Viscoelastic Metamaterials David M.J. Dykstra Institute of Physics University of Amsterdam Amsterdam, 1098 XH Netherlands E-mail: [email protected] Joris Busink Institute of Physics University of Amsterdam Amsterdam, 1098 XH Netherlands Bernard Ennis Materials Design Department Tata Steel Europe R&D Ijmuiden, 1970 CA Netherlands Corentin Coulais Institute of Physics University of Amsterdam Amsterdam, 1098 XH Netherlands E-mail: [email protected] Mechanical metamaterials are artificial composites with and inhomogeneous deformation rate, provoked by internal tunable advanced mechanical properties. Particularly in- rotations. Our findings bring a novel understanding of meta- teresting types of mechanical metamaterials are flexible materials in the dynamical regime and opens up avenues for metamaterials, which harness internal rotations and insta- the use of metamaterials for dynamical shape-changing as bilities to exhibit programmable deformations. However, well as vibration and impact damping applications. to date such materials have mostly been considered using nearly purely elastic constituents such as neo-Hookean rub- bers. Here we explore experimentally the mechanical snap- 1 Introduction through response of metamaterials that are made of con- stituents that exhibit large viscoelastic relaxation effects, en- Mechanical metamaterials exhibit a plethora of exotic countered in the vast majority of rubbers, in particular in mechanical responses. Static responses of interest span a 3D printed rubbers. We show that they exhibit a very strong wide range of tunable behavior, such as auxetic [1, 2, 3], sensitivity to the loading rate. In particular, the mechani- programmable [4, 5], shape-changing [6, 7], non-reciprocal [8] to chiral responses [9], often by harnessing nonlinear arXiv:1904.11888v1 [cond-mat.soft] 26 Apr 2019 cal instability is strongly affected beyond a certain loading rate. We rationalize our findings with a compliant mecha- mechanics and snap-through instabilities [4, 5, 10, 11, 12]. nism model augmented with viscoelastic interactions, which Interesting dynamical responses include shock absorption captures qualitatively well the reported behavior, suggesting [13,14,15,16] and soliton propagation [17,18] and transition that the sensitivity to loading rate stems from the nonlinear waves [19, 20]. Importantly, a compliant mechanism frame- work [4, 10, 21, 6, 8, 19, 20, 22, 17, 18] is often employed to (a) sponse of a metamaterial, that had been demonstrated earlier to exhibit a programmable hysteric response at slow loading w rate when produced from a nearly-ideal elastic rubber [4, 5]. We focus on metamaterials consisting of a 5 × 5 alternating D D 1 2 square pattern of circular holes, as seen in Fig. 1(a), which p h have been analyzed previously in the quasi-static regime us- ing a near ideally elastic material at low strain rates [4, 5]. y x It was found that when such samples were confined later- ally, as shown in Fig. 1(b), they would exhibit a snap- through response under compression, during which the cen- tral hole changes from an x-polarized state in Fig. 1(b) to a y-polarized state in Fig. 1(c), inducing a large reduction Confine in force. During unloading, a delayed snap-back instability (b) would occur for lower strain, inducing a large geometrically induced hysteresis. w cc However, if the sample is made instead of a viscoelas- d 1 tic rubber with a large stress-relaxation effect and we com- d2 press the sample quickly, no pattern change is provoked in Fig. 1(d). This suggests a nontrivial relation between the nonlinear response—induced by the elastic instability—and dissipation—induced by viscoelasticity. Fast The article is structured as follows. We first describe the sample and production process. Second, we show how Slow we calibrate the 3D printed material using a stress-relaxation (c) (d) v v test. We then present the experimental results, where we compress the sample using a variety of confinements across a wide range of strain rates, for which we systematically quantify the mechanical hysteresis and geometrical pattern change. In particular we observe an intricate balance be- tween the geometrically induced hysteresis and the vis- coelasticity induced hysteresis, such that optimal dissipation depends both on confinement and applied strain rate in a nontrivial manner. Finally, we present the soft viscoelastic mechanism model and its results, which we use to capture the mechanics of the interaction between viscoelasticity and Fig. 1. (a) Geometry of the metamaterial sample, with 5 × 5 al- mechanical instabilities. ternating holes, characterized by diameters D and D , hole spac- 1 2 Our study opens new a venues for the rational design ing p, sample height h and local width w. (b) Preconfined with of viscoelastic metamaterials, whose response drastically e = −18%, with the central hole initially in x-polarized position, x changes with loading rate or provide optimal energy absorp- with central hole major (d ) and minor (d ) axes. (c) Compressing 1 2 tion performance, combining geometrically induced hystere- slowly at a rate of e˙ = 9:25 · 10−5 s−1 leads to a snap-through y sis and viscous dissipation. instability , changing from x- to y-polarized. Repeating this process −1 fast, with e˙y = 0:3s suppresses the instability, with the sample instead remaining in an x-polarized position. 2 Sample fabrication, experimental methods and cali- bration We produce the sample in Fig. 1(a) using additive man- capture qualitatively the mechanical response and to explore ufacturing from a rubber-like PolyJet Photopolymer (Strata- the design space. sys Agilus 30) using a Stratasys Objet500 Connex3 printer. However so far, the effect of the constitutive materials’ The sample considered has hole sizes D1 = 10mm, D2 = 6 dissipation has been largely overlooked for nonlinear meta- mm, hole pitch p = 10 mm, local width w = 50 mm, height materials. As a matter of fact, extensive care has often been h = 54 mm and thickness t = 35 mm. We define the biho- devoted in the constitutive materials’ choice to avoid strong larity, c = jD1 − D2j=p = 0:3 [4]. The flat top and bottom dissipative effects. Viscoelastic effects have been consid- sides of the sample are then glued to two acrylic plates us- ered in dynamic regimes, but mainly for linear metamate- ing 2k epoxy glue that allow for clamping in our uniaxial rials [23, 24] and single snap-through elements [25, 26]. tensile tester, as performed previously [5]. To prepare the Here, we investigate the role of dissipation in nonlin- sample for testing: (i) we confine the sample laterally us- ear snap-through metamaterials. Specifically, we probe how ing 2 mm thick CNC machined steel U-shaped clamps as the constitutive materials’ viscoelasticity influences the re- shown in Fig. 1(b), where we define the strain of confine- Table 1. Fitted viscoelastic material properties for Agilus 30, with N = 1 and N = 3 NE0 h1 h2 h3 t1 t2 t3 [MPa] [s] [s] [s] 1 2.58 0.71 0.43 3 3.25 0.45 0.26 0.10 0.047 0.97 18 ties using a dogbone stress-relaxation test. In a single pro- cess, we 3D printed a sample with a central slender section Fig. 2. Stress-relaxation test for a dogbone sample of Agilus 30 3D of Agilus 30 and comparatively rigid top and bottom sides of printed material. Instantaneous Young’s modulus vs. relaxation time. Stratasys VeroBlackPlus, which are used for clamping in the The thick blue line indicates the test results, the thin dashed orange uniaxial testing device. The rigid sides are used to avoid pre- and green curves present fitted data with N = 1 and N = 3 respec- stress due to clamping. The Agilus 30 section has a length L tively. = 50 mm, depth d = 5 mm and width w = 10 mm. The sample is stretched quickly to a strain e = 20%, similar to the global strains induced on the main sample during confinement and ment as e = (w − w)=w, where w is the distance between x c c compression, at a strain rate of ˙ = 0:4s−1, after which the the clamps and w is the original local width. We choose only e force is allowed to relax for one hour. The data is measured to confine the even rows of our sample (2nd and 4th), as the with a frequency of 1000 Hz with t = 0s defined at the point odd rows differ in initial width; (ii) we leave the sample un- of highest load. confined. To describe the stress-relaxation response, we assume The sample is compressed uniaxially, both confined and the instantaneous time dependent Young’s modulus, E(t), unconfined, using a uniaxial testing device (Instron 5943), can be modelled using a neo-Hookean material model, which which controls the vertical motion of the top side of the sam- can then be characterized using Eqn. (2): ple with a constant controlled velocity v as shown in Fig. 1(c) and Fig. 1(d) from s = 0 to s = -12 mm, and immedi- ately unloaded from s = -12 to s = 0 mm at the same velocity, 3 F E(t) := ; (2) where we define the loading strain as e = s=h and strain rate 1 wd y l − l e˙y = jdey=dtj = jv=hj. The velocity is controlled down to ±0.1% of the set speed, while the force is monitored with where l is the applied stretch ratio and F is the measured a 500 N load cell with an accuracy of 0.5% down to 0.5 N. force. The load is calibrated to F = 0 N after attaching the sam- We characterize the viscoelastic properties by assuming ple to the load cell at the top, letting the sample relax for a a linear Maxwell-Wiechert viscoelastic material model, con- minute, after which the bottom is clamped to the test bench.