NonlocalHomogenizationModel for WaveDispersion andAttenuation in

Elastic andViscoelastic

Periodic LayeredMedia

Ruize HuDepartment of Civil and Environmental Engineering

Vanderbilt UniversityNashville, TN 37235

Email: ruize.hu@vanderbilt.edu

Caglar Oskay∗

Department of Civil and Environmental EngineeringVanderbilt UniversityNashville, TN 37235

Email: caglar.oskay@vanderbilt.edu

This manuscript presents a new nonlocal homogenizationmodel for wave dispersion and attenuation in elastic andviscoelastic periodic layered media. Homogenization withmultiple spatial scales based on asymptotic expansions of upto eighth order is employed to formulate the proposed non-local homogenization model. A momentum balance equa-tion, nonlocal in both space and time, is formulated consis-tent with the gradient elasticity theory. A key contributionin this regard is that all model coefficients including highorder length-scale parameters are derived directly from mi-crostructural material properties and geometry. The capa-bility of the proposed model in capturing the characteristicsof wave propagation in heterogeneous media is demonstratedin multiphase elastic and viscoelastic materials. The non-local homogenization model is shown to accurately predictwave dispersion and attenuation within the acoustic regimefor both elastic and viscoelastic layered composites.

Keywords: Homogenization; Higher-order models;Wave propogation; Band gap; Viscoelastic composites

∗Corresponding author address: VU Station B#351831, 2301 VanderbiltPlace.

1 IntroductionHeterogeneous materials exhibit complex response pat-

terns when subjected to dynamic loading due to the intrin-sic wave interactions induced by reflections and refractionsat material constituent interfaces. Controlling these inter-actions offer tremendous opportunities in many engineeringapplications. By tailoring the constituent material propertiesand microstructure, heterogeneous materials with favorableproperties within targeted frequency ranges, including cloak-ing [1, 2], energy harvesting [3, 4], vibration control [5] andimpact survivability [6], could be achieved.

Band gap phenomenon, i.e., complete attenuation ofwaves within certain frequency ranges, has been extensivelyinvestigated over the past two decades for composite ma-terials with periodic microstructures (i.e., phononic crystals[7, 8] and acoustic metamaterials [9, 10]). When the lengthof the traveling wave approaches the size of material mi-crostructure in a composite medium, the waveform starts tointeract with the microstructure through reflections and re-fractions at the interfaces of the constituent materials. Oneconsequence of this interaction is dispersion, which leads tothe distortion of wave shape and change of wave velocity.Another consequence is wave attenuation when the wave fre-quency is within stop bands. This manuscript is concernedwith dispersion and wave attenuation characteristics of com-posites with elastic and viscoelastic phases.

Early efforts of modeling dispersion and dispersion in-duced phenomena in heterogeneous elastic materials can betraced back to the classical works of Mindlin [11] , Achen-bach and Herrmann [12] among others. Analytical solutions(e.g., matrix transfer method [13] and plane wave expan-sions [14]) and direct numerical simulations using finite el-ement method [15] have been routinely employed to inves-tigate the characteristics of wave dispersion and band gaps.As an alternative, homogenization methods have been de-veloped over the past decades to understand and predict theoverall behavior of heterogeneous materials under dynamicloads. Asymptotic homogenization pioneered by Bensous-san et al. [16] and Sanchez-Palencia [17] was employedby Boutin and Auriault [18] to investigate wave dispersionin the context of Rayleigh scattering (i.e., long wavelengthregime) by exploiting the contributions from high order ex-pansions. Noticing the secular solutions of high order ex-pansions [19], Fish et al. [20, 21] proposed a nonlocal ho-mogenization model that combined the momentum balanceequations at the first three orders into a single macroscopicequation that captures wave dispersion in elastic compos-ites. Andrianov et al. [22] employed the asymptotic homog-enization approach to obtain the dispersion relation of elas-tic composites within the first Brillouin zone. Recent effortsby Hui and Oskay [23] focused on transient wave attenu-ation within the first band gap using a nonlocal homoge-nization model. By assuming harmonic wave propagation,the time dependent momentum balance equation reduces toHelmholtz equation and the steady state wave characteris-tics can be investigated in terms of frequency. Auriault andBoutin [24] studied the effective constitutive material prop-erties as a function of frequency for composites with highly

contrasted constituents. The high-frequency homogenizationmodel in Craster et al. [25] used asymptotic expansions inapproximating not only the displacement field, but also theeigenfrequency of the Helmholtz equation. This model cap-tures the dispersion curves near the edges of the first Bril-louin zone. In addition to the homogenization methods basedon asymptotic expansions, those that also employ Floquet-Bloch theory (e.g., [26, 27, 28, 29]) have been successful incapturing the dispersion relations of layered elastic compos-ites.

Multiscale methods that predict transient wave propa-gation in finite domains have also been developed in recentyears. Pham et al. [30] and Sridhar et al. [31] extendedthe first order computational homogenization framework todynamic problems. The authors studied the transient re-sponse of acoustic metamaterials in two dimensions by con-currently solving a set of fully coupled macroscale and mi-croscale balance equations. Filonova et al. [32] and Fafalisand Fish [33] investigated wave dispersion in one dimen-sional periodic structures using the concept of computationalcontinua, where the quadrature rules for the numerical inte-gration is adjusted as a function of the size scale ratio. Whilethe above-mentioned methods show good accuracy in cap-turing the transient wave dispersion and attenuation, solv-ing fully coupled momentum balance equations leads to pro-hibitive computational cost. Gradient elasticity modeling isan alternative approach to capture wave dispersion and bandgap formation. Askes et al. [34] and Metrikine [35] studiedthe dispersive wave propagation using the gradient elasticitymodels. Dontsov et al. [36] captured the formation of the firststop band by calibrating the length-scale parameters againstthe analytical dispersion relation. The identification and de-termination of the length-scale related model parameters inmultidimensional problems with inelastic materials remainsoutstanding [37].

Material damping provides an additional wave attenu-ation mechanism for composite materials with viscoelasticconstituents. The effects of viscoelasticity on the band gapsof phononic crystals include shifting of the band gap fre-quencies, changing bandwidth, and enhancing transmissionattenuation [38, 39, 40]. Modeling research of band gapsof viscoelastic composites typically focused on characteri-zation of the band gap structure in the context of dispersionanalysis [41, 42, 43], or wave transmission attenuation us-ing direct numerical simulations [38, 39, 40]. While manyhomogenization approaches are proposed for elastic com-posites, those for viscoelastic composites are relatively rare.Auriault and Boutin [24] studied the effect of viscoelasticityon the dynamic effective material properties. Hui and Os-kay [44, 45] derived a nonlocal homogenization model in theLaplace domain, which captures wave dispersion and atten-uation behavior in viscoelastic composites.

In this manuscript, we propose a new nonlocal homog-enization model that considers the asymptotic expansions ofup to eighth order for wave propagation in periodic layeredcomposites with elastic and viscoelastic phases. The result-ing macroscopic momentum balance equation of the pro-posed model is of the same structure as gradient elasticity

models (e.g. [35, 34, 36]), yet all the model parameters arecomputed directly from the microscale equilibrium equationsand dependent on the microstructural material properties andgeometry. Although the concept of applying high order cor-rectors to the first order asymptotic homogenization has beenpreviously proposed [46, 47, 18, 48] for problems with poorseparation of scales, the proposed model is unique in (1) itsattempt to link the asymptotic homogenization with the gra-dient elasticity models and; (2) it investigates transient wavepropagation and dispersion in viscoelastic materials. Theperformance of the proposed model is assessed by verify-ing the dispersion relation and transient wave propagation inelastic and viscoelastic composites against analytical solu-tions and direct numerical simulations.

The remainder of this manuscript is organized as fol-lows: Section 2 presents the description of the problemand the multiscale setting. Section 3 describes the generalasymptotic analysis for dynamic problems. Section 4 pro-vides the derivation of the nonlocal homogenization modelfor periodic layered composites with elastic and viscoelasticconstituents. Section 5 verifies the proposed model in twoexamples, i.e., elastic bilayer and viscoelastic four-layeredmicrostructure. The conclusions and future research direc-tions are provided in Section 6.

2 Problem setting

Consider a one-dimensional heterogeneous body withlayered microstructure (e.g., 1D phononic crystals [49]). Thedomain of the body, Ω = [0,L] is formed by repetition of mlocally periodic microstructures, Θ, with size l = L/m. Theunit cell of the microstructure domain consists of N materialphases, with the domain of phase j denoted by Θ j. x and y in-dicate the position coordinates at macro- and micro- scales,respectively, where the two coordinates are related to eachother by y = x/ζ, and 0 < ζ 1 is the small scaling parame-ter. In the context of dynamic analysis, the scaling parameteris defined as the ratio between the microstructure unit cellsize and the length of deformation wave (i.e., ζ = l/λ, whereλ is the deformation wavelength).

Consider an arbitrary response field, f ζ(x, t), which os-cillates in space due to fluctuations induced by material het-erogeneity. A two scale spatial decomposition is applied toexpress the response field in terms of both macroscale andmicroscale coordinates, f ζ(x, t) = f (x,y(x), t), where, super-script ζ indicates the dependence of the response field on themicrostructural heterogeneity. The spatial derivative of f ζ isobtained by applying the chain rule, f ζ

,x(x, t) = f,x(x,y, t)+1ζ

f,y(x,y, t), where, subscript comma followed by x and ydenote the spatial derivative with respect to the macroscaleand microscale coordinates, respectively. The response fieldsare assumed to be spatially periodic, f (x,y, t) = f (x,y+ l, t),where, l denotes the period of the microstructure in the mi-croscale coordinate (i.e., l = l/ζ). The response field of theheterogeneous body subjected to dynamic load is governed

by the momentum balance equation:

σζ,x(x, t) = ρ

ζ(x)uζ

,tt(x, t) (1)

in which, σζ and uζ are the stress and displacement fields,respectively; and ρζ denotes density.

The constitutive response is described by a generalizedlinear viscoelastic model:

σζ(x, t) =

∫ t

0gζ(x, t− τ)ε

ζ

,τ(x,τ)dτ (2)

where, gζ is the modulus function and εζ(x, t) = uζ,x(x, t) is

the strain. The dissipative process within viscoelastic con-stituents may lead to localized heating, which in turn changesthe constitutive behavior [50]. This thermal effect is not con-sidered in the current work. Elastic behavior can be recov-ered by setting gζ(x, t) = Eζ, where Eζ denotes the elasticmodulus. We assume the material properties of each con-stituent are of the same order of magnitude.

The dynamic load is applied in the form of prescribeddisplacement at the boundaries:

uζ(0, t) = 0; uζ(L, t) = u(t) (3)

where, u(t) is the prescribed boundary data. The initial con-ditions are:

uζ(x,0) = u0(x); uζ

,t(x,0) = v0(x) (4)

where, u0(x) and v0(x) are the prescribed data.

The particular forms of the generalized viscoelastic con-stitutive model and the momentum balance equation (Eq. 1)permit a simpler description of the governing boundary valueproblem in the Laplace domain. We introduce the key char-acteristics of the Laplace transform, recast the governing sys-tem of equations, and derive the nonlocal homogenizationmodel in the Laplace domain. By this approach, the convo-lution integral form of the viscoelastic constitutive relation istransformed to multiplication of strain and modulus function,which is dependent on the Laplace variable.

The Laplace transform of an arbitrary, real valued, timevarying function, f , is defined as:

F (s)≡L ( f (t)) =∫

∞

0e−st f (t)dt (5)

where, the Laplace variable, s, and the transformed functionin Laplace domain, F , are complex valued (i.e., s ∈ C andF : C→ C). The derivative rule for the Laplace transform

is given as:

L ( f, tt . . . t︸ ︷︷ ︸n times

(t)) = snF (s)− sn−1 f (0)− . . .− f, tt . . . t︸ ︷︷ ︸n−1 times

(0)

(6)For simplicity of the derivation in Laplace domain, stati-cally undeformed initial condition is assumed throughoutthis manuscript, i.e., any field variables (displacement andits derivatives) are initially zeros, u0(x) = 0 and v0(x) = 0,thus:

L ( f, tt . . . t︸ ︷︷ ︸n times

(t)) = snF (s) (7)

The convolution integral rule is given as:

L

(∫ t

0f1(t−ξ) f2(ξ)dξ

)= L ( f1)L ( f2) (8)

The momentum balance equation, Eq. 1, in the Laplacedomain is:

σζ,x(x,s) = ρ

ζ(x)s2uζ(x,s) (9)

Applying the convolution integral rule (Eq. 8) andderivative rule (Eq. 7) to the generalized linear viscoelas-tic model (Eq. 2), the viscoelastic constitutive relation in theLaplace domain is written as:

σζ(x,s) = Eζ(x,s)εζ(x,s) (10)

where, the modulus function, Eζ(x,s), in the Laplace domainis related to the modulus function in the time domain, gζ, asEζ(x,s) = sL

(gζ(x, t)

).

The boundary conditions in the Laplace domain aretransformed from Eq. 3, and written as:

uζ(0,s) = 0; uζ(L,s) = u(s) (11)

3 Asymptotic analysis with multiple scales

We perform a dimensional asymptotic analysis, as is fre-quently used in situations where only moderate contrast ispresent in material properties (e.g., [18, 20]). Based on thetwo-scale description, the displacement is approximated bythe following asymptotic expansion in the Laplace domain:

uζ(x,s)≡u(x,y,s) = u0(x,s)+8

∑i=1

ζiui(x,y,s)+O(ζ9) (12)

where, u0 denotes the macroscopic displacement field andis dependent on the macroscale coordinate only; and ui arehigh order displacement fields which are functions of bothmacroscale and microscale coordinates. The macroscopicnature of u0 is not an assumption, but a well-known con-sequence of the asymptotic analysis [20]. The strain field atany order is obtained as:

O(ζi) : εi(x,y,s) = ui,x +ui+1,y; i = 0,1, ... (13)

Employing Eq. 13 along with the constitutive relation(Eq. 10), the stress field at each order is obtained as:

O(ζi) : σi(x,y,s) = E(y,s)(ui,x +ui+1,y); (14)

Substituting the expanded displacement field (Eq. 12)and the constitutive equations at various orders (Eq. 14)into the momentum balance equation (Eq. 1), and collect-ing terms with equal orders yield the equilibrium equationsat each order of ζ:

O(ζ−1) : σ0,y(x,y,s) = 0 (15a)

O(ζi) : σi,x(x,y,s)+σi+1,y(x,y,s) = ρ(y)s2ui(x,y,s)(15b)

The classical homogenization models consider the twolowest order equilibrium equations (Eq. 15a and Eq. 15bwith i = 0), resulting in a macroscopic description that doesnot capture wave dispersion or attenuation [51, 52]. Thismodel is valid only when the deformation wavelength islarge enough that the local dispersion due to material het-erogeneities is negligible. Considering additional two equa-tions at O(ζ1) and O(ζ2), it is possible to capture dispersiveeffects and wave propagation in shorter deformation wave-length scenarios [20, 22, 44].

The equilibrium equations (Eq. 15a, b) are evaluated se-quentially by the decomposition of the corresponding dis-placement fields into macroscale components, which are in-dependent of the microscale coordinate, and influence func-tions that incorporate the effect of microstructures. The fol-lowing asymptotic procedure is well known and briefly ex-plained herein since it sets the stage for the proposed nonlo-cal model. We start by considering the following decompo-sition:

u1(x,y,s) =U1(x,s)+H1(y)U0,x(x,s) (16)

where, H1(y) denotes the first order microscopic influencefunction, and U1(x,s) is the first order macroscale displace-ment, and U0(x,s)= u0(x,s). Combining Eqs. 14, 15a and 16results in the linear equilibrium equation for the microscopic

influence function at O(ζ−1):

E(y,s)(1+H1,y),y = 0 (17)

The equilibrium equation for the N-layered unit cell isevaluated uniquely by imposing the following constraints[44]:

Periodicity : u1(y = 0) = u1(y = l);

σ0(y = 0) = σ0(y = l)(18a)

Continuity : Ju1(y =n

∑j=1

l j)K = 0;

Jσ0(y =n

∑j=1

l j)K = 0;

n = 1,2, ...,N−1

(18b)

Normalization : 〈u1(x,y, t)〉=U1(x, t)→〈H1(y)〉= 0(18c)

where, l j = l j/ζ denote the length of jth layer in microscalecoordinate. l j is the physical length of the jth layer. J·K de-notes the jump operator and y = ∑

nj=1 l j is the location of the

constituent interfaces; and 〈·〉 is defined as a spatial averag-ing operator over the unit cell:

〈 f 〉= 1l

∫ l

0f (x,y,s)dy (19)

The O(1) momentum balance equation is obtained byapplying the averaging operator (i.e., Eq. 19) to the corre-sponding equilibrium equation (Eq. 15b with i = 0) and con-sidering the local periodicity of σ1(x,y,s):

ρ0s2U0−E0(s)U0,xx = 0 (20)

where, ρ0 and E0(s) are the homogenized density and O(1)homogenized modulus, respectively:

ρ0≡〈ρ〉=1l

N

∑j=1

l jρ j (21)

E0(s) = 〈E(y,s)(1+H1,y)〉 (22)

The procedure to evaluate influence functions and derivemomentum balance equations presented above can be gener-alized for arbitrary orders. The recursive influence functiongeneration process is defined in Fig. 1. This process decom-poses the microscale dependent displacement ui(x,y,s) intomacroscale displacements and associated influence func-tions, and substitutes the decomposed displacements intoequilibrium equation at O(ζi−2). Considering the macroscale

Field decomposition: = + = 1 − , × , = 2, 3, …,

Influence function

Substitute − 2 , − 1 , and Eq. (14) into equilibrium equation at ( − 2 )

Momentum balance equation at ( ) , = 0,1, …, − 2

Equilibrium equation for influence function at ( ) ,

q = − 1, 0, …, − 3

Equilibrium equation for influence function at ( − 2 )

Periodicity: , − 1 ;Continuity: , − 1 ;Normalization:

Momentum balance equation at ( − 1 )

Average equilibrium equation at − 1 ;Periodicity:

Σ

Fig. 1: Recursive influence function generation procedure.

momentum balance equations and equilibrium equations forthe influence functions derived from lower orders, an equilib-rium equation for influence function at O(ζi−2) is obtained.By evaluating this equation with periodicity, continuity andnormalization conditions, the influence function Hi is cal-culated. The macroscale momentum balance equation atO(ζi−1) is then determined by applying the averaging op-erator and considering the periodicity condition of σi. Ex-amples of deriving the influence functions and momentumbalance equations can be found in [19, 20, 44].

4 Nonlocal homogenization model4.1 Model derivation

In this section, a novel nonlocal homogenization modelis proposed for elastic and viscoelastic periodic layered me-dia. A major contribution of this model is that we explorehomogenization up to O(ζ6) and derive the momentum bal-ance equations considering high order contributions. Fur-thermore, the proposed model is derived such that the mo-mentum balance equation shares the same structure as thegradient elasticity models [35, 34, 53, 36]:

u,tt − c20u,xx + c2

0l21u,xxxx− l2

2u,xxtt +l23

c20

u,tttt = 0 (23)

where c20 = E0/ρ0. This equation has been shown to accu-

rately capture the location and width of the first stop band,when the length scale parameters (l2

1 , l22 , l2

3 ) are calibratedagainst the analytical solution [36]. In contrast to the gra-dient elasticity models, the parameters of the nonlocal ho-mogenization model are derived directly from the constituentmaterial properties and microstructure.

As a result of the asymptotic analysis, the macroscale

momentum balance equations are obtained up to O(ζ6):

O(1) : ρ0s2U0−E0(s)U0,xx = 0 (24a)

O(ζ1) : ρ0s2U1−E0(s)U1,xx = 0 (24b)

O(ζ2) : ρ0s2U2−E0(s)U2,xx = Ed(s)U0,xxxx (24c)

O(ζ3) : ρ0s2U3−E0(s)U3,xx = Ed(s)U1,xxxx (24d)

O(ζ4) :ρ0s2U4−E0(s)U4,xx =

Ed(s)U2,xxxx + Eh(s)U0,xxxxxx(24e)

O(ζ5) :ρ0s2U5−E0(s)U5,xx =

Ed(s)U3,xxxx + Eh(s)U1,xxxxxx(24f)

O(ζ6) :ρ0s2U6−E0(s)U6,xx = Ed(s)U4,xxxx+

Eh(s)U2,xxxxxx + Ek(s)U0,xxxxxxxx(24g)

where, Ui (i = 0,1, ...,6) is the macroscale displacement atO(ζi); and, Ed(s), Eh(s), Ek(s) are the homogenized moduliat O(ζ2),O(ζ4),O(ζ6), respectively:

Ed(s) = 〈E(y,s)(H2 +H3,y)〉−〈θ(y)E0(s)H2〉 (25a)

Eh(s) =〈E(y,s)(H4 +H5,y)〉−〈θ(y)Ed(s)H2〉−〈θ(y)E0(s)H4〉

(25b)

Ek(s) =〈E(y,s)(H6 +H7,y)〉−〈θ(y)Eh(s)H2〉−〈θ(y)Ed(s)H4〉−〈θ(y)E0(s)H6〉

(25c)

For bilayer unit cell, the analytical expressions for E0, Ed andEh can be found in [19] and we provide Ek in Appendix C.

The boundary conditions at each order are prescribed as:

BCs : U0(0,s) = 0, U0(L,s) = u(s)

Ui(0,s) = 0, Ui(L,s) = 0 (i = 1,2, ...,6)(26)

Under the prescribed boundary conditions, the odd or-der macroscale displacements are trivial (i.e., U1(x,s) =U3(x,s) = U5(x,s) = 0). It is well known that solvingEqs. 24a, 24c, 24e, 24g sequentially at each order leads tosecular solutions [19, 54]. The secularity can be eliminatedby either introducing a slow temporal scale to the asymptoticanalysis [54] or employing the definition of a mean displace-ment, which combines macroscale momentum balance equa-tions at different orders into a single nonlocal homogenizedmomentum balance equation by considering the followingmacroscale displacement approximation [20]:

U (2n)(x,s) =n

∑i=0

ζ2iU2i(x,s)+O(ζ2n+2) (27)

where, U (2n) denotes the approximation to the macroscaledisplacement of O(ζ2n+2) accuracy.

Taking four spatial derivatives of Eq. 24a and substitut-

ing the resulting expression into Eq. 24e yields:

ρ0s2U4−E0U4,xx = EdU2,xxxx +Eh

E0ρ0s2U0,xxxx (28)

Inserting Eq. 24c into Eq. 28:

ρ0s2U4−E0U4,xx =EdU2,xxxx+

Ehρ0s2

E0Ed(ρ0s2U2−E0U2,xx)

(29)

Premultiplying Eq. 24a by Ehρ0s2/EdE0, and adding the re-sulting expression to the right hand side of Eq. 24c:

ρ0s2U2−E0U2,xx =EdU0,xxxx+

Ehρ0s2

E0Ed(ρ0s2U0−E0U0,xx)

(30)

Premultiplying Eqs. 30 and 29 by ζ2 and ζ4, respec-tively, adding the resulting equations to Eq. 24a, and usingEq. 27 result in:

ρ0s2U (4)−E0U (4),xx −EdU (4)

,xxxx +Eh

Edρ0s2U (4)

,xx −Eh

E0Edρ

20s4U (4)

=−ζ6(

EdU4,xxxx−Eh

Edρ0s2U4,xx +

Eh

E0Edρ

20s4U4

)(31)

where, Ed = ζ2Ed and Eh = ζ4Eh contain length scales l2 andl4, respectively, and are computed with the physical length ofthe unit cell. By truncating the O(ζ6) perturbations in Eq. 31,we obtain the fourth order nonlocal homogenization model(NHM4):

NHM4: ρ0s2U−E0U,xx−EdU,xxxx +Eh

Edρ0s2U,xx

− Eh

E0Edρ

20s4U = 0

(32)

The superscript (4) that indicates the order of approximationis dropped for simplicity. The second order nonlocal homog-enization model [44] (NHM2) and classical homogenizationmodel (CHM) are recovered by neglecting the last two termsand last three terms on the left hand side of Eq. 32, respec-tively.

In order to achieve higher accuracy for shorter wavelengths, we add higher order corrections by employingEqs. 24a-g. The procedure for obtaining the higher orderequation is similar to above and aimed at achieving the struc-ture of Eq. 23 through analysis of higher order momentumbalance equations. Consider the following decomposition of

Eq. 24g:

ρ0s2U6−E0U6,xx = EdU4,xxxx + EhU2,xxxxxx+

νEkU0,xxxxxxxx +(1−ν) EkU0,xxxxxxxx(33)

in which ν is the high order correction parameter. Taking twospatial derivatives of Eq. 24e, and substituting the resultingexpression for U0,xxxxxxxx in the third term of the right handside of Eq. 33:

ρ0s2U6−E0U6,xx =

(Ed−νE0

Ek

Eh

)U4,xxxx +ν

Ek

Ehρ0s2U4,xx

+

(Eh

Ed−ν

Ek

Eh

)EdU2,xxxxxx +(1−ν)EkU0,xxxxxxxx

(34)

Taking four spatial derivatives of Eq. 24c and substituting theresulting expression for U2,xxxxxx in Eq. 34:

ρ0s2U6−E0U6,xx =

(Ed−νE0

Ek

Eh

)U4,xxxx

+νEk

Ehρ0s2U4,xx +

(Eh

Ed−ν

Ek

Eh

)Ed

E0ρ0s2U2,xxxx

+

[(1−ν)Ek−

(Eh

Ed−ν

Ek

Eh

)E2

dE0

]U0,xxxxxxxx

(35)

Substituting Eq. 24e for U2,xxxx in Eq. 35, and taking six spa-tial derivatives of Eq. 24a and inserting the resulting expres-sion for s2U0,xxxxxx:

ρ0s2U6−E0U6,xx =

(Ed−νE0

Ek

Eh

)U4,xxxx

+

(2ν

Ek

Eh− Eh

Ed

)ρ0s2U4,xx +

(Eh

E0Ed−ν

Ek

E0Eh

)ρ

20s4U4

+

[(1−ν)Ek−

(Eh

E0Ed−ν

Ek

E0Eh

)(E2

d +E0Eh)

]U0,xxxxxxxx

(36)

Rewriting Eq. 24e in the following form:

ρ0s2U4−E0U4,xx =

(Ed−νE0

Ek

Eh

)U2,xxxx +ν

Ek

Ehρ0s2U2,xx

+νE0Ek

EhU2,xxxx−ν

Ek

Ehρ0s2U2,xx + EhU0,xxxxxx

(37)

Premultiplying Eq. 24c by s2, replacing the resultings2U0,xxxx by the expression resulting from taking four spatialderivatives of Eq. 24a, and substituting the resulting expres-

sion for U0,xxxxxx into Eq. 37:

ρ0s2U4−E0U4,xx =

(Ed−νE0

Ek

Eh

)U2,xxxx+(

νEk

Eh− Eh

Ed

)ρ0s2U2,xx +

Eh

E0Edρ

20s4U2

+νEk

Eh

(E0U2,xxxx−ρ0s2U2,xx

)(38)

In view of Eq. 24a and Eq. 24c, we have:

E0U2,xxxx−ρ0s2U2,xx =−EdU0,xxxxxx =

− Edρ0

E0s2U0,xxxx =

ρ0

E0s2(E0U2,xx−ρ0s2U2)

(39)

Substituting Eq. 39 into Eq. 38:

ρ0s2U4−E0U4,xx =

(Ed−νE0

Ek

Eh

)U2,xxxx

+

(2ν

Ek

Eh− Eh

Ed

)ρ0s2U2,xx +

(Eh

E0Ed−ν

Ek

E0Eh

)ρ

20s4U2

(40)

Considering Eq. 24a, we rewrite Eq. 24c as:

ρ0s2U2−E0U2,xx =

(Ed−νE0

Ek

Eh

)U0,xxxx

+

(2ν

Ek

Eh− Eh

Ed

)ρ0s2U0,xx +

(Eh

E0Ed−ν

Ek

E0Eh

)ρ

20s4U0

(41)

Premultiplying Eqs. 41, 40, 36 by ζ2, ζ4 and ζ6, respec-tively, adding the resulting equations to Eq. 24a and usingEq. 27:

ρ0s2U (6)−E0U (6),xx −

(Ed−νE0

Ek

Eh

)U (6),xxxx−(

2νEk

Eh− Eh

Ed

)ρ0s2U (6)

,xx −(

Eh

E0Ed−ν

Ek

E0Eh

)ρ

20s4U (6) =

+ζ6[(1−ν)Ek−

(Eh

E0Ed−ν

Ek

E0Eh

)(E2

d +E0Eh)

]U0,xxxxxxxx

−ζ8

[(Ed−νE0

Ek

Eh

)U6,xxxx +

(2ν

Ek

Eh− Eh

Ed

)ρ0s2U6,xx

+

(Eh

E0Ed−ν

Ek

E0Eh

)ρ

20s4U6

](42)

where, Ek = ζ6Ek has length scale l6 and is computed with

physical length of the unit cell. We note that Eq. 42 has thesame asymptotic accuracy as Eq. 32 for an arbitrarily cho-sen ν, due to the presence of O(ζ6) term on the right handside. In particular, when ν = 0, the left hand side of Eq. 42recovers Eq. 32 (i.e. NHM4). Therefore, Eq. 42 representsa one-parameter family of nonlocal homogenization modelsas a function of ν. O(ζ8) accuracy could be achieved by set-ting ν such that the coefficient of O(ζ6) term vanishes. Theexpression for the correction parameter is obtained as:

ν =Eh(E2

h E0 + E2d Eh−E0EdEk

)E3

d Ek(43)

By truncating the O(ζ8) perturbations in Eq. 42, we obtainthe sixth order nonlocal homogenization model (NHM6):

NHM6: ρ0s2U−E0U,xx−(

Ed−νE0Ek

Eh

)U,xxxx

−(

2νEk

Eh− Eh

Ed

)ρ0s2U,xx−

(Eh

E0Ed−ν

Ek

E0Eh

)ρ

20s4U = 0

(44)

The superscript (6) is dropped hereafter.

The Laplace domain derivation presented above appliesto both elastic and viscoelastic layered composites. For com-posites that only consist elastic phases, momentum balanceequations, NHM4 and NHM6, can also be derived in timedomain following a similar procedure and result in:

NHM4: ρ0U,tt −E0U,xx−EdU,xxxx +Eh

Edρ0U,xxtt

− Eh

E0Edρ

20U,tttt = 0

(45)

NHM6: ρ0Utt −E0U,xx−(

Ed−νE0Ek

Eh

)U,xxxx

−(

2νEk

Eh− Eh

Ed

)ρ0U,xxtt −

(Eh

E0Ed−ν

Ek

E0Eh

)ρ

20U,tttt = 0

(46)

The nonlocal homogenization model, NHM6, possessesthe same equation structure as the nonlocal equation (Eq. 23)of gradient elasticity models. One can easily observe the cor-responding relation of −

(EdE0−ν

EkEh

)and l2

1 ,(

2νEkEh− Eh

Ed

)and l2

2 , −(

EhEd−ν

EkEh

)and l2

3 , respectively. While the lat-ter parameters have to be calibrated in the gradient elasticitymodels, the former are fully determined through the homog-enization process.

4.2 High order boundary conditions

Due to the presence of fourth order derivative term, twoadditional high order boundary conditions, in addition toEq. 11, must be prescribed to solve the nonlocal momentumbalance equation (Eq. 44). The choice of high order bound-ary conditions to solve nonlocal equations of this type hasbeen extensively discussed in [37, 34, 55, 56, 57]. Most oftenthe second derivatives of displacement at the boundaries areprescribed [37]. The dispersion analysis of the nonlocal ho-mogenization model for elastic composites (Appendix A) in-dicates that non-physical solutions exist along with the phys-ical ones for steady-state wave propagation. In this study, thehigh order boundary conditions are prescribed such that thenon-physical solutions are suppressed:

U,xx(0,s) = 0, U,xx(L,s) = ξ2u(s), ω≤ ωen (47a)

U,xx(0,s) = 0, U,xx(L,s) = η2u(s), ω > ωen (47b)

where ξ and η are the solutions to the characteristic equationsof Eq. 44 and are given in Appendix B. ωen (see AppendixA) is the frequency at end of the first stop band.

For harmonic loads, the high order boundary conditionsare determined based on the loading frequency. For non-harmonic loads, a Fourier analysis of the applied load thatidentifies the frequency spectrum is performed apriori andthe high order boundary conditions are selected based on thedominant frequency components.

An alternative approach to obtain the high order bound-ary conditions is by deriving the expression for U,xx fromlower order models. Taking two spatial derivatives of themomentum balance equation of CHM:

ρ0s2U,xx−E0U,xxxx = 0 (48)

Substituting Eq. 48 into the governing equation of NHM4(Eq.32), an approximation to the fourth order differentialequation is obtained:

[(Eh

Ed− Ed

E0

)ρ0s2−E0

]U,xx +ρ0s2(1− Eh

E0Edρ0s2)U = 0

(49)From Eq. 49, we obtain the high order boundary conditionsfor NHM6 (Eq. 44):

U,xx(0,s) = 0, U,xx(L,s) =ρ0s2( Eh

E0Edρ0s2−1)(

EhEd− Ed

E0

)ρ0s2−E0

u(s)

(50)This set of high order boundary conditions does not dependon the frequency of the applied load, which makes it moreconvenient for viscoelastic composites, where solving forωen is usually non-trivial.

y

Θ1 Θ2

αl (1 − α) l

y

Θ1 Θ2 Θ3 Θ2 Θ1

αl βl

(a) (b)

Fig. 2: Unit cell of the microstructure: (a) elastic bilayer, (b) vis-coelastic four-layered.

5 Model verification

In this section, two examples, elastic bilayer and vis-coelastic four-layered microstructures are investigated todemonstrate the capability of the proposed nonlocal homog-enization model in capturing the characteristics of dispersionand transient wave propagation in periodic layered media.

5.1 Elastic bilayer microstructure

Consider a bilayer unit cell microstructure as illustratedin Fig. 2(a). The phases within the unit cell, Θ, is character-ized by the density, Young’s modulus and length. The vol-ume fraction of phase Θ1 is taken as 0.5. Aluminum andsteel are used for phases 1 and 2, respectively. The elasticmodulus and density are 210 GPa and 7800 kg/m3 for steel,and 68 GPa and 2700 kg/m3 for aluminum, respectively.

5.1.1 Dispersion relation

The dispersion relation of NHM6 and the mathemati-cal description of attenuation in the stop band are providedin Appendix A. Figure 3 shows the dispersion relation ofNHM6 and the reference dispersion relation (real part of thewavenumber solution is plotted). The horizontal and ver-tical axes are the normalized wavenumber and normalizedfrequency, respectively, and c0 is the homogenized wavespeed (i.e., c0 =

√E0/ρ0). The reference dispersion rela-

tion is plotted following Bedford and Drumheller [58]. Thetwo physical wavenumber solutions of NHM6, k1 and k2,match with the analytical wavenumber solutions in the firstpass band and second pass band, respectively. The non-physical solutions lead to negative group velocity, thereforenot characteristic of wave propagation in the homogenizedmedium. NHM6 captures the dispersion behavior accuratelyin the first pass band, and the initiation and end of the firststop band, where kl/(2π) = l/λ = ζ = 0.5, the limit of thefirst Brillouin zone [59]. The model starts to deviate fromthe reference solution as the frequency further increases andkl/(2π) approaches 1, where the second stop band initiates.As the scaling ratio reaches unity, the fundamental assump-tion of scale separation in homogenization is no longer valid[19, 60].

0 0.5 1.5

0.4

0.8

1.2

1.6

2.0

1.0Reference stop bandNHM6 stop band

kl/(2π)

ωl/(

2c 0)

k1 physicalk2 physical

k1 non-physicalk2 non-physical

Reference

Fig. 3: Dispersion relation of NHM6 and the referencesolution for bilayer microstructure.

5.1.2 Effect of material property contrast on model ac-curacy

A parametric study on the prediction error of NHM6in terms of capturing the initiation and end of the first stopband is conducted with a large number of constituent mate-rial property combinations for the unit cell. With the mate-rial phase 1 chosen as aluminum, the material property forphase 2 is sampled in two families of materials, i.e., met-als and polymers, with the material property domains Ωm =(E2,ρ2)∈R | 100≤E2≤ 400 and 4000≤ ρ2≤ 10000 andΩp = (E2,ρ2) ∈ R | 1 ≤ E2 ≤ 10 and 1000 ≤ ρ2 ≤ 2000,respectively. In each of the material property domain, 100samples are generated for E2 and ρ2, and the prediction erroris computed as |ωNHM6−ωre f |/ωre f at the stop band initi-ation and end frequencies for each sample. Figure 4 showsthe prediction error of NHM6 for the two sets of constituentmaterial combinations. Solid dots are the sampling pointsand the surface is created by quadratic fitting to the pointsto enhance visualization. The maximum error for stop bandinitiation prediction is approximately 2.5% for both Al-metaland Al-polymer microstructures. The error of stop band endprediction is relatively high compared to the initiation. InFigs. 4(b) and (d), it is noticeable that error grows as theYoung’s modulus ratio increases, especially for the caseswhich have low density ratio. The general trend of error in-dicates that NHM6 accurately captures the initiation of stopband for a large range of constituent material combinations.It predicts the end of the stop band reasonably well for lowstiffness contrast combinations, and starts to lose accuracyfor high stiffness contrast cases.

5.1.3 Models comparisonThe capability of NHM6 in capturing the dispersion and

band gap formation is due to the presence of high orderterms. In order to further study the effects of the high orderterms, we compare the dispersion curves of NHM6 (Eq. 46),NHM4 (Eq. 45), NHM2 and CHM in Fig. 5. While all mod-els capture the non-dispersive wave propagation character-istics at normalized frequency of less than 0.2, CHM and

65432123

0.010

0.020.03

40.01

0.015

0.02

0.025

65432123

0.10.15

0.050

4 0.02

0.06

0.1

0.14

7050301012

0.020.03

00.01

30.013

0.017

0.021

0.025

7050301012

0.150

0.450.3

3

0

0.15

0.3

0.45

εin

ρ2/ρ1 E2/E1(a)

(c)

εin

ρ2/ρ1 E2/E1

εen

ρ2/ρ1 E2/E1(b)

εen

ρ2/ρ1 E2/E1(d)

Fig. 4: Error in the prediction of the stop band. (a) Al-metal stopband initiation, (b) Al-metal stop band end, (c) Al-polymer stop

band initiation, (d) Al-polymer stop band end.

NHM2 deviate from the reference dispersion relation and donot capture the band gap. NHM4 captures the dispersion upto the onset of the first stop band, but not the bandwidth. Byincorporating the high order correction, NHM6 predicts thedispersion in the first and second pass band, as well as theonset and the width of the first stop band.

0 0.5 1.5

0.4

0.8

1.2

1.6

2.0

1.0kl/(2π)

ωl/(

2c 0)

Reference stop band NHM6 stop band

Reference NHM6

NHM4 NHM2 CHM

NHM4 stop band

NHM2 stop band

Fig. 5: Dispersion curves of NHM2, NHM4 and NHM6compared to the reference model.

The dispersion relations of Eq. 46 and the gradient elas-ticity model in Eq. 23 are shown in Fig. 6 using material pa-rameters provided in [36]. The two dispersion curves overlapwith each other except the minor difference at the end of thestop band. In addition, O

(EdE0−ν

EkEh

)= O(l2

1) = O(10−4),

O(

2νEkEh− Eh

Ed

)= O(l2

2) = O(10−5) and O(

EhEd−ν

EkEh

)=

O(l23) = O(10−4). The close match between the dispersion

curves and the length-scale parameters indicates that the pro-posed model could serve as an alternative method for gradi-ent elasticity models to determine the length-scale param-eters directly from material properties and microstructures,without referring to the analytical solution, which may notbe available in multidimensional problems or in the presenceof viscoelastic constituents.

0.5 10

0.4

0.8

1.2

kl/(2π)

ωl/(

2c 0)

GEM stop bandGEM NHM6

NHM6 stop band

Fig. 6: Dispersion curve of NHM6 compared to the gradientelasticity model.

DSCHMNHM6

U/M

0

x/L0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5t = 0.1 ms

x/L0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5t = 0.15 ms

Fig. 7: Responses at two time instances under step loading:(a) t=0.1 ms, (b) x=0.15 ms.

5.1.4 Transient wave propagationWe consider the transient response of a composite

structure with aluminum-steel bilayer microstructure (seeFig. 2(a)) subjected pulse loads. The overall length of thestructure is taken to be long enough, L = 0.5 m, such thatwave does not reflect at the fixed end within the time windowof interest. The unit cell is taken to have l = 0.01 m. Twopulse loads are considered, i.e., step load, u(t) = M0H(t),H(t) is the Heaviside function, and sinusoidal load, u(t) =M0 sin(2π f t). The study compares the responses computedby NHM6, CHM and the direct solution (DS) of the origi-nal problem, where each unit cell is resolved throughout theproblem domain. The time domain solution procedure forNHM6 is provided in Appendix B. The solutions for DS andCHM can be found in [44].

Figure 7 shows the normalized displacement along thelayered composite at T = 0.1 ms and T = 0.15 ms. Theresponses obtained from NHM6 with high order boundaryconditions, Eq. 47a, and DS both show dispersion inducedphase distortion, manifested by the oscillatory behavior be-hind the step wave front. Due to the presence of dispersion,both NHM6 and DS have peak amplitude of 1.2M0. NHM6shows good agreement with DS in terms of capturing the dis-persive response under the prescribed step load.

Figure 8 shows the displacement histories of x = 0.9Lfor 0.15 ms, with the structure subjected to sinusoidal loadsat frequencies 132.71 kHz and 221.18 kHz, corresponding tothe normalized frequencies (ωl/(2πc0)) of 0.3 and 0.5, re-spectively. At ωl/(2πc0) = 0.3, the response is in the firstpass band where dispersion occurs. NHM6 predicts the dis-

0 0.2 0.4 0.6 0.8 1-1.5-1

-0.50

0.51

1.5

0 0.2 0.4 0.6 0.8 1-1.5-1

-0.50

0.51

1.5

(a) f=132.71 kHz

(b) f=221.18 kHz

DS CHM NHM6

U/M

0

t/T

t/T

U/M

0

Fig. 8: Displacement histories at x=0.9L under sinusoidalloading at frequencies: (a) 132.71 kHz, (b) 221.18 kHz.

placement time history accurately compared to DS. The dis-persion is manifested by the distorted phase shape of the firsttwo cycles and slight phase shift of the dispersive models(NHM6 and DS) compared to CHM. As the frequency is in-creased to 221.18 kHz, the response resides in the first stopband. At this frequency, the displacement response is signif-icantly attenuated with peak normalized amplitude of 0.25,which asymptotes to complete attenuation in time.

Figure 9 shows the maximum transmitted wave ampli-tude at x = 0.9L within t/T ∈ [0.5,1] for sinusoidal loadsat a range of frequencies f ∈ [10,442.4] kHz. Solutions ofNHM6 with two high order boundary conditions, i.e., Eq. 52(BC-1) and Eq. 50 (BC-2), are compared to DS. BC-1 leadsto accurate wave amplitude for frequencies within the firstpass band and stop band. It starts to over predict the ampli-tude in the second pass band due to the increasingly poor sep-aration of scales at high frequencies. The transmitted waveamplitude computed by BC-2 is accurate up to f = 120 kHzin the first pass band and within the first stop band. As theapplied loading frequency is near the initiation and end of thefirst stop band, the result loses accuracy.

DSNHM6 BC-1

f (kHz)0 100 200 300 4000

0.5

1

1.5

2

NHM6 BC-2

Tran

smitt

ed A

mpl

itude

fen

Fig. 9: Transmitted wave amplitude at x=0.9L at frequencybetween 10 kHz and 442.4 kHz

5.2 Four-layered viscoelastic microstructureIn this section, the proposed model is assessed in the

context of a four-layered microstructure (see Fig. 2(b)) withboth elastic and viscoelastic phases. The unit cell is com-posed of elastic layers, epoxy (phase 1) and rubber (phase 3),with viscoelastic layers (phase 2) inserted in between. Thevolume fractions of phase 1 and phase 2 are 2α and 2β. Inthe following discussion, the volume fractions for phase 1and phase 2 are taken as α = 0.3 and β = 0.1.

The constitutive relation of the viscoelastic phase isgiven by Eq. 2, where the modulus function is expressed withProny series in time domain. After Laplace transformation,the modulus function is expressed as:

E2(s) = E∞

(1+

n

∑i=1

piss+1/qi

)(51)

where, the material properties of the viscoelastic phase is

Table 1: Material properties for elastic and viscoelasticphases.

Elastic phases

E1 [GPa] ρ1 [kg/m3] E3 [GPa] ρ3 [kg/m3]

1.0 1200 0.1 1100

Viscoelastic phase

E∞ [MPa] ρ2 [kg/m3]

67.2 1070

p1 p2 p3 p4

0.8458 1.686 3.594 4.342

q1 [ms] q2 [ms] q3 [ms] q4 [ms]

463.4 0.06407 1.163×10−4 7.321×10−7

provided in Table 1. The sizes of the macro- and micro-structures are identical to the example for the bilayer mi-crostructure.

Figure 10 shows the transmitted wave amplitude at x =0.9L for the four-layered viscoelastic composite structure un-der sinusoidal load. Loading frequency ranges from 1 to50 kHz. Solutions obtained by NHM6 using BC-1(a), i.e.,Eq. 47a, and BC-2 are compared to DS and CHM. Since theexplicit expression for ωen is not available for viscoelasticmicrostructures, Eq. 47b is not used for high frequency loads.The wave amplitude reduction in the first pass band causedby viscoelastic dissipation, which increases monotonicallyas the loading frequency increases, is observed in all models.As is for elastic composite, CHM does not predict dispersionand band gap formation of viscoelastic composite. BC-1(a)

0

0.4

0.8

1.2

Tran

smitt

ed A

mpl

itude

f (kHz)0 10 20 30 40 50

DS CHM NHM6 BC-1(a)NHM6 BC-2Tr

ansm

itted

Am

plitu

de

Fig. 10: Transmitted wave amplitude at x = 0.9L for thefour-layered viscoelastic composite at frequency between 1

kHz and 50 kHz.

leads to accurate solution up to the end of the first stop bandand deviates from the direct simulation in the second passband. BC-2 predicts the transmitted wave amplitude up tothe second pass band, except for the error near the edgesof the first stop band. With BC-1(a), NHM6 captures thedecreasing wave amplitude caused by dispersion in the firstpass band (15-18 kHz) which is not observed in the elasticcomposite example (Fig. 9). The phase distortion and phaseshift of the dispersive models (NHM6 with BC-1(a) and DS)in the first pass band are observed in Fig. 11(a). Within thestop band, near complete wave attenuation is present due tothe combination of viscoelastic dissipation and destructivewave interactions, as is shown in Fig. 11(b).

0 0.2 0.4 0.6 0.8 1-1.5-1

-0.50

0.51

1.5

0 0.2 0.4 0.6 0.8 1-1.5-1

-0.50

0.51

1.5

(a) f=15 kHz

(b) f=25 kHz

DS CHM NHM6

U/M

0

t/T

t/T

U/M

0

Fig. 11: Displacement histories of viscoelastic composite atx=0.9L under sinusoidal loading at frequencies: (a) 15 kHz,

(b) 25 kHz.

The effect of viscoelasticity is investigated by compar-ing the transmitted wave amplitude for the viscoelastic com-posite to an elastic counterpart where the modulus functionof the viscoelastic phase is replaced by its instantaneousmodulus, i.e., E2(s) = E2 = E∞ (1+∑

ni=1 pi). As is shown in

Fig. 12, viscoelastic dissipation results in enhanced wave at-tenuation within the stop band, compared to the elastic case.Moreover, the first stop band of the viscoelastic compositeis shifted towards lower frequency and its width becomesnarrower. Similar observation was reported in [42] for one-dimensional bilayer viscoelastic phononic crystals using theplane wave expansion method.

0

0.5

1

1.5

Tran

smitt

ed A

mpl

itude

f (kHz)0 10 20 30 40 50

NHM6 viscoelasticNHM6 elastic

Fig. 12: Transmitted wave amplitude of elastic andviscoelastic composite computed by NHM6.

6 ConclusionThis manuscript presented a nonlocal homogenization

model (NHM6) for the dynamic response of elastic and vis-coelastic periodic layered media. The proposed model isderived based on asymptotic expansions of up to the eighthorder. By introducing high order corrections, a momentumbalance equation that is nonlocal in both space and time isderived. The proposed model shares the same differentialequation structure as the gradient elasticity models, whereasall parameters are computed directly from the microscaleboundary value problems and dependent on the materialproperties and microstructure only. High order boundaryconditions to solve the nonlocal momentum balance equa-tion are also discussed.

The performance of the proposed model is assessed byinvestigating the wave propagation characteristics in elasticbilayer and viscoelastic four-layered composite structures,and verified against analytical solutions and direct numericalsimulations. NHM6 is accurate in predicting the dispersionrelation of elastic wave propagation within the first pass bandand the initiation and end of the first stop band for low mate-rial contrasts. The accuracy of predicting the end of the stopband decreases as the contrasts become high. In addition,NHM6 captures the transient wave dispersion and attenua-tion within the first pass band and stop band for elastic andviscoelastic composites. The procedure to compute modelparameters of NHM6 could serve as an alternative to derivethe gradient elasticity model length-scale parameters. Theeffect of viscoelasticity is observed to introduce viscoelasticdissipation that monotonically intensifies as the frequencyincreases, shift the first stop band towards lower frequencyand reduce its width.

The following challenges will be addressed in the nearfuture. First, the proposed model is developed in one-dimensional setting and it is necessary to extend NHM6 tomultiple dimensions. This will naturally contribute to theapplicability of NHM6 to wider range of problems with in-creased microstructural complexity. Second, the accuracy ofthe proposed model decreases as the contrasts of the phasematerial properties increase. Extension of the current modelto account for large material property contrasts will be per-formed to model the dynamics of metamaterials, where thestiffness of the material phases varies by orders of magni-tude.

AcknowledgementsThe authors gratefully acknowledge the financial sup-

port of the National Science Foundation, CMMI Mechanicsof Materials and Structures Program, and the Computationaland Data Enabled Science and Engineering (CDS&E) pro-gram (Grant No.: 1435115).

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Appendix A: Dispersion relations

The dispersion relation for the nonlocal homogenizationmodel defined by Eq. 44 (NHM6) is obtained by consider-ing a harmonic steady-state wave propagation in the homog-enized medium:

U(x, t) =U0ei(ωt−kx) (52)

where, U0 is the displacement amplitude, and ω and k arethe frequency and wavenumber, respectively. By substitutingEq. 52 into Eq. 46, the dispersion equation is obtained as:

(Ed−νE0Ek

Eh)k4 +

[(2ν

Ek

Eh− Eh

Ed)ρ0ω

2−E0

]k2

+(Eh

E0Ed−ν

Ek

E0Eh)ρ2

0ω4 +ρ0ω

2 = 0(53)

In general, this fourth order polynomial equation hasfour roots. Considering the elastic wave propagating in thepositive direction:

k =

√−B±

√B2−4AC

2A

A = Ed−νE0Ek

Eh

B = (2νEk

Eh− Eh

Ed)ρ0ω

2−E0

C = (Eh

E0Ed−ν

Ek

E0Eh)ρ2

0ω4 +ρ0ω

2

(54)

The wavenumber solutions for NHM4, NHM2 and CHM areobtained similarly. To seek the mathematical description ofattenuation in stop bands, we rewrite the wavenumber k inthe complex form, k = kre + ikim, where, kre and kim are thereal part and imaginary part of wavenumber, respectively.Applying the complex wavenumber expression to the dis-placement solution Eq. 52, the displacement reads:

U(x, t) =U0ei(ωt−krex)ekim (55)

in which, ei(ωt−krex) contains the oscillatory feature of thedisplacement, while ekim determines the solution to be atten-uating or amplifying depending on whether kim is negativeor positive. Stop bands are characterized as the frequencyranges within which the elastic waves are subjected to expo-nential attenuation due to negative imaginary wavenumberwith magnitude of strength proportional to |kim|. Within thepass bands, kim = 0.

The existence of complex-valued wavenumber k within

the first stop band requires −B±√

B2−4AC2A being complex, i.e.,

y(ω) = B2− 4AC < 0. Therefore, the initiation and end of

the first stop band are found by y(ω) = B2−4AC = 0:

ωin =

√−b−

√b2−4ac

2a, ωen =

√−b+

√b2−4ac

2a

a = ρ20

[(2ν

Ek

Eh− Eh

Ed)2−4(Ed−νE0

Ek

Eh)(

Eh

E0Ed−ν

Ek

E0Eh)

]b =−ρ0

[2E0(2ν

Ek

Eh− Eh

Ed)+4(Ed−νE0

Ek

Eh)

]c = E2

0(56)

where, ωin and ωen are the frequency that the first stop bandinitiates and ends, respectively.

Denote k1 =

√−B−√

B2−4AC2A and k2 =

√−B+√

B2−4AC2A .

While k1 and k2 are both the solutions of the dispersion equa-tion (Eq. 53), they have to be examined by physical consider-ations. Within the first pass band, when ω = 0, k1 = 0, whichrecovers the result of dispersion relation of infinitely longwave, i.e., k = w/c = 0; on the other hand, k2 =

√(E0/A),

which is not characteristic of long wave propagation. There-fore, k1 is used in the first pass band and stop band. In thesecond pass band, a typical observation (see Section 5.1.1)is that while k2 matches with the analytical solution well,k1 leads to negative group velocity followed by zero realwavenumber solution at higher frequencies.

Appendix B: Time domain solutionThis appendix provides the time domain solution for

the nonlocal homogenization model (NHM6). The boundaryvalue problem of NHM6 is evaluated in the Laplace domain.Time domain solution is then obtained by applying a numer-ical inverse Laplace transform [61]. Rewrite the nonlocalmomentum balance equation (Eq. 44):

(Ed−νE0Ek

Eh)U,xxxx +

[ρ0s2(2ν

Ek

Eh− Eh

Ed)+E0

]U,xx

+[ρ

20s4(

Eh

E0Ed−ν

Ek

E0Eh)−ρ0s2

]U = 0

(57)

The fourth order differential equation is evaluated byconsidering the following solution form:

U(x,s) =D1(s)sinh(ξx)+D2(s)sinh(ηx)+

D3(s)cosh(ξx)+D4(s)cosh(ηx)(58)

where ξ and η are the solutions of the characteristic equationof Eq. 57. With the boundary conditions Eqs. 11 and 52, thesolution for frequencies in the first pass band and stop bandis obtained as:

U(x,s) = D1(s)sinh(ξx) (59)

and in the second pass band:

U(x,s) = D2(s)sinh(ηx) (60)

The coefficients D1, D2, D3, D4 are:

D1(s) =u(s)

sinh(ξL); D2(s) =

u(s)sinh(ηL)

;

D3(s) = D4(s) = 0(61)

where:

ξ =

√−B+

√B2−4AC

2A; η =

√−B−

√B2−4AC

2A

A = Ed−νE0Ek

Eh

B = (2νEk

Eh− Eh

Ed)ρ0s2 +E0

C = (Eh

E0Ed−ν

Ek

E0Eh)ρ2

0s4−ρ0s2

(62)

Appendix C: Analytical expression for Ek

Ek =α2(1−α)2(E1ρ1−E2ρ2)

2E0 l6

60480ρ60(E1(1−α)+αE2)

6

[(1−α)4E4

1

(32(1−α)4

ρ42−160α(1−α)3

ρ1ρ32 +192α

2(1−α)2ρ

21ρ

22+

108α3(1−α)ρ3

1ρ2 +3α4ρ

41

)]+[

α(1−α)3E31 E2

(−160(1−α)4

ρ42 +944α(1−α)3

ρ1ρ32−

1580α2(1−α)2

ρ21ρ

22−12α

3(1−α)ρ31ρ2 +108α

4ρ

41

)]+[

α2(1−α)2E2

1 E22

(192(1−α)4

ρ42−1580α(1−α)3

ρ1ρ32+

3826α2(1−α)2

ρ21ρ

22−1580α

3(1−α)ρ31ρ2 +192α

4ρ

41

)]+[

α3(1−α)E1E3

2

(−160α

4ρ

41 +944α

3(1−α)ρ31ρ2−

1580α2(1−α)2

ρ21ρ

22−12α(1−α)3

ρ1ρ32 +108(1−α)4

ρ42

)]+[

α4E4

2

(32α

4ρ

41−160α

3(1−α)ρ31ρ2 +192α

2(1−α)2ρ

21ρ

22+

108α(1−α)3ρ1ρ

32 +3(1−α)4

ρ42

)](63)