1
Viscoelas ic S uc u al Damping Enables B oadband Low-
F equency Sound Abso p ion
Yanlin Zhanga,b, Junyin Lib,c, Qiongying Wud, Ma co Amabilib, Diego Misse onie, and Hanqing
Jiangb,c, *
aSchool o Ma e ials Science and Enginee ing, Zhejiang Uni e si y, Hangzhou 310027, China
bSchool o Enginee ing, Wes lake Uni e si y, Hangzhou, Zhejiang 310030, China
cWes lake Ins i u e o Ad anced S udy, Hangzhou, Zhejiang 310024, China
dZhejiang Sanlux Rubbe Co. L d., Shaoxing, Zhejiang 312031, China
eUni e si y o T en o, 38123 T en o, I aly
Resea ch Cen e o Indus ies o he Fu u e, Wes lake Uni e si y, Hangzhou, Zhejiang 310030,
China
*Email: [email p o ec ed]
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Abs ac
Low- equency sound abso p ion has adi ionally elied on ai - esonan s uc u es, such as
Helmhol z esona o s, which a e made o s i ma e ials ha unde go negligible de o ma ion. In
hese sys ems, ene gy dissipa ion a ises p ima ily om ai mo ion and he mal- iscous e ec s,
esul ing in inhe en ly na owband pe o mance and bulky, complex designs o b oadband
abso p ion. He e, we p esen ed a composi e acous ic me ama e ial ha eplaces he high-s i ness
neck o a Helmhol z esona o wi h a so , iscoelas ic cylind ical shell. This s uc u al
modi ica ion enables ma e ial de o ma ion and shi s he dominan ene gy dissipa ion mechanism
om ai esonance o in insic iscoelas ic damping. A single uni achie es o e 97% abso p ion
ac oss a b oad low- equency ange (227–329 Hz) wi h deep-subwa eleng h hickness (λ/15 a
227 Hz). We de eloped a disc e ized impedance model ha quan i a i ely links ma e ial p ope ies
and geome y o abso p ion beha io . Ou esul s es ablished a ma e ials-cen e ed design pa adigm
in which bo h ma e ial selec ion and geome y se e as coequal, unable pa ame e s o compac ,
b oadband low- equency sound con ol.
Keywo ds: Acous ic me ama e ials; Low equency; B oadband abso p ion; Viscoelas ic damping;
S uc u al ib a ions
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Signi icance
Con en ional acous ic me ama e ials ely on ai esonance and geome ically complex, high-
s i ness s uc u es o achie e low- equency abso p ion, ypically leading o negligible s uc u al
de o ma ion. These app oaches a e inhe en ly na owband. We p esen a composi e acous ic
me ama e ial ha eplaces he high-s i ness neck o a Helmhol z esona o wi h a so , iscoelas ic
shell, shi ing he dominan ene gy dissipa ion mechanism om ai -based esonance o s uc u al
damping go e ned by ma e ial iscoelas ici y. This ansi ion enables o e 97% abso p ion ac oss
a b oad low- equency band. A disc e ized impedance model links abso p ion pe o mance o
ma e ial and geome ic pa ame e s, es ablishing a p edic i e amewo k. This ma e ials-cen e ed
app oach e ames low- equency acous ics by embedding damping in o he ma e ial i sel ,
demons a ing ha bo h ma e ial selec ion and geome y se e as coequal ing edien s o acous ic
pe o mance.
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In oduc ion
Abso bing ai bo ne sounds, pa icula ly a low equencies, is essen ial o con olling noise
pollu ion and imp o ing acous ic en i onmen s (1-3). Con en ional po ous ma e ials achie e
e icien abso p ion only when hei hickness is compa able o a qua e o he acous ic wa eleng h
(4, 5), which becomes imp ac ically la ge a low equencies. In esponse, acous ic me ama e ials
ha e a ac ed conside able a en ion o e he pas decade o enabling subwa eleng h abso p ion
ia enginee ed esonan s uc u es (6-21). Example designs o hese me ama e ials include
Helmhol z esona o s (Fig. 1A) (13, 18, 22) ha ely on ai -column esonance wi hin a high-
s i ness neck whose de o ma ion is negligible, Fab y-Pé o channels (15, 19, 21), and
mic ope o a ed panels (20, 23, 24), which dissipa e sound ene gy h ough ai mo ion and
associa ed he mal- iscous e ec s.
Howe e , hese ai - esonan mechanisms emain inhe en ly e ec i e wi hin a na ow equency
band. To b oaden he abso p ion bandwid h, many ecen e o s ha e combined he e ogeneous
esona o s o a ying scales, o en a he cos o added complexi y and olume (15-21). A smalle
numbe o s udies ha e looked beyond ai esonance and explo ed he ole o lexible s uc u es.
Memb ane abso be s, o example, can enhance localized ene gy densi y and imp o e dissipa ion
(25, 26), bu hey equi e p ecise con ol o e memb ane ension, making hem en i onmen ally
sensi i e, p one o ins abili y, and di icul o une (27). O he app oaches include coupling lexible
ma e ials wi h open-ai en i onmen s (28-30) o eplacing he inne walls wi h so ma e ials (31-
33), aking ad an age o combined s uc u al-acous ic modes o inc ease damping. Howe e , hese
e o s o en ely on pa ame ic s udies ia simula ions o expe imen s, wi h limi ed sys ema ic
heo e ical amewo ks. In mos cases, abso p ion is s ill p ima ily d i en by ai mo emen a he
han he ma e ial i sel , necessi a ing complex mul i- esonan assemblies o b oadband
pe o mance.
This esea ch gap leads o an oppo uni y. So ma e ials, such as elas ome s and gels, o e
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dis inc i e ad an ages o acous ic applica ions, including low elas ic modulus ( anging om a
ew kPa o some MPa), high in insic damping (loss angen an δ ≈ 0.1–1.0 o highe ), and la ge
de o mabili y (34, 35). These p ope ies a e a ely used in acous ic sys ems. S ill, hey o e a
undamen ally di e en ou e o low- equency sound abso p ion: dissipa ing ene gy h ough
ma e ial de o ma ion, a he han mainly elying on ai low. Achie ing his, howe e , equi es
mo e han a ma e ials swap. I calls o an ad anced heo e ical app oach. This heo y mus ea
ma e ial dissipa ion, s uc u al dynamics, and acous ic coupling as coequal componen s in he
abso be ’s pe o mance.
In his s udy, we in oduced a composi e acous ic me ama e ial ha eplaces he high-s i ness neck
o a adi ional Helmhol z esona o (Fig. 1A) wi h a so , iscoelas ic cylind ical shell embedded
wi hin a sealed ca i y (Fig. 1F). This simple s uc u al change undamen ally al e s he ene gy
dissipa ion mechanism, om ai -based esonance o s uc u al damping go e ned by he ma e ial’s
in insic iscoelas ici y. As a esul , a single uni achie es o e 97% abso p ion ac oss a b oad low-
equency ange (227–329 Hz), wi h a deep-subwa eleng h hickness (λ/15 a 227 Hz).
Fu he mo e, o unde s and and op imize his beha io , we de eloped a disc e ized impedance
model ha links he ma e ial p ope ies (i.e., elas ic modulus, loss ac o , densi y) and geome y o
abso p ion pe o mance. We demons a ed ha his model no only cap u es he obse ed beha io
bu also se es as a p edic i e ool o uning abso p ion bands h ough ma e ial and s uc u al
design. Mo e b oadly, ou app oach e ames low- equency acous ic design, namely, a he han
elying on esonance s acking o complex a chi ec u es, we u n o he ma e ial i sel as he
dissipa i e medium. By in eg a ing iscoelas ic damping in o he co e a chi ec u e, we unlock a
new class o compac , unable, and b oadband low- equency sound abso be s, in which ma e ial
choice, no jus geome y, de ines pe o mance.
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Resul s and Discussion
Rigid s. Flexible-Neck Helmhol z Resona o
T adi ional acous ic me ama e ial abso be s o en assume ha he s uc u al componen s,
pa icula ly he esona o neck, a e made o ma e ials wi h su icien ly high elas ic moduli, so ha
hei de o ma ion unde acous ic exci a ion is negligible. We e e o his class o sys ems as Rigid-
Neck Helmhol z esona o s (RN-Helmhol z esona o s) h oughou his wo k. As illus a ed in
Fig. 1A, an RN-Helmhol z esona o ea u es a high-s i ness neck (Young’s modulus E = 2,650
MPa, densi y
s = 1,150 kg/m³, loss ac o
h
= 0.01) embedded in a sealed ca i y. Inciden sound
wa es p opaga e h ough he neck (indica ed by whi e a ows), exci ing esonance in he enclosed
ai column. Fini e elemen simula ions ia COMSOL Mul iphysics (see SI Appendix, Fig. S16 o
de ails) we e conduc ed o s udy he coupled acous ic-s uc u al esponse o his sys em. As shown
in Fig. 1B, he esul ing ai pa icle eloci y dis ibu ion is concen a ed wi hin he neck egion.
Thus, he ene gy is p ima ily dissipa ed h ough he mal- iscous ic ion a he ai - igid neck
bounda ies, while he displacemen o he neck is e ec i ely anishing (Fig. 1C). This mechanism
con ines signi ican abso p ion (de ined by abso p ion coe icien
a
> 0.8) o a na ow 18 Hz
bandwid h (141-159 Hz) cen e ed a he ai column’s esonance equency (150 Hz,
a
= 0.999)
(Fig. 1D). To b oaden he abso p ion bandwid h, mul iple RN-Helmhol z esona o s o a ying
dimensions (e.g., di e en ca i y olumes and neck leng hs) a e ine i ably combined (see SI
Appendix, Fig. S17 o de ailed geome ies). The esul ing mul i- esona o s uc u e (Fig. 1E)
achie es signi ican abso p ion ac oss a 210 Hz bandwid h (300–510 Hz), ea u ing i e dis inc
abso p ion peaks (quan i a i e da a in SI Appendix, Fig. S17). Consequen ly, his mul i- esona o
s a egy inc eases s uc u al complexi y and poses challenges o ab ica ion, uning, and
in eg a ion.
In ou wo k, we add ess hese limi a ions by eplacing he high-s i ness neck wi h a de o mable,
so cylind ical shell (Young’s modulus E = 120 kPa, densi y
s = 1,120 kg/m3, loss ac o
h
= 0.4)
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(Fig. 1F). We e e o his con igu a ion as he Flexible-Neck Helmhol z esona o (FN-Helmhol z
esona o ). Fini e elemen simula ions ia COMSOL Mul iphysics (see SI Appendix, Fig. S18 o
de ails) we e conduc ed o s udy he coupled acous ic-s uc u al esponse o his sys em. In his
con igu a ion, inciden sound wa es p opaga e h ough he lexible neck egion and also lead he
so shell o ib a e (indica ed by blue a ows). On one hand, ai -column esonance wi hin he neck
con inues o con ibu e o ene gy dissipa ion (Fig. 1G). On he o he hand, he so shell unde goes
subs an ial de o ma ion (Fig. 1H), wi h displacemen magni udes h ee o de s o magni ude highe
han hose o i s high-s i ness coun e pa (Fig. 1C), enabling e icien ene gy con e sion h ough
in insic iscoelas ic damping. This shi in ene gy dissipa ion, om p edominan ly ai -media ed
losses o iscoelas ic s uc u al ib a ion, ma ks a undamen al depa u e om con en ional
Helmhol z esona o beha io . As a esul , he FN-Helmhol z esona o achie es bo h b oadband
and high-e iciency sound abso p ion using a single uni , cha ac e ized by dual abso p ion peaks
a 320 Hz (
a
= 0.974) and 460 Hz (
a
= 0.967) and a signi ican abso p ion wi h an adequa e
bandwid h spanning 215 Hz (294-509 Hz) (Fig. 1I). Compa ed o b oadband abso be s cons uc ed
om a ays o RN-Helmhol z esona o s (yellow line in Fig. 1J, same as in Fig. 1E), his single-
uni pe o mance (blue line in Fig. 1J, same as in Fig. 1I) achie es a sligh ly wide e ec i e
bandwid h (215 Hz s. 210 Hz), while elimina ing he need o complex a ays o mul i- esona o
sys ems. The educ ion in s uc u al complexi y, enabled by his ma e ials-cen e ed design,
enhances manu ac u abili y and es ablishes a compac , high-pe o mance amewo k o low-
equency acous ic abso p ion.
Expe imen al and Nume ical Valida ion o Vib a ion-Induced Sound Abso p ion
To expe imen ally e i y he simula ion-based indings p esen ed in he p e ious sec ion, we
ab ica ed an FN-Helmhol z esona o , consis ing o a so cylind ical shell and a igid ou e
suppo s uc u e (Fig. 2A). The so shell was made om Eco lex-30 silicone ubbe (Smoo h-On
Inc., Young’s modulus = 90 kPa), and he ou e shell was 3D-p in ed using a pho osensi i e esin
wi h a Young’s modulus o 2,650 MPa. The de ailed geome y is p o ided in he cap ion o Fig.
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2A, and i s ab ica ion p ocess is ou lined in SI Appendix, Fig. S13. Dynamic mechanical analysis
(DMA 850, TA Ins umen s) was pe o med in comp ession mode o cha ac e ize he iscoelas ic
beha io o Eco lex-30. O e he es ing ange, i s s o age modulus E and loss ac o
h
app oach
app oxima ely 148.5 kPa and 0.2, espec i ely, a high equency (> 181 Hz) (Fig. 2B), as
suppo ed by ime- empe a u e supe posi ion (TTS) expe imen s (36), which ex apola ed he da a
up o 596 Hz (see SI Appendix, Fig. S7 o de ails).
No mal-incidence sound abso p ion coe icien s we e measu ed using an acous ic impedance ube
wi h an inne diame e o 29 mm (se up shown in SI Appendix, Fig. S14). The ab ica ed FN-
Helmhol z esona o exhibi ed b oadband abso p ion pe o mance, ea u ing wo p ominen peaks
a 330 Hz (
a
= 0.978) and 500 Hz (
a
= 0.996). Excep o a dip in he mid- equency ange a ound
400 Hz (
a
= 0.677), he abso p ion coe icien emained abo e 0.8 o e much o he ange om
307 Hz o 535 Hz (Fig. 2C, blue). We also pe o med coupled acous ic–s uc u al simula ions
using COMSOL Mul iphysics (see SI Appendix, Fig. S19 o de ails) o compa e wi h he
expe imen al esul s, using he iden ical geome y and ma e ial p ope ies as hose o he
expe imen . The simula ed abso p ion spec um (Fig. 2C, o ange) closely ma ched he
expe imen al da a, con i ming he accu acy o he model and he b oadband pe o mance o he
FN-Helmhol z esona o .
To elucida e he unde lying sound abso p ion mechanism, we measu ed he su ace ib a ion
displacemen o he cylind ical shell using a lase Dopple ib ome e (Poly ec PSV-500). Scans
o displacemen ampli ude and phase we e conduc ed along h ee axial lines loca ed a
ci cum e en ial angles o 0°, 120°, and 240° on he shell su ace (see SI Appendix, Fig. S15 o
de ails). The esul s e ealed nea ly iden ical displacemen p o iles ac oss he h ee di ec ions (SI
Appendix, Figs. S15D-G), indica ing ha he shell ib a es in an axisymme ic mode unde
acous ic exci a ion. A e aged expe imen al displacemen da a we e compa ed wi h simula ion
esul s, showing good ag eemen in bo h ampli ude and phase (Figs. 2D–G). In addi ion, he
simula ions di ec ly isualized he axisymme ic ib a ion modes a wo ep esen a i e abso p ion
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peaks (330 Hz and 500 Hz) (Figs. 2H and 2I), wi h he displacemen p o iles ma ching well wi h
he magni ude cu es in Figs. 2D and 2E, u he con i ming he mechanism.
Calcula ed dis ibu ions o dissipa ed powe densi y (W/m³) a 330 Hz and 500 Hz (Figs. 2J and
2K) show ha mos o he ene gy dissipa ion occu s in he silicone ubbe shell. We hen in eg a ed
he powe densi y o e he ubbe and ai egions sepa a ely o ob ain he o al dissipa ed powe
(W) in each domain. The equency-dependen esul s o bo h mechanisms a e compa ed in
Fig. 2L. No ably, he wo p ominen peaks in iscoelas ic damping ma ch well wi h he sound
abso p ion peaks. This co ela ion con i ms ha he dominan ene gy loss in he composi e
abso be comes om he ib a ion-induced damping o he so ubbe , wi h a seconda y
con ibu ion om ai - ela ed losses.
Equi alen Ci cui Analysis o Acous ic–S uc u al Coupling
To quan i a i ely analyze he sound-s uc u e in e ac ion mechanism, we de eloped a disc e ized
equi alen impedance model. As shown in Fig. 3A, his model disc e izes he cylind ical shell and
he enclosed ai column in o n equal-leng h segmen s (leng h = l/n) along he axial di ec ion.
Compa ed o he con igu a ion in Fig. 2A, we omi ed he ou e shell hickness and ocused only
on he ca i y bounda y dimensions (see SI Appendix, Fig. S1 o de ailed geome y). Fo each
segmen , he acous ic impedance o he ai column, 𝑍!(𝑥"), and he equi alen acous ic impedance
o he lexible cylind ical shell, 𝑍#(𝑥"), a e connec ed in pa allel, whe e 𝑥"="
$𝑙 ep esen s he
posi ion o he i h (𝑖=1,2,,…,𝑛) segmen along he shell axis. This se up cap u es he wo p ima y
pa hs o ene gy dissipa ion: sound a eling h ough he ai column and s uc u al ib a ions wi hin
he shell. In addi ion, he enclosing ca i y in oduces a sha ed acous ic eac ance 𝑍%, which is
connec ed in se ies wi h he coupled impedance o he luid-s uc u e sys em. Figu e 3B illus a es
he comple e acous ic impedance ci cui diag am. The o al acous ic esponse o he sys em is
he e o e go e ned by he combined e ec s o 𝑍!(𝑥"), 𝑍#(𝑥"), and 𝑍%. De ailed de i a ions o
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peak la gely unchanged (Fig. 4J). This decoupled in luence allows o independen ine- uning o
he wo abso p ion peaks.
By join ly uning ma e ial p ope ies (E, 𝜂, 𝜌2) and geome ic pa ame e s (a, l, ), along wi h
dynamic phase compensa ion h ough he ca i y olume V, he sys em’s acous ic impedance can
be syne gis ically op imized o achie e enhanced and highly cus omizable pe o mance ac oss
a ge equency anges.
In con as , inc easing he ca i y olume V alone, wi hou coo dina ed adjus men s o o he
pa ame e s, can shi he abso p ion spec um owa d lowe equencies bu may deg ade
pe o mance due o impedance misma ch (Fig. 4K). Likewise, simply enla ging he c oss-sec ional
a ea S o he impedance ube (modula ed by ube diame e d in Fig. 4L) can wo sen impedance
misma ch and educe abso p ion e iciency, as he abso be mus dissipa e ene gy o e a la ge
a ea. These esul s unde sco e he impo ance o sys em-le el op imiza ion: isola ed pa ame e
changes a e insu icien o main aining impedance balance and achie ing s ong, b oadband
abso p ion.
Expe imen al Implemen a ion: Ma e ial Op imiza ion and Cus omized Abso be
Designs
To alida e he p edic i e capabili y o he heo e ical model, we expe imen ally op imized he
iscoelas ic p ope ies o silicone ubbe and adjus ed geome ic pa ame e s o cus omized low-
equency abso p ion. Speci ically, we selec ed a so e silicone ubbe (Eco lex-20, Smoo h-On
Inc.) as he base ma e ial. By al e ing he p epolyme mixing a io (A:B = 1:2 by mass) and
in oducing ungs en (W) powde (pa icle size <1 µm) as a densi y-enhancing ille , we p epa ed
wo ma e ial a ian s: (1) a low-modulus, high-damping ubbe (Eco lex-20, A:B = 1:2) and (2) a
densi y-enhanced composi e (Eco lex-20 + W, A:B:W = 1:2:1.2).
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Dynamic mechanical analysis (DMA) in comp ession mode showed ha bo h ma e ials exhibi ed
equency-dependen iscoelas ic beha io (Figs. 5A and 5B). The s o age modulus inc eased wi h
equency and s abilized a app oxima ely 111 kPa, ep esen ing a 25.3% educ ion compa ed o
Eco lex-30’s 148.5 kPa. Meanwhile, he loss ac o ose signi ican ly o 0.385, ma king a 92.5%
inc ease o e Eco lex-30’s alue o 0.2. Impo an ly, he addi ion o W powde had minimal e ec
on he s o age modulus and loss ac o bu ma kedly inc eased he ma e ial densi y om
1,070 kg/m3 o 1,420 kg/m3 (Fig. 5C). This abili y o independen ly une iscoelas ici y and
densi y unde sco es he lexibili y o ma e ial design in achie ing a ge ed acous ic pe o mance.
Th ee abso be se s we e de eloped o exploi he imp o ed ma e ial p ope ies:
Se 1 used Eco lex-20 and coo dina ed shell dimensions (inne adius a = 4 mm, leng h l = 41.5 mm,
hickness = 1 mm), pai ed wi h a compensa o y ca i y heigh (h = 63 mm) o accommoda e he
ma e ial’s lowe s i ness. Expe imen al measu emen s (Fig. 5D, blue) demons a ed b oadband
abso p ion om 260–470 Hz (de ined by
a
> 0.83), wi h an a e age abso p ion coe icien o
0.912 (compu ed as he a i hme ic mean o e his ange). This downwa d shi om he o iginal
design (307–535 Hz, Fig. 2C) alida es he model’s sensi i i y o modulus educ ion (Fig. 4E).
Se 2 adop ed he densi y-enhanced Eco lex-20 + W composi e, while keeping he exac shell
dimensions as Se 1. To compensa e o he highe densi y, he ca i y heigh was inc eased o
h = 72 mm. As a esul , he abso p ion band (de ined by
a
> 0.9) shi ed u he o 248–405 Hz
(Fig. 5E, blue), wi h an a e age coe icien o 0.949, aligning wi h he p edic ed densi y-dependen
shi (Fig. 4F).
Se 3 also employed he Eco lex-20 + W composi e bu enla ged he shell dimensions (a = 4.5 mm,
l = 50 mm, = 1.5 mm) and expanded he ca i y heigh o h = 100 mm. This se up achie ed nea -
pe ec abso p ion om 227–329 Hz (de ined by
a
> 0.97) (Fig. 5F, blue), wi h an a e age
coe icien o 0.986, while main aining an o e all hickness o jus 1/15 o he wa eleng h a
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227 Hz.
Expe imen al esul s om all h ee se s showed excellen ag eemen wi h nume ical simula ions
(Figs. 5D–F, o ange cu es). This consis ency was u he suppo ed by he alida ion o
no malized acous ic esis ance and eac ance, p esen ed in SI Appendix, Fig. S21, ein o cing he
obus ness o he heo e ical model in cap u ing he ela ionship be ween ma e ial p ope ies,
s uc u al pa ame e s, and acous ic pe o mance. Toge he , hese indings demons a e a closed-
loop wo k low ha seamlessly in eg a es heo e ical modeling, ma e ial o mula ion, s uc u al
design, and expe imen al alida ion.
Discussion
This wo k in oduces a new s a egy o achie ing e icien low- equency and b oadband sound
abso p ion by e hinking he ole o s uc u al ma e ials in acous ic me ama e ials. T adi ional
Helmhol z esona o s ely on igid componen s, wi h ene gy loss coming mainly om ai mo ion
and he mo- iscous e ec s, which limi s hem o na ow equency bands. In con as , ou design
eplaces he igid neck wi h a so , iscoelas ic cylind ical shell, allowing he s uc u e i sel o
pa icipa e in he ene gy dissipa ion p ocess h ough ma e ial damping and s uc u al ib a ion.
This shi enables a single esona o uni o achie e o e 97% abso p ion ac oss a unable low-
equency ange (e.g., 227–329 Hz), wi h a hickness as small as 1/15 o he wa eleng h a he
lowes equency.
The b oadband pe o mance a ises om he combined e ec s o he shell’s iscoelas ic beha io
and i s dynamic in e ac ion wi h he su ounding ai . We de eloped a disc e ized impedance model
o desc ibe his coupled sys em, which enables us o di ec ly link ma e ial p ope ies (such as
modulus, damping, and densi y) and geome ic pa ame e s (including shell adius, leng h, and
hickness) o acous ic pe o mance. No ably, shell leng h and wall hickness o e decoupled
con ol o e he posi ion o abso p ion peaks, while ma e ial p ope ies can be independen ly uned
19
using mixing a ios o ille s. This p o ides he sys em wi h a high deg ee o lexibili y in mee ing
a ious design goals.
The concep o embedding he damping mechanism in o he s uc u e i sel , a he han elying
solely on ai esonance, opens he doo o mo e compac , ligh weigh , e icien , and unable
abso be s. The app oach also o e s a p ac ical pa h o wa d, u ilizing common so ma e ials such
as silicone ubbe and s aigh o wa d ab ica ion me hods. Mo e b oadly, i sugges s ha he
con en ional eliance on igid s uc u es and complex mul i- esona o assemblies o b oadband
pe o mance may no be necessa y. Ins ead, a ma e ials-cen e ed amewo k, whe e iscoelas ic
p ope ies a e delibe a ely enginee ed and ma ched wi h s uc u al geome y, can achie e high-
pe o mance abso p ion in a simple , mo e compac , and ligh weigh o m.
Looking ahead, his design app oach could be ex ended o sys ems wi h mul iple deg ees o
eedom, laye ed o hie a chical s uc u es, o ac i e unabili y h ough ex e nal s imuli like
empe a u e o elec ic ields. The same p inciple o s uc u e-enabled dissipa ion also holds
p omise in unde wa e acous ics. Fu he mo e, his concep also has po en ial applica ions beyond
acous ic abso p ion, such as in ib a ion damping, wa e il e ing, o p og ammable mechanical
me ama e ials. By combining ideas om ma e ial science, mechanics, and wa e physics, his wo k
lays he g oundwo k o a new class o lexible, e icien sound-abso bing ma e ials sui ed o he
needs o dense u ban a eas, anspo a ion sys ems, and indus ial noise con ol.
20
Acknowledgemen
We hank he Resea ch Cen e o Indus ies o he Fu u e (RCIF) a Wes lake Uni e si y, Zhejiang
Sanlux Rubbe Co. l d., and he Wes lake Educa ion Founda ion o hei suppo o his wo k. H.J.
acknowledges suppo om he Na ional Na u al Science Founda ions o China (G an s 12350003).
D.M acknowledges inancial suppo om he Eu opean Union, ERC g an HE GA 101086644 S-
FOAM. (Views and opinions exp essed a e howe e hose o he au ho (s) only and do no
necessa ily e lec hose o he Eu opean Union o he Eu opean Resea ch Council Execu i e
Agency. Nei he he Eu opean Union no he g an ing au ho i y can be held esponsible o hem),
and he Wes lake Fellows p og am. We hank Zhen Yang om Ins umen a ion and Se ice Cen e
o Physical Sciences a Wes lake Uni e si y o he assis ance wi h he DMA es and da a
in e p e a ion.
Au ho con ibu ions
Y.Z., Q.W. and H.J. designed esea ch; Y.Z. and J.L. pe o med esea ch; Y.Z., J.L., D.M., and H.J.
analyzed da a; and Y.Z., D.M., M.A., and H.J. w o e he pape .
Da a a ailabili y
Sou ce da a o all main and supplemen a y igu es a e p o ided in he accompanying Excel ile.
MATLAB code o calcula ing he sound abso p ion coe icien (including Full- ecu si e and
Closed- o m analy ical models) is a ailable in he Gi Hub eposi o y
(h ps://gi hub.com/yanlinZhang865/Da a-a ailabili y).
21
Ma e ials and Me hods
Ma e ial P epa a ion and Fab ica ion
Flexible cylind ical shells we e ab ica ed using silicone ubbe s (Eco lex-30 and Eco lex-20,
Smoo h-On Inc.). Eco lex-30 was p epa ed by mixing p epolyme s (Pa A:B) a a 1:1 mass a io,
while Eco lex-20 was p epa ed a a 1:2 a io (Pa A:B) o educe s i ness and enhanced damping.
Fo densi y modi ica ion, ungs en (W) powde (pa icle size <1 µm) was added o Eco lex-20
p epolyme s a a mass a io o A:B:W = 1:2:1.2. The mixed p epolyme s we e hen cas in o high-
p ecision CNC-machined aluminum molds (see SI Appendix, Figs. S13A-D o de ails) and cu ed
a 25 ℃ o 4 hou s. A e cu ing, he silicone cylind ical shell was demolded and immed o i s
inal dimensions (see SI Appendix, Fig. S13E). The ab ica ed shells exhibi ed excellen
dimensional ideli y, wi h de ia ions o less han 0.05 mm om he design speci ica ions. The
high-s i ness ou e shell was ab ica ed using 3D p in ing (pho osensi i e esin, C-UV9400A,
Young’s modulus: 2,650 MPa). The so silicone ubbe shell was hen embedded in o he esin
shell and bonded using silicone adhesi e (Sil-Poxy, Smoo h-On Inc.), o ming a comple e
composi e abso be (see SI Appendix, Fig. S13F). Fo isual cla i y, in Fig. 2A, Figs. 5D-F and
SI Appendix, Fig. S13F, we depic he ou e shells as anspa en ac ylic (PMMA) o e eal he
in e nal s uc u e, al hough he ac ual ab ica ed shells we e opaque.
Expe imen al Cha ac e iza ion
DMA.
Viscoelas ic p ope ies (s o age modulus and loss ac o ) we e measu ed using a TA Ins umen s
DMA 850 analyze in comp ession mode (expe imen al se up and sample geome y shown in SI
Appendix, Fig. S7). Squa e samples (18 × 18 × 6 mm³) we e es ed unde equency sweeps (1–
181 Hz, 10 Hz in e als) a 0.5% s ain and 25°C. The dimensions o he squa e samples we e
chosen o minimize da a luc ua ions a highe equencies ( ＞100 Hz). Compa isons wi h
al e na i e es ing me hods ( ime- empe a u e supe posi ion and can ile e beam esonance
22
me hod) a e discussed in de ail in SI Appendix, Sec ion 4.
Sound Abso p ion Measu emen .
No mal-incidence sound abso p ion coe icien s we e measu ed using an impedance ube
(AWA6290T Hangzhou Aihua Ins umen s Co., inne diame e : d = 29 mm) ollowing ISO 10534-
2 s anda ds. The expe imen al se up is shown in SI Appendix, Fig. S14. A 110 dB whi e noise
signal (50-6300 Hz) se ed as he exci a ion sound sou ce. Acous ic esponses we e eco ded by
wo mic ophones, and abso p ion coe icien s we e calcula ed using he ans e unc ion me hod
(40). Due o he low s i ness o he lexible cylind ical shell, ho izon al sample o ien a ion induced
sligh g a i a ional sagging (see SI Appendix, Fig. S14B). To elimina e his e ec , he impedance
ube was eo ien ed e ically (SI Appendix, Fig. S14C), minimizing s uc u al de lec ion du ing
es ing. Compa a i e analysis o abso p ion spec a om bo h ho izon al and e ical
con igu a ions (SI Appendix, Fig. S14D) showed nea ly iden ical esul s wi h o e lapping cu es,
con i ming ha g a i a ional sagging negligibly impac ed acous ic pe o mance unde he es ed
sample condi ions.
Vib a ion Displacemen Measu emen .
Su ace ib a ion displacemen o he cylind ical shell unde acous ic exci a ion a 330 Hz and 500
Hz was measu ed using a Poly ec-PSV-500 lase Dopple ib ome e (expe imen al se up shown
in SI Appendix, Fig. S15A). To ensu e unobs uc ed lase access o he shell su ace, he sample
was enclosed in a anspa en ac ylic enclosu e wi h a ec angula c oss-sec ion (SI Appendix, Fig.
S15B). This design minimized spu ious lase e lec ions om he cu ed cylind ical su ace. The
ec angula enclosed ca i y olume was designed o ma ch he cylind ical ca i y olume designed
in Fig. 2A. Axial ib a ion p o iles we e scanned along h ee axial lines a angula posi ions o 0°,
120°, and 240° (SI Appendix, Fig. S15C) wi h a spa ial esolu ion o 0.2 mm (along x axis) o
e alua e ib a ion symme y. The measu ed displacemen ampli udes and phase angles a bo h
23
exci a ion equencies (330 Hz and 500 Hz) exhibi ed consis en spa ial dis ibu ions ac oss all
angula o ien a ions (SI Appendix, Figs. S15D-G), con i ming he axisymme ic ib a ion
beha io o he cylind ical shell.
Nume ical Simula ions
Coupled acous ic-s uc u al simula ions we e pe o med in COMSOL Mul iphysics ( e sion 6.1)
using he P essu e Acous ic, The mo- iscous Acous ics and Solid Mechanics modules (SI
Appendix, Fig. S19B). The esin shell was modeled as igid, while he ai domain was ea ed as
a he mos- iscous luid. All silicone ubbe s we e assumed o be nea ly incomp essible (Poisson’s
a io: 0.49). A pe ec ly ma ched laye (PML) bounda y condi ion was applied a he ube ou le o
elimina e wa e e lec ions. The compu a ional domain was disc e ized using hexahed al meshes,
wi h bounda y laye meshes u he e ined in he mo- iscous egion. Bounda y condi ions
included ixed cons ain s a he silicone- esin in e ace and acous ic-s uc u e coupling a he
solid-ai in e ace.
Viscoelas ic damping in silicone ubbe was quan i ied by in eg a ing he dissipa ed powe densi y
(solid.Qh, W/m³) o e he ubbe domain. Simila ly, he mo- iscous losses in ai domains
( esona o ca i y and adjacen ube sec ion) we e e alua ed by in eg a ing he co esponding
dissipa ion densi y ( a.diss_ o , W/m³) o e luid olumes.
S abili y o Sound Abso p ion Pe o mance
Long- e m S abili y.
The FN-Helmhol z esona o sample (Se 3, Fig. 5F) was s o ed a 25°C ambien condi ions and
e- es ed a e 60 days and 80 days. The abso p ion coe icien , no malized acous ic esis ance, and
no malized acous ic eac ance we e measu ed using he same impedance ube se up. The
compa a i e esul s a e p esen ed in SI Appendix, Fig. S9.
Tempe a u e E ec s.
24
To e alua e empe a u e in luence on Eco lex-30 iscoelas ic p ope ies and acous ic pe o mance
o he FN-Helmhol z esona o , empe a u e-dependen da a om ime- empe a u e supe posi ion
expe imen s we e analyzed. Coupled acous ic-s uc u al simula ions compa ed abso p ion spec a
a 25°C (E = 152 kPa, 𝜂 = 0.19) and -10°C (E = 169 kPa, 𝜂 = 0.22). Resul s (SI Appendix, Fig.
S10) show ela i ely s able b oadband abso p ion ac oss a wide empe a u e ange.
Abso p ion Unde Oblique Incidence.
To assess pe o mance unde non-no mal incidence, abso p ion coe icien s we e calcula ed o
angles o 0°, 30°, and 60° using he ull ecu si e model wi h clamped- ee bounda ies. Fo a
locally eac ing su ace, he abso p ion coe icien 𝛼(𝜃) is calcula ed by modi ying he o mula
(Eq. 9) o accoun o he e ec i e impedance p ojec ion along he di ec ion o p opaga ion (5):
. [12]
whe e 𝜃 is he incidence angle measu ed om he no mal o he su ace. The esul ing spec a
(SI Appendix, Fig. S11) show angle-dependen a ia ions.
Abso p ion Unde High Sound P essu e. To e alua e po en ial in luences o geome ic o
ma e ial nonlinea i y a ele a ed inciden sound le els, he no mal-incidence sound abso p ion
coe icien o he ab ica ed FN-Helmhol z esona o was measu ed unde whi e-noise exci a ion
a 90-129 dB (0.632-56.2 Pa). All cu es a e nea ly iden ical ac oss his ange o inciden sound
p essu es (SI Appendix, Fig. S12), his indica es ha he abso be ope a es linea ly e en a high
exci a ion le els. Mo eo e , simula ions show maximal shell s ain ≤1.44% a 129 dB (SI
Appendix, Table S2), wi hin Eco lex-30's linea egime (<10% s ain) (41). These esul s suppo
he alidi y o he linea model used in ou s udy.
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25
Re e ence
[1] G. Ma, P. Sheng, Acous ic me ama e ials: F om local esonances o b oad ho izons. Sci. Ad .
2, e1501595 (2016).
[2] M. Yang, P. Sheng, Sound abso p ion s uc u es: F om po ous media o acous ic me ama e ials.
Annu. Re . Ma e . Res. 47, 83–114 (2017).
[3] S. Huang, e al., Sound-abso bing ma e ials. Phys. Re . Appl. 20, 010501 (2023).
[4] L. Cao, e al., Po ous ma e ials o sound abso p ion. Compos. Commun. 10, 25–35 (2018).
[5] J. Alla d, N. A alla, P opaga ion o sound in po ous media: Modelling sound abso bing
ma e ials. (John Wiley & Sons, 2009).
[6] S. A. Cumme , J. Ch is ensen, A. Alù, Con olling sound wi h acous ic me ama e ials. Na . Re .
Ma e . 1, 1–13 (2016).
[7] N. Gao, e al., Acous ic me ama e ials o noise educ ion: A e iew. Ad . Ma e . Technol. 7,
2100698 (2022).
[8] S. Qu, P. Sheng, Mic owa e and acous ic abso p ion me ama e ials. Phys. Re . Appl. 17,
047001 (2022).
[9] M. Yang, P. Sheng, Acous ic me ama e ial abso be s: The pa h o comme cializa ion. Appl.
Phys. Le . 122, 26 (2023).
[10] B. M. Assoua , e al., Acous ic me asu aces. Na . Re . Ma e . 3, 460–472 (2018).
[11] Y. Li, B. M. Assoua , Acous ic me asu ace-based pe ec abso be wi h deep subwa eleng h
hickness. Appl. Phys. Le . 108, 6 (2016).
[12] N. Jiménez, e al., Ul a- hin me ama e ial o pe ec and quasi-omnidi ec ional sound
abso p ion. Appl. Phys. Le . 109, 12 (2016).
[13] S. Huang, e al., Acous ic pe ec abso be s ia Helmhol z esona o s wi h embedded ape u es.
J. Acous . Soc. Am. 145, 254–262 (2019).
[14] K. Donda, e al., Ex eme low- equency ul a hin acous ic abso bing me asu ace. Appl. Phys.
32
equency-selec i e 𝜂(𝑓) peaking a 𝑓
* (pu ple). (E-J) Pa ame ic e ec s on 𝛼 (o he pa ame e s ixed;
𝑉 co-adjus ed pe Eq.11 o op imal impedance ma ching): (E) Young’s modulus E; (F) Ma e ial densi y
𝜌&; (G) Shell adius a; (H) Loss ac o 𝜂; (I) Shell leng h l; (J) Shell hickness ; (K-L) Pa ame ic e ec s on
𝛼 wi h isola ed uning: (K) Ca i y olume 𝑉; (L) Impedance ube diame e 𝑑.
Fig. 5. Ma e ial op imiza ion and cus omized abso be designs. (A) F equency-dependen s o age modulus E,
(B) F equency-dependen loss ac o
𝜂
, and (C) Ma e ial densi y
𝜌&
, o Eco lex-30 (b onze), Eco lex-20
(blue) and Eco lex-20 + W (black). (D-F) Expe imen ally measu ed and simula ed acous ic abso p ion
coe icien s o h ee esona o con igu a ions: (D) Se 1 (Eco lex-20; a = 4 mm, l = 41.5 mm, = 1 mm, h =
63 mm). (E) Se 2 (Eco lex-20 + W; a = 4 mm, l = 41.5 mm, = 1 mm, h = 72 mm). (F) Se 3 (Eco lex-20 +
W; a = 4.5 mm, l = 50 mm, = 1.5 mm, h = 100 mm).
a
V
simula ion expe imen simula ion expe imen simula ion expe imen
lh
Eco lex 30
(o iginal)
Eco lex 20
(A:B=1:2)
Eco lex 20
(A:B=1:2 + W)
350 350 350450 450 450550 550 550250 250 250
150 150 150
0 040 4080 80120 120160 160
equency (Hz)
equency (Hz) equency (Hz)
equency (Hz) equency (Hz)
se #1
Eco lex 20
(A:B=1:2), a=4 mm,
l=41.5 mm, =1 mm,
h=63 mm
Eco lex 20 (A:B=1:2) + W,
a=4 mm, l=41.5 mm, =1 mm,
h=72 mm
Eco lex 20 (A:B=1:2) + W,
a=4.5 mm, l=50 mm, =1.5 mm,
h=100 mm
se #2 se #3
0.2
0.05
0.2 600
0.1 300
0.10
0.3 900
0.4 1200
1070 1070
1420
0.15
0.5 1500
0.2 0.2
0.6 0.6 0.6
0.8 0.8 0.8
1.0
0.20
1.0 1.0
0
0 0 0
0 0
0.4 0.4 0.4
abso p ion, αs o age modulus, E
loss ac o , η
densi y, ρ (kg/m3)
abso p ion, α
abso p ion, α
D
A B C
E F
Eco lex 30 (o iginal)
Eco lex 20 (A:B=1:2)
Eco lex 20 (A:B=1:2) + W
260 Hz
248 Hz
227 Hz
470 Hz
405 Hz
329 Hz
