Stable propagation of mechanical signals in soft media using stored elastic energy
- aJohn A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138;
- bWyss Institute for Biologically Inspired Engineering, Harvard University, Cambridge, MA 02138;
- cGraduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125;
- dEngineering and Applied Science, California Institute of Technology, Pasadena, CA 91125;
- eDepartment of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland;
- fKavli Institute, Harvard University, Cambridge, MA 02138
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Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved June 27, 2016 (received for review March 24, 2016)

Significance
Advances in nonlinear mechanics have enabled the realization of a variety of nontraditional functions in mechanical systems. Intrinsic dissipation typically limits the utility of these effects, with soft polymeric materials in particular being incompatible with meaningful wave propagation. Here we demonstrate a nonlinear soft system that is able to propagate large-amplitude signals over arbitrary distances without any signal degradation. We make use of bistable beams to store and then release elastic energy along the path of the wave, balancing both dissipative and dispersive effects. The soft and 3D printable system is highly customizable and tunable, enabling the design of mechanical logic that is relevant to soft autonomous systems (e.g., soft robotics).
Abstract
Soft structures with rationally designed architectures capable of large, nonlinear deformation present opportunities for unprecedented, highly tunable devices and machines. However, the highly dissipative nature of soft materials intrinsically limits or prevents certain functions, such as the propagation of mechanical signals. Here we present an architected soft system composed of elastomeric bistable beam elements connected by elastomeric linear springs. The dissipative nature of the polymer readily damps linear waves, preventing propagation of any mechanical signal beyond a short distance, as expected. However, the unique architecture of the system enables propagation of stable, nonlinear solitary transition waves with constant, controllable velocity and pulse geometry over arbitrary distances. Because the high damping of the material removes all other linear, small-amplitude excitations, the desired pulse propagates with high fidelity and controllability. This phenomenon can be used to control signals, as demonstrated by the design of soft mechanical diodes and logic gates.
Soft, highly deformable materials have enabled the design of new classes of tunable and responsive systems and devices, including bioinspired soft robots (1, 2), self-regulating microfluidics (3), adaptive optics (4), reusable energy-absorbing systems (5, 6), structures with highly programmable responses (7), and morphological computing paradigms (8). However, their highly deformable and dissipative nature also poses unique challenges. Although it has been demonstrated that the nonlinear response of soft structures can be exploited to design machines capable of performing surprisingly sophisticated functions on actuation (1, 2, 9), their high intrinsic dissipation has prevented the design of completely soft machines. Sensing and control functionalities, which require transmission of a signal over a distance, still typically rely on the integration of stiff electronic components within the soft material (10, 11), introducing interfaces that are often a source of mechanical failure.
The design of soft control and sensing systems (and, consequently, completely soft machines) requires the ability to propagate a stable signal without distortion through soft media. There are two limiting factors intrinsic to materials that work against this: dispersion (signal distortion due to frequency-dependent phase velocity) and dissipation (loss of energy over time as the wave propagates through the medium). Dispersion can be controlled or eliminated through nonlinear effects produced via the control of structure in the medium (12). For example, periodic systems based on Hertzian contact (13⇓–15), tensegrity structures (16), rigid bars and linkages (17), and bistable elastic elements (18) can behave as nondispersive media, with the nonlinearity of their local mechanical response canceling out the tendency for the signal to disperse at sufficiently large amplitudes. However, dissipation is still an overarching problem. Structures designed to propagate elastic waves are typically built from stiff materials with low intrinsic dissipation (e.g., metals) and excited with small-amplitude excitation (to avoid plastic energy loss). This approach minimizes, but does not eliminate, dissipation. In soft, highly dissipative media, the problem is further exacerbated, and there is no robust strategy currently available to propagate signals in these systems.
Here we report an architected medium made of a highly dissipative, soft material that overcomes both dispersive and dissipative effects and enables the propagation of a mechanical signal over arbitrary distances without distortion. A stable mechanical signal can be transmitted over long distances through a dissipative medium only if additional energy is continuously supplied during its propagation. To achieve such behavior, we use bistable elastomeric beams that are capable of storing elastic energy in the form of deformation and then, stimulated by the wavefront, releasing it during the propagation of the wave, without the need of any external stimulus. Dissipation allows stable wave propagation by balancing the elastic energy release. The damping intrinsic to the soft materials removes all signals except the desired transition wave, which therefore propagates with high fidelity, predictability, and controllability. Furthermore, as observed for nondissipative (18) or minimally dissipative systems (19) made from stiff materials, a series of interacting bistable units can transmit nondispersive transition waves. By contrast, the proposed architecture is capable of propagating stable waves with constant velocity over arbitrary distances, overcoming both dissipative and dispersive effects, despite the soft, dissipative material of which it is composed. Together, these effects enable the design of functional devices such as soft mechanical logic elements. The ability to 3D print soft mechanical logic enables a higher degree of customizability and tunability relative to previous examples of mechanical logic (15, 20⇓–22).
System Architecture
The fundamental building block of our system is a bistable element (formed by two tilted beams) connected to a horizontal element with linear response, all made of elastomeric material (Fig. 1A). The tilted beams have aspect ratio
(A) The system consists of a 1D series of bistable elements connected by soft coupling elements. (Scale bar, 5 mm.) (B) The coupling elements are designed to exhibit a linear mechanical response, whereas (C and D) the bistable elements possess two stable states. The bistability originates from lateral constraint (d) on a beam pair that is displaced (x) perpendicularly to the constraint. The mechanical response is fully determined by the aspect ratio (L divided by the thickness of the beam) and d. The two stable configurations of the bistable element correspond to the displacements
Using different geometries for the linear coupling elements leads to different effective spring stiffnesses, which greatly affect the width and velocity of the propagating pulse. The stiffnesses were measured using an Instron 5566 in displacement control with a rate of 2 mm/min. The measured stiffnesses of the linear elements shown here were measured to vary from 30 to 2,100 N/m.
We start by characterizing the static response of both the bistable elements and the connecting horizontal elements. The force–displacement curve in Fig. 1B shows the linear response of the horizontal elements. In particular, for the element shown in Fig. 1A, with a zigzag morphology of length 10 mm, width 5 mm, and thickness 5.4 mm, we measure a stiffness
Response Under Large-Amplitude Excitations
Although the architected medium does not enable propagation of small-amplitude elastic waves over long distances due to the intrinsic damping of the polymer (Fig. S2), moderate- and large-amplitude excitation can lead to a very different response. If the bistable elements are initially set to their lower-energy (undeformed) stable configuration (
(A) The shaker (on the left) was attached to an accelerometer that was directly glued to the samples. The accelerator used to measure the output was glued to the other end of the sample. The acrylic braces (red) were used to hold the soft architecture at well-defined widths and were glued to the laboratory stands to prevent unwanted movement. (B) Small-amplitude, linear excitation from either end of the chain is rapidly dissipated due to the damping intrinsic to the polymer, as is particularly evident with increasing frequency, shown here for samples with 6, 15, and 50 bistable units.
Experimental Results
To characterize the propagation of such nonlinear waves experimentally, we used a high-speed camera and tracked the location of each bistable element along the chain as a function of time (Movie S1). Because at the wavefront the bistable elements transition from one stable configuration to the other, we monitor the displacement of each unit relative to its two stable configurations (
The speed of the nonlinear wave can be obtained by monitoring the evolution of the normalized distance
The transition wave can be initiated anywhere along the chain, with compressive and rarefaction pulses proceeding in opposite directions from the point of initiation (here
Another unique aspect of this system is that the propagation velocity and pulse shape are the same (within the margin of error) whether the wave is initiated in compression or tension, as revealed by comparison of the contour plots reported in Fig. 2A (for compression) and Fig. 2B (for tension). In both cases, the transition wave propagates with a constant velocity (3.4 m/s) and pulse width (approximately four elements). The propagation of rarefaction pulses is a rare find and thus a noteworthy feature of this system. Although compressive nonlinear solitary waves have been observed in nonlinear periodic systems as in, e.g., Hertzian contact-based chains (12, 32, 33) as well as in macroscopic nonlinear chains using magnetic connectors (19, 34), rarefaction pulses have not been found in those, due to the lack of stiffness in tension, among other reasons. Finally, we note that the transition wave can also be initiated at any intermediate location along the chain, in which case a compressive pulse travels in one direction and a rarefaction pulse travels in the other direction, both propagating outward from the point of initiation (Movie S3).
Numerical Results
We additionally characterized the transition wave propagation using a 1D mechanical model, in which the position
(A) An example of the beam deformation simulation is shown. All simulations were performed on only one half of the bistable element (i.e., on one tilted beam). Different configurations of the beam are shown as it is displaced from one stable configuration to another. The force at node B is measured (and doubled to account for bistable element consisting of two tilted beams). (B) The numerical, experimental, and best-fit force–displacement curves are shown for
(A) Experimental and (B) simulation results corresponding to
Control of Wave Propagation
The results reported so far were obtained for a system with connecting elements of stiffness
(A) The on-site potential as a function of x and d, as determined via quasistatic 1D displacement-controlled simulations of an individual bistable element. Simulated values of (B) pulse velocity and (C) pulse width as a function of end-to-end distance d and connector stiffness k. (D) The measured energy landscape (A) of the individual bistable elements is combined with the simulated pulse widths (C) to compute an approximate energy barrier
Experimental data obtained by directly measuring the force–displacement behavior of a single bistable element for different lateral constraints, d. The potential energy is calculated from this, showing a large effect of d on the energy barrier of the bistable elements.
Using these values for the on-site potential
(A) Experiments show that when d is small (17.5 mm here), the energy barrier between the two stable states is larger, and the wave propagation is slower. (B) When d is larger (18.6 mm here), the smaller energy barrier allows a larger propagation speed, as evidenced by the changed slope.
Because the system is deformable, different values of d can be used along the length of the system, resulting in spatially varying energy barriers to propagation; this can be used to vary the velocity along the length of the chain, as it is here for a gradient structure (d is about
(A) When k is high (2,100 N/m here), experiments show that both the pulse width and the pulse velocity (as determined by the slope) are much higher, even with the same value of d (18.6 mm), than (B) when k is low (80 N/m here). (C) The same comparison can be made by taking experimental snapshots of the two different systems (k = 80 N/m and k = 2,100 N/m, corresponding to the differences in morphology of these elements, as pictured in Insets).
The trends for velocity (Fig. 3B) and width (Fig. 3C) contours show a correlation with the energy barrier for different d values as shown in Fig. 3A. For a constant interconnecting element stiffness, the d values corresponding to high-energy barriers show lower velocity and width. This is because when the energy barrier is high, each element needs to absorb more energy to overcome the barrier, thereby causing a slower transition rate and therefore a lower wave speed and vice versa. Consequently, the energy barrier is the most important criterion in determining the transition speed and width of the displacement profile.
Because the N bistable elements that constitute a particular pulse are not simultaneously in morphologies that place them in the peak of their individual energy barriers, the total pulse energy barrier,
Tunable Functional Devices
Having demonstrated that the energy barrier for a transition wave to propagate can be controlled by tuning d and k, we now demonstrate how functional devices can be designed by carefully arranging the linear and nonlinear elements along the chain. To this end, it is critical to note that the pulse propagates independent of its initial conditions, so that it can be manipulated through entirely local geometric changes. This can be understood as a result of the high damping of the system, in which only the specific signal compatible with the local geometric parameters is able to propagate an appreciable distance.
For example, an accelerator can be designed by applying different values of d spatially along the length of the system to achieve a controlled variation in velocity. This can be done without fabricating a specifically graded system because the deformable architecture allows different values of d to be applied along the length of the system. The experimental results reported in Fig. S7 for a chain where d is ∼14.5 mm at one end and about 19 mm at the other show an evident change in slope of the interface between the pretransitioned and posttransitioned states (blue and red, respectively), indicating a variation in pulse velocity (the slope of the interface is inversely proportional to the speed). In particular, we observe a change in the wave speed by more than a factor of 6 from the left end of the chain to the right (0.8–5.2 m/s). The velocity can be seen to continually vary along the length of the chain, but at each location it matches the expected velocities from Fig. 3B.
Further, a mechanical diode can be designed as a heterogeneous chain with soft linear horizontal connecting elements (corresponding to a low-energy barrier) in one region and stiff ones (corresponding to a high-energy barrier) in another region. As an example, in Fig. 4 we show results for such a system set to
(A) A functional soft mechanical diode can be realized by creating a heterogeneous chain composed of a region with soft connectors and a small energy barrier (on the left) and a region with stiff connectors and a large energy barrier (on the right). A pulse initiated in the soft region (from the left) cannot pass into the stiff region due to the large energy barrier, causing the pulse to freeze indefinitely at the interface (A, iv–vi, and B). In contrast (A, vii–ix, and C), when the pulse is initiated in the stiff region, the propagation continues into the soft region and through the whole chain without interruption.
As seen above, the final equilibrium wave profile of the system depends on the internal equilibrium between the energy generated by the transition and the energy dissipated by the material damping and is therefore not dependent on the initial conditions. The latter only affect the transient time needed for the pulse to achieve its equilibrium configuration. Indeed, integration of Eq. 3 under the assumption of a smooth, stable propagating wave form shows that the total kinetic energy per mass, E, transported by the pulse and its velocity v are related by
Using similar principles, more complicated functional devices can be designed, such as mechanical logic gates (Fig. 5). One can define the high-energy state of the bistable element (
(A) A bifurcated chain demonstrating tunable logic in a soft mechanical system. The distance
Conclusion
We have designed, printed, and demonstrated a system that enables the propagation of a signal without distortion through soft, dissipative media over arbitrary distances. The soft medium damps out linear waves, leaving only the desired transition wave. The precisely architected system makes use of beam-based units exhibiting asymmetric bistabilities to achieve the propagation of nonlinear transition waves in which the dissipation inherent to the polymer is overcome by the local release of elastically stored energy during the transition of the individual bistable units from a high-energy to a low-energy configuration. The medium thereby undergoes a phase transition as the wave pulse propagates through it. The wave pulse itself locally stimulates the release of the stored elastic energy as it propagates. Although we have used beam pairs as a simple way to produce the (1D) asymmetric bistable potential that we use to store elastic energy along the path of the wave, other higher-dimensional arrangements of beams (5) and shells (31) would also exhibit asymmetric bistability and could therefore also be explored as alternative architectures for the phenomenon studied here.
Due to the intrinsically unidirectional transition from the high- to the low-energy state that each individual bistable unit undergoes during propagation, an external source of energy must be provided to reset the bistable elements to their higher-energy state if additional propagation events are desired [which, for example, could be provided pneumatically or via chemical reactions, as has been demonstrated in other soft autonomous systems (37)]. The high quality of the printed elastomer ensures that the system can be reused in this manner indefinitely, with a consistent response from cycle to cycle.
The soft system has the advantage of facile tunability (e.g., changing d) and control over wave speed, pulse width, and pulse energy, with pulse propagation independent of the initial conditions. Additionally, the linear coupling springs between the bistable units exert a large effect on pulse width and energy. A simple mechanical model was shown to accurately capture the wave characteristics and guide the design of functional soft logic devices, such as diodes, “or” gates, and “and” gates. This form of logic could be harnessed to introduce some level of feedback and control in truly soft autonomous systems (i.e., without the use of rigid electronics that introduce materials mismatches that can lead to failure). It is also unique in that the system undergoes relatively large-amplitude shape changes during its function, so that the process and output can be easily visualized. As discussed in our previous work (5), the mechanical response of the beams is scale-independent, and the elastic nature of the mechanism ensures a mechanical response that is independent of rate and loading history. Our findings can therefore be adapted to other scales and contexts.
Materials and Methods
The polydimethylsiloxane (PDMS) structures were produced using direct ink writing, an extrusion-based 3D printing approach (see Supporting Information and ref. 5 for more information). The soft architecture was connected to rigid epoxy supports, with the lateral distance between these supports, d, controlled by acrylic braces (thereby affecting the morphology of the soft architecture). Copper cylinders were press fit into the printed structure to enable optical tracking of periodic points along the structure. Measurements of the transition waves were made using a high-speed camera (Phantom v7.1), allowing the output of the positions for each element i for all time [
Experiments
Fabrication.
The structures were produced using direct ink writing, an extrusion-based 3D printing method, followed by an infilling step. A viscoelastic polydimethylsiloxane (PDMS) ink was used for 3D printing. This consisted of a shear-thinning PDMS material, Dow Corning SE-1700 (85 wt %), with a lower-viscosity PDMS additive, Dow Corning Sylgard 184 (15 wt %). The viscoelastic yield properties are tailored (see supporting information in ref. 5 for rheological characterization) to ensure that the uncured ink both flows readily during printing, yet maintains its shape until it is permanently cross-linked in a subsequent curing step (
To achieve a range of effective stiffnesses k, several different geometries were designed for the linear coupling elements that connect the individual bistable elements to one another. As shown in Fig. S1, we measured stiffness values ranging from 30 to 2,100 N/m (as measured with a commercial quasistatic test system, Instron 5566, in displacement control at a displacement rate of 2 mm/min). Additional intermediate values can be obtained by varying the translation speed of the printhead during the printing process.
Small-Amplitude Excitation.
To characterize the dynamic response of the system, we considered small-amplitude excitations with white noise up to 5 kHz generated by an electrodynamic shaker (model K2025E013; Modal Shop) directly connected to one end of the sample. We monitored the propagation of the mechanical signal using two miniature accelerometers (352C22; PCB Piezotronics) attached to both ends of the chain (Fig. S2A). Spectra were obtained for three different chain lengths (6, 15, and 50 bistable units in length) and were determined to be independent of d. The rigid epoxy supports were held apart at fixed distances by acrylic braces. These ensured that the morphology of the soft structure remained in a controlled configuration during the dynamic tests. The acrylic braces were in turn glued to steel laboratory stands on an optics table, to minimize undesired vibrations. As expected for a soft, dissipative material, the transmittance spectra [defined as the ratio between the measured output and input accelerations,
Measuring Transition Waves.
Measurements of the transition waves were made using a high-speed camera (Phantom v7.1). For systems with low wave speeds (usually
Control of Wave Propagation.
Although the results reported in Fig. 3 were obtained numerically, we also experimentally characterized the propagation of large-amplitude waves in systems characterized by different values of k and d.
First, to validate the numerical predictions for the on-site potential, we performed quasistatic 1D displacement-controlled experiments for different d values on an individual bistable element. The experimental results reported in Fig. S5 show a convincing agreement with the numerical results (Fig. 3A).
Next, we experimentally investigated the effect of d and k on both wave velocity and pulse width.
To explore the effect of d on the wave behavior, we tested the propagation of a transition wave through a system in which different values of d were assigned for the different experiments (Fig. S6). This can be done without fabricating a new sample for each experiment because different values of d can be achieved by applying a defined lateral displacement (
The stiffness of the linear connecting elements, k, also greatly affects the pulse propagation. Fig. S8 A and B show data for an experiment conducted on a system with stiff and soft connecting elements (2,100 N/m and k = 80 N/m, respectively; Fig. S8C, Insets). First, by comparing the slope of the boundaries in Fig. S8 A and B, it is evident that the stiffness of the connecting elements affects the pulse velocity. In fact, we find velocities of ∼18 and 3.4 m/s for
Simulations
Computation of the On-Site Potential.
One bistable beam element consists of two inclined beams with a mass placed at the center. Because the mass is rigid compared with the compliant beams, it is assumed that the force on the mass by the bistable structure is produced solely by the deformation of the beams. Due to the symmetry of the structure, the quasistatic deformation of only one tilted beam was modeled with appropriate boundary conditions. The beam was modeled using slender corotational beam finite elements (36) whose one-dimensional stretching and bending deformation are governed by a nonlinear Neo-Hookean material model with an initial slope of
Parameters.
The mass at each node was 0.419 g, with k ranging from 50 to 2,500 N/m and d ranging from 14.5 to 19.0 mm with
Acknowledgments
We thank Drs. Sicong Shan, Farhad Javid, and Daniele Foresti for valuable assistance. K.B. and J.A.L. acknowledge support from the Harvard Materials Research Science and Engineering Center (MRSEC) through Grant DMR-1420570. K.B. acknowledges support from the National Science Foundation (NSF) through Grant CMMI-1149456 Faculty Early Career Development (CAREER) Program. N.N. and C.D. acknowledge support from the NSF under Grant CMMI-1200319. D.M.K. acknowledges support from the NSF through CAREER Award CMMI-1254424.
Footnotes
- ↵1To whom correspondence may be addressed. Email: bertoldi{at}seas.harvard.edu, kochmann{at}caltech.edu, or jalewis{at}seas.harvard.edu.
Author contributions: J.R.R., C.D., D.M.K., J.A.L., and K.B. designed research; J.R.R. and N.N. performed research; J.R.R. contributed new reagents/analytic tools; J.R.R., N.N., C.D., D.M.K., J.A.L., and K.B. analyzed data; and J.R.R., N.N., C.D., D.M.K., J.A.L., and K.B. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1604838113/-/DCSupplemental.
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