Demonstration of accelerating and decelerating nonlinear impulse waves in functionally graded granular chains

Abstract

We propose a tunable cylinder-based granular system that is functionally graded in its stiffness distribution in space. With no initial compression given to the system, it supports highly nonlinear waves propagating under an impulse excitation. We investigate analytically, numerically and experimentally the ability to accelerate and decelerate the impulse wave without a significant scattering in the space domain. Moreover, the gradient in stiffness results in the scaling of contact forces along the chain. We envision that such tunable systems can be used for manipulating highly nonlinear impulse waves for novel sensing and impact mitigation purposes.

This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

1. Introduction

A granular crystal is a closely packed assembly of systematically arranged granules that generally interact as per the Hertzian contact law. These discrete platforms have become increasingly popular in recent years due to their intrinsic tunability in terms of supporting a wide range of interesting wave physics spanning across linear, weakly nonlinear and strongly nonlinear regimes. Consequently, based on the rich wave physics, researchers have proposed their use in a plethora of engineering applications, e.g. impact mitigation, signal filtering, energy harvesting and non-destructive evaluation [1–5].

The most fundamental granular crystal is a one-dimensional (1D) assembly of monodisperse spheres, i.e. a homogeneous spherical granular chain. If there is no initial compressive force applied to the chain, conventional linear elastic waves are forbidden to propagate due to the essential nonlinearity of the Hertzian contact law [6]. Nesterenko, in his pioneering work, showed that this ‘sonic vacuum’ rather supports a localized nonlinear travelling wave, i.e. solitary wave, under an impulse given to the chain [1]. The emergence of this wave solution is generally understood as an exquisite balance between nonlinearity and dispersion in the system. Since then, the propagation of strongly nonlinear impulse waves in many variants of this chain has been studied by several researchers (see some reviews, e.g. [2,7,8]).

Among such variants, we focus on a non-periodic granular system that is functionally graded in space. We note in passing that functionally graded architectures have been widely adopted in engineering, particularly for designing thermal barrier coatings. The feasibility of manufacturing such materials also motivated studies on wave propagation in functionally graded elastic solids [9,10]. While these previous studies employed continuum systems with gradient material properties, such as elastic modulus and density (volume fraction), we focus on discrete granular systems with a spatial gradient in mass or contact stiffness. Sen et al. first proposed such a system, which they called a ‘tapered’ chain [11]. This consists of a chain of spherical beads of decreasing radii. The mismatch of masses leads to the spreading of an impulse wave as it travels with the decreasing kinetic energy of the wave front. This eventually results in less contact force experienced in the latter part of the chain. This phenomenon was later verified by experiments as well [12–14]. In parallel, several studies have employed the collision models to gain theoretical understanding of energy propagation in such chains [15–19]. These previous studies have successfully demonstrated versatile wave dynamics in non-periodic, graded granular systems, such as impulse dispersion and shock mitigation, that are unprecedented in homogeneous granular chains. However, the physical systems that they employed have been limited primarily to sphere-based granular chains, which do not offer a rich design freedom to alter the system configurations in situ.

To tackle such a challenge, here we propose a functionally graded cylinder-based granular system, which provides extreme tunability by manipulating the stiffness gradient in 1D space. Note that the tapered granular chain proposed by Sen et al. results in a gradient of both mass and stiffness; however, a graded platform that can alter only one parameter freely—particularly stiffness—has not been thoroughly investigated. Therefore, a study of the wave propagation in such an ‘intermediate’ system, between Nesterenko's homogeneous granular chain and Sen's tapered granular chain, could shed new light on the strongly nonlinear wave phenomena in functionally graded systems. Furthermore, by leveraging the intrinsic tunability, this cylinder-based granular system might lead to interesting engineering applications for enhancing stress wave management.

The use of cylinders along with spherical granules was first proposed to enhance tunability in terms of independently changing particle masses [20]. However, later designs based solely on cylinders, interacting as per the Hertzian contact law, offered enhanced tunability. For example, contact stiffness can be tuned in situ simply by changing the contact angle between cylinders [21]. Moreover, for long cylinders, their bending can be invoked to include local resonance effects in wave propagation [22]. Therefore, such systems have recently become popular to demonstrate a wide span of wave phenomena. In linear regimes, these have been used for creating tunable bandgaps [22–24], topological defects [25], a mechanical non-reciprocal device [26,27] and Wannier–Stark ladders [28]. In addition, these systems have been used to manipulate strongly nonlinear impulse waves [21,29–31], along with possible extensions to three dimensions (3D) [32].

In this study, we create a table-top model of the functionally graded chain with an increasing and decreasing linear stiffness distribution in space by in situ tuning of contact angles between cylinders. We experimentally demonstrate that, without significant scattering of the wave front, the system can either accelerate or decelerate the impulse wave. We use numerical simulations to corroborate our findings. Moreover, we employ the binary collision approximation (BCA) [16,18], which agrees well with numerical and experimental results in predicting the speed of the wave front. By investigating the spectral characteristics of the velocity profile of the travelling wave, we observe that these graded systems support a cascading of frequencies as the wave travels along the chain; however, the wave front remains nearly localized (maintaining the spatial width) in the space domain, as there is no obvious cascading of wavenumber. Moreover, the system offers the scaling of the maximum contact forces depending on the stiffness variation along the chain. The numerical results for this scaling agree well with the BCA predictions.

2. Experimental set-up

Our set-up consists of 40 identical short cylinders (radius r = 9 mm and height h = 18 mm) made of fused quartz (Young's modulus Y = 72 GPa, Poisson's ratio ν = 0.17 and density ρ = 2200 kg m⁻³) stacked on top of each other to form a chain, as shown in figure 1. We designed 3D-printed enclosures to support the cylinders, and those can be independently rotated to change their contact angles, as shown in the lower left inset of figure 1. We dropped a spherical quartz striker (radius r = 9 mm, mass m₀ = 8.1 g and velocity v₀ = 0.8 m s⁻¹) on the top of the chain to induce an impulse wave in the system. We then measure the velocity of each cylinder one by one using a laser Doppler vibrometer (LDV). Note that the enclosures have dedicated holes in them for the passage of the laser beam. The cylinders interact through the contacts, and the contact stiffness can be controlled by altering the contact angles. We maintain contact angles ranging from α = 2° at one end to α = 90° at the other, and enforce a linear gradient in the contact stiffness coefficient β (detailed in the following section). These angles are simply reversed to test two cases, i.e. a positive and a negative gradient in contact stiffness.

Figure 1.

Experimental set-up: a vertical assembly of cylinders placed inside 3D-printed enclosures for support. Contact angles can be tuned in situ. After striking the chain, point-by-point measurement is performed by an LDV. (Online version in colour.)

3. Modelling

We model the system with the discrete element method [33] as a 1D chain, in which each cylinder is approximated as a lumped mass (m = ρπr²h) and is connected to neighbouring masses with a nonlinear spring element (upper left inset of figure 1). This spring follows the Hertzian contact law given by force F = β(α)[δx]ⁿ⁻¹₊, where β(α) is the contact stiffness coefficient dependent on contact angle α, δx is the relative displacement, n ( = 5/2) is the potential exponent, and the symbol [ · ]₊ indicates that the contact force is 0 for non-positive values of δx. One can calculate β as [6]

3.1

Here, e is the eccentricity of the elliptical contact area between two cylinders and equals , with a and b being the semi-major and semi-minor axis length, respectively. Moreover, the ratio can be expressed as b/a ≈ [(1 + cosα)/(1 − cosα)]^−2/3. K(e) and E(e) are the complete elliptical integrals of the first and second kinds, respectively. In this way we use the fact that a change in the contact angle leads to a change in the contact stiffness, and thus we can easily tune the stiffness gradient in the system.

For the first case of the functionally graded configurations considered in this study, we maintain a contact angle of 2° at the start of the chain (between the first and second cylinders from the top) and an angle of 90° at the end of the chain (between the 39th and 40th cylinders from the top). The rest of the contact angles are chosen such that the contact stiffness coefficient β from equation (3.1) varies linearly along the chain from 7.27 × 10¹⁰ N m^−1.5 to 4.68 × 10⁹ N m^−1.5. The higher the contact angle is, the lower the stiffness becomes. Therefore, this represents a case of a linear decrease in stiffness along the chain. For the second case, we simply reverse the contact angles along the chain, so that it represents a case of a linear increase in stiffness. We ignore any material dissipation in the system. We also neglect the effect of gravity in this short chain, as the maximum compressive load due to gravity is about 1.6 N at the bottom of the chain, which is about two orders of magnitude less than the contact forces induced by the impulse, as shown in the later sections.

To solve this nonlinear system numerically, we write the following equation of motion for the cylinders (with position index i = 1, 2, 3, …)

3.2

where x_i denotes the dynamic displacement, β_i is a simplified notation for β(α_i) used hereafter, and indexing i starts from the top of the vertical chain. Note that β₀ = 3.98 × 10⁹ N m^−1.5, which is calculated by considering a sphere-to-cylinder contact between the striker and the first cylinder. We solve this system of equations using the ODE45 solver in Matlab with the initial condition given in the form of striker velocity v₀ = 0.8 m s⁻¹.

To get further insights into the wave dynamics of this graded system, we employ the BCA [16]. This method considers the interaction of only two masses at a time. First, by applying the conservation of energy in our dissipation-less model, we calculate the time it takes for the ith mass to completely transfer its momentum to the (i + 1)th mass. We use the exact solution for this ‘residence time’ derived in ref. [18] to write

3.3

where μ_i = m_im_i+1/(m_i + m_i+1) denotes the reduced mass of interacting masses m_i and m_i+1, and v_i is the maximum velocity of the ith mass. This velocity of the ith cylinder can be calculated from the velocity of its previous (i − 1)th cylinder simply by applying the conservation of linear momentum. Therefore

3.4

We know that all the cylinders are identical in mass (m_i = m ∀ i≥1), thus the velocity of all the cylinders would be the same. However, due to the mass mismatch between the first cylinder (i = 1) and the striker (i = 0), we would have the following relation:

3.5

Once we have the velocity of each mass, we can calculate the residence time for each cylinder from equation (3.3). This can be used to calculate the wave speed V at the ith cylinder as

3.6

Moreover, the total time taken by the wave to pass the ith cylinder can be calculated as

3.7

4. Results and discussion

(a). Accelerating and decelerating nonlinear waves

We show the spatio-temporal evolution of the cylinders' velocities in figure 2. Numerical and analytical results for the chain with decreasing stiffness along its length are shown in figure 2a. We notice that, as the wave front evolves over time, it slows down by curving upwards in the spatio-temporal map. Moreover, the wave front does not show significant scattering. As a result, the velocity amplitude of the cylinders does not show a large variation along the chain, complying well with the assumption in the BCA that binary collisions between cylinders transfer the same velocity along the chain (equation (3.5)). Therefore, the analytical solution obtained from equation (3.7) shows a good agreement with the numerically obtained wave front. We also notice that this wave front is followed by other pulses whose characteristics are similar to the former. These waves emerge because of the large stiffness mismatch between the first contact (between the striker and first cylinder) and the second contact (between the first and second cylinders) in the chain. In figure 2b, we show the experimentally obtained velocity map for this graded chain. A good match with the numerical results is evident. This, therefore, demonstrates the impulse deceleration capability of our system.

Figure 2.

Spatio-temporal velocity map. (a) Numerical velocity map of the graded chain with angles varying from 2° to 90° along the chain. Wave trajectory obtained through analytical method (black line) is superimposed on the map, closely following the wave front. (b) Experimentally obtained velocity map for the same case. (c) Numerical and analytical results for the graded chain with angles varying from 90° to 2° along the chain. (d) Experimental velocity map for the same case. (Online version in colour.)

Figure 2c shows numerical and analytical predictions for the reverse chain with increasing contact stiffness. We clearly see the acceleration of the wave front in this case, with a good agreement with the BCA predictions. Note that this case does not cause multiple wave fronts, as shown in figure 2a. This is because of the small stiffness mismatch between the first two contacts in the chain. We see that these numerical predictions are in agreement with experimental results, as shown in figure 2d. This demonstrates impulse acceleration in our tunable system. Minor scattering shown in experimental data (figure 2b,d) is due to variations in the setting of contact angles in the experiment set-up.

In order to get further insights into the system, we extract more information, such as residence time and wave speed, from the velocity maps shown in figure 2. We calculate residence time analytically using equation (3.3) for every cylinder location. For comparison, we extract the same for the wave front in numerical and experimental maps in the following way. Given the velocity profiles, we define the residence time for the ith cylinder as the interval between tⁱ_i and t^f_i. Here, tⁱ_i is the time at which the velocity of the ith cylinder exceeds the velocity of the (i − 1)th cylinder, and t^f_i is the time at which the velocity of the (i + 1)th cylinder exceeds the velocity of the ith cylinder. We plot the results in figure 3a,b for the chains with decreasing and increasing stiffness, respectively. The decrease in the stiffness along the chain (figure 3a) is clearly reflected in the increase of the residence time. Similarly, the increase in the stiffness along the chain (figure 3b) is reflected in the decrease of the residence time. We observe that the experimental data follow the trend predicted by the analytical and numerical methods.

Figure 3.

(a) Residence time for the wave front calculated along the chain (2° to 90°). Increasing residence time indicates stiffness decrease along the chain. (b) The same calculated for the reverse chain. (c,d) Wave speeds obtained from the corresponding residence times. A power-law fit is used to obtain smooth experimental data. (Online version in colour.)

We calculate wave speed from residence time by using equation (3.6) and show this in figure 3c,d for the respective chains. We observe that the experimental data have a large variation in its values along the length of the chains. So we use a power-law fit to better interpret the experimental wave speed. This makes sense because we can deduce V_i∝β^1/n_i from equation (3.3) and equation (3.6). The linearly varying β_i in our system can be expressed as β_i = b₁ + b₂i. Therefore, we use the relation V_i = (b₁ + b₂i)^1/n to obtain a fit for experimental wave speeds. We note that this wave speed closely follows the trends predicted by the analytical and numerical methods. However, we observe that it is relatively small in value. This may be because of several factors, including (1) the presence of dissipation in experiments, which is not accounted for in the numerical and analytical calculations, and (2) variation in the power law (value of n) for cylindrical contact as one goes from small angles to high angles [21].

One may now ask the following question: How can this acceleration and deceleration of the impulse wave be enhanced further in the context of any practical applications? One of the solutions is to introduce an inertial mismatch along the chain. The cylinder-based system provides a unique advantage in this regard. The heights of the cylinders can be varied independently, so that we introduce a mass gradient without changing the stiffness gradient. Therefore, by following the conservation of linear momentum, it is straightforward to conclude that the wave deceleration caused by a decreasing stiffness in figure 3c can be enhanced by setting an increasing mass gradient. Similarly, the wave acceleration caused by an increasing stiffness in figure 3d can be enhanced by setting a decreasing mass gradient (see appendix A for some exemplary cases analysed numerically).

(b). Spectrum

In this section, we discuss the spectral characteristics of the nonlinear impulse propagating in the graded chain. We use only numerical simulations hereafter. For better data visualization, we consider a longer chain with 200 cylinders. Moreover, to get rid of multiple waves emerging from the striker impact as shown in figure 2a, we use a striker with the same mass as that of cylinders, and equate its contact stiffness (β₀) to β₁. Figure 4 shows the velocity maps for these long chains and frequency/wavenumber extracted from the same by employing the fast Fourier transformation. For the case of decreasing stiffness (figure 4a), we see that the impulse slows down as it propagates along the chain. We plot its frequency spectrum in figure 4b. It shows a clear cascading of frequencies, i.e. spectral energy shifts to lower frequencies along the chain. This is a reflection of higher residence time observed in figure 3a due to the decrease in stiffness. However, the wavenumber calculated at each time instant in figure 4c shows an interesting trend. There is no such clear cascading. As we have already discussed that the height of the wave front (i.e. velocity amplitude) remains nearly intact (equation (3.5)), the absence of cascading in wavenumber means that the wave front is nearly localized in space and maintains its spatial width without significant scattering (see the velocity–time history of the 100th cylinder in the inset of figure 4a). Similarly, the results for the reverse chain can be interpreted as in figure 4d–f . Unlike the case of decreasing stiffness, we observe the spreading of the energy to a broader spectrum of frequency in this increasing stiffness case (compare figure 4b,e). However, in the wavenumber domain, the localization trends are similar between the two cases. Previous studies on tapered granular chains [11] showed acceleration and deceleration of the wave front; however, the width and the kinetic energy of the wave front change due to scaling in both mass and stiffness along the chain. Unlike the tapered chain, our current system offers the tunability of impulse speed with contact force scaling (discussed next) but preserving the spatial width and the kinetic energy of the wave front.

Figure 4.

Spectral characteristics of graded chain. (a) Numerically obtained spatio-temporal velocity map for a longer graded (2° to 90°) chain. Inset shows velocity–time profile of 100th cylinder. (b) Its spectrum calculated in the time domain, i.e. frequency, along the chain length. (c) Its spectrum calculated in the space domain, i.e. wavenumber, against time. (d–f) The same for the reverse chain. (Online version in colour.)

(c). Contact force scaling

We extract the maximum contact force values (CF) at each cylinder location for the system discussed in figure 4 and plot them in figure 5. We clearly notice that this graded chain provides varying contact force along the chain depending on the stiffness gradient in figure 5a,b. Also note in the inset a typical force–time history profile at the middle of the chain. To evaluate the dependence of this contact force on stiffness coefficient, we plot them in a logarithmic scale in figure 5c,d, which show a linear trend. This power-law scaling can be understood from the BCA predictions. One can calculate the maximum compression at the ith contact as [18]. This means that the maximum contact force CF from the contact law, and therefore, CF. In figure 5c,d, we plot a black line with the slope equal to 1/n = 2/5, to show the remarkable agreement of numerical simulations to the scaling law predicted by the BCA.

Figure 5.

Scaling of contact force. (a) Numerically obtained maximum contact force variation along the chain for the same system as shown in figure 4. Inset shows force–time profile at the contact between 99th and 100th cylinders. (b) The same for the reverse chain. (c,d) Maximum contact force versus contact stiffness (circles) for the respective chains shows a power law. A black line with slope equal to that power guides the eyes. Note the reverse values on the x-axis for (c). (Online version in colour.)

5. Conclusion

We have demonstrated the impulse acceleration and deceleration capability of cylinder-based functionally graded granular chains. The architecture allows us to have in situ control over stiffness distribution, and therefore it offers extreme tunability. We showed that space–time evolution of the wave front in the experiments is well predicted by numerical and analytical calculations. The spectral characteristics obtained from the velocity profile of the wave propagating in such systems show a clear cascading of frequencies, but not of wavenumbers. This means that, even though the impulse accelerates/decelerates along the chain, it does not show significant scattering and remains localized in the space domain by retaining its spatial width. Finally, we verified that the contact forces along the chain follow a power-law scaling depending on the stiffness distribution, and this complies well with the analytical predictions. The proposed system can be potentially used for manipulating stress waves under an impact in various engineering applications. Further studies include the effect of plasticity and viscoelasticity in such a system, and also its generalizations to build tunable architectures in higher dimensions.

Appendix A. Effect of inertial mismatch in accelerating and decelerating impulse waves

In this section, we numerically study the effect of introducing inertial mismatch (i.e. gradient in mass) in our system in addition to a stiffness gradient. To that end, we keep all the conditions the same as those in figures 2 and 3, but introduce a linear gradient in mass such that m_i = m(1 + γi) ∀ i∈[1, 40]. Note that both positive and negative gradient (γ) can be introduced, irrespective of the stiffness gradient, by changing the heights of the cylinders.

Figure 6 shows some exemplary cases, in which the introduction of mass gradient in the system affects the impulse deceleration and acceleration phenomena. In figure 6a, decreasing mass along the chain forces the wave front to travel faster, and therefore the wave deceleration caused by decreasing stiffness is weakened (compare figure 6a and figure 2a). However, increasing mass in figure 6b complements the wave deceleration capability of the system with decreasing stiffness (compare figure 6b and figure 2a). We calculate the wave speed numerically and analytically as shown in figure 3c, and further extract the quantity V_out/V_in, i.e. the ratio of wave speeds at the output and the input of the system. In figure 6c, we plot this wave-speed ratio as a function of mass-gradient index (γ). It is clear that a ratio less than 1 indicates the wave deceleration along the chain; however, it is dependent on mass-gradient index. Therefore, positive mass gradient can further enhance wave deceleration for this stiffness configuration of the system. The same procedure is followed for the configuration with the increasing stiffness along the chain and plotted in figure 6d–f . We observe from figure 6f that the impulse wave can be further accelerated with negative mass gradient. It would be an interesting exercise for the future to see to what extent the mass gradient can be introduced in real experimental set-ups. This is because increasing heights of the cylinders would also introduce local resonance effects [22,29], and that can potentially have qualitative effects on the impulse wave propagation.

Figure 6.

Effect of mass gradient. (a) Numerical velocity map of the graded chain with angles varying from 2° to 90° and with a decreasing mass gradient (γ = − 0.01). Spatio-temporal evolution of the wave front is also well captured by the analytical method (black line). (b) Keeping the same stiffness gradient but reversing the mass gradient (γ = 0.01). (c) Wave-speed ratio (V _out/V _in) calculated at multiple mass-gradient values indicating the effect on impulse wave deceleration. (d–f) The same but with reverse stiffness gradient, and the effect on impulse wave acceleration. (Online version in colour.)

Data accessibility

The data set analysed during the current study is available from the corresponding author on reasonable request.

Author's contributions

E.K. and J.Y. conceived and designed the study. E.K. performed the experiments. R.C. and E.K. performed numerical and analytical calculations, performed data analysis and interpreted the results. R.C. drafted the manuscript, and all the authors were involved in finalizing the manuscript.

Competing interests

The authors declare that they have no competing interests.

Funding

J.Y. gratefully appreciates the support from the NSF (CAREER-1553202) and AFOSR (FA9550-17-1-0114). E.K. acknowledges the support from the National Research Foundation of Korea (NRF) grant funded by the Korean government (NRF-2017R1C1B5018136).