Nonlinear wave scattering at the interface of granular dimer chains and an elastically supported membrane

International Journal of Solids and Structures 182–183 (2020) 46–63

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Nonlinear wave scattering at the interface of granular dimer chains and an elastically supported membrane Qifan Zhang a,b,∗, Randi Potekin c, Wei Li a, Alexander F. Vakakis b a

School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China Department of Mechanical Science and Engineering, University of Illinois, Urbana, IL 61801, USA c Department of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA b

a r t i c l e

i n f o

Article history: Received 2 November 2018 Revised 13 July 2019 Accepted 1 August 2019 Available online 1 August 2019 Keywords: Granular dimer chains Flexible boundary Nonlinear wave scattering Solitary wave Transient breathers

a b s t r a c t In this work we study computationally nonlinear wave scattering at the flexible interfaces of 1D dimer granular chains with a square membrane with a linear uniform elastic foundation. A computational algorithm that combines successive iterations and interpolations is developed to accurately model the highly nonlinear and non-smooth scattering of impeding pulses from the granular chains to the flexible boundary. Energy localization (through the excitation of local transient breathers), intense wave transmission or reflection, as well as strong pulse scattering in the frequency/wavenumber domain are detected for varying mass ratios of the dimers and stiffness of the elastic foundation of the membrane. Moreover, it is found that the realization of resonances or anti-resonances in the dimer granular chains at different mass ratios has significant effects on the nonlinear wave scattering at the flexible boundary. Interestingly enough, an inverse relation between the foundation stiffness and the residual energy transferred to the membrane from the impulsively excited dimer is found. Finally, we show that the energy exchanges between two granular chains interacting through the flexible foundation strongly depend on the distance between them. The presented results and the associated computational method discussed in this work contribute to the predictive modeling and design of granular media with flexible interfaces. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Wave propagation in granular media composed of ordered or disordered arrays of discrete particles (granules) has attracted considerable attention from both theoretical and practical points of view. Due to their highly tunable and tailorable acoustic properties, granular media have been considered for various potential applications, such as nonlinear acoustic lenses (Spadoni and Daraio, 2010; Donahue et al., 2014), shock and energy absorbing layers (Melo et al., 2006) and passive acoustic filters (Lawney and Luding, 2014). The pioneering research in the field of wave propagation in one-dimensional (1D) ordered granular media was carried out by Lazaridi and Nesterenko (1985), Nesterenko (1983), who discovered analytically, in numerical simulations and in experiments a new type of spatially localized, coherent, and shape preserving strongly nonlinear solitary wave (Nesterenko, 2001). The speed of these strongly nonlinear solitary waves has a stronger dependence on their amplitude than for weakly nonlinear Krteweg-de

∗ Corresponding author at: School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China. E-mail address: [email protected] (Q. Zhang).

https://doi.org/10.1016/j.ijsolstr.2019.08.001 0020-7683/© 2019 Elsevier Ltd. All rights reserved.

Vires (KdV) type solitary waves, which renders the former passively tunable with respect to energy (Nesterenko, 2001). Based on the earlier works by Coste et al. (1997) performed several experiments and observed a good agreement between experimental measurements and theoretical predictions. It is worth mentioning that the acoustics of 1D homogeneous granular chains (i.e., composed of a number of identical spherical elastic granules) depend heavily on the ratio of the amplitude of the wave and the pre-compression force between granules (Nesterenko, 2018). Indeed, in the case of no pre-compression, the essentially nonlinear inter-particle Hertzian interactions and possible separations along with ensuing collisions between adjacent granules yields complete absence of linear acoustics and zero speed of sound compression waves (as defined in classical acoustics), resulting in a “sonic vacuum” (Nesterenko, 2001; Nesterenko, 1994). In this case Nesterenko solitary waves are the main disturbances propagating in the sonic vacuum, and are characterized by strong dependence of their speed on amplitude, instead of the sound waves (phonons) that are the main disturbances in classical linear wave dynamics with speed independent of amplitude (Nesterenko, 2001). Moreover, the class of granular or non-granular nonlinear “sonic vacua” possesses highly complex nonlinear dynamics and acoustics, such as propagating solitary pulses (Jayaprakash et al., 2011a), travel-

Q. Zhang, R. Potekin and W. Li et al. / International Journal of Solids and Structures 182–183 (2020) 46–63

ing waves (Starosvetsky and Vakakis, 2010), a mixture of solitary and nonlinear shear waves (Zhang et al., 2015), frequency bands or breathers (Hasan et al., 2015), extreme nonlinear energy exchanges and “energy explosions” (Zhang et al., 2018), and strongly non-reciprocal acoustics (Zhang et al., 2016). Conversely, in the regime of strong pre-compression between granules (wave amplitude is very small in comparison with precompression stress), the dynamics and acoustics of granular media become weakly nonlinear, linearizable (i.e., a speed of sound in the sense of classical acoustics can be defined) and smooth (Nesterenko, 2001; Lazaridi and Nesterenko, 1985). It follows that depending on the ratio of the wave amplitude to the precompression stress, the dynamics and acoustics of 1D granular chains can be highly tunable, ranging from strongly nonlinear and non-smooth, to weakly nonlinear and smooth (Nesterenko, 2018). Extensions to higher dimensional ordered granular media were reviewed in Starosvetsky et al. (2017), Tiwari et al. 2017). In addition to 1D homogeneous chains, the acoustics of 1D diatomic (dimer) granular chains have also been studied (Jayaprakash et al., 2011b, 2012a,b, 2013a,b; Herbold et al., 2009; Kim et al., 2015; Potekin et al., 2013; Rosas and Lindenberg, 2017), i.e., of ordered granular chains consisting of repetitive pairs of “light” and “heavy” granules. Theoretical analysis and experimental studies (Kim et al., 2015; Potekin et al., 2013) revealed that these non-dissipative chains with different mass ratio can support countable infinities of resonances and anti-resonances, yielding to either drastic attenuation of propagating pulses, or propagation of undistorted propagating solitary pulses, respectively. Typically, the dynamics and acoustics of 1D granular chains are studied using free or fixed boundary conditions (Job et al., 2005). It is clear, however, that to incorporate ordered granular media into practical applications such as granular interfaces, it is important to model, study and understand how these highly discrete and strongly nonlinear media interact with flexible boundaries, e.g., when they are supported by spatially continuous structures or media. A natural step in this direction is to study the nonlinear interactions of ordered granular media with “non-standard” interfaces (boundaries), e.g., strings, membranes, shells or plates. There is a limiting number of previous works that considered the effects of linear flexible boundaries with ordered granular media. Yang et al. (2011) theoretically, numerically and experimentally studied a 1D homogeneous granular chain in contact with a cylindrical elastic medium and found that the formation and propagation of reflected solitary waves from the interface is strongly influenced by the elastic modulus and geometry of the adjacent linear medium. In another work by Yang et al. (2012) a combined discrete element and finite element model was constructed to computationally investigate the interaction between a 1D homogeneous granular chain with a thin plate; in the same work a series of experiments was performed to confirm the computational predictions. The experimental results along with the numerical simulations confirmed the conclusion that the amplitude and speed of propagation of reflected solitary waves depend on the plate thickness and the size of the granules. In a recent work by Potekin et al. (2016), an iterative numerical algorithm was developed and applied to investigate the nonlinear wave scattering at the interface of 1D dimer granular chains with a linear string on an elastic foundation. Energy conservation within the system provided validation of this new numerical approach and thus ensured the accuracy of the numerical predictions. Following this, the effects of resonances or antiresonances in the granular dimer chains on the transmission and reflection of energy at the interface with the flexible boundary were studied and the physics of nonlinear wave scattering at the flexible boundary was investigated in detail. Motivated by Potekin et al. (2016), in this work we study the nonlinear wave scattering at the interface of finite 1D granular

47

dimer chains with a (linear) 2D rectangular membrane on elastic foundation. Apart from developing the computational algorithm that will enable the performance of this study, of emphasis will be the investigation on the effects of resonances and antiresonances in the dimers on the transmitted and reflected waves at the 2D flexible interface, as well as the study of possible localization phenomena (such as the formation of transient breathers (Potekin et al., 2016) in the neighborhood of the interface. The manuscript is structured as follows. In Section 2, we present the governing equations of the integrated granular dimer–membrane system and express them in non-dimensional form. In Section 3, we study the highly nonlinear and discontinuous wave scattering phenomena realized at the flexible boundary between dimer chain and the membrane. The effects of the dimer mass ratio and foundation stiffness on energy transmission, reflection and scattering at the chain-membrane interface are investigated also. In Section 4, we study the energy transmission between two granular chains through waves propagating in the flexible boundary. Finally, in Section 5 we summarize our observations and discuss possible future work. 2. Theoretical modeling 2.1. Mathematical model In this work, we consider initially uncompressed 1D dimer (diatomic) granular chains composed of N linearly elastic, spherical, alternating “heavy” and “light” granules with Hertzian interaction, in contact to a membrane on a linear elastic foundation. The system depicted in Fig. 1 is considered in the initial part of this work where a computational algorithm is developed for studying the strongly nonlinear wave scattering at the flexible interface of the granular dimer with the membrane. At a later part of this work multiple granular chains in contact with the membrane are considered. The Hertzian contact assumptions (Nesterenko, 2001) are made to facilitate the corresponding analysis. In addition, gravity is neglected, and no plasticity or any dissipative effects are taken into consideration in our analysis. Also, in the absence of precompression granule separations and ensuing collisions between them can occur, rendering the acoustics highly discontinuous and strongly nonlinear. When the spherical, linearly elastic granules are in Hertzian contact, the nonlinear Hertzian interaction force can be mathematically modeled by, F = k3+/2 , where k is the coefficient of Hertzian interaction dependent on geometrical and material parameters of the interacting granules,  ≥ 0 is the distance approach between the centers of the interacting granules, and F the resulting con-

Fig. 1. Schematic of a dimer granular chain in contact with a membrane on elastic foundation.

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tact force. This nonlinear relationship determines the strong nonlinearity in the acoustics of the granular systems. Moreover, the subscript (+) in the previous Hertzian law denotes that the expression has meaning only if ࢞ ≥ 0, and is zero otherwise, indicating absence of tensile force between granules, and, hence, models both the nonlinear Hertzian interaction under compression of the interacting granules, as well as the possible granule separation in the absence of compressive forces and loss of contact; as mentioned previously, this provides an additional source of strong nonlinearity in the dimer chain. Moreover, the left boundary of the dimer chain is free (and is the point where an external impulsive excitation is applied), whereas its right boundary is in point contact with a rectangular membrane – cf. Fig. 1. The membrane provides a flexible but discontinuous right boundary for the dimer chain (since separation and loss of contact between the Nth granule of the dimer chain and the membrane may occur). The rectangular, homogeneous, linearly elastic membrane is subject to constant, uniform tension, is supported by a uniformly distributed linear elastic foundation, and has all edges clamped. Assuming infinitesimal elasticity theory and neglecting axial deformations, all material points of the membrane undergo purely transverse deformations (i.e., in the z-direction), and it is assumed that the dimer chain is initially in contact with the membrane at point (x0 ,y0 ) – cf. Fig. 1. Moreover, we assume that, following the application of the impulsive excitation, the elastic deformations of the membrane and all granules are sufficiently small, so that rotations of all granule can be neglected and that the dimer chain is aligned perpendicular to the membrane at all time instants. The strongly nonlinear and discontinuous equations of motion governing the axial deformations of the N granules of the dimer chain, uj (t),j = 1, ... , N, are given by,

 4 ∗√ ∗ E R −(u1 − u2 )3+/2 3 ···  4 √  m j u¨ j = E ∗ R∗ (u j−1 − u j )3+/2 − (u j − u j+1 )3+/2 3 ···  4 √  mN u¨ N = E ∗ R∗ (uN−1 − uN )3+/2 − FC 3 m1 u¨ 1 =

the right-hand sides of the last of Eqs. (1) and (2), the coupling force FC = FC (t) models the Hertzian contact force between the Nth granule and the membrane (represented for simplicity as a function of time), and is expressed as,

 2  2

∂w ∂w ϕ T + + K w2 dS ∂x ∂y S  2  1 ∂w κ= ρm dS 2 S ∂t 1 = 2

 2  ∂ 2w ∂ w ∂ 2w ρm 2 − T + + Kw = FC δ (x − x0 ) δ (y − y0 ), ∂t ∂ x2 ∂ y2 w(0, y, t ) = w(a, y, t ) = 0 w(x, 0, t ) = w(x, b, t ) = 0 (2)

In Eq. (1) the axial deformations of the granules are measured from their corresponding equilibrium positions, m is the granule mass, E∗ and R∗ are effective elastic modulus and effective radius between interacting granules, respectively (these are defined below), and overdot denotes differentiation with respect to time. With regard to Eq. (2) for the membrane, ρ m is the mass per unit area, T the tension per unit length, K the distributed stiffness per unit area of the linear elastic foundation, and w = w(x, y, t) denotes the transverse displacement at position (x, y). Correspondingly, on the left-hand side of Eq. (2) transverse inertia effects (the first term), elastic effects due to the internal tension T (the second term), and stiffness effects due to the external elastic foundation (the third term) are included. It is noteworthy that the stiffness term introduces dispersion into the dynamics of the membrane, which means different harmonic components of the wave with different wavenumbers will not propagate with the same speed as they do for classic wave equation where  all harmonic components propagate with constant speed c = T /ρm . On

 



(4)

Throughout this study, we assume that the total number of granules in one dimer chain is an odd number equal to N, and the first and last granules are heavy ones with radius R1 . The radius of light granules is R2 . All granules are composed of the same material labeled by “b”, while the material of the homogeneous membrane is labeled by “m”. The parameters Eb , ν b , Em , and ν m are the elastic modulus and Poisson ratio, respectively, of the granules and the membrane. Accordingly, the effective elastic modulus and radius in the governing Eqs. (1) and (3) can be written as:

Ec∗ =

where as the transverse deformation of the membrane by Rao (2007), Graff (2012),

(3)

where w(x0 ,y0 ,t) denotes the membrane’s displacement at the contact point, Ec∗ and R∗c are the effective elastic modulus and effective radius, respectively, between the membrane and its neighboring granule. The combination of Dirac functions in Eq. (2) models the point of contact where the interaction force is applied. Finally, the displacement response along all four edges vanishes. The potential and kinetic energies of the membrane with its elastic foundation over the entire area S, φ and κ , respectively, can be expressed as (Rao, 2007):

E∗ =

(1)

4 ∗ 3/2 E Rc [uN − w(x0 , y0 , t )]+ 3 c

FC (t ) =

Eb 2 ( 1 − νb )

2

,

R∗ =

R1 R2 R1 + R2

Eb Em E b ( 1 − νm ) + E m ( 1 − νb ) 2

2

R∗c =

,

R1 Rm R1 + R˜m

∼ = R1

(5)

Here we set Rm → ∞ by assuming that the membrane has infinite local radius of curvature at the point of contact with the Nth granule and obtain the approximation RC∗ ≈ R1 . Introducing the dimensionless parameter ε = R32 /R31 denoting the mass ratio between the light and heavy granules of the dimer chain, considering the additional normalizations for the dependent variables (with tilde denoting non-dimensional variables),

u˜ =

u , R1

x˜ =

x , R1

y˜ =

y , R1

˜ = w

w R1

(6)

and normalizing the time variable (by introducing the characteristic time A which allows for the reduction of Eq. (1) to a much simpler form) and the nonlinear contact force as follows,

τ

t = =t A

E∗ π R21 ρb



ε 1/3 1 + ε 1/3

1 / 2 ,

F˜C =

3A2 FC 4π ρb R41

(7)

the governing equations of motion can be expressed in the following non-dimensional form,

ε1 u˜ 1 = −(u˜1 − u˜2 )3+/2 ···

ε j u˜ j = (u˜ j−1 − u˜ j )3+/2 + (u˜ j − u˜ j+1 )3+/2 ,

j = 2, 3, 4, · · · , N − 1

···

εN u˜ N = (u˜20 − u˜21 )3+/2 − F˜C (τ )

(8)

Q. Zhang, R. Potekin and W. Li et al. / International Journal of Solids and Structures 182–183 (2020) 46–63

and,





∂ 2 w˜ ∂ 2 w˜ ∂ 2 w˜ ˜ = ζ F˜C (τ ) δ (x˜ − x˜0 ) δ (y˜ − y˜0 ), −β + + μw ∂τ 2 ∂ x˜2 ∂ y˜2 ˜ (0, y˜, τ ) = w ˜ (a˜, y˜, τ ) = w ˜ (x˜, 0, τ ) = w ˜ (x˜, b˜ , τ ) = 0 w

β=

A2 T , R21 ρm

μ=

A2 K

ρm

,

ζ=

4 π ρb R 1 3 ρm

(9)

˜ =w ˜ (x˜, y˜, τ ) and, where w

γ [u˜N (τ ) − w˜ (x˜0 , y˜0 , τ )]3+/2 ,

−1  2 E b ( 1 − νm ) −1 / 3 γ = 2 (1 + ε ) 1+ Em (1 − νb2 )

49

equation subject to an arbitrary inhomogeneous term by first computing the response to a unit impulse, i.e., the Green’s function, and then, by the principle of linear superposition, constructing the solution via the convolution of the resulting Green’s function with the given inhomogeneous term. It follows that, if we compute the membrane’s response to a unit impulse, we should be able to compute its response to the interaction force F˜C (τ ). Hence, the analytical solution of Eq. (9) can be expressed in the form of the following convolution,

˜ (x˜, y˜, τ ) = ζ g(x˜, y˜, τ ) ∗ F˜C (τ ) w

F˜C (τ ) =

(10)

In the normalized Eqs. (8-9) prime denotes differentiation with respect to the normalized time τ ; ε j denotes the mass ratio between the jth granule and a heavy granule, i.e., ε j is equal to ε if j is even (for the light granules), and is unity otherwise (for the heavy granules); γ is a coefficient denoting the stiffness mismatch between the last heavy granule of the dimer chain and its neighboring flexible boundary; and β , μ and ζ are determined by the material properties of the dimer chain and membrane, as well as the prescribed tensionin the membrane and stiffness of the elastic foundation. Note that β is the normalized wave speed in the corresponding classical wave equation of the membrane without elastic foundation. From the non-dimensional equations above, we deduce the dimensionless parameters governing the nonlinear wave scattering at the flexible interface of the granular dimer. In particular, the mass ratio ε which determines the nonlinear acoustics of 1D granular chains (Jayaprakash et al., 2013a,b; Kim et al., 2015; Potekin et al., 2013), and the normalized stiffness parameter μ which plays an important role in the dynamics of the membrane, are used as control parameters in the following computations that consider the strongly nonlinear wave scattering at the flexible interface. The interaction of the dimer chain with the dispersive elastic membrane at its boundary is governed by the previous equations. However, given the possibility of separations (loss of contact) and ensuing collisions between the right-most heavy granule of the dimer with the flexible boundary, and considering the highly nonlinear and discontinuous nature of the resulting interaction force, special consideration should be given to its accurate computation. To this end, we extended to two-dimensions the computational algorithm developed in (Potekin et al., 2016) where the nonlinear interaction of a granular dimer with a finite-length 1D elastic string resting on an elastic foundation. The computational algorithm in (Potekin et al., 2016) utilizes iteration and interpolation at successive time steps, to accurately compute (and ensure convergence of) the highly discontinuous contact forces and associated displacements at the flexible interface. Whereas the full details of this algorithm can be found in (Potekin et al., 2016), in the Appendix we provide a brief overview of its main structure and discuss some issues related to its present extension to two dimensions. It suffices to state at this point that a key issue is to accurately and reliably solve the partial differential equation in Eq. (6) subject to a time-variant interaction force, which is discussed in the following section. 2.2. Green’s function of the membrane Considering the rectangular membrane which forms the flexible boundary of the dimer, given that the governing linear partial differential equation in Eq. (9) contains the inhomogeneous term ζ F˜C (τ ) δ (x˜ − x˜0 ) δ (y˜ − y˜0 ), it is natural to consider the method of Green’s functions for computing its response. The Green’s function method computes the solution of an inhomogeneous differential

(11)

where g(x˜, y˜, τ ) is the Green’s function for the membrane and is computed by solving the following auxiliary problem,

 2  ∂ 2g ∂ g ∂ 2g − β + + μg = δ (x˜ − x˜0 ) δ (y˜ − y˜0 ) δ (τ ), ∂τ 2 ∂ x˜2 ∂ y˜2

g(0, y˜, τ ) = g(a˜, y˜, τ ) = g(x˜, 0, τ ) = g(x˜, b˜ , τ ) = 0

(12)

Using the technique of the generalized functions Vladimirov, 1976), system (12) is equivalent to the following system,

 2  ∂ 2g ∂ g ∂ 2g −β + + μg = 0 , ∂τ 2 ∂ x˜2 ∂ y˜2  ∂ g(x˜, y˜, τ )   = δ (x˜ − x˜0 ) δ (y˜ − y˜0 ), ∂τ t=0

g(0, y˜, τ ) = g(a˜, y˜, τ ) = g(x˜, 0, τ ) = g(x˜, b˜ , τ ) = 0

(13)

whose solution can be determined by applying the method of separation of variables. It follows that the Green’s function is expressed as, ∞ ∞ 4   −1 g(x˜, y˜, τ ) = ωij sin a˜b˜ i=1 j=1



sin

iπ x˜ a˜





sin



jπ y˜ b˜

iπ x˜0 a˜





sin



jπ y˜0 b˜

sin ωij τ



(14)

where,

   2  2

 iπ jπ ωi j = β + + μ, i, j = 1, 2, . . . a˜

b˜

are the countable infinity of natural frequencies of the rectangular membrane. It can be noted that the influence of the elastic foundation on the dynamics of membrane mainly lies in the natural frequencies. Higher stiffness means higher frequencies, which in turn lead to smaller amplitudes. The deflection of the membrane at the contact point (x˜0 , y˜0 ) can be computed by substituting its coordinates into Eq. (14). However, an ideal membrane without elastic foundation cannot support a force concentrated at a point since the corresponding deformation at the point of contact will be theoretically infinite no matter how small the applied force is (Morse, 1936). If the membrane is supported by a distributed elastic foundation (as in our model), then a point force will not produce an infinite displacement, but unless the stiffness is considerable, the displacement of the point of application will be considerably larger than the displacement of the rest of the membrane. It follows that the numerical computation of the membrane response at the contact point obtained by Eq. (14) may either converge extremely slowly or even fail to converge. This poses a computational problem when computing the associated Green’s function. Accordingly, to compute the Green’s function required for our nonlinear wave scattering problem, we replaced the Dirac function in Eq. (11) by an approximation. We note at this point that since

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Q. Zhang, R. Potekin and W. Li et al. / International Journal of Solids and Structures 182–183 (2020) 46–63

the Dirac function δ (x) is a generalized function, it may be approximated numerically in different ways. One common and simple approach is to use instead the rectangular function,



δrec (α ) =

1

( 2α )

|x| ≤ α |x| > α

,

0,

(15a)

in the limit as α → 0. Based on (15a) the first initial condition in Eq. (13) can be approximated by,

 ∂ G(x˜, y˜, τ )   ∂τ



≈

1 , 4h 2

|x˜ − x˜0 | ≤ h, |y˜ − y˜0 | ≤ h

0

τ =0

(15b)

where h is a sufficiently small number and G(x˜, y˜, τ ) denotes the approximate Green’s function. It is worth noting that this approximation has certain physical meaning since the approximate initial velocity of the membrane is now uniformly distributed over a (non-dimensional) small square area S = 4h2 instead of a single point, and the dimensionless length h can be the regarded as the ratio between the lateral size of this area and the diameter of the Nth (last) heavy granule of the dimer chain. Furthermore, when transforming to dimensional coordinates and considering the contact between the last granule and the membrane, this operation approximates the contact surface of the sphere as a flat surface 2hR1 × 2hR1 during mutual interaction, since the radius of curvature is assumed to be large in comparison to the area of contact, providing h is small enough. Based on the approximation of the Dirac function (15) the approximate time-dependent Green’s function of the membrane following an impulse excitation at (x˜0 , y˜0 ) can be expressed as follows,

g(x˜, y˜, τ ) ≈ G(x˜, y˜, τ ) =

M  M  

4 h2

π

2

i j ωi j

−1

sin

i=1 j=1

iπ h iπ x˜0 jπ h sin sin a˜ a˜ b˜

jπ y˜0 iπ x˜ jπ y˜ sin sin sin sin ωi j τ ˜b a˜ b˜

(16)

where M is the truncation order satisfying certain convergence criterion. This solution describes the non-dimensional free response of the rectangular membrane when it is excited at (x˜0 , y˜0 ) by a uniform excitation over a sufficiently small area, in such a way that this area experiences an evenly distributed velocity at τ = 0. However, since the contact area is now taken as a surface instead of a point, and this area is sufficiently small, here we use the average displacement of all material points in this area as the corresponding response of the membrane at (x˜0 , y˜0 ) particularly. It follows that the Green’s function (16) is further approximated by the following average expression:



g(x˜0 , y˜0 , τ ) ≈ G¯ (τ ) =

G(x˜, y˜, τ )dS

S

4 a˜ b˜ = 4 4 h π



2

sin

M  M 

2

i j

i=1 j=1

iπ x˜ a˜



sin

2

ωi j

 2

−1

jπ h b˜

 sin

2



iπ h a˜

 2

sin



jπ y˜ b˜

 sin ωi j τ

(17)

We demonstrate the convergence properties of the approximation (17) by setting h = 0.01, a˜ = b˜ = 100, μ = 0, β = 0.005, for the impulse applied at the center of membrane. In Fig. 2 we depict the convergence of the exact Green’s function computed from Eq. (14) and the average expression from Eq. (17) for different truncation orders M. Note that the results depicted are the responses at a very short time following the impulse excitation. It can be observed that the exact Green’s function (14) does not converge

Fig. 2. Response of the center of the membrane (which coincides with the point of the impulse excitation) as function of τ for different truncation order M: (a) absence of convergence of the exact Green’s function g(x˜0 , y˜0 , τ ), Eq. (14), (b) convergence of the average approximation for the Green’s function G¯ (x˜0 , y˜0 , τ ), Eq. (17).

with increasing M (cf. Fig. 2a) due to its singularity at τ = 0. Indeed as M  1 the peak of the response at the point of application of the impulse increases tending to infinity. On the contrary, the average Green’s function (17) converges well with increasing truncation order M. Moreover, it is worth mentioning that the center of the membrane falls back toward its equilibrium position after it reaches a maximum. In fact, for the membrane with an initial velocity impulse, a displacement pulse is generated at the excited point and then spreads outward from that location as an ever-increasing circle (before the wave reaches a boundary), while the amplitude decreases as the circular wavefront expands due to conservation of energy. The waves traveling on the membrane will leave a wake behind them as the circular wavefronts expand outward (Morse, 1936; Graff, 2012). In the following computations we use the approximation for the Green’s function. Based on this approximation, we are now in the position to accurately and efficiently compute the transient nonlinear coupling force FC (τ ) at the right boundary of the dimer granular chain with the membrane, as well as the elastic deformations of the material points of the membrane and the granules of the dimer. To this end, we extend the iterative-interpolative computational algorithm developed in (Potekin et al., 2016) for the membrane, as described in the Appendix, and apply convolution of the converged contact force and the corresponding Green’s functions.

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3. Nonlinear wave scattering at the flexible interface of the granular dimer In this section, the nonlinear wave scattering at the flexible interface of the dimer chain of Fig. 1 is studied by performing parametric numerical experiments. Motivated by the findings in (Potekin et al., 2013), (Jayaprakash et al., 2013a,b) and summarized in (Starosvetsky et al., 2017), the number of granules of the dimer chain throughout this work is fixed to N = 21. Wave propagation in 1D ordered granular chains has been theoretically, numerically, and experimentally studied (Nesterenko, 2001; Lazaridi and Nesterenko, 1985; Nesterenko, 1983; Coste et al., 1997; Jayaprakash et al., 2013a, b; Kim et al., 2015; Potekin et al., 2013; Sen et al., 2008). Homogeneous chains correspond to mass ratio ε = 1 (i.e., they are composed of identical heavy granules), and support the propagation of solitary waves (the so-called Nesterenko solitary waves) due to the counterbalance of dispersion and strong nonlinearity from Hertzian granular interactions (Nesterenko, 2001). Considering dimer chains corresponding to 0 < ε < 1, it has been shown that a nonlinear 1:1 resonance phenomenon is realized for the mass ratio ε ≈ 0.5 (the actual value depends on the number of granules) yielding strong passive attenuation of propagating pulses due to intense energy scattering at the interfaces between heavy and light granules (Jayaprakash et al., 2013b; Potekin et al., 2013). Conversely, a nonlinear anti-resonance phenomenon is realized in the dimer chain with ε ≈ 0.125, yielding undistorted solitary pulse propagation (as in the homogeneous chain) and unhindered energy transmission in this highly heterogeneous medium (Jayaprakash et al., 2013b; Potekin et al., 2013). In fact a granular dimer supports countable infinities of such resonances and antiresonances at discrete sequences of the mass ratio ε . These nonlinear acoustical phenomena affect drastically the nonlinear wave scattering at the flexible boundary of the dimer chain. Meanwhile, previous discussion in Section 2.2 has shown that the elastic foundation of the membrane also has significant influence on this nonlinear wave scattering. Accordingly, in the following simulations dimer chains with different mass ratios ε = 1, 0.7, 0.5, 0.25 and 0.125 combined with membranes with different elastic foundations are considered. The aim is to numerically investigate wave transmission and reflection at the highly discontinuous interface of the dimer with the membrane, which is caused by the possible repetitive loss of contact (and subsequent gain of contact) of the last granule of the dimer with the flexible boundary. Throughout this study, all granules in the dimer chains are made out of stainless steel with density ρ b = 7985 Kg/m3 , elastic modulus Eb = 193 GPa, and Poisson’s ratio ν b = 0.3. The radius of the heavy granule is set to R1 = 12.7 mm, whereas the radius of the light granule varies depending on the selected mass ratio ε . The same initial velocity as in Potekin et al., 2016) (due to the applied impulse) of the first heavy granule of the chain is taken, V0 = 172.9 mm/s. For convenience purposes, the membrane used in our simulations is a square steel membrane having dimensions a = b = 1.27 m = 100R1 with thickness 2 mm. The corresponding density, elastic modulus, and Poisson’s ratio of the membrane are ρ m = 15.92 Kg/m2 , Em = 210 GPa, and ν m = 0.3, respectively. Based on these material parameters and dimensions, the normalizations in Eqs. (5)-(10) are determined. In the normalized equations, the radius of the heavy granule is equal to unity, and the membrane measures 100×100. The parameter h used in our approximation is 1/100, so that the contact area between the granule and the membrane are much smaller than the size of the heavy granule. In addition, we assume that the dimer chain is in contact with the center of the square membrane at the position x˜0 = y˜0 = 50, which creates symmetry about the diagonal of the membrane. Hence, due to symmetry the dynamics of the membrane can be fully captured by considering

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only a triangular subsection (1/8) of the membrane. For the simulations we select the truncation order M = 3 × 104 as a reasonable value in terms of the convergence and accuracy as well as the computational time required for the computations. Moreover, taking into the “slow” interaction between the granular chain and the membrane, we set β = 0.005 where the tension in the membrane is physically implementable. In this case, the normalized time required for a nondispersive wave to travel from the contact point (its center) to the nearest  point on the boundary and then reflect back is τ = 100/ β = 1414. Since we want to study the effects of the elastic foundation of the membrane on the nonlinear wave scattering, we perform a set of simulations for eight different normalized values of μ = 0, 10−6 , 10−5 , 10−4 , 10−3 , 10−2 , 10−1 and 1. For each value of the normalized elastic foundation stiffness μ, we consider all the five dimer chains corresponding to the five different mass ratios mentioned above. Moreover, a total simulation time window of normalized time equaling 2100 is considered in all numerical experiments, which has is sufficient to fully capture the nonlinear wave scattering phenomena under investigation; that is, we consider the nonlinear scattering of primary and secondary (reflected) pulses in the granular chain and wavefronts in the membrane until they decrease to low levels with increasing time. Finally, no dissipative effects are taken into consideration in this work. The deformation results of all granules and the membrane obtained in all the simulations are verified to be small and within the requirements of Hertzian elastic contact law and within the approximations of pulse propagation in the dimer chain (Starosvetsky et al., 2017). To gain better insight into the nonlinear wave scattering at the flexible interface as well as the discontinuous features of this interaction, the time series of the non-dimensional elastic deformations of the centers of the last two granules (the 20th and 21st) of the dimer and the center of the membrane, the normalized contact force FC (τ ) between the last granule and the membrane, and the total energy of the integrated dimer chain – elastically supported membrane are discussed hereafter. Given that in the absence of dissipative effects the total energy of the integrated system should be conserved, this last measure is numerically computed to access the accuracy of the numerical simulations over the time window of interest. In addition, all energy measures discussed below are normalized as percentages with respect to the total initial input energy Einput = m1V02 /2. For better depiction of the nonlinear scattering at the flexible interface, and to reveal the highly discontinuous features of the scattering phenomena, we also plot the spatiotemporal evolution of the velocities of the granules of the dimer chain and of the membrane material points lying on the half diagonal connecting its center to one of its corner; these plots clearly and accurately depict the resulting nonlinear and discontinuous wave interactions. In these 2D plots, the section of the membrane is rotated appropriately in order to align it with the axis of the granular chain (cf. Fig. 3). In Fig. 4, the nonlinear wave scattering of solitary pulses propagating in the homogeneous granular chain (with ε = 1) at the flexible interface with the membrane without elastic foundation (μ = 0) is depicted. Note that in the absence of elastic foundation the membrane is a non-dispersive elastic medium and supports the propagation of undistorted elastic waves. Immediately following the impulse excitation, a primary solitary pulse is generated in the chain and propagates from its left end towards the right boundary. This solitary pulse is the main mechanism of energy transmission in the homogeneous chain (Jayaprakash et al., 2011b; Starosvetsky et al., 2017), and its velocity is dependent on the energy; in fact, the energy levels of different solitary waves in the granular chain can be inferred directly from the slopes of their corresponding spatio-temporal velocity plots (where they are represented by approximate straight lines – cf. Fig. 4d). The primary

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Fig. 3. Schematic of the coordinates applied in the spatio-temporal plots.

Fig. 4. Wave scattering at the interface of the dimer chain with ε = 1 and the membrane with μ = 0: (a) Converged normalized deformations of the center of the membrane and the last two granules, (b) converged normalized contact force FC (τ ), (c) energies (%) in the membrane and the granular chain and total energy, and (d) spatio-temporal evolutions of the normalized velocities of the granules and of points on the diagonal half-line of the membrane.

solitary pulse reaches the flexible boundary at around τ = 140, which causes the first large jump in the contact force (cf. Fig. 4b). Afterwards, the last granule of the chain separates from its neighboring (20th) granule and keeps moving forward, forcing the membrane to move in the same direction, and initiating the first (primary) propagating wave front in the membrane. In the meantime, the 20th granule is left behind while still moving at a positive constant velocity (i.e., towards the membrane). At about τ = 270, the last granule and membrane reach their maximum axial deformations simultaneously, and then begin to move backwards, with the last granule remaining continuously in contact with the membrane. It can be observed from the backward motion of the membrane that the primary wave on the membrane leaves behind

a wake as it propagate outward. However, at τ = 357 a collision between the last granule and its neighboring one occurs, which initiates the first reflected (backward) solitary pulse in the homogeneous chain. It is worth noting that, when the reflected pulse reaches the left end of the chain, the first granule of the chain “takes off” due to the free boundary condition (cf. Fig. 4d). The membrane is subsequently compressed by the last granule, which leads to a second peak in the contact force. Following this, the last heavy granule and the membrane, while still maintaining contact, move forward and then turn back together again until τ = 517. At that time instant, the membrane separates from the last granule due to its decreasing velocity, while that granule gets into a state of uniform motion. Consequently, the contact

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force becomes zero at this time. However, the 20th granule having gained a positive velocity from the collisions with the 19th granule, collides with the last granule inevitably at τ = 532, initiating the second reflected solitary pulse in the homogeneous chain and pushing the 21st granule once again back towards the membrane, interacting with it again at τ = 600 and causing a third peak in the plot of the nonlinear contact force. A sequence of separations and collisions between the last two granules is realized thereafter yielding further small jumps in the contact force (cf. Fig. 4b). Due to this sequence of collisions and separations, strongly nonlinear wave scattering occurs between the homogeneous granular chain and the membrane resulting in intense energy exchanges between them, with the reflected solitary pulses in the chain and the generated wavefronts in the membrane getting increasingly weaker. At later times there is a short separation between the last granule and the membrane from τ = 1380 to τ = 1415, and after that, they move together again. However, since all edges of this square membrane are clamped, the waves propagating in the membrane will be reflected back by the fixed boundaries. Hence, after τ = 1558 the primary reflected wavefront returns to the center of the membrane and excites the granular chain, pumping a part of energy back into the chain in the form of a backward solitary pulse and causing a relatively higher jump in the contact force. During this time, the last heavy granule continues to be in contact with the membrane until τ = 1875. Thereafter the last granule and the membrane separate from each other and will not get in touch again. Throughout the simulation window we establish that the total energy of the combined granular chain – membrane system is conserved (with the exception of a small error in the initial high-energetic regime of the response), which affirms the computational accuracy of the simulation. The maximum error 1.13% occurs at around τ = 270 when the center of the membrane reaches the maximum displacement. At this time, the displacement gradient close to the contact point is relatively large due to the intense wave interaction, so most of the energy in the membrane is stored in the form of strain energy and is concentrated over a small area close to this point. In turn, the local energy concentration or large gradient leads to numerical inaccuracies in the computation of energy integrals (4). This error can be eliminated by taking more discretization points for the membrane into account, at the cost of correspondingly longer computational time. It is noted that nearly all of the energy imparted in the homogeneous granular chain by the applied impulse is transmitted to the membrane soon after the primary pulse interaction, but as time progresses, more than half of this energy backscatters eventually to the chain. Specifically, the maximum energy transferred to the membrane during primary pulse scattering is 96.5% of the total energy in the system, whereas the remaining residual energy that is eventually transferred to the membrane (measured at the end of the simulation) is 37.91% of the total energy (cf. Fig. 4c). It is clear that the nonlinear wave scattering phenomena at the discontinuous flexible boundary of the granular chain is greatly affected by its mass ratio ε . Indeed, it is apparent that the fact that the homogeneous chain supports the propagation of undistorted solitary pulses affects drastically the transmitted and reflected waves at the point of contact with the membrane, as well as the resulting energy exchanges. To investigate the effects of strong pulse dispersion in the granular chain on wave scattering at the flexible interface, we consider next the case of a dimer chain in 1:1 resonance with mass ratio ε = 0.5 (Jayaprakash et al., 2011b; Starosvetsky et al., 2017) in contact with a membrane without elastic foundation, μ = 0. In fact, for this specific mass ratio there occurs maximum pulse dispersion and attenuation in the dimer, with energy of the propagating pulse being scattered from low to high frequencies in the form of highly nonlinear traveling “wavetails” in

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the wake of the propagating pulse (Kim et al., 2015; Starosvetsky et al., 2017). This results in substantial attenuation of the propagating pulse, contrary to the case of the homogeneous chain, where the energy is localized in the propagating solitary pulse without distortion. The results for this case are depicted in Fig. 5. It can be observed that the nonlinear wave scattering phenomena that occur at the flexible interface in this case significantly differ compared to the previous homogeneous case. After the application of the impulse to the first heavy granule, the primary pulse generated in the dimer chain suffers strong dispersion and scattering during its propagation through the chain (cf. Fig. 5d), and at τ = 120 arrives at the right flexible boundary with a greatly distorted waveform. This initiates a primary wavefront in the membrane, and corresponds to the first jump in the contact force (cf. Fig. 5b). At τ = 275, the membrane and the last granule reach their maximum deformations and start to move backwards for the first time. Shortly after that, at τ = 285, the last heavy granule collides with its neighboring light granule, leading to a reflected pulse back into the dimer chain. It is worth noting that separations and collisions between the last two granules take place repeatedly as a result of the large mass mismatch between them. The light granule, in particular, undergoes relatively fast oscillations between its neighboring heavy granules. These collisions force the last granule to gradually compress the center of the membrane, and cause several jumps in the contact force. This initiates a sequence of propagating wavefronts in the membrane and of backward reflected pulses in the dimer chain. Due to the excitation of the transient breather at the site of the last light granule the intense interaction between the 21st granule and the membrane lasts for a relatively longer time, compared with the homogeneous chain. At τ = 900, the last granule loses contact with the membrane permanently. One notable difference here is that, even when the primary reflected wavefrom from the fixed boundaries of the membrane arrive to the point of contact with the dimer at τ = 1545, no contact with the last granule of the dimer occurs. In fact, the local oscillations of the last light granule due to the excitation of the transient breather drastically affect the pulse propagation and energy transmission in the dimer chain. For example, after some local oscillations the dimer chain splits into two subsets at around τ = 210, with the subset of the ten left granules (1 to 10) moving backwards, while the rest of the granules (11 to 21) move forward, i.e., towards the membrane. When the first reflected pulse from the flexible boundary of the dimer chain arrives at the contact point and initiates a reflected backward pulse in the dimer, the splitting of the dimer chain and the residual local oscillations induce rapid dispersion of this pulse. These strong attenuation and dispersion phenomena, in turn, drastically influence the energy transmission and reflection at the flexible boundary of the dimer. Specifically, the peak energy transferred to the membrane is now reduced to 62.2% of the input energy of the system, which is much smaller compared to the previous homogeneous chain. Moreover, the residual energy that is retained by the membrane is now 29.04% of total input energy, which, again, is smaller compared to the homogeneous case; and this result occurs, in spite of the fact that the primary wave interaction between the dimer chain and the membrane lasts a longer period of time. Additionally, the maximum contact force also decreases compared to the homogeneous case. Again, the reduction in transferred energy and contact force is mainly attributed to the realization of resonance in this dimer chain, since the resulting low-to-high frequency energy scattering in the tail of the primary propagating pulse (Kim et al., 2015) results in less energy being transmitted to the location of contact compared to the homogeneous chain case. Finally, the total energy of the entire system is also numerically conserved, while the maximum error is only 0.6%, smaller

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Fig. 5. Wave scattering at the interface of the dimer chain with ε = 0.5 and the membrane with μ = 0: (a) Converged normalized deformations of the center of the membrane and the last two granules, (b) converged normalized contact force FC (τ ), (c) energies (%) in the membrane and the granular chain and total energy, and (d) spatio-temporal evolutions of the normalized velocities of the granules and of points on the diagonal half-line of the membrane.

than the homogeneous chain case. This can be attributed to the “milder” interaction between the dimer chain and the membrane. This is an interesting result from a practical standpoint (especially for the shock mitigation point of view), since it shows that granular media can be designed to inherently attenuate propagating pulses and, thus, significantly reduce contact forces and transmitted energy across their flexible boundaries. The results for the case of the dimer granular chain with mass ratio ε = 0.125 and a membrane without elastic foundation (μ = 0) are depicted in Fig. 6. For this mass ratio the dimer is in a state of anti-resonance (Starosvetsky et al., 2017) and supports the propagation of undistorted solitary pulses (albeit of a different waveform compared to the homogeneous case with ε = 1). In this case we note that a primary solitary pulse is generated immediately after the impulsive excitation at the left boundary of the dimer, which arrives at the right end of the chain at around τ = 110 forcing both the last heavy granule and the center of the membrane to move forward and initiating the primary propagating wavefront in the membrane. The maximum deformations of the last granule and the membrane (i.e., the maximum compression of the membrane) are realized at around τ = 234, after which they start to move backwards and separate temporarily at τ = 1338 (cf. Fig. 6a). Meanwhile, the primary wavefront propagates through the membrane and is reflected back by the clamped boundary. At τ = 1606, this reflected wavefrom reaches the center of the membrane so that the membrane gets into contact with the last heavy granule again.

And shortly afterwards, at τ = 1724, they lose contact permanently. Whereas the pattern of primary pulse propagation in this dimer resembles the homogeneous case (as, despite the large mass inhomogeneity, it supports the propagation of undistorted solitary pulses), in this case there are two important new features that affect the nonlinear wave scattering at the interface with the membrane (cf. Fig. 6d). First, an intense transient breather (Starovetsky et al., 2012) is generated at the site of the 20th (light) granule as it undergoes a sequence of collisions and “free flights” between its neighboring heavy granules; this is caused by the large mass mismatch between the heavy and light granules in this dimer, and as discussed in (Starovetsky et al., 2012) yields rapid scattering of energy in higher frequencies and slow “energy leakage” to the rest of the dimer. Specifically, the energy entrapped by the transient breather is eventually released back to the rest of the dimer chain and the membrane, but at a slower time scale compared to the “fast” oscillating frequency of the localized oscillations of the transient breather. This slow energy transmission from the transient breather to the rest of the system can be clearly discerned in Fig. 6d. Moreover, the excitation of the transient breather results in a series of small peaks in the contact force (cf. Fig. 6b), and initiates a sequence of secondary weak wavefronts in the membrane (from τ = 320 to τ = 950) which continuously radiate energy from the dimer to the membrane. Second, the excitation of the localized transient breather completely changes the number and intensity of

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Fig. 6. Wave scattering at the interface of the dimer chain with ε = 0.125 and the membrane with μ = 0: (a) Converged normalized deformations of the center of the membrane and the last two granules, (b) converged normalized contact force FC (τ ), (c) energies (%) in the membrane and the granular chain and total energy, and (d) spatio-temporal evolutions of the normalized velocities of the granules and of points on the diagonal half-line of the membrane.

the reflected solitary pulses in the dimer compared to the homogeneous chain. Indeed, each collision during the breather initiates a reflected solitary pulse back to the dimer (since this is the only way that energy can be transmitted in this dimer (Starosvetsky et al., 2017), but with decreasing amplitude as time increases (this is verified by studying the velocities of the reflected solitary pulses in Fig. 6d). This “fan” of reflected solitary pulses was completely absent in the homogeneous chain (cf. Fig. 4d), and is a finding that highlights once more the significant effect of the transient breather on the nonlinear acoustics of this system. In this case, the maximum energy transferred to the membrane is 91.27% of the total input energy, while the retained energy in the membrane is 36.94%. We note that compared to the homogeneous granular chain, the reduction in transferred energy and contact force in this case is significantly smaller than the case of dimer chain with ε = 0.5, which can be attributed to more efficient and intense energy transmission by the undistorted propagating primary solitary pulses due to the realization of anti-resonance in this dimer chain. The simulations for other values of mass ratios ε and non-zero elastic foundation of the membrane μ>0 are analyzed in the same way. In Fig. 7 we depict the spatiotemporal evolutions of the normalized velocities of granular dimers with ε = 1, 0.5, and 0.125 with μ = 10−3 . Compared to the non-dispersive membrane without elastic foundation (μ = 0) of the previous cases, distributed elastic foundation increases the stiffness of the flexible boundary, yield-

ing an increase of the natural frequencies of the membrane and naturally bringing about faster acoustics (i.e., the nonlinear acoustical phenomena occur on faster time scales). Moreover, strong dispersion of propagating waves now occurs in the membrane, leading to absence of coherent wavefronts in this case. Considering the demonstrated numerical convergence of the total energy computation in the previous integrated dimer chain – membrane systems, to save computational time in the following computational results, the energy of the membrane and its elastic foundation is estimated based on the difference between the input energy and the instantaneous energy of the dimer chain, so that the total energy of the system is always conserved. Considering the simulations depicted in Fig. 7, for the homogeneous chain and the dimer chain with ε = 0.125 which support undistorted solitary wave propagation, there occur relatively large peaks of the contact force FC (τ ), and nearly the entire input energy is transferred to the membrane and its elastic foundation after the primary wave scattering interaction. For the dimer chain with ε = 0.125, a major part of the input energy gets scatters in the interior of the dimer due to 1:1 resonance, leading to a reduced peak of the contact force. However, the notable difference in these cases compared to the previous simulations with nondispersive membrane is that most of the transferred energy to the membrane shortly backscatters to the granular chains, and only a very small proportion of input energy is retained eventually by the membrane. This is solely attributed to the elastic foundation which

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Fig. 7. Wave scattering at the interface of the dimer chain with (a) ε = 1, (b) ε = 0.5, and (c) ε = 0.125, and a the membrane with μ = 10−3 : The spatio-temporal evolutions of the normalized velocities of the granules and of points at the diagonal half-line of the membrane are shown.

Table 1 Peak energy transferred to the flexible boundary as a percentage of input energy.

Table 2 Residual energy retained by the flexible boundary as a percentage of input energy.

Stiffness μ

ε=1

ε = 0.7

ε = 0.5

ε = 0.25

ε = 0.125

Stiffness μ

ε=1

ε = 0.7

ε = 0.5

ε = 0.25

ε = 0.125

μ=1 μ=10−1 μ=10−2 μ=10−3 μ=10−4 μ=10−5 μ=10−6 μ=0

94.97 95.61 95.96 96.26 96.47 96.50 96.50 96.50

76.91 77.45 77.75 77.99 78.16 78.19 78.19 78.19

59.41 60.91 61.62 62.03 62.17 62.19 62.19 62.20

87.29 87.73 87.99 88.24 88.42 88.43 88.44 88.45

90.17 90.59 90.84 91.07 91.24 91.26 91.27 91.27

μ=1 μ=10−1 μ=10−2 μ=10−3 μ=10−4 μ=10−5 μ=10−6 μ=0

0.00 0.03 0.46 5.32 25.94 37.39 37.90 37.91

0.00 0.03 0.41 3.43 19.93 29.77 33.92 34.44

0.00 0.02 0.27 1.70 9.23 23.19 28.43 29.04

0.00 0.03 0.38 2.32 20.66 30.33 34.71 34.90

0.00 0.04 0.24 2.60 23.62 33.37 36.53 36.94

stiffens the membrane interface and also causes strong dispersion of transmitted wavefronts. A series of simulations were performed for additional mass ratios and stiffnesses of the elastic foundation and the results are summarized in Table 1 where the maximum normalized energy transferred from the granular chain to the membrane is given, and Table 2 where the residual normalized energy retained by the membrane is listed. Both energy measures are given as percentages of the input energy of the impulse, and the residual energy is measured at the end of simulation when the granular chain and the membrane have completely separated. From Table 1 we note that for fixed μ the maximum energy transmitted to the membrane is strongly dependent on the mass

ratio of the dimer chain, being highest for ε = 1 (homogeneous chain) and lowest for ε = 0.5 (case of 1:1 resonance). However, considering the results of Table 2 we conclude that the residual energy retained eventually by the membrane and its elastic foundation appears to depend more on the foundation stiffness than the mass ratio of the granular chain, and is inversely related to the stiffness μ. For the cases with μ = 1, although a large proportion of energy is initially transferred into the membrane and its elastic foundation, nearly all (almost 100%) of this energy is eventually backscattered into the granular chain. It appears that in this case, the elastic foundation is so stiff that the membrane behaves like a fixed right boundary for the granular chain.

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Fig. 8. Comparison of peak contact force between the membrane and the granular chain.

The maximum contact forces between the last granule and the membrane for all performed simulations are compared in Fig. 8. It is evident that there is a positive correlation between the maximum contact force and the stiffness of the elastic foundation of the membrane, with the stiffest foundation yielding the largest contact force. Meanwhile, for fixed elastic foundation, the homogeneous chain would yield the largest peak contact force due to the energy localization in the propagating solitary wave that impedes on the membrane, while the dimer chain with ε = 0.5 leads to the smallest values of the contact forces because of interior energy scattering of propagating pulses. 4. Energy transmission between two granular chains and flexible boundary A natural extension of this study is to investigate wave scattering and energy transmission in the nonlinear acoustics of multiple granular chains with the previous flexible boundary. Accordingly, in this section, we consider the impulsively excited system composed of two granular chains and the square membrane (cf. Fig. 9). Both granular chains are aligned orthogonally to the membrane, with one being a dimer chain labeled as “chain 1 that makes contact with the center of the membrane at point 1 with normalized coordinates (x˜1 , y˜1 ) = (50, 50 ), and the other, a homogeneous chain placed on the other side of the membrane and interacts with it at point 2 with normalized coordinates (x˜2 , y˜2 ). We assume that an impulse excitation of intensity mV0 is applied to the free end of chain 1 at time τ = 0, generating strongly nonlinear pulses that propagate towards the flexible membrane. Depending on the mass ratio of the dimer chain 1 and the stiffness of the elastic foundation of the membrane, a portion of the energy induced by the impulse is transferred to the membrane in the form of propagating wavefronts, and the remaining into energy is reflected back to the dimer. After the propagating wavefronts in the membrane arrive at point 2, the homogeneous chain 2 is excited, initiating Nesterenko solitary pulses in it. Eventually, chain 2 detaches from the membrane and its energy is retained permanently. It is anticipated that the strongly nonlinear wave scattering that occurs at the highly discontinuous interface between the directly excited dimer chain 1 and the membrane, as well as the distance between the two contact points 1 and 2 should drastically affect the energy transmitted from the dimer chain 1 to the homogeneous chain 2. A set of simulations for varying mass ratio of chain 1, elastic foundation of the membrane, and distance D were carried out to study the effects of these parameters on the energy

Fig. 9. Schematic of: (a) the two granular chains at a normalized distance D interacting with the rectangular membrane, and (b) the reference case for D = 0.

transmission between the two granular chains through the connecting membrane. To this end, the iterative-interpolative computational algorithm developed for the simulations of Section 3 was extended for the system of Fig. 9, as described in the Appendix. In the present case the response of the membrane can be expressed as a superposition of the two responses due to the contact force F˜C1 (τ ) at point 1, and the contact force F˜C2 (τ ) at point 2,

˜ (x˜, y˜, τ ) = ζ G1 (x˜, y˜, τ ) ∗ F˜C1 (τ ) + ζ G2 (x˜, y˜, τ ) ∗ F˜C2 (τ ) w

(18)

where G1 (x˜, y˜, τ ) and G2 (x˜, y˜, τ ) are the Green’s functions of the membrane due to unit impulses applied at points 1 and 2, respectively. To avoid numerical instabilities due to singularities in the Green’s functions, they were approximated as described in Section 3. For example, the approximate Green’s function at point 2 due to a unit impulse applied at point 1 is given by,

G¯ 12 (τ ) =

M M 4a˜b˜    2 2 −1 iπ h iπ x˜1 iπ x˜2 i j ωi j sin 2 sin sin a˜ a˜ a˜ h4 π 4 i=1 j=1

× sin

2

jπ h jπ y˜1 jπ y˜2 sin sin sin ωi j τ b˜ b˜ b˜

(19)

where the notation of Section 3 was utilized. In the following simulations we consider three different mass ratios for the dimer chain 1, ε = 1, 0.5 and 0.125, and two values for the elastic foundation of the membrane, μ = 0 corresponding to a membrane without elastic foundation, and μ = 10−4 . Moreover, the distance D between contact point 1 (fixed at the center of the membrane x˜1 = y˜1 = 50) and contact point 2 is varied by changing the location of point 2. In this work, point 2 is placed along the diagonal of the membrane, and D is varied from 0 (corresponding to x˜2 = y˜2 = 50) up to 35.35 (corresponding to x˜2 = y˜2 = 75). The system with D = 0 is the special case since point 2 coincides with point 1, and will be referred to as the “reference system” (cf. Fig. 9b). The parameters used in the simulations are given in Section 3, and the total simulation time window is similarly set to 2100, when both granular chains 1 and 2 have separated from the membrane and do not interact with it again. Again, no dissipative effects are taken into consideration. As a check on the physical validity of our results, we verify the elastic deformations of the granules and the membrane are small and fulfill the restrictions of

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Fig. 10. Reference system (D = 0): Spatio-temporal evolutions of the normalized velocities of all granules of chains 1 and 2 for a membrane with μ = 0 and a dimer chain 1 with (a) ε = 1, (b) ε = 0.5, and (c) ε = 0.125.

the Hertz contact law and the acoustical modeling of the granular chains (Starosvetsky et al., 2017). In Fig. 10 we depict the spatiotemporal evolutions of the normalized velocities for the reference case (D = 0) and a membrane without elastic foundation (μ = 0). We note that although the two contact points have the same coordinates, the two granular chains can only interact indirectly through the membrane. Based on our discussion in Section 3, the membrane would impede the forward movement of the last granule of chain 1. Therefore, all granules in chain 1 will move backwards eventually, which can be deduced from the results of Fig. 10. For ε = 1, when the solitary pulse propagating in the homogeneous chain 1 reaches the membrane, a considerable portion of the impulsive energy is transferred to the homogeneous chain 2 in the form of propagating solitary wave, resulting in separation between chain 2 and the membrane (cf. Fig. 10a). Meanwhile, the wave scattering at the flexible interface yields a set of weak reflected solitary pulses in homogeneous chain 1. In this case, 94.17% of the input energy is transferred in chain 2, whereas chain 1 retains 4.69% and the membrane only 1.14% of this energy. For ε = 0.5 (cf. Fig. 9b) there is intense internal pulse dispersion due to 1:1 resonance (Starosvetsky et al., 2017), so the energies retained by chain 1, chain 2 and the membrane are 28.07%, 68.96% and 2.97% of the input energy, respectively. For ε = 0.125 solitary pulses propagate in chain 1 due to anti-resonance

(Starosvetsky et al., 2017), so chain 1, chain 2 and the membrane retain 6.49%, 92.28% and 1.23% of the input energy. It is of interest to study how wave scattering at the flexible interface changes when the distance between the two chains increases from zero. The spatio-temporal normalized velocity plots for the two granular chains and points on the membrane’s diagonal section between two the contact points 1 and 1 are presented in Fig. 10 for a membrane without elastic support (μ = 0), D = 0.72 (corresponding to x˜2 = y˜2 = 50.5), and dimer chain 1 with ε = 1, 0.5 and 0.125. In these three cases, the wavefronts propagating on the membrane provide the means to transfer energy from chain 1 to chain 2. Since the amplitudes of propagating wavefronts in the membrane attenuate drastically away from contact point 1, the transferred energy to chain 2 is expected to be quite small. Accordingly, the velocity contours for chain 2 in Fig. 10 are four times magnified for presentation purposes; in addition, the non-dimensional length of the depicted diagonal section of length D that connects the two chains is stretched five times, since D is relatively small compared to the lengths of the two chains. Referring to the simulations of Fig. 11, we note that the acoustics of chain 1 share similarities to the results for the single dimer chain in Section 3. Indeed, the energy localization in Nesterenko primary solitary pulses for ε = 1, the dimer splitting for ε = 0.5, and the excitation of a strong transient breather close to the interface

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Fig. 11. Spatio-temporal evolutions of the normalized velocities of all granules and of material points on the diagonal segment connecting the two contact points 1 and 2, for μ = 0 and D = 0.72, for chain 1 with (a) ε = 1, (b) ε = 0.5, and (c) ε = 0.125; for clarity the velocities in chain 2 are magnified (×4), and the length of the section between contact points is magnified (×5).

for ε = 0.125 can be clearly deduced. Moreover, it is clear that a sequence of Nesterenko solitary waves is generated in chain 2 for all the three mass ratios. These solitary pulses are not as strong as their counterparts in the reference cases of Fig. 10, as can be deduced from the velocities of the solitary pulses in the spatiotemporal plots of Fig. 11. This indicates the non-zero distance between the contact points of the two chains has significantly reduced the energy transferred to chain 2. In these simulations the residual energy transferred in chain 2 at the end of the simulation window is 5.62% (for ε = 1), 7.30% (for ε = 0.5) and 5.71% (for ε = 0.125), which are much smaller than the corresponding energy measures for the reference case. It is interesting to note that for the case ε = 0.5 the interaction between the first granule of chain 2 and the membrane lasts longer than the cases for ε = 1, and ε = 0.125, which explains the increased energy transfer in chain 2 in this case. A series of additional simulations were performed for different values of elastic foundation μ and mass ratios ε . The plot in Fig. 12a compares the residual energy in chain 1 and in Fig. 12b the transferred energy in the membrane and its foundation for varying distance D. Note that the horizontal axis in these plots is in logarithmic scale, and, accordingly, we give a small artificial coordinate shift of 0.001 to the value D = 0 in order to include that singular value in the logarithmic axis.

Similar to the observations in Section 3, the results for ε = 0.125 (case of anti-resonance) are always close to the results of the homogeneous case, due to solitary pulse propagation. For fixed distributed stiffness μ, chain 1 retains the highest energy for ε = 0.5 (case of resonance) and the lowest energy for ε = 1. On the contrary, the energy transferred in the membrane and its elastic foundation is highest for ε = 1 and lowest for ε = 0.5. Moreover, the introduction of elastic foundation with μ = 10−4 leads to an increase in the stiffness of the flexible boundary, and results in higher residual energy in chain 1 and less energy transferred in the membrane. It is clear that the energy transferred in the flexible boundary also depends strongly on the distance D, with more energy transferred with increasing D. However, we note that the energy measures in Figs. 12a,b reach saturation levels when D is sufficiently large (e.g., when D>10 in this work). This is reasonable, since the interaction effects between the two chains are expected to be weak for sufficiently large distance between them, so that chain 2 hardly affects the overall acoustics of wave scattering. Fig. 13 compares the energy transferred in chain 2 at the end of simulation window for varying distance D (the case D = 0 is excluded). It is obvious that the energy in chain 2 is strongly dependent on D. When chain 2 is sufficiently close to chain 1, up to 28% is transferred, whereas for D>10 this energy is negligible indicating negligible interaction between the two chains.

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5. Conclusions

Fig. 12. Residual energy in chain 1 and energy transferred in the membrane for varying mass ratio ε , foundation stiffness μ and distance D; the left-most data point corresponds to D = 0 and was shifted to fit in the logarithmic scale.

We studied nonlinear wave scattering at the flexible interface between one or more granular dimer chains and a membrane resting on a linear elastic foundation. The governing equation of motion is transformed into non-dimensional form, so that the motion of the system depends on several dimensionless parameters. A computational algorithm that combines successive iterations and interpolations is applied to accurately analyze the highly nonlinear and discontinuous interactions of the granular chains with the flexible boundary. The computational algorithm relies on the accurate computation of the highly discontinuous contact forces between the granular chains and the membrane, which, in turn, enable the computation of the transverse responses of the material points of the membrane through a Green’s function formulation. By performing parametric numerical simulations, we studied the effects of two non-dimensional parameters, namely the mass ratio ε of the impulsively excited dimer chain, and the stiffness μ of the elastic foundation), on the dynamics of the nonlinear wave scattering at the flexible interface of this system. For the case of a single granular chain, the homogeneous chain with ε = 1 supports the propagation of Nesterenko solitary pulses, and, hence, yields intense wave scattering between the chain and the membrane, through which more energy is transferred into the flexible boundary. However, for the dimer chain with ε = 0.5, the peak contact force between the chain and membrane as well as the energy transferred to flexible boundary are significantly smaller than that of the homogeneous chain, due to realization of 1:1 resonance in the dimer (Starosvetsky et al., 2017) and strong internal dispersion of propagating pulses. Moreover, the dimer chain with ε = 0.125 gives rise to similar results to the homogeneous chain since it also supports the propagation of (undistorted) solitary pulses due to anti-resonance (Starosvetsky et al., 2017). In additional, it is observed that the energy that is eventually transferred to the flexible boundary is inversely proportional to the stiffness of the elastic foundation. These observations share some similarities with the findings in (Potekin et al., 2016) for a granular dimer interacting with a flexible string at its boundary. For the case of two granular chains, the distance D between the two chains is a major factor governing the wave scattering at the interface. When D is sufficiently large homogeneous chain 2 has little or even negligible effects on the acoustics of the system. However, when the two chains are close enough, the mass ratio of the dimer chain 1, along with the stiffness of the elastic foundation of the membrane play an important role in the energy exchanges between the subsystems. More importantly, the methods and results of this work contribute towards the accurate modeling of strongly nonlinear and discontinuous nonlinear acoustical systems having linear flexible boundaries and is a further step towards the predictive design of granular interfaces with complex flexible boundaries as shock and vibration mitigators of practical acoustical systems. Acknowledgment This work was supported in part by the China Scholarship Council (Grant 201706160084) which funded the visit of Qifan Zhang to the University of Illinois at Urbana – Champaign.

Fig. 13. Energy transferred in chain 2 for varying mass ratio ε , foundation stiffness μ and distance D.

Appendix. Extension of the computational algorithm of (Potekin et al., 2016) for the membrane Considering the model in Fig. 1, for a sufficiently small time interval the integrated chain-membrane system could be approximately “decoupled” into two sub-systems: (i) The granular chain in contact with a fictitious “rigid wall” with a certain prescribed

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Fig. A-1. Qualitative depiction of the converged contact force and deformation of the membrane at the contact points obtained by iterations at successive time steps; linear interpolation of the force is applied within each time step ࢞τ .

displacement, and (ii) the membrane with its elastic foundation excited by the estimated contact force at the contact point. The motion of all granules in the chain-wall system can be obtained by computationally solving the nonlinear equations (8), while the deformation of the membrane at the point of contact can be calculated by means of the convolution (11) using the Green’s function. At this point, the key to the problem is to accurately compute the coupling terms between the two sub-systems, i.e., the deflection of the last granule and the membrane at the contact point, as well as the corresponding contact force. In the work by Potekin et al. (2016), a computational algorithm valid for this issue was developed and applied to accurately compute the nonlinear wave scattering at the interface of the 1D dimer granular chain and a linear string on an elastic foundation. Through iteration and interpolation at successive time steps, this algorithm ensures simultaneous convergence of the contact force and the displacements of the granules of the chain, as well as the deflection of the string. The full details of this algorithm can be found in (Potekin et al., 2016). Here we provide a brief discussion of its present extension to our model with multiple granular chains. Taking the system with two granular chains as an example (cf. Fig. 9a), the chains are in contact with the flexible boundary at contact points 1 and 2. The total time window of interest in our simulations is discretized into p sub-time intervals having a uniform small time step of ࢞τ (cf. Fig. A-1). The contact forces between the membrane and its neighbouring granules at points 1 and 2 are denoted by F1 and F2 , respectively. Similarly, we denote the deflections of these two contact points of the membrane by w1 and w2 . The axial displacements of the arbitrary j − th granule in chain 1 and chain 2 are uj and vj , respectively. The iteration number within one time step is denoted by k. On the k − th iteration at the (i + 1) − th time interval, i.e., for τ i ≤ t ≤ τ i + 1 , the computed displacements of the j − th granules in the two chains are denoted by (u j )ki+1 and (v j )ki+1 , respectively, while the contact forces and

displacements of the membrane at the two contact points are denoted by (F1,2 )ki+1 and (w1,2 )ki+1 , respectively,

(u j )ki+1 ≡ (u j )k (τi+1 ), (

)

w1,2 ki+1

(F1,2 )ki+1 ≡ (F1,2 )k (τi+1 ),

≡ (w1,2 ) (τi+1 ) k

(A-1)

The convergent iteration number is denoted by k = C when either one of the following convergence criteria is satisfied,

 C  (F1 )i+1 − (F1 )C−1  < εabs , i+1    (F1 )Ci+1 − (F1 )C−1  i+1  or  C−1  < εrel , (F1 )i+1

 C  (F2 )i+1 − (F2 )C−1  < εabs i+1    (F2 )Ci+1 − (F2 )C−1  i+1   C−1   < εrel (F2 )i+1

(A-2)

where ε abs is the absolute tolerance and ε rel is the relative tolerance. Note that the value of C is expected to vary for different time intervals. At each time step of width ࢞τ , the three sub-systems shown in Fig. A-2 are considered separately. In order to achieve simultaneous convergence of the coupling terms, a sequence of iterations is performed successively until the contact forces meet either one of the convergence criteria. A schematic for the kth iteration in the (i + 1)th time interval is shown in the plots of Fig. A-2. Suppose that the converged displacements of the  i C chains ir=0 (u j )Cr , r=0 (v j )r , and the displacements of the memi C brane r=0 (w1,2 )r , as well as the corresponding contact forces i C r=0 (F1,2 )r of previous time steps have been obtained. At this time interval, the initial conditions of the two chains (at time instant τ i ) are the converged displacements (u j )Ci , (v j )Ci and velocities of the granules (u j )Ci , (v j )Ci calculated at the previous ith time interval. In addition, the initial prescribed displacements of the fictitious “rigid walls” for this particular iteration are set equal −1 to (w1,2 )ki+1 , which are computed in the previous (k − 1)th iteration. For the chain – fictitious “rigid wall” systems, the responses of the chains at this time interval are computed by numerically solving Eq. (8). Accordingly, the forces applied to the fictitious “rigid

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(w1,2 )ki+1 = (wˆ 1,2 )ki (τi+1 ), which in turn, become the prescribed displacements of the fictitious “rigid walls” for the next iteration (cf. Fig. A-2). This iteration-interpolation scheme continues until the convergence criterion (A-2) is satisfied, after which we proceed to the next time step. Through the repetition of this iterationinterpolation process at each time step, we can finally achieve the converged results for the entire time interval of interest. This computational algorithm can be easily extended to a more general class of granular interfaces composed of multiple 1D ordered or disordered chains or even 2D networks, as long as the required Green’s functions of the flexible boundaries at the points of contact can be numerically or explicitly computed. References

Fig. A-2. Schematics of the auxiliary sub-systems with regard to the contact forces, the displacements of the granular chains, and the deflections at the contact points of the membrane for the kth iteration at the (i + 1)th time interval; the chains are in contact with fictitious “rigid walls” with prescribed corresponding displacements determined by the previous iteration, so one can compute the estimated contact forces for this iteration, which, in turn, are applied as the excitations of the membrane to get the corresponding deflections for the next iteration.

walls” by the two chains, (F1,2 )ki+1 , can be directly computed by Eq. (10) and subsequently examined by (A-2). If (F1,2 )ki+1 does not meet either one of the convergence criteria, more iterations need to be performed. Then, we linearly interpolate the forces between the time instants τ i and τ i + 1 using the following relation:

(Fˆ1,2 )ki (τ ) =

τi+1 − τ τ − τi (F1,2 )Ci + (F )k , τi ≤ τ ≤ τi+1 τ τ 1,2 i+1 (A-3)

The interpolated forces in this iteration, combined with the  −1 converged forces from previous time steps, ( ir=1 (Fˆ1,2 )Cr ) ∪ (Fˆ1,2 )ki are applied to the membrane at the contact points as depicted in Fig. A-2. Thus, the corresponding deflections of the membrane at contact points 1 and 2 in this iteration can be computed using numerical convolution,  

  

   i −1  i −1  C i −1  C C  k  k  k ˆ1 w



r=1 i −1 r=1



ˆ2 w

r

C r

ˆ1 ∪ w





ˆ2 ∪ w

i

k i

Fˆ1

=

 =

r=1 i −1 r=1

r

 C Fˆ1

r

∪ Fˆ1



i

 k

∪ Fˆ1

i

∗ G¯ 11 +



Fˆ2

r=1

 ∗ G¯ 12 +

i −1

r=1

r

 C Fˆ2

r

∪ Fˆ2



i

 k

∪ Fˆ2

i

∗ G¯ 21

 ∗ G¯ 22

(A-4) where G¯ 11 and G¯ 12 are the converged Green’s functions of the membrane at contact points 1 and 2 due to a unit impulse applied at point 1, cf. Eqs. (17) and (19), and G¯ 21 and G¯ 22 are the Green’s functions of points 1 and 2 due to an impulse applied at point 2. Following this computation, the deflections of the contact points of the membrane at the time instant τ i + 1 are given by

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