THROUGH A SOFT MATERIAL

RELATED APPLICATION

[0001] The present patent document claims the benefit of priority under 35 iiS.C. 119(e) to U,S. Provisional Patent Application No. 62/246,433, filed on October 2§ _!: 2015, which Is hereby incorporated by reference in its entirety.

TECHNICAL FIELD

[ 082] The present disclosure is related generally to soft architectures and more particularly to bistable structures formed from soft materials that may be useful for wave propagation.

BACKGROUND

[0003] Soft, highl deforrnable materials have enabled the desig of new classes of tunable and responsive systems and devices, including bioinspired soft robots, seif-rogul&tlng niicroflu dics, ada tive optics, reusable energy- absorbing systems, structures with highly programmable responses, and mofipbologicai computing paradigms. However, their highly deformabls and dissfpatlve nature also oses unique challenges. Although It has been

demonstrated that the nonlinear response of soft structures can be exploited to design machine capable of performing surprisingly sophisticated functions on actuation, their hig intrinsic dissipation has prevented the design of completely soft machi es. Se sing and control functionalities, which require transmission of a signal over a distance, still typically rely on the Integration of stiff electronic components within the so material, introducing interfaces that are often a source of mechanical failure _*

[0004] The design of soft control and sensing systems {and, consequently, completely sot machines) requlms the ability to ropagate: a stable signal without distortion through soft media. There are two limiting factors intrinsic to materials that work against this: dispersion (signal distortion doe to frequency-dependent phase velocity) and dissipation (loss of energ 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 th medium. For example, periodic systems based on Hertzian contact, tensegrlty structures, rigid bars and linkages, and bistable elastic elements can behave as nondispersive media, with the nonlfneartty of their local mechanical response canceling out the tendency for the signal to disperse at sufficiently large amplitudes. However, dissipation m 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-amplitud excitation (to avoid plastic energy loss). This approach minimizes, but does not eliminate, dissipation., in soft, highly dlsslpattve media, the problem is farther exacerbated, and there is no robust strategy currentl available to propagat signal© in these systems,

BRIEF SUMMARY

£09Θ5| An architecture for achieving stable wave propagation through a sof material Includes a plurality of bistable elements arranged In series. Each bistable element comprises a soft .material and h s two stable configurations _* a higher energy configuration and; a lower energy configuration.. The bistable elements are configu ed to snap from the igher energy configuration to the lower energy configuration sequentially when stimulated by a mechanical wave, thereb allowing the mechanical wave to propagate therethrough.

ΪΟ0093 A mechanical logic device comprises a first input chain comprising a first plurality of bistable elements arranged in series, a second input chain comprising a second plurality of bistable elements arranged in series, and an output chain connected to the first and second Input chains and comprising a third plurality of bistable elements arranged in series. Each bistable element comprises a soft material and has two stable configuratio s _* a higher energy configuration and a lower energy configuration. The bistable elements are configured to snap from the highe energy configu ation to the lower energy configuration sequentially when stimulated by a mechanical wave, thereby allowing fie mechanical wave to propagate therethrough. The mechanical logic device may be configured as a mecha ical OR gate or a mechanical AMD gate, £0097| A method of propagating mechanical signals m a soft material

comprises providing a plurality of bistable elements arranged in series,, where each: bistable element comprises a soft material nd has two stable

configurations, a higher energy configuration and a lower ene gy configuration. The first bistable element Is simulated by a mechanical wave and Induced to snap from the hig er to the tower e e g configuration. The mechanical wave thereby passes through the first bistable element and stimulates arc adjacent bistable element. The stimulation of adjacent bistable elements continues until the mechanical wave has propagated through ail of the plurality of bistable elements.

£00383 A method of fabricating an architecture for achieving stable wave propagation through a soft material includes extruding an ink formulation from deposition o zle moving relative to a substrate, wher the ink formulation comprises an unaired polymer. One or mom continuous filaments comprising the ink formulation are deposited In a predetermined pattern on the substrate. Thus, an as-printed architecture comprising a plurality of printed structures arranged In series is formed,, where each printed structure comprises the uncured polymer. The uncured polymer is cured so as to form, from the printed

structures _* a plurality of bistable elements arranged i aeries, where each bistable element is In a km energy configuration and comprises a soft material... The bistable elements are then deformed to have a deformed geometry, such that each bistable element adopts a higher energy configuration► The bistable elements are configured to snap from the higher energy configuration to the low energy configuration sequentially when stimulated by a mechanical wave, thereby allowing the mechanic l wave to propagate therein rough.

BRIEF DESCRIPTION OF THE DRAWINGS

0i | Fie. 1 A shows an exemplary soft architecture for achieving wave propagation that includes a plurality of bistable elements arranged in series. The bistable elements have two stable configurations _* a lower energy configuration and a higher energy configuration. The bistable elements are in the lower energy configuration in FIG. 1 A..

(0010| FIG. 1 B shows a close-up view of three of the bistable elements of FIG. 1A The bistable elements are physically connected fey coupling, elements having the form of linear springs. £0011! FIG, 1 C shows a ciose-u vie® of t ree of the bistable elements of FIG. 1A in a higher energy, deformed configuration,

[0012] FIG. 2A shows force F and potential energy V versus d splacement ¾ for the coupling elements of RGs< 1A~1C,

[0013] FIG, 28 shows a portion of an exemplary soft architecture Including a plurality of bistable elements in series, where the linea coupling elements have a different geometry than the springs shown in FIGs, 1A-1C.

[001 ] FIG, 2C shows a plot of force F versus displacement * for linear coupling elements of various sizes and geometries.

[001 S] FIG. 3 shows force F and potential energy V versus displacement x for the bistable elements of FIGs,. 1 A-1G.

£ 0 0 FIGs, 4A and 4B show force F and potential energy V as a function of displacement* and end-fo-end distance dfor bistable elements ha ing the geometry shown in FIGs. 1A-1C.

£0017! FIGs, 5A-5B show simulated values of pulse velocity and puls width, respectively, as a function ofend-to-end distance d a d coupling element stiffness

[0010] FIG. 5C shows an approximate ene gy barrier E _m for an entire propagating pulse, which is a function of both d and k, by combining the

measured energy landscape of the individual bistable elements (FIG. 40) with tie simulated pulse- widths of FIG, 58,

£0019] FIG, 6 is a contour map showing normalized displacement For eac bistable unit of a sertes having a variation I d along the lengt , from distance dt ^™ 14,5 mm at one end to a distance d2 ^™ 19.0 mm at the other end, A wave propagating through such an accelerator may exhibit a β-fold increase in wave p ed moving from the first end to the second end of the series,

[0020] FI , ? shows n exemplary soft mechanical diode co structed from a heterogeneous chain (series) of bistable elements, where a first region of the series Includes soft coupling: elements (e.g., k - 60 Ni m) inking adjacent bistable elements, and a second region of the series includes stiff coupling elements (e.g., k - 2,100 N/m) linking adjacent bistable elements. |0 21| FiGs. 8A and SB show optical images and a contour map, res ectively, of fie behavior of a puis© oliated in the soft region of the mechanical diode of FIG, ? when i! encounters the soft-stiff boundary.. (Propagation is halted).

[0022] FIGs, 9A and 98 show optical images and a contour map, respectively, of the behavior of a pulse foliated in the stiff region of the mechanical diode of FIG., 7 when It encounters the stiff-soft boundary. (Propagation continues).

[0023] FIG. 10 shows an exemplary soft mechanical logic device that includes a first input chain comprising a first plurality of bistable elements arranged in series, a second input chain comprising a second plurality of bistable elements arranged in series, and an output chain connected to the first and second: input chains, where the output chain comprises a third plurality of bistable elements, arranged in series,

[0024] FiGs.. 11 A and 118 show the soft mechanical logic device of FIG, 10 configured as an AND gate.

[O025J F!Gs. 12A and 12B show t e soft mechanical logic device of FIG. 10 configured as an OR gate.

[002$] FIG, 3 Is a schematic of a 3D printing process in which a filament comprising a viscoeiastlc ink composition is extruded through a moving deposition i ozzl and deposited on an underlying substrate,

[ 027] FIG,. 14 illustrates propagation of a stable nonlinear transition ave through an exemplary architecture, with each bistable element undergoing a displac ment from x ^ x to x « x* The instability ¾ may propagate with constant velocity and geometry,; enabled by both {ή the balance of nonlinear and dispersive effects and ) the balance of dissipation and energy release.

Snapshots of the evolving state of She ch in are shown, ith: |. _t ~ 0.128 s, ~ 0.194 s, and ™ 0.252 s relative to the start of the experiment, in this case wlh « 18.8 mm,

[0020] FIGs, 15A and 158 sho contour maps of an exemplary architecture during wave propagation, where the wave pulse Is in compression (FIG. 1SA) or tension (FIG. 158). 029] FIGs. ISC and ISO show optical Images obtained from a high-speed camera during wave propagation, corresponding to the experimental data in

F!Gs. ISA a d SB, respectively,

03393 FIGs, 151 and 15F show simulated compression-initiated nd tension- initiated pulses, respectively:, which sho excellent agreement with the experimental data shown In FIGs. 15 arsd 15B,

DETAILED DESCRIPTION

[00313 Described herein is an architected mediom. made of a highly

dlssipat!ve, soft material that can overcome both dispersive nd d!ssi ative effects and enable 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 suppled during its propagation. To achieve such behavior, bistable elements comprising a sot material that are capable of storing elastic energy in the form of deformation are used. When stimulated by the wave-front, the bistable elements release the energy as the wave propagates, without the need of any externa! stimulus. Dissipation allows stable wave propagation by balancing the elastic energy release. The damping intrinsic to the soft materials may remove all signals except the desired transition wave, which therefore can propagate with high fidelity, predictability, and controllability.

[§032] The architectures described herein, which can be fabricated and customized by 30 printing, ar capable of propagating stable waves with constant velocity over arbitrary distances, overcoming both dissipative and dispersive effects, despite th soft, dissipailve material of which they are

composed. Such architectures may serve as the ack one of functional devices, such as soft mechanical logic devices,

[00333 FIG. 1A shows an exemplary architecture 100 fo achieving stable wave propagation through a soft material The architecture 100 includes a plurality of bistable elements 102 arranged in series,. Each bistable element 102 comprises a soft material, such as an elasto eric material or other polymer, and has two stable- configurations, higher energy configuration 106, as shown in FIG,. 1C _S and a lower energy configuration 104,. as shown In FIG. B, When simulated by a mechanical wave, the istable elements 102 snap torn the higher energy configuration 106 to the lower energy configuration 104 in a sequential fashion, thereby allowing the mechanical wave to propagate th ough the

architecture Thus, energy issipation in the soft material may fee countered by the energy released as each testable element 102 transitions to the tower energy configuration 104, n this exemplary architecture 100, each: bistable element 102 includes a first beam portion 110 attached to a second be m portion 112, thereby defining a junction or node 114 in a mldportfon of the bistable element 102, as described further below,

[00343 Adjacent bistable elements 102 may be coupled together by a physical connection or by a force fi d (e.g.,, a magnetic -field), in the architecture of FIGs, 1A-1C, coupling elements 108 exte d between and physically connect adjacent bistable elements 102, Th functions or nodes 114 e ween the first and second beam portions 110,112 can serve as connection points for the coupling elements 08, The nodes 114 in the architecture of FIGs. A~ G include small metallic cylinders press fit into the soft material to add a mass concentration; at each node and to provide optical contrast for tracking wave propagation during eriments. The bistable elements 102 may be arranged in series along a longitudinal axis U, as shown, but generally speaking the serial arrangement of bistable elements need not: follow a straight line. For example, the bistable elements 102 may be seriall arranged along a curved line,

[00353 Each bistable element includes at least one constrained portion that is constrained from motion, while an unconstrained portion of the bistable element is free to snap (transition) from the higher to the lower energy configuration when stimulated by a mechanical w ve. As shown in FIGs, 1 -1C, both ends

102a, 102b of the bistable elements 102 may be constrained from, motion while mtdportions 102m thereof (which Include the junctions 4} are free to transition from the higher energy configuration 06 to the lower energy configuration 104, The ends 1 ^: G2a,1Q2b of each bistable element 102 may be fixedly or translatably secured to constraints 1 6 positioned adjacent to fie plurality of bistable

elements 102 in both the lower and higher energy configurations 104,106, The constraints 116 may be attached to or may be pari of longitudinally oriented bars 118, as shown In FIG. 1A. The constraints 116 may be spaced apart a. distance as illustrated In Fie, 18, The distance tf may be constant or variable along the series,, depending on the desired behavior of the wave.. T us, the longitudinally oriented bars 1 18 may e oriented either substantially parallel to or nonp ra!lel to the longitudinal axis L _s ,

|δ038] Irt the lower energy configuration 104 shown In FIG. 1 B _s the first and second beam portions 110,112 re substantially straight a d have opposing life with respect to he longitudinal axis L _s . In this exam e, the first beam portion 110 has a negative till with respect to the longitudinal axis L _¾ and the second beam portion 112 has a positive tilt with respect to the longitudinal axis L* The first, and second beam portion 110,112 of this example have a teng!tvto- thickness aspect ratio of 18, where the length is J mm, in the higher energ configuration 106 sh wn In FIG, 1C _t the first and second team portions 110,112 have a deformed (e.g., non-straight) geometry, which Is capable of storing energy. Other geometries are possible also, as long as each bistable element i cludes stable higher and lower energy configurations that allow energy to be stored In the architecture,

[0037 The coupling elements 108 of FIGs. 1A*1C take the form of springs having a zigzag geometry. The springs xhibit a linear relationship between force F and displacement χ ₍ m shown in FIG, 2A,. where the slope of the line represents the stiffness k Thus, the coupling elements 108 may be referred to as linear coupling elements. In this example, the springs have a length of 0 mm _:f a width of 5 m, a thickness of 5,4 mm, and a stiffness k of 80 N/m, it should be noted that suitable linear coupling elements 108 may have any geometry that exhibits linear behavior in respon e to an applied force,, such as the cellular structures show in FIG, 2B. Using different geometries for the linear coupling elements 108 may lead to different effective stiffnesses, which can influence the characteristics of the propagating mechanical wave, as discussed below. FIG, 2C shows force versu s displace men! data for exemplary linea r coupling elements 108 of different sizes and geometries (springs and cellular structures). The measured ¹ stiffness values range from about 30 N/m to about 2,100 N/m. p038] The coupling elements 108 may alternative^ be designed to e ibi a nonlinear relationship between force and displacement Such coupling eiements may be referred to as nonl near coupling elements, Exemplary nonlinear coupling elements Include spheres Sr Hertzian contact, where force F{x)

Increases according to or magnets that couple the bistable elements via a magnetic fi ld, as discussed in Nadfcaml et at, Physical R tew Letters i 16, 244501 (2016),

Like the bistefeie eiements 102, the coupling element 108 may comprise a soft material As indicated above, tie soft material may comprise one or more polymers, such as an elastomer (&.g ,, a silicone, polyuret ane, palybutadiene, or neoprene). Specific examples of suitable elastomers include poiydimeti ylsHoxane PDMS) and Dragon Skin® . The soft material may also or alternatively comprise a *smarf material, such as a shap memory polymer, that can return to a previous "remembered" shape when triggered by a stimulus, such as a temperature change or light exposure. Use of such a material may permit the bistable element to be reset from the low energy configuration to the high energy configuration after transmission of a wave through the soft architecture. The soft material of the coupling eiements 108 may he the same as or different f om that of the bistable elements 102, It is also contemplated that the cou ling elements 108 may co ise a hard material {e.g», a non-polymeric solid) such as a metal, alloy or semiconductor. Hard materials may exhibit elastic moduli much higher than the elastic moduli of soft materials. Advantageously, the bistable elements 102 and the coupling elements 1 8 are each formed of a material that may be 3D printed, a discussed further below.

[00491 Toe stiffness k may be nfluenced by the geometry of the coupling elements 1 8 and also by their chemical composition (e.g., hard or soft material), in the case of 3D i ted architectures, process conditions ( .g extrusion/print speed) also may play a role in determining the stiffness t Typically, when soft materials ar utilized, the stiffness k of th (linear) coupling elements 108 is in the range from about 1 Him to out 3,00 Him, or from about 33 Him to about 2,100 Him, £O041| FIG. 3 shows force F and potential ene gy F as a function of displace merit x for the bistable elements W2 of FfGs. 1 It can be observed that the bistable elements 102 exhibit a highly nonlin ar response when a fore© fs applied, with a region of negative in emeriia! stiffness (negative slope}:. The associated instability can lead to a rapid shape change that has been studied In the context of both natural and synthetic stems. Th potential energy r"( v) Is characterised by two local minima at x ^™ * ^! ~ 0 and ,* - corresponding to the two stable configurations shown in FIGs. 1 B a d 1C. The potential energy is defined such that d¥~8x ~ -Fand is calculated by first itiing a fifth-degree polynomial io the measured force-displacement data and then integrating. The stable conjuration at x* (corresponding to the higher energy configuration 108) Is characterised by an energ state higher than that of the stable configuration at x*'∞ 0 (corresponding to the lower energy configuration 104). Therefore,, similar to a. phase transition, the transition {or snapping) between the two stable states involves a net change in stored potential energy. Depending on the director! of the tra sition energy is either absorbed or released. The release of energy associated with a transition from the higher to the lower energy state can be exploited to overcome dissipation: and propagate a mechanical signal over arbitrary distances, as Indicated above, enabling the design of soft and highly tunable devices, such as the mechanical logic elements described below; The architecture 100 typically includes at least six bistable elements and may include- as many as (or more than) 100 bistable elements. The bistable etemenis may in some cases be myfiistabie elements that have more than two stabl

configurations,

[0O42| Due to the intrinsic damping of the soft material, the architecture may not enable propagation of small-amplitude elastic waves over long distances; however, moderate* and large* amplitude excitation can lead to a very different response, if the bistable elements are initially set to thei lower*energy

(yndeformed) stable e»«f¾ymtlon (x ~ ^s *™ 0 In FIG. 3, corresponding to FIG. 18), even a large amplitude displacement may not lead to a transition wave due to the energetically unfavorable {energy-absorbing) transition of each bistable element. Small-amplitude linear modes may also disintegrate because of dissipation; fhus, there may be no stable modes of energy transport whan the elements are in the low-energy state. However, f the bistable elements are initially set to their higher-energy (deformed) stable configuration (a? - ⁹ in FIG, 3, corresponding to FIG, 1C) _» a sufficiently large displacement applied to any of the bistable elements cam cause the displaced bistable element to transition states, producing a nonlinear transition wave that propagates Indefinitely outward from the paint of initiation with constant speed and shape. This can be attributed to both (0 an between dispersive and nonlinear effects of the architecture and (« a release of energy that overcomes the effects of dissipation as each of the bistable elements, stimulated by th wsvefront, transitions from its higher- to lower-energy stable state (i,e,, _: from x « xf to x « ^¾f » G).

£0O43| Consequently, a method of propagating mechanical signals In a soft material may entail: providing a plurality of bistable elements arranged in series, where each bistable element comprises a soft material and has two stable configurations, a higher energy configuration and a lower energy configuration; and stimulating: a first bistable element by a mechanical wave, causing the first bistable element to snap (transition) from the higher to the lower energy configuration. The transition allows the mechanical wave to pass through the first bistable element and to stimulate an adjacent bistable element, which also snaps (transitions) to the lower energy configuration . The stimulation of adjacent bistable elements continues until the mechanical wave has propagated through all of the bistable elements. The bistable elements may be coupled together by a physical connection or by a force field (e.g., a magnetic Held), as described above.

[00443 Tlie first bistable lement m y be positioned at an end of the ply r lly of bistable elements, and the mechanical wave may propagate In a single direction (e.g., toward the other end). Alternatively, the first bistable element may be positioned at an intermediate location in the plurality of bistable elements, and the mechanical wave may propagate in more than one direction simultaneously {&.g, _f tow rd both ends), The mechanical wave may propagate as a compressive wave pulse in one direction and as a ra refraction wave pulse In the other direction. P&4SJ As indicated abo e, the soft material may comprise a smart mater al, such as a s pe memory polymer, either alone or in combination with another polymer. By incorporating a smart material into the bistable elements, it ma be possible to reset each bistable element from the lower energy configuration to the higher energy configuration after propagation of t e mechanical wave. The smart material may e ogr mmed to return to a "mmembered" configuration upon exposure to a suitable stimulus, as kno n: n the art and discussed in, for example, M, Be hi and A. Lendleln, "Sh e Memory Polymers " terials Toda , 10, 4, 200?, pp. 20-28. Thus, the method may further include, after propagation of Hie mechanical wave through tie plurality of bistable elements, exposure of the smart material to a stimulus In order to activate the smart, material, thereby retumsng each bistable element from the lower energy configuration to the higher energy configuration. The stimulus m y be a change in temperature or exposure to ultraviolet f ^: UV) radiation, for example. In this way, the soft architecture may be reset and used again for wave propagation

£00463 The width of the pulse and it propagation velocity can be greatly changed by manipulating the nonlinear response of the .bistable elements: 102 and/or (in the case of linear coupling elements) the stiffness k. Although changes in k require fabrication, of new units with different morphology, as shown by the data of FIG, 20, it is possible to exploit the high doformabi ty of soft materials to tune the nonlinear response of the bistable elements. For example, the application of small lateral loads can change the ertd-to*end distance d

[004?] Simulations are used to systematically investigate the effects of the parameters d and k on the behavior of the propagating wave. The obtained force- displacement corves ere subsequently integrated to determine the on-site potential, and an eighth-order polynomial fit Is used as an approximation fo V(x), as shown for four representative values of d in FIGs, 4A and 48, It can be observed that d has a large effect on the energy harrier separating the two stable configurations, which is reflected also In changes in the peak forces during transition- Furthermore, d strongly affects the displacement necessary to obtain snap-through from the high-energy state to the low-energy state (and thus to initiate th transition during wave propagation). The effects of 4 and k on wave propagation velocity and pulse width are shown In P!Gs, BA and 5B and discussed further below. FfG. SO provides an a rox m te energy barrier i¾ _; for an .entire propagating poise, which is a function of both d and k, by combining the measu ed energy landscape- of the individual bistable elements (FIG,. 48) with fit simulated puts© widths of FIG, SB.

£06 81 Given that the energy barrier for propagation of a transition wave can be controlled by tuning; d and &, the inventors recognized that functional devices can be designed by carefully arranging the linear and nonlinear elements in the series (ί&,, along the chain). Since pulse propagation is substantially or entirely independent of initial conditions, the wave poise can be- manipulated through entirely local geometric changes. Dye to fee high damping of the system, only the specific signal compatible with the local geometric parameters may he able to propagate an appreciable distance.

00 13 Several exemplary mechanical devices based on soft architectures comprising series arrangements of bistable elements are described* including a wave accelerator, a mechanical diode and mechanical logic gates,

£60501 In a first example, an accelerator can be designed by applying different values of d spatially along the length of the architecture to achieve a controlled variation In wave velocity. This can be done without fabricating a specifically graded system because the deformable architecture allows different values of d to he applied as a function of length. The bistable elements may be arranged In series along a longitudinal axis and each bistable element may be secured

(fixedly or translaiabiv) to longitudinally oriented constraints positioned adjacent to the plurality of bistable elements. The distance of between the longitudinally oriented constraints may be varied: as desired along the longitudinal axis (®.g _tf linear Increase or decrease, exponential increase or decrease, ste wise change, etc).

£60511 Experimental results are shown m FIG, 6 for a series of bistable elements, where d varies linearly from about 14.5 mm at one end to about 19 mm at the other end. The results show a change In siope of the interface between the pre-transitioned and post-transltfone states (dark and light regions,

respectively), indicating a variation in pulse velocity,, as the slope of the interface is Inversely proportional to the speed ¹ . In this example, a change in the wave speed by more than a factor of 6 from the left: end of the series to the right end can be achieved (from 0.8 mis to 5.2 mis,). The velocity can he seen to

continually vary along the length of the chain, but at each location it {matches the expected velocities from: FIG, 5A,

£0652| *n another example, a mechanical diode is designed as a

heterogeneous chain or series of bistable elements, where a first region Includes soft coupling elements linking adjacent bistable elements and a second region includes stiff coupling elements linking adjacent bistable elements. Specifically, t e coupling elements n the first region of the aeries have a stiffness k d the coupling elements In a second region of the series have a stiffness A¾ where ¾ >

|0653! FIG. 7 shows an example of such an architecture Stat Includes 25 bistable elements wi h soft coupling elements (ki ~ 80 N/m) and 25 bistable elements with stiff coupling elements (!¾ ~ 2,100 N/m), with d - 17.5 mm. As shown in FIG. SC _t propagating poises In these two distinct portions of the architecture encounter very different energy barriers. Consequently, the architecture can functio as a mechanical diode,

£0O54| Referring to FIGs, 8A and 8B _S when a pulse is initiated in the soft region (k - 80 Him), where it possesses a small width -4 units) and a resulting tow energ barrier (E _m 0.2-0,3 mJ _< it is unable to continue propagating when it reaches the stiff region (k - 2.100 N/m), where a wide pulse {-2 units) and high- energy barrier are encountered (E * 1 md}.. As a result, the pulse freezes indefinitely at the soft-stiff boundary, with the wave energy that has not already been dissipated being stored In the elastic deformation of tie local structure, pOSSl In contrast, referring to FIGs, 9A and 9B, a pulse initiated in the stiff region readily propagates through the sot region as well, although a kink in wave velocit is observed at the transition between stiff and soft, as a result of ihe change in L The fact that the wave velocity rapidly changes at the boundary is a manifestation of the system's insertsi!ivity to initial conditions. Also, the large* amplitude wave appears to be essenti lly Insensitive to any fabrication-induced imperfections i the architecture. SSSJ A mom complex logic device is shown in FIG. 10. This device, a mechanical logic gate, includes a first Input chain comprising a first plurality of bistable elements arranged in series, a second Input chain comprising a second plurality of bistable elements arranged in series, and an output chain connected to the first and second Input chains, where the output chain comprises a third plurality of bistable elements arranged In series. Each bistable element comprises a soft material and has two stable configurations, a higher energy configuration and a lower energy configuration:. The bistable elements are configured to snap from the higher energy configuration to the lower energy configuration sequentially when stimulated by a mechanical wav , thereby allowing the mechanical wav to propagate therethrough.

iueSTJ One can define the high-energy state (e,g,., .x. * .r* ⁹ in FIG. 3} of the bistable eleme ts as logical state 0 d the low-energy state ( ~ in FIG. 3) as logical stats 1 and then design systems that predictably control pulse propagation in accordance with the energy barrier relationships in FIG. SC. The logic device may be configured as an AND gate or an OR gate,

Cif 8] For example, the logic device may f urther comprise a first set of constraints positioned along both sides of the first input chain adjacent to the first plurality of bistable elements; a second set of constraints positioned along both sides of the second input chain adjacent to the second plurality of bistable elements; and a third set of constraints positioned along both sides of the output chain adjacent to the third plurality of bistable elements. Ends of eac bistable element may be fixedly secured to trie respective adjacent constraints, and a distance d between the first set of constraints may be substantially the same as the distance d between the second set of constraints.: if a distance between the third set of constraints is less lhan the distance d, then the logic device may function as a mechanical AND gate. Alternatively, if the distanc d _m between the third set of constraints is greater than the distance d, then the logic device may function as a mechanical OR gate.

[0fK?9] The logic gate of FIG. 10 includes coupling elements ail having the same value of k (80 N/m i this example), with the first and second Input chains having d « 17,5 mm. Referring ow to FIGs, 11A and 118, when the end-to-end distance, d _&ll ¾, of the bistable elements In the output chain is small in comparison to d (e.g., o s ~ 16,? mm), the energy barrier is sufficiently igh fiat both Input chains need to be activated by propagating: transition waves n order to achieve wave propagation through t e output chain. Such a configuration yields a soft mechanical logical AND gate,

£06ββ| However, the architecture can become a logical OR gate when is increased sufficiently relative to d to ~ 18.6 mm), as shown in FiGs, 12A and 12B. in this case, because the energy barrier of the outpot chain is smaller (less than 0,1 mJ), if either of the input chains has propagated the transition wave, the wave can propagate through the output chain. Similar behavio can be obtained by other combinations of bistable element geometries and linear cooping eleme ts, using FIG, 5C as a guide.

[0661] Ail of the architectures described in this disclosure may be fo med by 3D printing, an extrusion-based printing process that may also be referred to as direct ink writing or direct write fabrication. Referring to FIG, 13, 3D printing:

entails flowing a rheoiogicaliy-iaiared ink formulation through a deposition nozzle that is moving relative to a substrate _* such that a filament comprising the Ink composition may be deposited on the substrate in a. predetermined 2:0 or 3D pattern.. In this way 30 printing; may be employed to build up 3D structures layer by layer. During 30 printing, the deposition nozzle may be moved at: a constant or variable print speed V while the substrate remains stationary, as shown In FIG, 13, Alternatively the substrate may be moved while the deposition nozzle remains stationary _* or both the deposition noz le and the substrate may be moved. While the substrate is typically a solid, 3D printing may be carried out using a gel or liquid as a substrate.

[fl0i.2| Advantageously, the ink formulation extruded through the deposition nozzle is viseoelastic with a strain-rate dependent viscosity, The ink formulation ma also be said to be shear-th inning. As shown in FIG. 13, the filament

deposited on the substrate may have a sufficient stiffness to retain its shape and a h ¾ht h. An exemplary ink formulation and 3D printing process is described below.. Further descriptions of 3D printing may he found in U.S. Patent

Application Serial No, 14/900,860, Hied December 22, 2015;;; US, Patent Application Serial No. 15/146,613 _* fi ed May 4, 2016: U,S. Patent Application Serial No. 16/036,937, fi ed M y 18, 2018,; and U.S. Patent Application Serial No. 15/117,623, filed August 9, 2018, ail of which are hereby incorporated by reference In their entirety,

§β§3] Generally speaking, a method of fabricating an architecture fo

achieving stable w ve propagation -through a soft: material includes extruding an ink formulation from a deposition nozzle moving relative to a substrate, where the ink formulation comprises an uncured polymer, One or mom continuous filaments comprising the ink formulation are deposited in a predetermined pattern on the substrate, as illustrated in FIG, 13. Thus, an as-printed architecture comprising a plurality of printed structures arranged in series s formed, where each printed structure comprises the unoured polymer. Typically after depositing the one or more continuous filaments, the uncured polymer is cured to form (from the printed structures) a plurality of bistable elements arranged in series, where each bistable element Is in a low energy configuration and comprises a soft material (e.g., a polymer as described previously), The bistable elements are deformed to have a deformed geometry, such that each bistable element adopts a higher energy configuration. The bistable elements are configured to snap from the higher energy configuration to the low (lower) energy configuration

sequentially when stimulated by a mechanical wave, thereby allowing the mechanical wave to propagat therethrough. The architecture and serially- arranged bistable elements formed as described above may have any of the features described elsewhere in this disclosure, Including, for example, coupling elements, constraints, etc.

£096 | The exemplary architectures ma b produced using direct ink: writing, an extrusion -based 3D printing method, which may be followed by an Infilling step. A visooeiastic poiydimethyisiioxane (POMS) ink formulation is used for 3D printing. This ink formulation includes a she -thinning POMS material, Dow Coming SE-1700 (85 wt.%), with a Sower-viscosity POMS additive, Dow Coming Sylgard 184 (IS wt.%}. The viscoelaatio yield properties are theologically tailored to ensure that the uncured ink lows readily during printing yet maintains its shape until it is permanently cross-linked In a subsequent curing step {e,g, _s 100 ^* G for 30 rain). The ink formulation Is extruded through a tapered noz^e (e.g?,, 20Ό μι ϊ inner diameter, obtained from Nordson EFD) during programmed translation; of the nozzle over a fixed substrate {e.g., PTFE-coated aluminum), as shown schematically in FIG. 13. Ink extrusion may be controlled via. fixed pressure (Nordson EFD Ultimus V pressure box), with the ozzle precisely positioned using a custom 3D positioning stage (Aerotech).

[iu6§] After printing and curing of the m formulation, two regions parallel with and adjacent to the functional region of wa e propagation may be infilled wit* epojcy (e.g., Momentiv© Epon 828) to prevent undesired structural ^; bending that could inhibit measurements.. The lateral distance, d, between* these longitudinal constraints or supports may be defined by acrylic braces of precise dimensions, which are made using an Epilog Laser Mini cutting s stem. The acrylic braces also serve to elevate the soft structure (via the longitudinal supports) without contacting to eliminate any interactions between the wave pulse and the table surface.

[i06S| A cylindrical copper rod $.g, _t 3. 75 mm diameter) is cut to pieces (e.g., 5,1 ? mm length, giving; a mass of -0.4? gj, which can be press fit Into the printed bistable elements to enable optical tracking of portions of the architecture during wave propagation. For example, top surfaces of the copper cylinders are painted with flat white paint to m&xirnke light contrast for visualization of the transition wave propagation.

[0067] To achieve a range of effective stiffnesses I, several different

geometries are designed for the linear coupling elements that connect the individual bistable elements together. As- shown i FIG, 2C, stiffness values range from 30 to 2 _* 100 N/m (as measured with a commercial quastsfatic test system, Instron 5586, in displacement control at a displacement rale of 2

mm min). Additional intermediate values can fee obtained by varying the translation speed of the printbead during the printing process.

0 haraste.ozgtio. . of Wave . Fro agaion

0068] To characterize the propagation of nonlinear waves experimentally, a high-speed; camera is used to track the location of ach bistable element In the series as a function of time. Because at the wavefronf the bistable elements tran ition from one stable configuration to the other, the d splacement of each unit is monitored relative to Its two stable configurations and For the unit two ced:

[00703 «, being the position of the P bistable element in the series, FIG _* 14 illustrates the propagation of the nonlinear wave by showing for each unit lis

e nearest stable configuration

[ W2] at different times. If & » 0, the * ^¾' element ss In ether of its two stable configurations, whereas ·¾ > 0 indicates that the unit Is passing through the energy barrier separating them. The experimental data of FIG, 1 dearly show that at ie wavefront a few bistable units {in this case, about four) are

undergoing a change from one stable state to the other at any give time and that the transition sequentially propagates through the elements in the series _* This transition wave is found to propagate with a constant shape, deafly indicating that both dispersive and dissipat effects are overcome in the architecture, 00733 The speed of fie nonlinear wave can he obtained by monitoring the evolution of the normalized distance a: ^sti; for each bist ble unit during the entire experiment:, as In IG _* 15A< Because in th s contour map the dark and light regions indicate bistable units in the high-energy and iow-energ stable configurations, respectively, the sequential change of each of the elements along the chain from one stable state to the other is evident, Furthermore, the constant slope of the boundary between the pretransllon (dark) a d post-transition (light) regions reveals a constant propagation velocity (In this case, 3,4 ± 0>1 m/s). Note also that the pulse width for any time can be extracted from the ma by taking a. horizontal slice of the plot ( e, _s a fixed time) and measuring the number of bistable elements in the midst of transitioning between the solid dark and solid Sight regions (approximately four elements in width). p074] Another uni ue aspect of this system is that the propagation velocity and pu!se shape are the same (w thin the margin of error) whether the wave is initiated in compression or tension, as revealed S y comparison of tri contour plots reported in FIG, 15 A. (for compression) and Fie, 158 (for tension}.,, in both cases, the transition wave propagates with a constant velocity (3,4 m s In this example) and pulse wtdlh (approximately four elements). The propagation: of rarefaction pulses Is rare find and thus a noteworthy feature of this system. Finally, It is noted ¹ that a wave can be initiated at any intermediate location in the series., in which case a compressive pulse travels in one direction d:

rarefaction pulse travels in the other direction, both propagating outward from the point of initiation.. FIGs. ISC and 5D show optical images obtained during wave propagation from a high-speed camera _* corresponding to the data in FIOs. 15A and 158, respectively,

[0075J Transition wave propagation is also characterized using a 1D mechanical model, in which the position of the middle of the - bistable element

$0 SJ

(0077J where F is the quasi- 1 D on-site potential of each bistable element, γ is a linear damping parameter, and k is the coupling element (&.,g., spring) stiffness. The linear damping model is a leading-order approximation to the complex dsss!paflve nature of elastomers. The bistable potential V is numerically computed by nonlinear flnite-eiernent simulations of a qyasisfaticsl!y deforming, co- rofafional, linear elastic beam; In 2D, The numerical force-displacement curves are validated tsy comparison with the experimental data shown in FIG, 3, To simulate the response of the system under iarge÷ampiitude excitations, initially, all nodes are placed in the high-energ configuration.- The first node is then excited by displacing: it from the high-energy stable point to the low-energy one, and the system response In time Is solved using a Newmark-β scheme.. The only unknown model parameter, γ, is determined by fling experimental wave speed data for a particular combination of geometric parameters (k ^™ 80 Him and ά ~ 17.5 mm). With all model parameters thereby determined,, architectures with different combinations of geometric parameters c n he examined ^' . As an example, FIGs. 15E and 1SF show simulated compression-initiated and tension- initiated pulses, respectively, which show excellent agreement with the

experimental data discussed earlier in regard to FK3s> 15A and 15Β»

£00783 Using values obtained for the o -site potential !¾i, as shown in FIG, 48, simulations are performed to predict the wave characteristics for different coupling element stiffnesses, &, and ©ncMo-end distances, d The wave speed, which is computed by tracking the point of maximum particle velocity, increases monotonically with increasing t _s shown in FIG,. SA, However, the effect of d on the wave velocity is more complicated. The wave velocity Is highly sensitive to changes In when d Is small but much less sensitive when is large. The associated width of the transition wave is also noted.,, defined as the number of nodes that simyiteneousiy have displacements between 10% and 90% of the transition displacement. The results reported in Fig. SB indicate that the pulse width increases with Increasing k, showing the same trend as t.he velocity. For constant coupling element stiffness value k _s d does not have an effect on the pulse width fo lower stiffnesses but shows a similar variation as the velocity for higher stiffnesses. Experimental measurements of architectures wi h different values of d and k match these numerical results with excellent quantitative agreement, confirming the validity of our simulations.

[0079J The trends for velocity (FIG. 5A) and pulse width (FIG. 58) contours show a correlation with the energy barrier for different d values, as shown in FIG, 48, For a constant coupling 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 need to absorb more energ to overcome the barrier, thereby causing a slower transition: rate and therefore a. lower wave speed and vice versa, Consequently, the energy barrier appears to be the most important criterion in determining the transition speed and width of the

displacement profile. £0080 Because tie 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, E _ws is calculated as

[00821 where V and N are determined from the simulation results (FIGs. 48 and FIG. SB, respectively) and the .¾ values are a proximated by c islrlb ing them equally between and x ⁵ ^ FIG, SC shows fie total potential energy barrier (E _m ), associated with: the tr nsition events of the individual bistable elements from their higher-energy slate (¾ - .¾ά to their Sower-energy stale {x ~ x _e1 ). As expected, as the number of elements in the pulse (N) and the energy barrier for the Individual eiemeiits Increase, so does the total energy barrier required to Initiate tie pulse along a given portion of the series (or chain) because the tot l energy barrier is the sum of the transient barriers of the individual elements undergoing transition at a given time,

0 831 Although the present invention has bee described i considerable detail with reference to certain embodiments theieof, otser embodiments are possible without departing from the present invention. The spirit and scop© of the appended claims should not be limited, therefore, to the description of th preferred embodiments contained herein. Ail embodiments that come within the meaning of the claims, either literally or by equivalence, are Intended to be embr ced: therein. Furthermore, the advantages described above are not necessarily the on y advantages of the invention, and It Is not necessarily expected that all of the described advantages i! be achieved with every embodiment of the invention.