ZHANG Hang，WU Jun，ZHANG Yihui*，F(xiàn)ANG Daining

1.Applied Mechanics Laboratory，Department of Engineering Mechanics，Tsinghua University，Beijing 100084，P.R.China；2.Center for Flexible Electronics Technology，Tsinghua University，Beijing 100084，P.R.China；3.Institute of Advanced Structure Technology，Beijing Key Laboratory of Lightweight Multi?functional Composite Materials and Structures，Beijing Institute of Technology，Beijing 100081，P.R.China

Abstract: Over the past decade，multistable mechanical metamaterials have been widely investigated because of their novel shape reconfigurability and programmable energy landscape. The ability to reversibly reshape among diverse stable states with different energy levels represents the most important feature of the multistable mechanical metamaterials. We summarize main design strategies of multistable mechanical metamaterials，including those based on self-assembly scheme，snap-through instability，structured mechanism and geometrical frustration，with a focus on the number and controllability of accessible stable states. Then we concentrate on unusual mechanical properties of these multistable mechanical metamaterials，and present their applications in a wide range of areas，including tunable electromagnetic devices，actuators，robotics，and mechanical logic gates. Finally，we discuss remaining challenges and open opportunities of designs and applications of multistable mechanical metamaterials.

Key words：multistable mechanical metamaterials； self-assembly； snap-through； structured mechanism；geometrical frustration

0 Introduction

Multistable mechanical metamaterials［1-30］are artificial structures with two or more different stable configurations that can be switched reversibly among each other. From the perspective of energy landscape，these stable configurations correspond to the minima of the system energy，and the energy barrier between different stable states confers the stability of multistable mechanical metamaterials at these configurations. Due to the need to be capable of switching among various geometric configura?tions， the multistable mechanical metamaterials mostly belong to the class of soft（or flexible）me?chanical metamaterials with a high level of deform?ability. During the last decade，many intriguing sys?tems of soft mechanical metamaterials［31-34］have been developed to offer unusual mechanical proper?ties，such as J-shaped stress-strain curves，negative Poisson’s ratios，negative thermal expansion and anisotropic swelling under large strains. Since many mechanical properties（e.g.，phononic band gaps，stress-strain curves and Poisson’s ratios）depend highly on the microstructure geometries of soft me?chanical metamaterials，many researchers have in?vestigated the tunability of these properties by me?chanical loads，as well as by light，heat or magne?tism. In many of these studies，the deformed config?urations of mechanical metamaterials cannot be maintained after mechanical loads（or other stimuli）are removed，which sets certain limitations to practi?cal applications. Developments of multistable me?chanical metamaterials can overcome these limita?tions，since these metamaterials can maintain di?verse stable configurations after removal of external loads/stimuli. As such， multistable mechanical metamaterials could significantly broaden the appli?cation fields of soft mechanical metamaterials，and enable novel developments of functional systems for energy absorption［17，19，22，35-37］， logical opera?tions［9，13-14，38-39］，elastic wave control［13，15，40-46］，soft robotics［47-55］，and etc.

A diversity of multistable mechanical metama?terials［1，3-5，8，18，20-21，26，35-36，44］have been reported in the past decade. According to the different design strate?gies，these multistable mechanical metamaterials can be classified into four categories，including those based on the self-assembly scheme［4-5，20］，snapthrough instability［8，18，44］， structured mecha?nism［21，35-36］and geometrical frustration［1，3，26］，as shown in Fig.1. This paper aims to present an over?view of state-of-art multistable mechanical metama?terials，covering these four important classes of de?sign strategies. It begins with the discussions on the self-assembly scheme，followed by the introduction to snap-through instability，structured mechanism and geometrical frustration. Then，we present vari?ous applications of multistable mechanical metama?terials in soft robotics and mechanical devices Final?ly，we provide an outlook on the existing challenges and open opportunities.

1 Self?assembly Scheme

Self-assembly schemes usually rely on rolling，folding and buckling deformations of patterned 2D precursor structures to form well-controlled 3D ar?chitectures. In this process，different loading paths could result in different 3D stable configurations for specially engineered 2D precursor structures. This section focuses on discussions of three representa?tive self-assembly schemes（including origami，kiri?gami and compressive buckling） that have been widely used to form 3D multistable mechanical metamaterials［2，4-7，10-12，16，20，24，25，30，38，52，54，56-63］.

1.1 Origami

With strategic designs of creases and folding deformations，a flat sheet can be transformed into 3D multistable metasurfaces. Fig.2（a1） demon?strates the configuration space of a foldable planar circular sheet and provides the schematics for fold?ing motions［25］. The flat sheet here is composed of four rigid plates connected by four folds（marked by red，orange，purple and green in Fig.2（a1）with common endpoints. There is only one degree of free?dom in the origami metamaterials，and the folding angles ρ2，ρ3and ρ4can be represented by ρ1，as shown in the top-right panel of Fig.2（a1）. The sta?ble configurations can be achieved by changing the folding angle（ρ1）. Differently，F(xiàn)ig.2（a2）utilizes the square twist whose crease pattern has zero de?gree of freedom as the 2D precursor of multistable mechanical metamaterials［4］. The dark red and blue in Fig. 2（a2） represent the mountain and valley creases，respectively. By allowing bending deforma?tions of facets，the square twist turns out to be fold?able，and it offers bistability when the bending-tocrease energy ratio is large enough（i.e.，＞1 000）.Through periodic tessellations of origami miltistable units（Fig.2（a3）），a multistable metasurface that can be transformed into hyperbolic paraboloid is pre?sented［10］.

Origami techniques can be also utilized to de?sign 3D multistable metamaterials with more com?plex geometric configurations（Fig.2（b））. Usually，a 3D multistable or monostable origami block is ad?opted as the building-block structure. Fig.2（b1）shows the stacking and bonding of translational peri?odic origami sheets（i.e.，the classic Miura-ori）to form a space filling architecture［38］. The adjacent ori?gami sheets are regarded as basic bistable unit cells that have stable convex and concave configurations.Instead of stacking the origami metasurface，Over?velde et al.［2］directly utilized space-filling tessella?tions of polyhedra to create 3D multistable mechani?cal metamaterials（Fig.2（b2））. The cardboard and double-side tapes are used as the rigid face and flexi?ble hinges，respectively. The prepared basic triangu?lar，hexagonal，octahedral and cuboctahedra prims are also rigid. The tessellations of those prims deter?mine the reconfigurability of architected metamateri?als. Specially，the metamaterials in the top-right panel of Fig.2（b2）follow from a combination of oc?tahedra and cuboctahedra，showing a completely rig?id configuration. However，by combining the trian?gular and hexagonal prisms，the architected materi?als can be reconfigured by bending the edges，lead?ing to two stable configurations. To enhance the multistability of such architected metamaterials，de?formable prismatic structures are adopted as the ba?sic building-block structures［11］. In Fig.2（b3），a multistable cuboctahedron is achieved by extruding the edges of the convex structure perpendicular to the faces. Then cubic tessellations of the prims are adopted to form multistable mechanical metamateri?als with three additional stable states.

1.2 Kirigami

Kirigami represents another effective method to design multistable mechanical metamaterials.Here，the multistability mainly results from new de?formation modes induced by geometrical cuts. Yang et al.［62］introduced periodic cuts in a flat sheet to form a stretchable metamaterial composed of repeat?ing unit cells. Changing the kirigami patterns leads to different deformation modes，either with symmet?ric or antisymmetric deformed configurations. At the critical kirigami parameters，the structure of lo?cal unit cells is energetically metastable with two lo?cal minima，and the symmetric and antisymmetric states can coexist（Fig.2（c1））. Except for the ho?mogeneous sheet，“kirigami composites”were pro?posed to enable adaptive and shape-morphable com?posites［57］，as well as achieving complicated shape change and mechanical property adaptations（Fig.2（c2））. The“zig-zag”pattern splits the composite paper into a row of kirigami module that consists of two patches with asymmetric fiber arrangements.The two stable configurations of the unit are shown in the bottom panel of Fig.2（c2）. Combining kiriga?mi with origami represents another effective strate?gy to design multistable mechanical metamaterials.For example，Bobbert et al.［60］used kirigami cut pattern to create bistability in a flat sheet，and then folded them into deployable implants（Fig.2（c3））.Due to the multistability，the deployable implants can switch freely between the compact retracted state and the deployed state，and maintain in a speci?fied state，showing potential applications in ortho?paedic surgeries.

1.3 Compressive buckling

Fig.2 Self-assembly scheme

Mechanically guided assembly of 3D structures are of growing interest due to their widespread appli?cations in bio-integrated electronics and optoelec?tronic devices. This method provides a new design strategy of multistable mechanical metamaterials that have significant applications in reconfigurable and tunable devices. For the cross-shaped 2D ribbon precursor［30］shown in Fig.2（d1），the simultaneous compression in x and y directions leads to a pop-up configuration（shape I）while the sequential com?pression（x direction first，then y）leads to a popdown configuration（shape II）. Luo et al.［30］estab?lished a finite-deformation model to analyze the sta?bility of different buckling modes by using a pertur?bation method. Fig.2（d2）demonstrates a type of membrane-shaped 2D precursors that can be recon?figured into different stable states［5］. Here，shapes I and II correspond to simultaneous and sequential re?lease，respectively. More interesting 3D multistable structures can be found in Ref.［5］. In addition to the translational displacements，Zhao et al.［12］uti?lized kirigami substrates to introduce local twisting deformations to form chiral 3D structures that can?not be realized by controlled compressive buckling.

2 Snap?Through Instability

Negative stiffness structure is a type of special?ly engineered structure exhibiting increasing dis?placement with decreasing force，which is contrary to traditional structures. In the load-displacement curve of such structures，there are one or more mini?mum points，in addition to the starting point. Snapthrough instability occurs in the structure，when the minimum force of load-displacement curve becomes negative，providing another effective way for design?ing multistable mechanical metamateri?als［8-9，13-14，17-19，22，27-29，37，39，41-44，46，55，64-84］.

Fig.3（a1） introduces a typical design of bi?stable negative stiffness structure by combining“＜”shaped topology with beam microstructures［17］.When a vertical displacement is applied，the straight tilted elastic beam bends，leading to the drop of the force. As the displacement increases further，the curved beam gradually becomes straight，and the force starts to increase，as shown in the top panel of Fig.3（a1）. Shan et al.［17］stacked the bistable units through direct ink writing to form multistable me?chanical metamaterials for trapping elastic strain en?ergy. The middle panel of Fig.3（a1）demonstrates the deformation process of the metamaterial，and the bottom panel shows fabricated structures at dif?ferent length scales. However，due to the homoge?neous design，it is difficult to control the deforma?tion sequence of bistable units at each layer，leading to the uncertainty of deformed configurations. Che et al.［73］introduced small variations in the unit cell geometry（i.e.，the width of inclined beams in“＜”shaped topology）to obtain a deterministic deforma?tion sequence for such multistable mechanical meta?materials. Specifically， in the top-left panel of Fig.3（b2），the beam widths of each layer from bot?tom to top are 0.90，1.02，1.08，1.15 and 0.96 mm.When the metamaterial is under a vertical compres?sive loading，each layer snaps into another stable state，in the order from small widths to large widths. Note that many reported multistable me?chanical metamaterials show multistability in only one direction（e. g.，the vertical direction of 3D structures），referred to as 1D multistable mechani?cal metamaterials in this paper. Great challenges ex?ist in preparing mechanical metamaterials with con?trollable multi-directional multistability due to the complex topology. Instead of stacking“＜”shaped bistable units in the vertical direction，F(xiàn)ig.3（a3）shows an array of bistable units in 3D space and use rigid frames to fix and connect these bistable units［80］，similar to the design proposed in Ref.［17］.The realized 2D/3D multistable mechanical meta?materials show various unusual mechanical proper?ties，including robust shape-reconfigurability and ze?ro Poisson’s ratio at large deformations. Also，the deterministic deformation sequence is achieved by using a gradient design in the geometry or material.However，it is challenging to control the deformed configuration of each bistable unit precisely. The lo?cal instability could be easily triggered in such meta?materials under compressive loadings，due to the in?dependent flexible deformations of each“ ＜”shaped bistable unit，which represents certain limita?tions for applications. Experimental demonstrations of multistable mechanical metamaterials with numer?ous controllable stable states also remain challeng?ing.

Based the“＜”shaped topology，another two types of bistable units can be achieved by replacing the beam microstructure with membrane［14，19，37，83］or spring microstructures［43-44，78，81］，as shown in Fig.3（b）and Fig.3（c1）. For example，Pan et al.［19］uti?lized“flexible drinking straw”to construct a type of multistable mechanical metamaterials capable of shape-reconfigurability and programmability. As shown in Fig.3（b1），the metamaterial is designed by parallel multistable mechanical pixels，which are constructed with a hollow multistable structure（i.e.，drinking straw，architected by a series of bi?stable units）and a guide bar. Similarly，F(xiàn)ig.3（b2）utilizes a bistable hemispherical membrane to sepa?rate two cylindrical chambers［14］. The pressure input（PM）at the top of the cylindrical chamber controls the switch of the bistable membrane. By inserting two pathways for airflow，the bistable structure can be used as a soft valve，where the output Pout（i.e.，Pin，1or Pin，0）can be controlled by PM. Fig.3（c1）shows 2D multistable metamaterials with spring mi?crostructures［44］. Utilizing the nonlinear bistable springs，the frequency and direction of elastic wave propagation can be tuned by switching the metama?terial among different stable configurations. More re?configurable phononic crystals［43］are realized by adopting“＜”shaped bistable elements with spring microstructures.

Since the mechanical response of structured materials usually depends on the boundary condi?tions，porous materials［8，27-28］could show multistabil?ity by exploiting special confinements. For exam?ple，the confined biholar sheets［27］are exploited to design programmable mechanical metamaterials with different types of mechanical behavior. When compressed，the holes in sheet undergo large defor?mations and change from circles into ellipses.Fig.3（c2）shows the force-displacement curves of the sheet with horizontal confinement and presents several configurations marked on the curves. The negative force Fysuggests the existence of multista?bility in this confined biholar sheet. Furthermore，Zhang et al.［8］investigated the mechanical response of porous rubber metamaterials with a regular array of elliptic holes. By applying pre-compression to the metamaterials in the vertical direction，the cells change from monostability to bistability in the hori?zontal direction. A theoretical model is also estab?lished to explain the unusual mechanical behavior.

Fig.3 Snap-through instability

3 Structured Mechanisms

Fig.4 Structured mechanism and geometrical frustration

Another common strategy to design multistable mechanical metamaterials relies on the development of structured mechanisms［15，21，23，35-36，40，45，85-87］，where self-locking usually occur by utilizing living hinges，frictional forces and magnetic forces. Fig.4（a1—a3）shows three representative examples that adopt liv?ing hinge to realize multistable mechanical metama?terials［15，21，87］. Specifically，Haghpanah et al.［21］uti?lized bistable triangular units as the building block，and realized 2D multistable mechanical metamateri?als with large numbers of stable states based on hinge mechanisms（Fig. 4（a1））. Except for the beam-hinge design，the panel-hinge design also pro?vides an effective route to multistable mechanical metamaterials，but with denser configurations（bot?tom panel of Fig.4（a1））. Furthermore，Jin et al.［15］introduced a novel building-block design strategy composed of four triangular units，also based on the panel-hinge mechanisms. Differently，the buildingblock structure（top panel of Fig.4（a2））is monosta?ble，while the configuration（bottom panel of Fig.4（a2）） with a 3×3 tessellation of building-block structures is multistable. Mao et al.［87］utilized shape memory polymers as living hinges to realize selffolding structures. By controlling the heating posi?tion，they controlled the sequence of self-folding and formed a self-locked box（Fig.4（a3））capable of maintaining its configuration after cooling. Fu et al.［35］utilized stretchable components to connect rough rigid cylinders/spheres to form multistable granular metamaterials （Fig.4（a4））. When the structure is compressed， the elastic band is stretched，and the structure would recover to the ini?tial configuration，if there is no friction between the cylinders. In this design，the frictional forces be?tween rough cylinders limit the mutual movement between the cylinders，and lock deformed configura?tions（left column in Fig.4（a4））. The same selflocking mechanism exists in granular metamaterials in middle and right columns of Fig.4（a4）. Using functional materials that respond to external stimu?lus in metamaterial designs，various novel multi?stable mechanical metamaterials controlled by multi?physical fields were proposed［36，87］. For example，Tan et al.［36］introduced magnetic systems in multi?stable mechanical metamaterials for trapping energy（Fig.4（a5））. Specifically，the bistable unit consists of three magnets，including two outer magnets and one central magnet，and the repulsive magnetic force exists between the central and outer magnets with appropriate arrangement. It is well known that the direction of magnetic field force is always consis?tent with the connection between the central and out?er magnets. As the middle magnet crosses the line of two outer magnets，the repulsive magnetic force would push central magnets move further until an?other stable configuration is reached. Based on the design of structural mechanisms，various mechani?cal multistable metamaterials with abundant control?lable stable states can be achieved.

4 Geometrical Frustration

Geometrical frustration mainly refers to the use of geometrical constraints to break the propagation of the local order in lattice structures［1，3，26］. The lo?cal“defect”in the lattice structure leads to a diversi?ty of local deformations，resulting in the multistabili?ty. Fig.4（b）shows two representative examples of multistable mechanical metamaterials based on geo?metrical frustration［3，26］. Compared with the square cell，the edges of the triangle cell（Fig.4（b1））can?not be bent in the same mode，due to geometric con?straints. Kang et al.［26］investigated the two deforma?tion modes（i.e.，symmetric and chiral configura?tions）of triangular cell and analyzed the deforma?tion behavior of geometrically frustrated triangular lattice materials. The design strategy of multiple de?formation modes provides a useful guideline for de?signing multistable mechanical metamaterials. Dif?ferently，Coulais et al.［3］changed the way of stack?ing cells in cubic lattice to realize multiple deforma?tion modes. Each flexible building block structure can be oriented independently，and by frustrated stacking（Fig.4（b1））of building block structures，texture metamaterials with multistability and pro?grammability were realized.

5 Unusual Mechanical Properties and Applications

In previous sections，we introduce various de?sign strategies of multistable mechanical metamateri?als. This section discusses the unusual mechanical properties［13，15，17-19，22，35-37，40-46，57-58，65，67-68，79-80，83］of these multistable mechanical metamaterials and demon?strates their representative applica?tions［5，9，13-14，16，38，47-55］.

5.1 Unusual mechanical properties

For 1D multistable mechanical metamaterials with“＜”shaped bistable elements［18］，they expand layer by layer under stretch（Fig.5（a1））. Each layer of bistable units shows a special load-displacement response where the minimum force is negative as shown in Fig.2（a1）. Consequently，stacking of them leads to a stepped load-displacement curve of lattice materials. By tuning the geometric parame?ters of bistable building-block structures，the num?ber of steps in the stress-strain curve and effective elastic modulus can be tuned in this type of multi?stable mechanical metamaterials. By 3D stacking of“＜”shaped bistable elements，multistable mechani?cal metamaterials［78］with zero Poisson’s ratio and controllable thermal expansion can be realized as well（Fig.5（a2））. The interlocking assembly meth?od with multimaterials and bistable unit is utilized to ensure completely symmetric multistable mecha?nisms，resulting in a robust shape-reconfigurability.It is known that multistable mechanical metamateri?als have complex energy landscape，and each stable state corresponds to an energy minimum. The meta?materials need to absorb energy from the external system during the process of switching from low-en?ergy stable state to high-energy stable state.Fig.5（b1—b3）shows three representative exam?ples on using multistable mechanical metamaterials to absorb impact energy［17，19，22］. They all adopt“＜”shaped bistable units as the basic building-block structures，and stack them in the height direction.Different from traditional energy-absorbing materi?als based on dissipation or destruction，when the metamaterials are impacted，each layer of bistable units sequentially switches to high-energy stable states，and the impact energy can be stored in the structure in the form of elastic energy. Compared with Fig.5（b1），F(xiàn)renzel et al.［22］reduced the lattice constant by two orders of magnitude by laser lithog?raphy，and realized a faster energy absorption while maintaining a much lower effective mass density（Fig.5（b2））. Pan et al.［19］utilized flexible drinking straw to realize low-cost multistable mechanical metamaterials with a good capacity of energy ab?sorption（Fig.5（b3））. On the contrary，when the multistable mechanical metamaterials are switched from high-energy stable states to low-energy stable states，they release energy. Based on this mecha?nism，Raney et al.［13］realized the stable propagation of mechanical signals in soft dissipative systems.Specifically，they used a series of“＜”shaped bi?stable elements（Fig.5（c1））and set all of them at high-energy stable state. The input signal pushed the first bistable unit back to low-energy state and the released energy pushed the next bistable unit to low-energy state. Continuing to cycle until the end，the stable nonlinear transition wave could propagate through the system with a high fidelity and controlla?bility. Furthermore，the shape reconfigurability of multistable mechanical metamaterials allows applica?tions in controlling wave propagations. Specifically，Jin et al.［15］adopted tessellated multistable units to tailor propagating transitions fronts. The defects and boundary conditions of structures show combined in?fluences on the propagations of transition wave（Fig.5（c2））. Meaud et al.［42］investigated the pho?nonic properties of multistable mechanical metama?terials at different stable configurations （Fig. 5（c3））. They observed that the low frequency band gaps that do not exist in the initial configuration could emerge in deformed configurations.

Fig.5 Unusual mechanical properties of multistable mechanical metamaterials

5.2 Applications

Considering the abundant unusual mechanical properties of multistable mechanical metamaterials，they have shown promising potential for applications in diverse fields. Fig.6（a）demonstrates two repre?sentative electromagnetic devices［5，16］realized by multistable mechanical metamaterials. Wang et al.［16］periodically arranged split-ring resonators on Miura-ori chiral metamaterials. As shown in Fig.6（a1），the flat sheet can be reconfigured into two different 3D metasurfaces with parallel and antiparallel net electric and magnetic dipoles of splitring resonators，resulting in strong chiral responses.Therefore，the electromagnetic properties of the metamaterials can be easily controlled by tuning the configurations. Fu et al.［5］utilized the compressive buckling to realize morphable 3D optoelectronic de?vices and electromagnetic devices. As shown in top panel of Fig.6（a2），a flat precursor can be trans?formed into two different stable states（i.e.，con?cave or convex configurations）by controlling the loading path，resulting in two different working state（i.e.，on or off states）of the optoelectronic de?vice. By integrating antennas on the 2D precursor，electromagnetic device with reconfigurable shielding capability is realized（bottom panel of Fig.6（a2））.In addition，multistable mechanical metamaterials are widely used to design actuators and robots（Fig.6（b））. Many mechanical actuators have been reported based on the mechanism of energy release from high-energy states to low-energy states. Since this type of mechanical actuators show low efficien?cy，many responsive materials are introduced in the design of multistable mechanical metamaterials. For example，Jeong et al.［47］extended bistable struc?tures into quadristable rotational structures，and uti?lized shape memory polymers as the four connec?tions（i.e.，two rigid and two rubbery beams）be?tween central rotational units and outer cycle. By controlling the heating temperature and time，the shapes of multistable structures could be changed，and various thermal actuation behavior could be ob?tained for tailored design parameters. Pagano et al.［54］utilized multistable mechanical metamaterials based on the origami to realize a crawling robot. An origami tower was formed by folding sheets and can produce longitudinal and rotational deformation. As shown in Fig.6（b2），two origami towers were con?nected by another origami paper，and DC motors were utilized to actuate the deformation of the origa?mi robots. Since multistable mechanical metamateri?al could be designed to offer a rich number of con?trollable stable states，they show significant applica?tions in information processing and logic opera?tions［9，13-14，38］. For example，Raney et al.［13］merged two series of bistable units into output bistable units.These bistable units are initially in a high-energy state（i.e.，logic state 0）and the states of input chains have a combinational effect on the output. By changing design parameters of the bistable units at the output，the energy landscape of output bistable units can be tuned to realize different logic gates.Specifically，for bistable units with a high energy barrier at the output，the released energy could acti?vate the output chain，only when both input chains are activated，behaving as an AND gate（top panel of Fig.6（c1））. For bistable units with low energy barrier at the output，the output chain is activated as long as one input chain is activated（bottom panel of Fig.6（c1））. However，it is difficult to switch the in?put/output chain from logic state 1（i.e.，low-ener?gy state）to logic state 0（i.e.，high-energy state），due to the flexible connections between bistable units. Preston et al.［14］introduced a novel design of soft valve based on single bistable structures. They used the pressure to control the switch of stable states，thereby controlling the on/off state of the air?flow inside the bistable valves. In addition to the three basic logic gates（i.e.，AND，OR and NOT gates）and their combinations，more complex me?chanical devices，including set-reset latch，shift reg?ister，leading-edge detector and digital-to-analog converter，were realized by the bistable valves（Fig.6（c2））. Note that most of the previous me?chanical logic devices are all in centimeter or even larger scales. Based on the additive manufacturing technology，Song et al.［9］has realized the fabrica?tion of micro-mechanical logic gates（Fig.6（c3））based on bistable structures. Since more complex logic operations require more stable states，it re?mains challenging to effectively integrate basic logic gates in a small space for complex logic operations.

Fig.6 Applications of multistable mechanical metamaterials

6 Conclusions

This paper provides an overview of recent de?velopments of multistable mechanical metamateri?als. Four different classes of multistable mechanical metamaterials are highlighted，according to their de?sign strategies. The unusual mechanical properties of these multistable mechanical metamaterials en?able many novel applications，as discussed in this re?view.

Despite this significant progress，several chal?lenges remain in the designs and applications of mul?tistable mechanical metamaterials，as detailed in the following.

（1）For multistable mechanical metamaterials based on the self-assembly mechanism，the number of stable states in the demonstrated experiments is not very high，due to the limited scalability and yield of the fabrication process. Development of high-precision， micro/nano-scale self-assembly technology with improved yield is essential to solve this problem. The resulting periodical array of 3D multistable structures with hundreds of unit cells has promising potentials for uses in tunable electromag?netic/mechanical devices whose properties are ad?justable to meet requirements of different applica?tion scenarios.

（2）For multistable mechanical metamaterials based on the snap-through instability，the reported structures are mostly based on“＜”shaped bistable structures and show multistablility mostly in a single direction. Additionally，the flexible connections are required between bistable elements to ensure a large number of stable states. However，these flexible connections make it challenging to individually ad?dress/switch the stable state of each unit cell in the multistable mechanical metamaterial. Novel designs of multistable building-block structures are needed to realize multistable mechanical metamaterials with massive，controllable stable states in multiple direc?tions.

（3）Most of the existing multistable mechani?cal metamaterials did not incorporate functional ma?terials as their component materials. Thereby，the switch of stable states is mostly through mechanical loading，and the reported unusual properties are mainly mechanical properties， including tunable stress-strain curves，unusual Poisson’s ration，ener?gy absorption and elastic wave propagation control.By adding active materials that respond to external stimuli，multistable mechanical metamaterials could be designed to offer more different types of unusual physical properties（e.g.，tunable thermodynamic，electromagnetic or acoustic properties）.

（4）Currently，a majority of current research in this area focus on the metamaterial design and dem?onstration of their multistability. Although Fig.6 in?troduces some representative applications of multi?stable mechanical metamaterials， there are still many open opportunities for future developments.Considering the freely switchable stable states，mul?tistable mechanical metamaterials have broad oppor?tunities in the design of adjustable meta-devices that exploit the properties of metamaterials. Also，the ample numbers of stable states ensure the applica?tions of multistable mechanical metamaterials in in?formation processing，memory and transmission.Achieving a huge quantity of controllable stable states in a limited space is the key to this application.