Independent control of dynamic material properties by exploiting structural hierarchy and intrinsic structural gradients

Journal Pre-proof Independent Control of Dynamic Material Properties by Exploiting Structural Hierarchy and Intrinsic Structural Gradients David Murgado, Ramathasan Thevamaran

PII:

S2352-4928(19)31593-4

DOI:

https://doi.org/10.1016/j.mtcomm.2019.100865

Reference:

MTCOMM 100865

To appear in:

Materials Today Communications

Received Date:

4 December 2019

Accepted Date:

12 December 2019

Please cite this article as: Murgado D, Thevamaran R, Independent Control of Dynamic Material Properties by Exploiting Structural Hierarchy and Intrinsic Structural Gradients, Materials Today Communications (2019), doi: https://doi.org/10.1016/j.mtcomm.2019.100865

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Independent Control of Dynamic Material Properties by Exploiting Structural Hierarchy and Intrinsic Structural Gradients

Department of Engineering Physics, University of Wisconsin - Madison

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David Murgado and Ramathasan Thevamaran

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539 Engineering Research Building, 1500 Engineering Drive, Madison, WI 53706.

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Abstract

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Achieving high damping and stiffness is challenging in common materials because of their interdependent scaling. Controlling extreme mechanical waves requires synergistically enhanced damping and stiffness. We demonstrate superior damping and stiffness in vertically aligned carbon

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nanotube (VACNT) foams that are also independently controllable by exploiting their synthesistailored structural hierarchy and structural gradients. They exhibit frequency- and amplitudedependent responses with dramatically tunable dynamic stiffness while maintaining constant damping. Developing independent control over critical dynamic properties via engineered hierarchical and gradient materials will enable the creation of non-Hermitian metamaterials and

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active control of mechanical waves and vibrations.

Keywords:

Extreme damping and stiffness, Hierarchical materials, Structural gradients,

Nonlinear tunability, Lightweighting, Carbon nanotube foams

Introduction 1

Controlling mechanical waves and extreme vibrations in air, space, underwater, and arctic environments require lightweight materials that have superior strength, stiffness, and damping that are robust across broad frequencies and temperatures. However, those critical mechanical properties are often found to be mutually exclusive [1,2]. For example, open and closed cell polymeric foams and rubbers that are commonly used for vibration damping and impact mitigation applications do not provide sufficient stiffness and strength, and importantly, are not functional in low and high temperatures because of the drastic adverse variations in their properties across glass-

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transition temperature [3–5]. Designing materials with structural hierarchy and gradient functional properties enables unprecedent control over mechanical properties while simultaneously reducing density [5,6]. Taking inspiration from the high hardness, strength, and toughness induced by

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structural hierarchy in biological materials—e.g. bones, seashells, beaks, etc.,—synthetic materials have been designed to amplify properties beyond the composite rules of mixtures [6–12].

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Structural gradients where mechanical properties vary gradually over space has also been utilized in addition to structural hierarchy to achieve unique functionalities in both natural and synthetic

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materials—e.g. bridging soft body and stiff beak tip (rostrum) in squids [13], shock attenuation in woodpecker beak via stiffness gradients[14], increased contact resistance in mechanical

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components via compositional and structural gradients[15], and synergistically improved strength and toughness in metals via gradient plasticity[16,17]. The vertically aligned carbon nanotube (VACNT) foams present both structural hierarchy

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and intrinsic structural gradients that can be tailored by synthesis to achieve a broad range of desirable bulk mechanical properties [18–23]. The structural hierarchy of the macroscale VACNT foams, which spans nanometers to millimeter-scales, constitutes of nanoscale multiwalled carbon nanotubes (MWCNTs) that entangle with each other to form a forest like system in the microscale, and vertically aligned bundles in the mesoscale (Fig.1(a)). Their bottom-up chemical vapor

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deposition (CVD) synthesis also results in a varying entanglement morphology with mass density increasing from the bottom to the top of the sample. When such VACNT foams are compressed quasistatically, the strain localizes in the lowest density region at the bottom and form collective buckles that progress sequentially upward governed by the density gradient [24–26]. VACNT foams with MWCNTs often have the ability to recover near-completely from large compressive strains up to 80% upon unloading [18,24,27,28]. These intriguing deformation mechanisms lead to a nonlinear stress-strain response (Fig.1(b)) with hysteretic energy dissipation that is more than 2

200 times higher than the commercial protective foams of comparable densities [24,28,29]. Unlike the polymeric foams, the mechanical properties of VACNT foams have been shown to be robust across broad temperature range, over millions of fatigue cycles, and multiple striker impacts [26– 28,30]. In dynamic regime subjected to striker impacts, the VACNT foams exhibit a behavior similar to that of the quasistatic regime up to a critical velocity of impact, and support shock formation at velocities higher than the critical velocity—which is also a function of the bulk

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density of the VACNT foam [28]. In linear dynamic regime subjected to low-amplitude harmonic excitations, various non-aligned CNT-sponges and CNT-composites have been investigated and

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shown to exhibit storage and loss moduli that vary with excitation parameters. For example, random network of CNTs excited in torsional-mode dynamic mechanical analysis (DMA) between

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1 and 100 Hz showed temperature and frequency-invariant viscoelasticity that is robust across 196 to 1000 °C and a million loading cycles [30]. The VACNT-reinforced epoxy composites, tested in shear-mode DMA, exhibited frequency-invariant (1-10 Hz) and amplitude-dependent

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(shear strains up to 20%) storage and loss moduli [31]. A densified (0.47 g cm-3) and interlocked random network of CNTs showed temperature-invariant, but frequency-dependent (1-200 Hz)

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storage and loss moduli that reached over 10 GPa in tension-mode DMA[32]. Amorphous-carboncoated VACNTs tested up to 10 Hz and at amplitudes up to 100 nm in compression-mode DMA exhibited storage and loss modulus that are invariant over frequency, but nonlinearly depend on

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amplitude [33]. An interconnected VACNT foam tested in compression-mode DMA over 1-25 Hz showed enhanced elasticity with significant loss of damping (tan 𝛿 < 0.01) [34]. Regardless of these studies, the linear dynamic behavior and the bulk dynamic properties of VACNT foams over broad frequency regime relevant to applications and the effects of intrinsic structural gradients

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remain elusive.

Here, we show that a high dynamic stiffness and damping can simultaneously be achieved

with independent control over a three-orders-of-magnitude-broad frequencies in VACNT foams by exploiting the nonlinearities arising from the structural-hierarchy and the intrinsic-structuralgradient-induced deformation mechanisms. Results and Discussions

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We developed an experimental apparatus to perform dynamic mechanical analyses in compression mode over 1 Hz to 1.5 kHz and excitation amplitudes up to 45 µm (see Materials and Methods and Fig.S1 in Supplementary Information). When harmonically excited by the actuator, the VACNT foam samples (bulk density: 0.1550.005 g cm-3) exhibit a hysteretic stress-strain response that is linear in nature (elliptical) up to 1.5 µm excitation amplitudes (Fig.1(c)). They additionally exhibit strain softening with increasing excitation amplitude (Fig.1(d)) suggesting the emergence of weak nonlinearities (see Fig.S2 in Supplementary Information). The stress-strain

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hysteresis shows an increase in enclosed area with increasing excitation frequency at constant excitation amplitude implying an increase in energy dissipation (Fig.1(c)). The samples also

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exhibit substantial stiffening with increasing static precompression at constant excitation

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amplitude and frequency (Fig.1(e)).

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Fig.1. The dynamic response of VACNT foams: a. Optical and SEM images showing the hierarchical structure of VACNT foams, b. Characteristic stress-strain response in quasistatic compressive loading-unloading cycle, c. Characteristic stress-strain responses at various excitation frequencies with 0.1% strain amplitude (a/h, a = displacement amplitude, h = sample thickness) and 10% static precompression strain (u/h, u = precompression displacement, h = sample thickness), c. Characteristic stress-strain responses at strain amplitudes 0.04% to 0.1% with 100 Hz excitation frequency and 10% static precompression strain, d. Characteristic stress-strain responses at static precompression strains 0.2% to 39.7% with 100 Hz excitation frequency of and 0.1% strain amplitude.

We calculate the dynamic mechanical properties from the measured stress-strain responses.

The damping by linear viscoelastic materials is typically measured by loss tangent (tan 𝛿)[2], which relates the energy dissipated to the energy stored during dynamic loading-unloading cycle as, tan 𝛿 =

𝑊𝑑 2𝜋𝑊𝑠

. The total energy dissipated by the material in a cycle is given by, 𝑊𝑑 =

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2𝜋⁄𝜔

∫0

𝜎

𝑑𝜀 𝑑𝑡

𝑑𝑡 = 𝜋 𝐸 ′′ 𝜀0 2 , and the maximum elastic energy stored in the material which occurs in 𝜀

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quarter cycle is given by, 𝑊𝑠 = ∫0 0 𝐸 ′ 𝜀𝑑𝜀 = 𝐸 ′ 𝜀0 2 . Here, the storage modulus 𝐸′ and the loss 2

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modulus 𝐸 are the real and the imaginary parts of the complex modulus of a viscoelastic material, given by 𝐸 ∗ = 𝐸 ′ + 𝑖𝐸 ′′ , 𝜀0 is the strain amplitude, and 𝜎 is the stress [2]. The slope of the dynamic stress-strain curve represents the dynamic stiffness of the material, |𝐸 ∗ | = √𝐸 ′ 2 + 𝐸 ′′ 2 . The loss tangent can also be expressed in terms of storage and loss moduli as tan 𝛿 =

𝐸 ′′ 𝐸′

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The loss tangent and the dynamic stiffness of the VACNT foams at 10% static precompression strain as functions of frequency are shown in Fig.2(a-b). Both loss tangent and

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dynamic stiffness exhibit high values reaching over 0.8 and 47 MPa, respectively. For three different representative samples (height = 1.350.07 mm), we observe the variations in properties

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to be minimal—for example, loss tangent and dynamic stiffness are 0.230.03 and 15.044.97

Fig.2. The variation of (a) loss tangent, (b) dynamic stiffness, (c) storage modulus, and (d) loss modulus of VACNT foams with frequency at three different excitation amplitudes (strain amplitudes 0.04%, 0.08%, and 0.1%) and at 10% applied precompression strain.

MPa, respectively at 100 µm precompression, 1.5 µm excitation amplitude, and 100 Hz frequency. 5

For a given excitation amplitude and precompression, our VACNT foams exhibit nearly constant loss tangent up to 50 Hz, similar to previous studies at low frequencies [33]. Beyond 50 Hz, the loss tangent increases nonlinearly to more than 0.8 with increasing frequency up to 1.5 kHz. The dynamic stiffness exhibits a frequency response similar to that of the loss tangent. The observed increases in loss tangent and dynamic stiffness with frequency primarily arise from the increase in loss modulus rather than the storage modulus. The storage modulus remains nearly constant across the range of excitation frequencies (Fig.2(c)) while the loss modulus remains constant initially and

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then nonlinearly increases beyond 50 Hz (Fig.2(d)). We hypothesize that this increase in loss modulus as a function of increasing frequency is due to the increasing participation of smaller lengthscales associated with collective CNT fiber interactions in dissipating energy. Previous

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studies on individual CNTs have shown that nearly a micron long isolated CNT fiber has characteristic resonance between 250 to 400 MHz depending on whether it is straight, slack, or

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coiled CNT [35,36]. They also exhibited nonlinear response and the physical origins of such nonlinearity has been attributed to the geometric nonlinearities [35]. Since such CNTs with high

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characteristic frequencies self-organize into a hierarchical structure that spans from a few nanometers to millimeter lengthscales, the damping can be expected to increase at higher

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frequencies of excitations that excite smaller lengthscale resonant structural features. We examined the variations in loss tangent and dynamic stiffness as the excitation amplitude is increased. The Figs.3(a-b) show that the loss tangent slightly increases with increasing

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amplitude while the dynamic stiffness significantly decreases nonlinearly. This strain softening is likely due to the compressive weakening as the vertically aligned CNT fibers in the strain-localized region buckle as a function of increasing excitation amplitude. Fibrous networks often exhibit stiffening in tension and show compressive weakening in compression that can occur at two orders of magnitude smaller strains than the strain that causes tensile stiffening [37]. From Fig.3(c-d), it

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is also evident that the storage modulus decreases with increasing amplitude across all frequencies, while the loss modulus remains constant across amplitude at low frequencies up to 100 Hz and shows a monotonic decrease with increasing amplitude at higher frequencies. This suggests that the weak lossy nonlinearities emerge in higher frequencies in the hierarchical VACNT foams.

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Fig.3: The variation of (a) loss tangent, (b) dynamic stiffness, (c) storage modulus, and (d) loss modulus of VACNT foams with excitation amplitude at six different excitation frequencies and at 10% applied precompression strain.

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When the applied static precompression is increased, we observe that the dynamic stiffness increases because of the intrinsic density gradient present in the material (Fig.4(b)) [28]. We have previously measured [28] the intrinsic density gradient in similar VACNT foams synthesized using

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the floating-catalyst thermal chemical vapor deposition process with similar synthesis conditions using synchrotron X-ray scattering and mass attenuation techniques. The mass density in those samples were observed to increase from less than 0.1 g cm-3 at the bottom region of the sample (adjacent to synthesis substrate) to over 0.35 g cm-3 in the top dense region, across a height of 1 mm [28]. This density gradient has also been correlated to the increase in elastic modulus

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(measured at unloading) and the progressive sequential buckling that was measured using highspeed microscopic imaging [28]. The loss tangent (Fig.4(a)) shows mild decrease (~0.1) with increasing precompression at low frequencies up to 100 Hz. At high frequencies, e.g. 1000 Hz in Fig.4(a), we observe a large increase of loss tangent by ~0.3 at ~160 µm precompression. This increase is likely due to the resonance of the bundles of buckled VACNTs. This is also evident in the increased loss modulus and decreased storage modulus at this precompression Fig.4(c-d). Enhanced damping has previously been observed even in individual MWCNTs that were

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compressed through a buckle instability [38]. Both storage and loss moduli increase with increasing precompression (Fig.4(c-d)). However, it is interesting to note that, at low frequencies, the loss tangent decreases as a function of increasing precompression (Fig.4(a)) because of the higher rate of increase in the storage modulus (Fig.4(c)) with precompression compared to the loss modulus (Fig.4(d)). The loss tangent remains nearly constant at 1000 Hz (Fig.4(a)) because of the

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comparable rate of increase in both storage and loss moduli (Fig.4(c-d)).

Fig.4: The variation of (a) loss tangent, (b) dynamic stiffness, (c) storage modulus, and (d) loss modulus of VACNT foams with applied static precompression strain from 0.2% to 39.7% at 1.5 µm excitation amplitude and at four different excitation frequencies.

To comprehensively investigate the effects of control parameters on the linear dynamic

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properties, we plot loss tangent, dynamic stiffness, storage modulus, and loss modulus in color densities as functions of amplitude and frequency (Fig.5), and precompression and frequency (Fig.6). The Fig.5a shows that the loss tangent remains nearly constant across frequency up to 200 Hz and all amplitudes and reaches significantly higher values above 0.8 in high frequency regime. It shows a similar behavior as a function of precompression (Fig.6a). Within the high frequency regime, it is interesting to note the emergence of a characteristic frequency at which the loss tangent is significantly enhanced, which is also dependent on the static precompression (see the red region near 1000 Hz in Fig.6a). For a precompression of 130 m shown in Fig.5, the highest 8

dynamic stiffness of over 47 MPa occurs at the low amplitude-high frequency regime, and the

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stiffness gradually decreases as the amplitude is decreased (Fig.5b).

Fig.5. The color density plots of a. loss tangent, b. dynamic stiffness, c. storage modulus, and d. loss modulus of VACNT foams as functions of excitation amplitude and frequency at 130 m precompression.

The dynamic stiffness dramatically increases over 275 MPa when the precompression is

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increased to 500 µm because of the intrinsic density gradient and its associated stiffness gradient in the material (Fig.6b). Note that this precompression (strain ~0.4) is much less than the densification strain (~0.7) [18,24,28] of VACNT foams. Therefore, the stiffening response is not due to the densification of the entire foam like in conventional cellular foams and arises from the intrinsic density gradient present in the material. The storage modulus increases with increasing frequency but decreases with increasing amplitude in consistent with the amplitude-dependent softening behavior (Fig.5c). However, with increasing precompression, the storage modulus 9

increases by nearly 50 times (Fig.6c). The loss modulus remains approximately constant up to 200 Hz and then increases at higher frequencies (Fig.5d), and shows significantly higher values in the regime of high frequency and high precompressions (Fig.6d). These comprehensive property maps as functions of tunable parameters of interest shown in Figs.5 and 6 would allow identifying

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appropriate mechanical design regimes for specific engineering application.

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Fig.6. The color density plot of a. loss tangent, b. dynamic stiffness, c. storage modulus, and d. loss modulus of VACNT foams as functions of applied static precompression and excitation frequency at 1.5 m excitation amplitude.

To examine the dynamic stiffness-damping properties of VACNT foams in comparison to

other material systems, we plot the stiffness-loss map of various materials (Fig.7). This map demonstrates the mutual exclusiveness of the stiffness and damping—for example the structural metals have high stiffness but very low damping, while rubbers have high damping but low stiffness. Because of this nature of the two properties, most of the materials often obey an interdependent stiffness-damping scaling law and fall below a ‘stiffness X damping’ product of 0.65 10

(the line on the top right corner of the Fig.7) [2]. Exceeding this 0.65 limit requires judiciously designed composite materials with phase transforming constituents or instabilities [2,10,39,40]. Our VACNT foams with ~0.15 g cm-3 density shows both high stiffness and damping compared to other materials with similar densities. Noteworthy is that the open cell polymeric foams with comparable densities used for protective applications do not even have sufficient stiffness (> 1 MPa) to be included in Fig.7. In addition to these superior dynamic properties, an unconventional characteristic of the VACNT foams is the ability to control their properties nearly independently

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over a broad range. For example, the damping increases with increasing frequency with approximately no change in stiffness while the excitation amplitude softens the materials with minimal change in damping. The static precompression provides dramatic control over the

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stiffness by exploiting the intrinsic structural gradient in the material while maintaining the damping characteristics constant. These unique functionalities are possible in VACNT foams

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because of the presence of both a structural hierarchy and an intrinsic density gradient—where the multiscale interactions from the hierarchically organized structural features enable high dissipation

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tunability independent of the damping.

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across broad frequency range while the intrinsic density gradient allows dramatic stiffness

Fig.7. The dynamic stiffness-loss map of VACNT foams compared to other common structural materials.

Our study suggests that the dynamic properties of VACNT foams can further be controlled over a broad range by tailoring their intrinsic density gradient and structural hierarchy or by effectively tailoring the bulk density that would stiffen the bulk material. We estimate that a change 11

in bulk density from 0.15 to 0.25 g cm-3 itself will show a dramatic increase in dynamic stiffness while maintaining/enhancing the loss tangent, there by exceeding the natural ‘stiffness X damping’ bound of 0.65. Micro-architecting techniques can additionally be used to create favorable structural features at mesoscale to dramatically tailor the VACNT foam’s properties with significant reduction in bulk density [41,42]. Conclusions Using a broad-band dynamic mechanical characterization, we reveal that the floating-

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catalyst CVD grown VACNT foams exhibit frequency- and amplitude-dependent dynamic response with high damping and stiffness between 1 Hz and 1.5 kHz. The material damping (loss

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tangent) increases with increasing frequency and remain approximately constant over the excitation amplitudes. The VACNT foams additionally exhibit a strain-softening response with

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increasing excitation amplitude. We also show that the dynamic stiffness of the VACNT foams is dramatically tunable using a static precompression that exploits the intrinsic functional property

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gradients present in the material while maintaining near-constant damping. Our study shows several fundamental mechanisms that exploit structural hierarchy and intrinsic structural gradients

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to achieve synergistically enhanced dynamic stiffness-damping properties with their independent control. Such independently controllable properties—especially the nonlinearly tunable damping characteristics—can be used for developing novel non-Hermitian mechanical metamaterials for

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frequency-preserving asymmetric wave transport [43]. Using static precompression, which imposes different stress states on the material by exploiting intrinsic structural gradients, the VACNT foams can potentially be used for actively tunable damping systems where the stiffnessdamping characteristics can spontaneously be tuned in response to an extreme mechanical wave or vibration that is incident on the material system.

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Materials and Methods

The VACNT foam synthesis. We synthesized the VACNT foams using a floating-catalyst

thermal CVD process in a horizontal tube furnace at 827 °C and atmospheric pressure. A feedstock solution consisting of ferrocene (catalyst precursor) and toluene (carbon source) mixed at 0.01 g ml-1 is injected at 0.5 ml min-1 into a carrier gas containing 95% Ar and 5% H2 flown at 800 sccm into the reactor tube. Silicon substrates with a thermal oxide layer placed inside the furnace tube facilitate the VACNT growth to thicknesses between 1 and 1.5 mm and average bulk density of 12

0.1550.005 g cm-3. Standalone 5 mm diameter cylindrical samples have been extracted from the substrate for dynamic experiments using a custom-made core drill. The dynamic mechanical analyzer and data reduction. The dynamic testing apparatus consists of a piezoelectric actuator with 1.8 nm resolution strain sensor (Physik Instrumente model: P-843.60; resonant frequency: 6 kHz; max travel distance: 90 µm; max push force: 800 N), a high power signal amplifier (Physik Instrumente model: PI E-505; voltage: -30 V-130 V), a dynamic force sensor (PCB model 208C01; sensitivity: 112.2 mV/N; max dynamic compression force: 44.8

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N; charge discharge time > 500 s) and a signal conditioner, and a data acquisition board (National Instruments modules NI-9269 and NI-9215) and a MatLab instrument control and data processing

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program for generating, measuring, and analyzing signals. Both force sensor and actuator are mounted on 4-inch steel blocks and the block are rigidly mounted on a noise-isolating optical table.

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A manual actuator with microscale resolution is attached to one of the steel blocks to apply static precompressions on the sample placed between actuator and force sensor. Nominal stress-strain responses are calculated from measured force normalized by the sample’s cross-sectional area and

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imposed displacement normalized by sample’s height. We subject the samples to twenty cyclic loadings to ensure steady state response, because VACNT foams typically exhibit preconditioning

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Declaration of interests

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effects during first few cycles [27,28].

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Conflict of Interest.

Authors declare no conflict of interest

Acknowledgements 13

Support for this research was provided by the University of Wisconsin-Madison, Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin

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Alumni Research Foundation. We also thank Prof. Roderic Lakes for the useful conversations.

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