Mere tension output from spring-linkage–based mechanical metamaterials

Abstract

Metamaterials whose properties are inaccessible with conventional materials offer powerful tools for unprecedentedly manipulating physical signals. However, an effective design strategy of metamaterials still remains a challenge for changing the compression or tension characters of stress waves during forward propagation. Here, we introduce a class of spring-linkage–based metamaterials exhibiting mere tension output at the distal end, no matter that the input is an axial impact, a sudden tension, or even alternating tension-compression. The metamaterials can turn compressive waves into pure tension and filter them out from the tension-compression mixed ones while allowing tensile signal stably propagating in soliton form. This is achieved by combining nonuniform and nonlinear properties of the proposed cells. In particular, these extraordinary functions of the metamaterial can be turned on or off and adjusted by tuning a key switch cell; thus, it is anticipated to serve as a start for more complex manipulation and utilization of mechanical signals.

A compressive wave can be flipped into tension or filtered out from mixed waves by designed metamaterial with an on-off switch.

INTRODUCTION

A long-standing challenge in fundamental and applied sciences is to devise advanced tools to manipulate physical signals. Metamaterials are enabling the unprecedented properties that are inaccessible with conventional materials, whose behaviors are governed by structure rather than composition. They have shown powerful capacities in manipulating various physical signals, such as in electromagnetics (1–2), optics (3–5), acoustics (6–10), thermology (11), and magnetics (12), making unexpected functions achieved like perfect absorption (1, 7), invisibility cloaks (2, 5, 9–12), hyperlens (4–6), and holographic imaging (3, 8). Moreover, metamaterials also open a door to abnormal mechanical behaviors, not only realizing negative Poisson’s ratio (13–15), negative compressibility (16, 17), tension-torsion coupling effect (18, 19), and mechanical cloak (20, 21) in static but also making possible for unidirectional propagation (22, 23), stable propagation (24), and vibration attenuation (25, 26) of mechanical waves. However, it is difficult to change the compressive or tensile character of a longitudinal wave during forward propagation, which will break the direct consequence of the classical stress wave theory, as the foundation of impact engineering, and thereby opening up inspiring opportunities for dynamic systems.

Recently, it has been theoretically and numerically predicted that this wave invariance principle is expected to be broken by the strain-softening discrete system, in which tensile rarefaction solitary waves followed by oscillatory tail can form under the input of compressive wave after propagating several hundreds of cells (27–29). Because of the challenges in fabricating an effective strain-softening medium for waveguiding purposes, the rarefaction solitary wave was experimentally observed only in the origami-based metamaterials under the input of a major compressive displacement followed with a minor tensile one (30). Despite these important progresses, an effective design strategy of metamaterials capable to flip compressive wave into pure tensile one still remains a challenge.

Besides, the screening of one type from tension-compression mixed stress waves is also expected for inspiring prospects, so as to achieve a similar effect of electronic diode on alternating current rectification. A kind of hierarchical mechanical metamaterial capable to filter out tensile displacement from the low-frequency vibration was reported (31). Under the input of displacement, the huge difference between the tensile and compressive modulus of the metamaterial makes the input tension far less than the pressure and thus almost blocking the transmission of tensile displacement. However, for this reason, it will be difficult to realize filtering function under input of alternant compression-tension force with similar amplitude. Up to now, it still remains elusive to design medium for the sake of filtering stress wave from mixed ones.

Here, we propose an efficient strategy to design metamaterials for the purpose of flipping and screening stress wave by modulating the cells’ nonlinearity and setting a key switch cell. The introduced spring-linkage–based metamaterials are capable to turn compressive stress waves into pure tensile ones and filter them out from tension-compression mixed ones while allowing a tensile signal stably propagating in soliton form. Thus, mere tension signal can output from the metamaterial, no matter that the input is a compressive pulse, a tensile wave, or tension-compression mixed ones, as shown in Fig. 1A. Significantly, the unique flipping and screening functions of the proposed metamaterial can be turned on or off and adjusted by tuning the key switch cell, thus setting a powerful designable platform for the manipulation of mechanical signals and paving novel avenues to impact mitigation, energy harvesting, soft robotic, signal conversion, and long-distance delivery.

Fig. 1. Unique functions and design strategy.

(A) Mere tension output from the spring-linkage–based metamaterial rod. Blue (red) represents compressive (tensile) wave. (B) Metamaterial rod and unit cell. (C) Nonlinear force-strain and (D) stiffness-strain curves of the two types of unit cell. Both compressive and tensile curves are plotted in the first quadrant for the convenience of comparison. The solid (dashed) lines represent experimental (theoretical) results, and the shadows indicate the SD of three repeated experiments. The short flat segment at the beginning of each tensile curve was caused by assembly gap of unit cell in experiments. (E) Schematic of turning compressive wave into tensile one. Arrows represent wave traveling directions. E_I and E_II denote stiffness.

RESULTS

Metamaterial fabrication and design strategy

Our spring-linkage–based unit cell is composed of a four-bar linkage, two springs, and two connecting plates with additional weight (Fig. 1B, fig. S1, and the “Fabrication of unit cell” section). The mechanical behavior of the unit cell exhibits strong nonlinearity (the “Quasi-static experiments on unit cell” section and fig. S2). Under compression, its stiffness gradually decreases to 0 and even becomes negative with the increase of strain, while the stiffness rapidly increases with tensile strain (Fig. 1, C and D). The nonlinearity is controlled by the bar angle θ, while the cell stiffness is directly proportional to the spring stiffness k (fig. S3; see the “Static theoretical analysis” section in the Supplementary Materials for details). Two types of cells are designed with different stiffness. Cell I with weaker spring is the key cell and assembles as the first cell of the rod to bear external load, while the other 15 cells adopt the strong one (named cell II). In total, 16 cells are assembled to form the spring-linkage–based metamaterial rod. It should be noted that the key weak cell is essential for the desired behaviors as discussed in the following. Two steel guide rails symmetrically passed through the connecting plates of all cells and are fixed on the supports (Fig. 1B). Under external force, the metamaterial rod will deform along the rails with the distal end fixed on the support.

When a compressive wave inputs from left to the first weak cell (cell I), the stiffness of cell I decreases to approximately 0 and far less than that of the second cell (cell II), preventing the compressive wave transmitting into the second cell (state 2 in Fig. 1E). Hence, the compressive wave is reflected from the I-II interface and travels back to the left of the first cell (a free end), producing a tensile wave due to the free-end reflection (state 3 in Fig. 1E), which subsequently propagates to the right. Then, the stiffness of cell I quickly increases under tension and approaches to that of cell II, making the tensile wave easily transmitting to cell II from cell I (state 4 in Fig. 1E). In this way, most of the input compressive wave has been turned into a tensile one transmitting to the second cell. Then, in the subsequent propagation from the second cell, the less transmitted compressive wave is diffused by the nonlinearity and dispersity of the metamaterial, so it disappears quickly under frictions from the rails, while the tensile wave propagates in soliton form (verified below), resulting in mere tensile wave output.

To determine the detailed parameters of cells in the spring-linkage–based metamaterial rod, a two-dimensional (2D) mathematical particle-bar-spring model considering both transverse and longitudinal inertia is first established on the basis of the Lagrange equation (fig. S4; see the “Theoretical analysis on metamaterial rod” section in the Supplementary Materials for details), which can predict the performance of the metamaterial rod in manipulating stress waves. On the basis of extensive theoretical calculations, the cells’ parameters are determined. The length of each linkage bar is L = 20 mm. The total stiffness of the two springs in cell I is k₁ = 0.27 N/mm and that in cell II is k₂ = 0.9 N/mm, and the bar angles of the two types of cell are θ₁ = 35° and θ₂ = 32°, respectively.

Performance of manipulating stress waves

To examine the wave flipping performance of the metamaterial rod, we launched a plate by a spring to impinge the left end of the metamaterial rod with a speed of 0.8 m/s (movie S1, the “Dynamic experiments on metamaterial rod” section, and figs. S5 and S6A). As seen from Fig. 2 (A and B), a large compressive strain was generated in the first cell at t = 0.1 s after impact, but it was switched to a tensile strain at t = 0.2 s with rare compressive strain being observed in the subsequent cells. This indicates that only a small part of compressive wave transmitted to the second cell, while most turned into tensile wave. Afterward, the rare compressive wave was rapidly attenuated during the subsequent propagation on account of the metamaterial’s nonlinearity and dispersion together with the external friction force. At t = 0.37 s, only tensile wave remained, resulting in a pure tension signal output at the distal end (i.e., the 17th plate) of the rod (Fig. 2C).

Fig. 2. Performance of manipulating stress waves.

Tensile strain or force is positive, and compressive one is negative. (A to C) Impinge metamaterial rod by a plate launched with speed of 0.8 m/s. (D and E) Apply a compressive pulse, or (F and G) periodic mixed waves, or (H and I) a tensile pulse. (A) Snapshots of the metamaterial rod after impact. The instant of the first cell beginning to deform is defined as t = 0 s. Blue (red) arrow lines indicate the compressive (tensile) displacement of plates. (B), (D), (F), and (H) Cells’ strain history monitored in experiments. Three curves in (B) are shifted upward by 0.6, 0.4, and 0.2 strain, respectively. Red marked lines indicate theoretical predictions, showing a good agreement with experiments. (C), (E), (G), and (I) Force-time curves at four plates of plates 1, 3, 8, 13, and 17. Red dashed lines indicate theoretical predictions, showing a good agreement with experiments. The input and output comparisons are experimental results. (G) The output curve is offset to compare the frequency with the input one.

We also applied a compressive pulse to the metamaterial rod by a shaker (movie S2, fig. S6B, and the “Dynamic experiments on metamaterial rod” section). Obvious tensile wave was observed because of the third cell with less compressive signal remaining (Fig. 2, D and E). Then, the compressive wave almost disappeared after transmitting to the 15th cell, and only tensile stress was detected at the output end, exhibiting excellent flipping function of turning compressive wave into tensile wave (Fig. 2E).

To demonstrate the distinctive screening capability on stress waves, we applied periodic mixed waves to the metamaterial rod by a shaker (movie S3, fig. S6C, and the “Dynamic experiments on metamaterial rod” section). Both enhanced tensile waves and attenuated compression waves were observed in the third plate, and the compression wave dissipated faster than the tensile wave in subsequent propagation and disappeared after propagating around 15 cells (Fig. 2F). Thus, merely periodic tensile waves were detected at the output end with similar frequency to the input ones (Fig. 2G). These measured results have proven the extraordinary stress wave screening function of the metamaterial rod. Besides, better filtering function can be expected for inputting tension-compression alternant displacement, because the input compressive force is less than the tensile one due to the nonlinearity of metamaterials and thus disappearing more quickly.

When we applied a tensile pulse to the metamaterial rod, it stably propagated to the fixed end with little compressive signal detected after propagating eight cells (Fig. 2, H and I; movie S4; and the “Dynamic experiments on metamaterial rod” section), which is understandable because the tensile wave is easy to transmit from cell I to cell II. Our interests are in the soliton propagation (32, 33) of the tensile wave within the metamaterial, which is derived as an isolated wave solution of the KDV (Korteweg–de Vries) equation by ignoring the friction and the mass of springs and four-bar linkages (see the “KDV Equation” section in the Supplementary Materials for details; in experiments, the total mass of springs and the four-bar linkages is only about 4% of the connecting plates; see table S1). This is attributed to the nonlinearity of the metamaterial, which makes the tensile wave converging and balances the dispersion effect, resulting in a soliton propagation. To demonstrate the soliton character more clearly, we applied a tensile pulse to a uniform rod composed of only cell II. Both experimental and theoretical results show that the tensile pulse propagates stably within the rod and agrees well with the solutions of KDV equation except for a little attenuation by friction and transverse inertia, exhibiting solitary wave characters, as shown in Fig. 3A. If the friction and the transverse inertia are small enough, then the formed solitary wave can remain its waveform and amplitude during propagation even after travelling 100 cells (Fig. 3B). The soliton characters ensure the stable propagation of tensile wave, which not only contributes to the flipping and screening functions but also paves a way to long-distance delivery of mechanical signals. Similar optical soliton has demonstrated bright prospects in communications (34, 35).

Fig. 3. Tensile solitary wave in uniform spring-linkage–based metamaterial rod.

(A) Cells’ strain history. Three curves are shifted upward by 0.3, 0.2, and 0.1 strain, respectively. Both experimental and theoretical results agree well with the solutions of KDV (Kdv.sol) equation. (B) Force-time curves at plates predicted by theoretical model with frictions and transverse inertia ignored, showing a stable propagation of tensile stress wave.

Nonreciprocity and on-off switch of unique functions

The extraordinary flipping and screening functions on stress wave were unidirectional and nonreciprocal, which vanished when we loaded from the 16th cell with the first cell serving as the output end (Fig. 4, A and B). Similar nonreciprocity has shown a strong application prospect in the fields of optics (36, 37), acoustics (38–40), mechanics (41–43), and so forth.

Fig. 4. Nonreciprocity and switch cell of the spring-linkage–based metamaterial rod.

Cell I and cell II are indicated by red spring and gray spring, respectively. The positive (negative) force means tension (compression). Orange solid lines and black dashed lines represent experimental and theoretical results, respectively, showing a good agreement. (A) Flipping and screening functions under forward input. External force was applied to the first cell with the 16th cell fixed. (B) Lose the special functions under reverse input. External force is applied to the 16th cell with the first cell fixed. (C) Switch off the special functions by replacing the first cell (cell I) by cell II. Nondispersion of output compressive pulse in (C) is resulted from short flat segment at beginning of cell’s force-strain curve, as discussed in the “Influence of short flat segment in the force-strain curve” section and fig. S8 of the Supplementary Materials.

It should be emphasized that the first weak cell acts as a switch to realize the flipping and screening functions on mechanical wave. When we replaced the first cell (cell I) by cell II, it was found that the uniform rod cannot realize these unique functions (Fig. 4C). More detailed elucidation on the pivotal role of the first weak cell is presented in the “Effect of key cell” section and fig. S7 of the Supplementary Materials. That means that these functions can be turned on or off by designing the key switch cell, like key genetic switches on a living organism’s DNA (44).

On the basis of the theoretical model verified by plentiful experiments (Figs. 2 to 4), the switched-on or switched-off scopes of the flipping and screening functions depend on both the cells’ design and the external load, as well as the cell number. When the stiffness of the key cell and that of the rear ones are severely mismatched under external compression but well matched under the induced or imposed tension, the unique functions can be aroused up, and the flipping and screening efficiency can reach as high as 70%, as shown in the red and yellow zones in Fig. 5. Both the insufficient mismatch under compression (green and blue zones in the top left portion of Fig. 5, B and D) and the insufficient match under tension (green and blue zones in the bottom right portion of Fig. 5, B and D) of the cells’ stiffnesses will reduce the efficiency and even dispel these functions, as represented by the white zones in Fig. 5.

Fig. 5. On-off switch and efficiency of flipping and screening functions.

Blank (i.e., white) regions indicate the functions being turned off. The efficiency is defined on the basis of the metamaterial rod with critical number of cells. m is the mass of cell’s connecting plate. The sliding friction between each connecting plate and the rails is set to be 0.2 N. (A and B) Flipping efficiency on compressive waves, defined as the negative momentum ratio of half output tensile impulse at the fixed end to the external input compressive impulse. (A) θ₁ = θ₂ = 32°. The y axis represents the normalized external impact momentum by a plate of mass M launched with velocity v. (B) Mv = 0.045 kg·m/s, θ₂ = 32°. (C and D) Screening efficiency on periodic mixed waves, defined as the momentum ratio of half output impulse at the fixed end to the total external input tensile impulse. (C) θ₁ = θ₂ = 32°. The amplitude of input period mixed waves is 3.14 N. The y axis represents the normalized frequency of external load with f denoting the frequency. (D) External load is 2.5 sin (20πt) N.

It should be noticed that the stiffness mismatch of the key cell with the regular ones also depends on the key cell’s strain (Fig. 1D), which increases with the external load (fig. S9). To turn on the flipping and screening functions, the key cell has to be sufficiently compressed under external load with the strain >25% in flipping case and >3% in the screening one while not causing collision of the two plates adjacent to the key cell with strain <100%, as depicted in fig. S9. With reference to the maps on the functions’ efficiency (Fig. 5, A and C) and the key cell’s strain (fig. S9), it is evident that high efficiency would likely be achieved when the strain of the key cell is large.

In the switched-on cases, there exists a critical cell number, defined as the least sufficient cell number to realize the flipping and screening functions. The critical cell number generally decreases with k₂/k₁ and increases with θ₂/θ₁, as shown in fig. S10. By properly tuning k and θ of cells, the unique functions can appear within 10 cells. In the whole, the established theoretical model and the predicted diagrams (e.g., Fig. 5 and figs. S9 and S10) have provided a fundamental design guide for tailoring the metamaterials.

DISCUSSION

We have introduced an effective strategy to change the compression or tension characters of stress waves during forward propagation. The designed spring-linkage–based metamaterials realized unprecedented flipping and screening functions on mechanical waves, enabling mere tension output regardless of input wave being compressive, tensile, or tension-compression mixture. These original results break the direct consequence of the classical stress wave theory. The turned or extracted tensile waves would enable the metamaterial to move toward the loading end even under impact, inspiring the reverse motion robots accordingly. The flipping and screening on compressive waves result in no impact being “felt” at the distal end, which opens a novel avenue to affect mitigation (45, 46) and energy harvesting. In addition to the spring-linkage cells reported here, cells with other configurations can also achieve similar nonlinear behaviors (27–30). By ingeniously designing the key switch cell based on the proposed strategy, these extraordinary functions of flipping and screening mechanical waves can be appropriately turned on. It should be noticed that the spring-linkage–based metamaterials are superior in large deformation and full recoverability, which is advantageous under impact conditions. Although the proposed metamaterial is in 1D setting, it is expected to be further extended to multidimensions due to the high design performance. We anticipate that this work will serve as an initiative for more complex manipulation and utilization of mechanical signals. The revealed principles and the design strategy may also shed light on inspiring way in manipulating various physical signals of acoustics, optics, and electromagnetics.

MATERIALS AND METHODS

Fabrication of unit cell

The unit cell was composed of a four-bar linkage, two springs, and two connecting plates with additional weight (fig. S1). Some detail components were pasted by HY-303 glue, such as rolling bearings with inclined bars, springs with fixed blocks, and sliding bearings with connecting plates. The inclined bars were jointed with the springs and the connecting plates, respectively, by M2 screws. The connecting plates, the inclined bars, and the fixed blocks were 3D printed with 9400 resin. Springs of 18 mm in height were customized by spring steel with two kinds of stiffness: 0.135 and 0.45 N/mm, respectively. The sliding and rolling bearings were type LMUT13 and type MF52ZZ, respectively.

Quasi-static experiments on unit cell

The load-displacement curves of the two types of unit cells (cell I and cell II) shown in Fig. 1C were obtained by using the universal testing machine. The top and the bottom plates of the unit cell were, respectively, connected with the indenter and the fixed workbench of the machine by bolts; hence, the top plate was able to move up and down (fig. S2). The compressive or tensile loading was applied to the cell at a speed of 5 mm/min. Three experiments were conducted for each cell to verify the repeatability.

Dynamic experiments on metamaterial rod

Red dots were marked on the side of plates to track the motion of each cell by using a high-speed camera with wide-angle lens (fig. S5A). The 3rd, 8th, and 13th plates were made of two thin pieces with LDT1-028K polyvinylidene fluoride (PVDF) sensors pasted between them to measure the force-time curves (fig. S5B). The mass sum of the two pieces of thin plates was equal to that of an integral connecting plate. Besides, a PVDF sensor was glued on the surface of the first plate with covered by two small sheets to measure the input pulse (fig. S5A). Another PVDF sensor was inserted between the last plate and the fixed support to measure the output force. The PVDF sensors were successively connected to the KD5008C charge amplifiers and a TDS 2014C oscilloscope. Then, the corresponding force-versus-time curves were obtained by the measured voltage signals.

The first plate was divided into two equal pieces and bonded by magnets (fig. S5A). One piece was used as impact plate, which was launched by a spring to impinge the other piece with a speed of 0.8 m/s (fig. S6A). After collision, the two plates moved together due to the magnets’ bonding. The launch spring with stiffness 1 N/mm was fixed on the support.

A JZQ-50 shaker was used to produce compressive pulses or periodic mixed waves to the metamaterial rod. The shaker head was firmly connected with an extension bar, through which the pulses were delivered to the metamaterial rod, as shown in fig. S6 (B and C). To apply a periodic sinusoidal excitation, the extension bar was connected to the first plate by strong magnets (fig. S6C). In the case of compressive pulse, an extra stick was put between the extension bar and the first plate, ensuring only compression applied on the metamaterial rod, as the stick would fall under tension (fig. S6B). A tensile pulse was applied to the metamaterial rod by pulling a string tied to a magnet, which is attracted by another magnet fixed on the first plate (fig. S6D). By pulling the string quickly, the two magnets separated, so a tensile pulse was suddenly applied to the first plate.

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S10

Legends for movies S1 to S4

Table S1

Other Supplementary Material for this manuscript includes the following:

Movies S1 to S4