalsurface. The radius of the spherical surface has been chosen as 700 μm, which isbased on the SEM observation of the scale-curvature morphology. Due to thesize of the unit cell relative to the wavelength, the dynamics of the unit cellcan be neglected in our finite element simulation and hence it is treated as apure stiffness element. The effective material stiffness matrix is assigned to thesingle-scale model, and a modal analysis is conducted to obtain the resonancesand the mode shapes of the single scale. When doing the modal analysis, thescale pedicel end is fully clamped, while all other edges are free. The scale wasinserted in a socket structure in the wing membrane. The socket’s structureand its mechanical degree of freedom, to the authors’ knowledge, has notbeen reported. To simplify the experimental and numerical calculation study, amechanically clamped boundary condition was used in this study.

Scanning Laser Doppler Characterization of Scale Vibration. The vibrationalbehavior of the single scale was characterized using an LDV (Polytec PSV-400;POLYTEC GmbH). Amicroscan lens PSV-A-CL-80 was used, which can focus thelaser spot diameter to be as small as 10 μm.

The moth scale was mechanically clamped at one end using tweezers andmounted at a distance 13 cm from the lens. The vibration of the scale wasexcited by sound pressure from a custom-made speaker based on an electretfilm (HS-02-Film; Emfit Ltd.), which was mounted at a distance of 7 cm and anangle of θ= 40° below the specimen (SI Appendix, Fig. S1A). The speaker waslocated out of the optical path so that it would not block the path; therealized angle was as close as possible to normal incidence. We used a high-voltage amplifier (PZD350; TREK Inc.) with the speaker. The SPL at thesample position was calibrated using a calibration microphone (1/8-inchMicrophone Type 4138, with Dual Microphone Supply Type 5935L as theamplifier; Brüel & Kjær). The calibration was performed with the scale inplace and the acoustic axis of the microphone aligned with that of thespeaker. The scale and the microphone were on the same plane perpen-dicular to the laser-beam path and maintained the same distance with thespeaker. The SPL at 100 kHz was 59.0-dB SPL (SI Appendix, Fig. S1B). Thevibrational spectrum from 20 to 180 kHz was obtained by amplitude aver-aging the displacement spectra of all of the scanning points (Fig. 4 A–C). Thescanning area is a fan-shaped area defined on the blade part of the scale(Fig. 4). The scanning point grid has a scanning step of 12 μm.

Modeling of Scale Damping Coefficients and Absorption Coefficients of theScaled Wing. Two models were built in COMSOL to explore the dampingeffect of the scale and the ultrasonic properties of the moth wing composed

Fig. 6. (A) Calculated displacement spectra vs. the measured displacementspectra from 20 to 80 kHz. The calculated spectra were under the dampingratio of 4.5%. (B) Calculated reflection, transmission, and absorption coef-ficients of the moth-scaled wing. The absorption coefficient of a single wingmembrane layer was also plotted for comparison.

12204 | www.pnas.org/cgi/doi/10.1073/pnas.1810025115 Shen et al.

of such scales. The first model contained a single scale with one end fullyclamped (SI Appendix, Fig. S2A). The scale was immersed in an air chamberand had a 40° oblique angle with respect to the horizontal plane. Incidentplane waves were assigned in the upper air chamber to mimic the wavesgenerated by the speaker. The incident wave amplitude was based on thecalibrated SPL spectrum (SI Appendix, Fig. S1B). Two perfect matched layerswere added on the top and bottom of the model to absorb the reflected andtransmitted waves so that the model mimicked an open space condition.A fan-shaped area reflecting the LDV scanning was defined on the scale.A frequency-domain analysis was conducted with frequencies spanning from20 to 80 kHz, which focused on the first resonance and represents the mostrelevant bat biosonar range. The displacement spectra were calculated byperforming amagnitude average on the scanning area following the frequency-domain simulation.

A Rayleigh damping model was adopted to phenomenally describedamping in this model. The two Rayleigh damping coefficients, α and β, wereparameter swept in COMSOL and their values were determined when thecalculated displacement spectra matched with the LDV detected spectra (α =1.51 × 104 and β = 8.30 × 10−8).

A second model (SI Appendix, Fig. S2B) was built to calculate the ab-sorption coefficient of the scaled array and Rayleigh damping was added tothe material. The leaf-shaped scale (length = 295 μm, width = 189 μm,thickness = 3.86 μm) was attached on a 3-μm-thick wing membrane made ofchitin. The scale had a 25° insertion angle with the wing membrane. Periodicboundary conditions were assigned to the vertical unit cell walls to expand

such a unit cell to a 2D array. The transmission coefficient RΠ, reflectioncoefficient TΠ, and absorption coefficient α of the scale array were calculatedby the following formulae (28):

RΠ =jpr j2jpi j2

, [5]

TΠ =jpt j2jpi j2

, [6]

α=1−RΠ − TΠ, [7]

where pr = the reflected acoustic pressure, pi = the incident acoustic pres-sure, and pt = the transmitted acoustic pressure. The numerators and de-nominators in the above equations were calculated by averaging over thetwo planes located above and below the scale (SI Appendix, Fig. S2). Forcomparison, the absorption coefficients of a single layer of wing membranewere also calculated.

ACKNOWLEDGMENTS. The project Diffraction of Life (BB/N009991/1) is fundedby the Biotechnology and Biological Sciences Research Council UK. The authorsacknowledge the discussion with Dr. Mihai Caleap, Dr. Rob Malkin, andDr. Alberto Pirrera and the support from the COMSOL Support Center for theclarification of the mechanical behavior and the modeling work.

1. Corcoran AJ, Hristov NI (2014) Convergent evolution of anti-bat sounds. J CompPhysiol A Neuroethol Sens Neural Behav Physiol 200:811–821.

2. Miller LA, Surlykke A (2001) How some insects detect and avoid being eaten by bats:Tactics and countertactics of prey and predator. Bioscience 51:570–581.

3. Dowdy NJ, Conner WE (2016) Acoustic aposematism and evasive action in selectchemically defended Arctiine (Lepidoptera: Erebidae) species: Nonchalant or not?PLoS One 11:e0152981.

4. Kawahara AY, Barber JR (2015) Tempo and mode of antibat ultrasound productionand sonar jamming in the diverse hawkmoth radiation. Proc Natl Acad Sci USA 112:6407–6412.

5. Barber JR, Conner WE (2007) Acoustic mimicry in a predator-prey interaction. ProcNatl Acad Sci USA 104:9331–9334.

6. Clare EL, Holderied MW (2015) Acoustic shadows help gleaning bats find prey, butmay be defeated by prey acoustic camouflage on rough surfaces. eLife 4:e07404.

7. Zeng J, et al. (2011) Moth wing scales slightly increase the absorbance of bat echo-location calls. PLoS One 6:e27190.

8. Hunt S (1971) Composition of scales from the moth Xylophagia monoglypha.Experientia 27:1030–1031.

9. Ghiradella H (1991) Light and color on the wing: Structural colors in butterflies andmoths. Appl Opt 30:3492–3500.

10. Dhungel B, Otaki JM (2014) Morphometric analysis of nymphalid butterfly wings:Number, size and arrangement of scales, and their implications for tissue-size de-termination. Entomol Sci 17:207–218.

11. Siddique RH, Vignolini S, Bartels C, Wacker I, Hoelscher H (2016) Colour formation onthe wings of the butterfly Hypolimnas salmacis by scale stacking. Sci Rep 6:36204.

12. Shawkey MD, Morehouse NI, Vukusic P (2009) A protean palette: Colour materialsand mixing in birds and butterflies. J R Soc Interface 6(Suppl 2):S221–S231.

13. Tanaka H, Shimoyama I (2010) Forward flight of swallowtail butterfly with simpleflapping motion. Bioinspir Biomim 5:026003.

14. Roush W (1995) Wing scales may help beat the heat. Science 269:1816.

15. Kristensen NP, Simonsen TJ (1999) Handbook of Zoology, Lepidoptera, Moths andButterflies, 2: Morphology Physiology, and Development (De Gruyter, Berlin).

16. Cox TJ, D’Antonio P (2017) Acoustic absorbers and Diffusers (Taylor & Francis, BocaRaton, FL), 3rd Ed.

17. Jing X, Wang X, Sun X (2007) Broadband Acoustic liner based on the mechanism ofmultiple cavity resonance. AIAA J 45:2429–2437.

18. Yang M, Sheng P (2017) Sound absorption structures: From porous media to acousticmetamaterials. Annu Rev Mater Res 47:83–114.

19. COMSOL (2012) Structural Mechanics Module User’s Guide (COMSOL Inc., Stockholm),Version 4.3.

20. Ma G, Yang M, Xiao S, Yang Z, Sheng P (2014) Acoustic metasurface with hybridresonances. Nat Mater 13:873–878.

21. Roeder KD (1963) Echoes of ultrasonic pulses from flying moths. Biol Bull 124:200–210.

22. Dinwiddie A, et al. (2014) Dynamics of F-actin prefigure the structure of butterflywing scales. Dev Biol 392:404–418.

23. Kelly P, (2018) An introduction to solid mechanics, online book. Available at homepages.engineering.auckland.ac.nz/∼pkel015/SolidMechanicsBooks/Part_I/index.html. AccessedFebruary 1, 2018.

24. Webb DC, Kormi K, Al-Hassani STS (1995) Use of FEM in performance assessment ofperforated plates subject to general loading conditions. Int J Pressure Vessels Piping64:137–152.

25. Slot T (1972) Stress Analysis of Thick Perforated Plates (TECHNOMIC Publishing Co.,New York).

26. Mishra N, Krishna B, Singh R, and Das K (2017) Evaluation of effective elastic, pie-zoelectric, and dielectric properties of SU8/ZnO nanocomposite for vertically in-tegrated nanogenerators using finite element method. J Nanomater 2017:1924651.

27. Vincent JFV, Wegst UGK (2004) Design and mechanical properties of insect cuticle.Arthropod Struct Dev 33:187–199.

28. Pierce AD (1989) Acoustics-An Introduction to Its Physical Principles and Applications(Acoustical Society of America, New York).

Shen et al. PNAS | November 27, 2018 | vol. 115 | no. 48 | 12205

BIOPH

YSICSAND

COMPU

TATIONALBIOLO

GY