Qianli ChenDepartment of Civil and Environmental

Engineering,

University of Illinois at Urbana Champaign,

Champaign, IL 61801

Ahmed ElbannaDepartment of Civil and Environmental

Engineering,

University of Illinois at Urbana Champaign,

Champaign, IL 61801

Modulating Elastic Band GapStructure in Layered SoftComposites Using SacrificialInterfacesA wide range of engineered and natural composites exhibit a layered architecturewhereby individual building blocks are assembled layer by layer using cohesive interfa-ces. We present a novel mechanism for evolving acoustic band gap structure in a modelsystem of these composites through patterning the microstructure in a way that triggersnonplanar interfacial deformations between the layers as they are stretched. Through thecontrolled deformation and growth of interlayer channels under macroscopic tension, weobserve the emergence of multiple wide band gaps due to Bragg diffraction and local res-onance. We describe these phenomena in details for three example microstructures anddiscuss the implications of our approach for harnessing controlled deformation in modu-lating band gap properties of composite materials. [DOI: 10.1115/1.4034537]

1 Introduction

Layered composites are frequently found as building blocks oftough biological structures [1–4]. Among those, bone is a typicalexample exhibiting remarkable combination of stiffness andtoughness [5]. The basic building block of bone at the microscaleis the mineralized collagen fibril (MCF) in which the soft collagenmatrix is reinforced by stiff hydrated calcium phosphates [6]. TheMCFs are glued together using interfaces of noncollagenous pro-teins with sacrificial bonds and hidden length to form a layeredstructure [7–10]. Another example of a layered structure is sea-shell nacre which endows dramatic improvement of stiffness andtoughness compared to its constituent phases independently [11].The enhancement of toughness in those biological materials ispartially attributed to wavy surfaces and cohesive interactionalong the interfaces between bulk materials [12–14]. These inter-faces are sacrificial in the sense that they represent weak spotswhere energy gets dissipated by progressive accumulation of dam-age while the rest of the composite remains intact and elastic. Thenaturally optimized properties of stiffness, strength, and tough-ness, suggest that there is a link between geometric patterning,sacrificial interfaces, and improved mechanical performance.More recently, it has also been hypothesized that the origin ofthese improved properties, in a composite like bone, for example,may be linked to the ability of generating multiple band gaps [15].

Composite materials with periodic microstructure are capableof generating a band gap structure through which elastic andacoustic wave propagation may be controlled [16–20]. Waveswith frequencies falling into the band gap are barred by the peri-odic structure. The band gap phenomenon in composite materialis caused by to two mechanisms: Bragg diffraction and local reso-nance [21,22]. The Bragg diffraction occurs when the elasticwavelength is comparable to the periodicity of the microstructure.The position and the width of the band gap may be controlled bycareful design of the microstructure materials [23–26]. Moreover,mechanically triggered large deformation [27] and instability-induced interfacial wrinkling [28,29] are found of capacity to tunethe evolution of band gap profile. Local resonance happens whenthe phases of composite material have strong elastic propertiescontrast. Banded frequencies caused by local resonance are

generally lower than those caused by Bragg diffraction [30] andmay exist even in the absence of periodicity and symmetry[21,31]. Designing such band gap structures enables the applica-tion of wave filtering, wave guiding, acoustic mirrors and vibra-tion isolators [32–38].

We showed in an earlier paper that a bone-inspired composite,with patterned heterogeneity, exhibits the capacity of generatingtunable corrugation under externally applied concentric tension[39]. By stacking 1D fibers, with the designed microstructure, toform layered composites, these corrugations collectively lead to theformation of interlayer channels with shapes and sizes tunable bythe level of stretch (Fig. 1). In this paper, we demonstrate an appli-cation of harnessing this design in evolving the elastic band gapstructure in these composites. A unique feature of our design is thedynamic evolution of the structural composition as a function ofstretch and inclusion distribution. In particular, an additional phase,namely voided channels, gradually emerge (disappear) withincreasing (decreasing) stretch despite being absent at zero stretch.

2 Model and Results

We consider layered composites made up of stacks of fibrilsthat are glued along their longer dimension as shown in Fig. 1.We choose fibril thickness h ¼ 30 lm, length of inclusionLi ¼ 75 lm, inclusion thickness t ¼ 7:5 lm, inclusion spacingS ¼ 40 lm, and inclusion eccentricity from center line of fibrilc ¼ 4:5lm [39]. Three examples of inclusion patterns are dis-cussed: (a) nonstaggered aligned inclusions (Fig. 1(a)), (b) stag-gered identical inclusions (Fig. 1(b)), and (c) staggered symmetricinclusions (Fig. 1(c)). An extensive analysis of the effect of geom-etry and material properties on the response of these compositesunder longitudinal (along x-axis in Fig. 1) stretch are reported inour previous paper [39]. In general, the corrugation amplitudeincreases with the stiffness of inclusion and its eccentricity fromthe local tension axis.

The unit cell is discretized using eight-node bi-quadrilateral ele-ments (Q8). We run the basic stretch analysis and band gap calcula-tions in an in-house MATLAB code for finite-deformation elasticity.The dynamic wave propagation verification simulation is run bythe commercial finite-element software ABAQUS. The two phases aremodeled using Neo-Hookean hyperelastic material model. The sili-con inclusion has Young’s modulus of Ei ¼ 170 GPa, Poisson’sratio �i ¼ 0:064, and density qi ¼ 2300 kg=m3. The matrix mate-rial is modeled as polydimethylsiloxane (PDMS) with initial

Contributed by the Applied Mechanics Division of ASME for publication in theJOURNAL OF APPLIED MECHANICS. Manuscript received June 13, 2016; final manuscriptreceived August 23, 2016; published online September 9, 2016. Assoc. Editor:Kyung-Suk Kim.

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Young’s modulus of Em ¼ 750 kPa, Poisson’s ratio of 0.495, anddensity qm ¼ 1000 kg=m3. This combination of materials wereused in our previous experiment [40]. Reduced integration isapplied for quadrature and the nonlinear problem is solved withfull Newton Raphson method. Elastic band structure calculations areexecuted for a unit cell in the deformed configuration, and thus, thedimensions of the Brillouin zone are updated accordingly withCraig–Bampton mode decomposition, and Bloch boundary conditionsare then applied [41]. The formulas for finite element method (FEM)and block wave calculations are summarized in the Appendix.

The interaction between two composite fibrils is simulated by atraction–separation law with equivalent Young’s modulus ofEcoh ¼ 1 kPa, Poisson’s ratio �coh ¼ 0:495, and initial thicknesstcoh ¼ 1lm. Then, we have equivalent normal stiffness Kn ¼Ecohtop=tcoh and shear stiffness Ks ¼ Ecohtop=½2tcohð1þ �cohÞ�,where top is the out-of-plane thickness of fibril. In this paper, we

use the bilinear law (Eq. (A7)) for simplicity but the qualitativenature of the results depends weakly on the specific form andproperties of the cohesive law. To investigate the influence of theinterfaces, the responses of both monolithic (i.e., no interfaces)and layered systems (with interfaces) under uniaxial stretch alongx-axis in Fig. 1 are simulated and compared.

Three monolithic composites with inclusions are modeled andserved as reference groups for comparisons. Unlike the layeredsystem there are no interfaces in the monolithic case. The mono-lithic nature of matrix material stiffens the overall compositeresponse. The Bloch wave analysis reveals that no elastic band-structure behavior is found except in case (c) where a narrowband gap of width 0.008 MHz appears near 0.26 MHz (Fig. 2).

The introduction of interfaces enables more complex deforma-tion patterns and richer elastic band gap structure. In the case ofthe nonstaggered inclusion pattern (case a), the fibrils shrink in

Fig. 1 Representative assembly of (a) composite with nonstaggered aligned inclusionpattern, (b) composite with staggered identical inclusion pattern, and (c) composite withstaggered symmetric inclusion pattern (cohesive interfaces do not exist in monolithicmodels). The corresponding unit cells are shown to the right, while the deformed shapesof the composite under uniaxial stretch along x-axis in Fig. 1 are shown below the unde-formed ones. Overall dimensions are marked in the unit cell in Fig. 1(a). Inclusion dimen-sions and spacing are the same in all cases.

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the transverse direction, within the region of pure PDMS, morethan in the regions where the silicon inclusions exist. In cases (b)and (c), corrugation develops due to the eccentricity of inclusions.Channels, of different shapes and orientations, form as stretchlevel increases because of different stacking patterns. All threecases show deformed wavy surface under uniaxial stretch along x-axis in Fig. 1, and this alters the dispersion property of unit cell.

Unlike in the monolithic cases, the layered composites haveband gaps at the undeformed state and are able to develop richerbehavior after stretch.

2.1 Case (a) Composite With Nonstaggered Inclusions.Figure 3(a) reveals that there exist multiple wide elastic bandgaps at zero strain. As stretch increases, some of the smallerbandgaps disappear and other bandgaps become even wider (Fig.4(a)). To understand the origin of the band gaps in this case, it isilluminating to analyze the eigenmode deformation. For that pur-pose, selected eigenmodes are plotted to the right of the dispersioncurves. Most of these eigenmodes suggest that the deformation ismainly concentrated within soft matrix material representing alocal resonance behavior. Few exceptions exist. For examplemodes D, F, and D’ show distributed deformations where theinclusions also vibrate due to Brag scattering. While in mode F’the deformation is localized in the matrix but the influence of thechannel is apparent: the deformation is concentrated close to thechannel tips. Furthermore, the dispersion lines along the verticalpropagation direction Y � C are almost flat signifying very lowgroup velocities. This is due to the existence of the horizontalcohesive interfaces with low stiffness values that slow down wavepropagation across them. Finally, eigenmodes of the composite inthis inclusion pattern keep their general shape as stretch grows butthe corresponding eigenfrequencies change.

2.2 Case (b) Composite With Identical Staggered Inclu-sions. Figure 3(b) shows examples of the band gap structure inthis case. Fewer band gaps exist compared to case (a) and theymostly arise due to local resonance in the PDMS matrix. Onceagain, the effect of channel development is apparent where defor-mations are banded parallel and normal to their locations. Further-more, the dispersion relations are altered significantly bydeformation, unlike in case (a). This is primarily due to the com-plex growth of the interfacial channels.

2.3 Case (c) Composite With Symmetrically StaggeredInclusions. Figure 3(c) shows examples of the band gap structurein this case. Unlike the previous cases, no band gaps develop atzero strain. Also the dispersion lines along the vertical propaga-tion direction Y � C are not as flat as in cases (a) and (b) above.This is attributed to the development of contact between the layersupon bending. The composite is thus continuous through contactregion and waves may propagate across the horizontal interfacesat the speed allowed by the matrix material. We also note that theeigenmode analysis reveals that the deformation at the boundariesof the band gap is localized near the nonplanar features of thechannels, signifying the influence of the complex evolution of theinterfacial separation.

The band gap evolution as a function of stretch is shown inFig. 4. As previously discussed, multiple band gaps develop asstretch level increases. The composite with nonstaggered inclu-sions (case (a)), shows multiple wide band gaps above 0.13 MHzat all levels of stretch. For case (b), four clusters of band gaps ini-tially exists near frequencies 0.16, 0.23, 0.33, and 0.35 MHz,respectively. As stretch increases, the band gap near 0.16 MHzbecomes wider and moves to higher frequency while the otherthree close. Near stretch levels of 1.15 and 1.17, two band gapsopen at frequency 0.26 and 0.28 MHz, respectively, but they aregenerally smaller than the lower frequency band gap. For case (c),a band gap appears after stretch of 1.05 near 0.2 MHz and opensup to width of 0.04 MHz as the stretch level increases.

The opening and closure of band gaps as a function of deforma-tion may be explained as follows. Due to the hyperelastic natureof the matrix material, the matrix material undergoes stiffnesssoftening as stretch grows. Stiffness changes nonuniformly amongthe bulk constituents with stretch since the matrix experiences themost changes while the silicon inclusion remains almost linearelastic with constant stiffness. The variability of the band gapwidth is due to the competition between stiffness changes as thehyperelastic material is changed and the evolving geometry due tothe nonuniform deformation of the interfaces which lead to com-plex channel shapes and scattering response.

As a verification of band gap behavior, we examine the trans-mission plots for the composite with staggered inclusion pattern(case (b)) as an example. Nine unit cells are connected together inthe longitudinal direction, and periodic boundary condition isapplied to the upper and lower boundaries of those cells to create

Fig. 2 Dispersion plots for three monolithic composites corresponding to the three inclusionpatterns illustrated in Fig. 1 ((a) nonstaggered, (b) staggered identical, and (c) staggered sym-metric) and corresponding Brillouin zones

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a semifinite composite system. We connect one end of the com-posite system to infinite elements to absorb waves and applystretch and vibration at the other end. The finite-element analysisproceeds as follows. We first stretch the composite to the required

level. Then, we apply a small amplitude vibration (less than 1% ofthe stretch amplitude) at the stretched end. We compute the trans-mission ratio as the ratio of the vibration amplitude at the inter-face of the composite with the infinite elements to the applied

Fig. 3 Dispersion relations (blue lines), band gap (gray region), and critical eigen-modesof the three composites fibril systems corresponding to the three inclusion patterns ((a)nonstaggered, (b) staggered identical, and (c) staggered symmetric) at stretch level of1.00/1.01 and 1.20

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vibration amplitude at the stretched end. Two scenarios are pre-sented in Fig. 5, one at stretch of 1.10 where no band gap hasshown up between 0.2 MHz and 0.3 MHz, and one at stretch of1.30, where there are two band gaps near 0.26 MHz and0.28 MHz. From Fig. 5(b), we may observe that for both stretchlevels, shear waves are attenuated above 0.25 MHz. However, forthe longitudinal waves (Fig. 5(a)), the attenuation is most noticea-ble below 0.24 MHz and above 0.29 MHz for the stretch level of1.10 but it is coincident with the band gap region at stretch levelof 1.30. That is, while longitudinal and shear waves may eachattenuate at frequencies outside the band gap, it is only within theband gap that the amplitudes of both types of waves are signifi-cantly reduced simultaneously.

3 Discussion

We have numerically demonstrated the capability of modulat-ing band gap structure in soft composites using sacrificial interfa-ces. While several approaches have been used previously toachieve this purpose, including designing of hierarchical

composite structure [23,37], triggered instabilities in monolithicperiodic material [27,42], and instability-induced interfacial wrin-kling [28,29], this is the first time, to the best of our knowledge,that sacrificial interfaces with controlled damage behavior is usedto tune the band gap behavior.

The introduction of interfaces to a monolithic compositeenriches the deformability of the system and provides a mean tocreate and alter the band gap structure. The interfacial deforma-tion along composite fibrils is controlled by the inclusion pattern.Nonstaggered (case a) and staggered symmetrical (case c) inclu-sion patterns result in the formation of parallel horizontal array ofholes in aligned and staggered manners, respectively, while thestaggered inclusion pattern (case b) leads to oblique orientation ofholes. In all cases, the composite geometry is evolving gradually.The layered nature of the composite makes the group velocity ofvertical wave approach zero in case (a) and (b). In case (c), thecontact between the fibrils promote relatively faster groupvelocities.

The eigenmodes of unit cell indicate that the band gap evolu-tion in our system is facilitated by both Bragg diffraction and local

Fig. 4 Band gap evolutions corresponding to three inclusion patterns ((a) nonstaggered, (b) staggered identical, and (c) stag-gered symmetric)

Fig. 5 Transmission ratio of composite fibril system corresponding to the case (b) of stag-gered identical inclusion pattern under: (a) longitudinal wave and (b) shear wave

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resonance. The interfacial channels represent a new periodicinclusion of degrading stiffness and irregular boundary geometry.The periodicity of the channels provides additional diffractionpaths for wave propagations. Moreover, wave reflection andrefraction at the channel/matrix interface is opposite in polarity towave dynamics at the matrix/stiffer inclusion interface (channelstiffness<matrix stiffness< inclusion stiffness) and this contrib-ute to local resonance.

In the current study, the composite is elastic throughout thedeformation, and damage is only localized to the interfacesbetween fibrils. With the use of self-healing interfaces [43,44], thephenomena we are describing here are completely reversible. Fur-thermore, even if the interface remains damaged, the channelsnucleate and grow to complex irregular shapes that interfere withthe wave propagation and cause more complex behavior and pos-sibly more frequency band gaps than composites with regularholes or cuts [42,45,46].

In general, band gaps of the composite fibril system showsrich and controllable band gap behavior than the monolithiccomposite where no band gap shows up even with the sameinclusion pattern as the layered composite (except for a tinyband gap in case (c)). It is the interaction between the inclusionpattern and the sacrificial interfaces which manipulate wave dif-fraction and interference leading to the creation of multiple fre-quency band gaps. For the cases considered in this paper, weshow that the band gaps exist at initial stretch-free state and canbe modulated through stretch level. Multiple wide band gaps areobserved in nonstaggered inclusion pattern (case a) making itvery efficient for wave attenuation. For the other two cases, bandgap are also wide but not as rich as the case (a). For all threecases, the introduction of interfaces and deformation-causedstiffness redistribution plays essential roles in the evolution ofthese band gaps.

Future extension of this study will involve investigation of fur-ther applications and optimization of the tunable band gap struc-ture. For example, the emergence of multiple band gaps underlarge stretch may have applications in wave filtering, sound isola-tion, and vibration damping. Furthermore, the width of the bandgaps may be maximized through topology optimization so that thetunable band gaps, observed in the current study, may extend overwider regions of frequencies. Finally, the connection betweenband gap structure and the effective toughness of the compositematerial is a topic that requires further exploration. In particular,it may be possible, through careful designing of the inclusion pat-tern and sacrificial interfaces, to be able to control crack nuclea-tion and propagation by varying the band gap structure in thecomposite as a function of deformation.

Acknowledgment

This work was supported by the National Science Foundationthrough Grant No. CMMI-1435920 and UIUC Start-up funds.

Appendix: Constitutive Laws

The material deformation is described as a mapping from refer-ence configuration X to the deformed configuration x

x 5 vðX;tÞ (A1)

The displacement of a material point form is

u 5 x� X 5 vðX;tÞ � X (A2)

The deformation gradient tensor is defined as

F 5@x

@X¼ I1

@u

@X(A3)

where I is the identity matrix, and its Jacobian is J ¼ detðFÞ.

Two different soft materials are building up the composite (Fig.1). Each material is assumed to be homogeneous and isotropicwith hyperelastic response. We use the generalized Neo-Hookeansolid [47,48] model with strain energy density proposed by

W ¼ l2

�I1 � 3ð Þ þ K

2J � 1ð Þ2 (A4)

where �I1 ¼ I1=J2=3 making it more convenient for nearly incom-pressible materials, l and K are the shear modulus and bulk mod-ulus of the material, respectively.

The stress–strain relation and tangent stiffness follow as

r 5l

J5=3B� 1

3I1I

� �þ K J� 1ð ÞI (A5)

and

Cijkl ¼l

J2=3dikBjlþ djkBil �

2

3dklBijþ dijBkl

� �þ 2

3

I1

3dijdkl

� �

þK 2J� 1ð ÞJdijdkl (A6)

Cohesive Interaction for the Layered Composite. The inter-action between two composite fibrils is simulated by atraction–separation relation. In this paper, we use the bilinear lawfor simplicity but the qualitative nature of the results dependsweakly on the specific form of the cohesive law. The two-dimensional bilinear cohesive law is defined by [49,50]

Tn;s ¼

Kn;sdn;s; d � dc

Kn;sdcd� df

dc � df; dc � d � df

0; d � df

8>>><>>>:

(A7)

where Tn;s, Kn;s, dn;s is the normal or tangent traction, stiffness,and separation, respectively. d is the magnitude of total separation

given by d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidn

2 þ ds2

p. dc and df determine the damage initial-

ization and complete failure, respectively (Fig. 6).Figure 7 shows the effect of varying the cohesive interface

equivalent Young’s modulus on the band gap profile of the com-posite with the nonstaggered inclusion pattern (Case a) at astretch-free state. In this case of zero elongation, since the cohe-sive material remains intact along the whole interface, the bandgap profile is only affected by the elastic normal and shear stiff-ness of the layer and not its softening response. The results in Fig.7 suggest that the band gap width decreases as the equivalentYoung’s modulus of the cohesive interface increases The disper-sion lines plot converges to the monolithic composite case (Fig.2(a)) as the modulus increases. Indeed, when Ecoh is infinitely

Fig. 6 Bilinear cohesive law of inter-fibril interaction

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large (which is equivalent to imposing a rigid constraint betweenthe two points on the opposite side of the interface), we will getexactly the same result in Fig. 2(a) where interface no longer exists.

Wave Propagation. The equations of motion are given by

@r@x¼q

@2u

@t2(A8)

in absence of body force, where r is the Cauchy stress, and q isthe density which varies for different material in composite.

The propagation of elastic wave in periodic composites maybe analyzed by Bloch’s theorem [51] and further developedin the following way [52]. We start by decomposing the solu-tion as

uðX;tÞ5 UðXÞe�iwt (A9)

where UðXÞ is a complex valued function of X.Accordingly, we also have the following decomposition for

stress

rðX;tÞ5 RðXÞe�iwt (A10)

Equation (A8) now becomes

@R@x¼qx2U (A11)

Due to periodicity, the displacement is constrained by

UðXÞ5 UðX1LÞe�ikL (A12)

where L is the vector connecting equivalent points in periodicstructure, k is the wave vector contain the information of wave-length and direction.

Similarly, the stress also satisfies

RðXÞ5 RðX1LÞe�ikL (A13)

which is satisfied by applying the constrain of displacement underFEM discretization.

The wave vector is chosen from the first Brillouin zone in recip-rocal lattice for the representative unit cell as shown in Fig. 7. Thereciprocal lattice is defined by base vectors R1 and R2 and

R1 ¼ 2pA2 � e

a; R2 ¼ 2p

e� A1

a(A14)

where A1 and A2 are base vector in direct lattice (Fig. 1),a ¼ kA1 � A2k, and e ¼ ðA1 � A2Þ=a so that

RiAj ¼ 2pdij (A15)

It is suggested in the literature that the band gap informationmay be obtained by surfing only the points along boundaries ofthis zone [53]. While this is not rigorously proven, we adopt thesame procedure in the current paper (Fig 8).

The wave propagation response is analyzed by solving theeigenvalue problem of

K� x2M ¼ 0 (A16)

The matrices K and M are the tangent stiffness and mass matri-ces generated from the standard finite-element nonlinear analysisat different stretch level. The displacement constrain of Eq. (A12)should be applied to both K and M before solving the eigenvalueproblem. Mode decomposition can then be used to accelerate thecomputation process [41].

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