GB2452089A - Reversibly deformable structure - Google Patents

Abstract

The invention relates to a reversibly deformable structure comprising a flexible substrate, preferably in the form of a strip, wire or plate, and at least one multistable element connected to and spaced apart from the substrate via a support arrangement of triangular trusses which may flex in use. The multistable element preferably changes its length resulting in the substrate curving. The multistable element has at least a first passively stable configuration and a second passively stable configuration which is different to the first configuration. When the multistable element is arranged from a first configuration towards the second configuration, a corresponding change in the shape of the substrate occurs. The multistable element and the substrate together provide a first stable structure configuration and at least a second stable structure configuration. At least five multistable elements may be provided on the substrate. Later embodiments relate to a method of forming multistable elements by injection moulding, a configurable apparatus and use of said configurable apparatus.

Description

Reversibly deformable structure The present invention provides a reversibly deformable structure and an apparatus incorporating such a deformable structure.

It is well known that some materials can be plastically deformed. This allows the possibility of forming a structure in a first configuration, and allowing the structure to be deformed into a second configuration. The second configuration may be stable in the sense that no external force is needed in order to maintain the structure in the second configuration. However, it is difficult to reverse the deformation so that the first configuration is reached again. Due to work-hardening, continued reconfiguration between the first and second configurations becomes more and more difficult.

Shape memory alloys are known that can be formed into a structure so as to be reversibly configurable between two different shapes, typically due to a change in temperature.

Such materials typically rely on crystallographic martensitic phase transformations to accommodate the shape change without permanent, non-recoverable deformation.

However, such materials are extremely limited in their temperature operation range and the effect is only seen for very particular alloy compositions and is typically triggered by a temperature change. Furthermore, the incorporation of a "memorised" shape for a shape memory alloy device requires complex and repeated mechanical Plante et al (J S Plate, M Santer, S Dubowsky and S Pellegrino, "Compliant bistable dielectric elastomer actuators for binary mechatronic systems" Proceedings of IDECT/CUE 2005: ASME Mechanism and Robotics Conference, 24- 28 September 2005 Long Beach, California, USA) disclose a device in which two opposed actuators operate a bistable element back and forth between a first configuration and a second configuration. The bistable element in this device consisted of a frame, a central movable member and four curved snap-through trusses connecting the central movable member with the frame, one pair on either side of the central movable member. The bistable element disclosed was symmetrically bistable. At a force above a threshold level, applied to the central movable member, the bistable element can be re-configured from the first configuration to the second configuration by appropriate elastic deformation of the four curved snap-through trusses.

The present inventors have realised that bistable (or, more generally, multistable) elements may be useful in providing apparatus whose shape can be stably configured without the need for active elements such as actuators.

In the present context, multistable means the ability to rest in more than one discrete configuration, corresponding to a local energy minimum, without application of external force, but excludes the ability to rest in continuously variable configurations. An example of a multistable element is discussed above, with reference to Plante et al. An example of a device having the ability to rest in continuously variable configurations is a spring tensioned by a threaded bolt and nut - threading of the nut down the bolt may increase tension in the spring, but the device remains at rest when the nut is stationary. However, this device is not multistable because there is not more than one discrete configuration at which the device has a local energy minimum.

In a first preferred aspect, the present invention provides a reversibly deformable structure including a flexible substrate and at least one multistable element connected with and spaced apart from the substrate via a support arrangement, the multistable element having at least a first passively stable configuration and a second passively stable configuration different to the first configuration, wherein arranging the multistable element from the first configuration towards the second configuration causes a corresponding change in shape of the substrate, the multistable element and the substrate together providing a first stable structure configuration and at least a second stable structure configuration.

Further preferred aspects of the invention, and preferred and/or optional features of the invention, are set out below. The preferred and/or optional features are combinable either singly or in any combination with any aspect of the invention, unless the context demands otherwise.

The rnultistable element, in isolation from the flexible substrate and/or the support structure, may have different first and/or second passively stable configurations compared with when the multistable element is part of the structure defined above. The reason for this is that the flexible substrate and/or the support structure typically elastically deform in use, and therefore provide some restoring force to the multistable element in at least one of the stable structure configurations.

The mechanical separation of the multistable elements using the support structure allows the use of relatively weak multistable elements to support the chosen shape of the structure in each stable configurations, the separation typically providing a mechanical advantage.

Preferably, a difference in length of the at least one multistable element between the first passively stable configuration and the second passively stable configuration is L. Preferably, the length of the multistable element in its first configuration is L, being the separation distance of connections between the multistable element and the support structure in the direction of L (or parallel to the direction of L) . Preferably, the ratio L/L is at least 0.01. More preferably, the ratio L/L is at least 0.02, at least 0.03, at least 0.04, at least 0.05, at least 0.06, at least 0.07, at least 0.08, at least 0.09 or at least 0.1. In some circumstances, the ratio �L/L is higher, e.g. at least 0.15, at least 0.2, at least 0.25 or at least 0.3.

Preferably, the at least one multistable element is spaced apart from the substrate, via the support arrangement, by a distance xL in at least one of the first and second configurations. x is preferably at least 1. More preferably, x is at least 2, at least 3, at least 4 or at least 5. For example, xLL may be at least L/3, more preferably at least L/2 and typically at least 2L/3.

Preferably, the support arrangement provides at least three connection locations to the substrate, by which the change in shape of the multistable element can affect the shape of the substrate. Preferably, a length change of the multistable element is operable to cause movement of at least one of the at least three connection locations relative to the others, thereby causing a change in curvature of the substrate.

Preferably, hinge connections are provided between the support arrangement and the substrate. Hinge connections may be provided between the support arrangement and the at least one multistable element. Preferably, the hinge connections are living hinges. The advantage of this is that the structure can be easier to manufacture, optionally monolithically, such as by injection molding, or by assembly of a few monolithic parts.

Preferably, the support arrangement includes trusses defining a substantially triangular connection of the at least one multistable element, the apex of the triangle being located at the substrate. Preferably, the support arrangement consists (at least in effect) of tessellating triangles, the points of the triangles being defined by the connection locations between the support structure and the multistable elements, and between the support structure and the substrate.

Preferably, the trusses are flexible and flex in use. It is preferred that the trusses have a greater stiffness, however, than the living hinges.

The substrate may be a strip or wire, in the event that it is preferred to have a substantially one-dimensional structure. However, in the preferred embodiments, the substrate is a plate. The curvature applied to the plate, however, is preferably cylindrical-type curvature, i.e. a developable curvature.

Preferably, further multistable elements are provided in the structure. These multistable elements may form a chain of multistable elements spaced apart from and connected to the substrate by a corresponding support arrangement. The chain of multistable elements may include at least two or three multistable elements, but preferably there are more, e.g. at least four, at least five, at least six, at least seven, at least eight, at least nine or at least ten.

Preferably, the multistable elements are bistable elements.

Bistable elements provide simplicity of structure in comparison with, for example, tristable elements, and where more than one bistable element is provided, the number of stable configurations for the structure increases.

Preferably, the actuation of a single multistable element between the first and second configurations imparts a change of curvature (by way of rotation) to the flexible substrate of at least 10. Preferably, this change of curvature is at least 2°, at least 3°, at least 4°, at least 50, at least 6°, at least 7°, at least 8°, at least 9° or at least 100. For example, the change in curvature may be between 100 and 200.

Preferably, the material of at least the multistable element is a viscoelastic material. Typically the multistable element is formed from plastics, such as nylon or polypropylene. A benefit of using viscoelastic materials is that, some time after moving the structure from the first configuration to the second configuration, relaxation of some of the restoring stresses in the structure may occur. The result of this may be that the force required to move the structure from the second configuration back to the first configuration may more closely match the force required to move the structure from the first configuration to the second configuration. It is not essential, however, that the multistable element is viscoelastic. In that case, the multistable element may be formed of a non-viscoelastic material such as metallic materials (e.g. steel) or fibre-reinforced composite (e.g. GFRP) In a second preferred aspect, the present invention provides a method of forming a structure according to the first aspect, including the step of injection moulding the at least one multistable element. The support structure may also be injection molded. It is possible for the support structure and the at least one multistable element to be injection molded together. However, in some embodiments it may be preferred to fit the at least one multistable element into the support structure after injection molding, e.g. by a positive location fit.

In a third preferred aspect, the present invention provides a configurable apparatus including a plurality of reversibly deformable structures according to the first aspect, sharing a common substrate. In this aspect, the common substrate may be formed separately from the remainder of each reversibly deformable structure, and the reversibly deformable structures fitted to the common substrate.

In a further preferred aspect, the present invention provides a configurable apparatus including a plurality of reversibly deformable structures according to the first aspect, and further including a flexible plate to which each substrate of the reversibly deformable structures is attached. Each reversible deformable structure may include its own substrate, and the flexible plate of the configurable apparatus may in effect provide a further, common substrate. For example, the reversibly deformable substrates may fit to the common substrate by fixing (e.g. by adherence) . This allows the reversibly deformable structures to be modular, in that they can be fitted as desired to the flexible plate in order to achieve the required configurations from the apparatus.

Preferably, the configurable apparatus of the third or fourth aspect includes at least 10 multistable elements.

In general, the overall shape of the apparatus depends on the configuration of each multistable element, and the number of permutations of configurations of all of the mult.istable elements is preferably at least 10.

Preferably, the configurable apparatus of the third or fourth aspect is selected from a paper sorting device, an item of medical furniture, and an adjustable splint.

In a fifth preferred aspect, the present invention provides use of a configurable apparatus according to the third or fourth aspect in which the shape of the configurable apparatus is stably altered, the use including the step of applying force to the substrate in order to reconfigure at least one of the multistable elements in at least one of the reversibly deformable structures. Alternatively, force may be applied to the support structure, or directly to the multistable elements, in order to provide reconfiguration to the apparatus.

Preferred embodiments of the present invention will now be set out, by way of example, with reference to the accompanying drawings, in which: Figs. la and lb show schematic models of a basic embodiment of the invention, at the conceptual level.

Figs. 2a-2c show free body diagrams illustrating the determination of the forces due to: (a) the flexible plate; (b) the living hinges; Cc) the internal forces in the structure.

Fig. 3a shows an illustration of a compliant mechanism with a living hinge subject to an end moment.

Fig. 3b shows the pseudo-rigid-body equivalent mechanism to Fig. 3a.

Fig. 4 illustrates an arbitrary discrete strain loading history.

Fig. 5 shows a schematic viscoelastic bistable truss.

Figs. 6a and 6b show analytically predicted force responses for a truss formed from the material Si50, with Lo = 50 mm and a = 11.5°, subject to a (a) fast and (b) slow load rate.

Fig. 7 shows an analytically predicted equilibrium path of a viscoelastic snap-through truss subject to four spaced force oscillations following a slow transition.

Figs. 8a-8d show analytically-predicted oscillation responses for four different cases.

Fig. 9 illustrates the response due to creep buckling of a viscoelastic bistable truss.

Fig. lOa shows a schematic view of a reversibly deformable structure according to a preferred embodiment of the invention.

Fig. lOb shows a more detailed view of the bistable elements and support structure of Fig. lOa.

Fig. 11 shows a finite element model of the structure analysed using finite element modelling.

Fig. 12 shows an analytical response of a reversibly deformable structure formed from unrelaxed Si50 material.

Fig. 13 shows an analytical response of a reversibly deformable structure formed from relaxed Si50 material.

Fig. 14 shows a viscoelastic analytical response of a reversibly deformable structure formed from Si50 material.

Figs. 15a-15d show comparisons of experimental and analytical responses of a multistable structure model, viewed in a direction perpendicular to the flexible plate and to the direction of rotation of the flexible plate.

Figs. 15a and b are reproduced from photographs of the actual model and Figs. 15c and d are graphical representations of the analytical response.

Fig. 16 shows top views of the bistable element of the preferred embodiment, in Fig. 16a the element is in the closed configuration and in Fig. l6b the element is in the open configuration.

Fig. 17 shows a detailed view of an alternative embodiment of the rnultistable structure, incorporating snap-through truss transverse restraints.

Figs l8a-18f show side views, reproduced from photographs, of the structure of Fig. 17, illustrating the effect of sequential actuation of the five bistable elements in the structure.

In the preferred embodiments of the present invention, there is provided a structure that is stable in a number of distinct geometric configurations. Such a structure may be utilized as an adaptive structure. As the number of stable states possessed by a multistable structure increases, the structure can appear to deform continuously and remain in its desired configuration even when the actuation that effected the transition to this configuration is removed.

It has been shown by Wicks and Guest (Wicks, N., and Guest, S., 2004. "Single member actuation in large repetitive truss structures", International Journal of Solids and Structures, 41(3-4)) and by dos Santos e Lucato (dos Santos e Lucato, S., Wang, J., Maxwell, P., McMeeking, R., and Evans, A., 2004. "Design and demonstration of a high authority shape morphing structure", International Journal of Solids and Structures, 41, pp. 3521-3543) that when elements within a Kagome-lattice-based plate structure are actuated such that they undergo a change in length, a linear extensional deformation of the plate can result.

When the plate is combined with additional structure to provide out-of-plane rigidity, the linear extension of the plate can result in a curvature of the total structure.

In the preferred embodiments of the present invention, actuation occurs in a support structure rather than the plate itself.

The structure of the preferred embodiments of the present invention is a linear multistable structure including a large number of bistable units. The actuation of each bistable unit imparts a localized, smooth change of curvature to the structure. As many individual bistable units are actuated, the initially straight structure can become very tightly folded and compact.

It is preferred that the plate is an elastically-compliant plate. When the plate is folded, the attached multistable support structure provides stability which allows it to remain in the desired configuration. Provided the multistable structure consists of a sufficiently large number of bistable units, the resulting structure is stable in both flat and tightly-folded, and many intermediate configurations.

In this disclosure, the design of a suitable multistable structure is presented in which linked bistable units are offset from a flexible plate. Initially, a chain of three bistable units is considered as this is sufficient to capture the behaviour, although the modular nature of the design allows this number to be easily extended and/or scaled up. The design is for a compliant, monolithic structure, which enables it to be manufactured in a single stage. Polymeric materials are well-suited to the fabrication of such structures.

The majority of polymers, however, exhibit strongly viscoelastic behaviour. This has important corollaries for the design of multistable structures, as conventional definitions of stability are based on the assumption of conservative, elastic behaviour. For polymer-based designs, such as the one set out in this disclosure, an alternative definition of stability suitable for viscoelastic, non-conservative structures should be used, as discussed below.

In addition, design specifications must be given not only of peak actuation forces and displacements, but also the rate' at which they occur.

There are many suitable applications for the structure of the preferred embodiments. These tend to benefit from the fact that a continuous (or quasi-continuous) curvature of a flexible plate is achievable in a large number of distinct stable configurations. For example, in the paper industry there is currently a limit on the weight of paper that may be used in copiers and printers, as the discontinuity resulting from hinged sorting mechanisms causes the paper to jump, resulting in a jam. The continuous curvature provided by the proposed structure would significantly reduce this problem. A similar benefit is expected to be provided to assembly line processes. The present technology is also suitable for integration with medical furniture, such as operating tables, to allow patient repositioning whilst retaining a comfortable, smooth profile.

It is of interest to design a structure which can perform a similar function to the actuating Kagome-lattice plate structure but which incorporates multistability, thus allowing the plate to remain flexed without continued actuation. Preferred requirements for the structural concept are enumerated below: 1. The structure should be lightweight.

2. The structure should have a completely flat stable configuration and a stable configuration in which it is folded to the smallest possible radius of curvature.

3. The structure should remain in any of its stable configurations for as long as required.

4. The actuation of the structure between stable states should require forces easily achievable by the human hand.

5. The structure should be scalable to allow it to be minimized as available material properties improve.

6. The flexible plate is only required to deform into a developable surface i.e. with zero Gaussian curvature.

In order to realize this concept, a multistable compliant mechanism fabricated as a single monolithic entity is proposed in the preferred embodiment. The folding of the compliant plate is achieved by linking several bistable elements which each impart a local, continuous curvature.

The initial step in the design of a compliant mechanism is often to carry out a simplified analysis based on a pseudo-rigid model i.e. one based around rigid components connected by conventional hinges. This initial model is then modified to account for the flexure of individual components and convert conventional to compliant hinges.

With some modifications described below to account for the continuous curvature of the flexible plate, this is the method that was adopted for the design of the preferred multistable plate structure.

The structure is envisaged as a chain of bistable actuating elements, each element operating to impart local curvature to the flexible plate upon actuation. The pseudo-rigid model used is the repeating triangular truss illustrated in Fig. 1(a). Although the preferred embodiments use a support structure having a long chain of actuating elements, the analysis of only the three adjacent units shown in the figure needs to be considered in order to capture the behaviour.

In Fig. 1(a), the structure is shown in its flat configuration. The structure consists of a truss containing equally sized isosceles triangles. The two equal sides of the triangle have length a, and the third side has length L. This length is the initial length of a bistable actuating element 10, represented by the dark rectangle. The internal angles of the triangular elements are 01 and 2p as shown. The thick line offset from the actuator by the rigid truss elements represents the two dimensional view of the flexible plate 12.

When the central actuator 10 connecting hinges 1 and 2 is increased in length by L, represented by the light rectangle in Fig. 1(b), a kink is formed in the bottom face of the truss representing the flexible plate. It can be seen that when an actuator in a single triangle changes length all the other triangles in the structure are unaffected. The effect of actuation is to change the internal angles of the associated triangle to 02 and 2Cp2.

In reality, the compliant plate does not kink at a single point, but instead distributes the rotation by forming a constant (or approximately constant) curvature as shown in Fig. 1(b) by the dotted line.

Initially, and perhaps most importantly, it is of interest to know the restoring force applied to the bistable actuator in its longer configuration due to moments in both the flexed plate and in the compliant hinges for a given initial geometry. If this force exceeds the snap-through force of the bistable actuator, then the structure will not remain in the folded (or curved) configuration and is therefore not multistable. It can be seen from the pseudo-rigid model that hinges 1, 2 and 3 are all subject to rotation upon actuation of the bistable unit. As an initial approximation, however, it was decided to neglect the restoring moments due to hinges 1 and 2 because of the relatively small moment arm leading to restoring forces in the actuator. For the majority of geometries considered, these rotations were also comparatively small which justifies their neglect. Therefore, as a first approximation, all restoring force on the bistable actuator following actuation is assumed to come from the flexible plate 12 and hinge 3. Their contributions are analysed separately and then superposed.

First the restoring force in the central bistable actuator is estimated due to the flexure of the plate. The curvature of the plate is estimated by taking the post-actuation location of the three bottom hinges and fitting a circular arc through these points. The plate curvature is estimated by first evaluating the angles e1, p1, (p2 and 4' as defined in Figs. 1(a) and (b) 1 =arccos(L12a) (1) (2) 2 =arcsin((L+AL)/2a) (3) (4) In order to keep the analysis purely geometrical (i.e. material independent) it is necessary to make the assumption that the difference in length between the curved plate and the approximating rigid elements is negligible and that both have a length equal to L. This enables the values of q and which define the plate curvature, defined in Fig. 1(b) to be determined according to qLsin (5) 8=Lcosyi (6) Hence there can be obtained a value for the curvature of the plate x due to the actuation of a single bistable element R 52+q2 (7) K=1/R5 (8) The plate therefore provides a restoring moment M5 D5K5 in which D5 is the plate bending stiffness of the plate per unit width, i.e. D= Et3 (9) 12(1-v2) multiplied by the width of plate which contributes to the restoring moment. In the case where there is only one continuous backing structure, this corresponds to the full width of the plate.

The force in the bistable actuator P due to restoring moment can be determined by means of the free body diagram in Fig. 2(a). It can be seen that it is necessary to determine the force F in the diagonal member. This is carried out via the structures shown in Fig. 2 (c) . The structure on the left hand side of the figure corresponds to half of the structure. The plate element is assumed to be built-in, and all other connections to be pin-jointed.

The plate-element and the two diagonal members form a fully-triangulated structure, but at the built-in end the connections are shown separated for clarity. If a positive force F is assumed in the right-hand diagonal member, the forces and moments in all the remaining members may be determined as shown in the figure. A particularly important result is that the force in the left-hand diagonal member is -F.

The forces in the members of the complete structure are shown on the right hand side of Fig. 2(c) . It can be seen that the bending moment in the plate has a triangular distribution. In Eqn. 7 it is assumed that the bending moment in the plate is constant, resulting in a constant curvature. This would be achieved if units immediately to the left and the right of the actuation unit (not shown in the figures) were also actuated. The member forces in this case are found by the superposition of the structure shown, with the same structure shifted one unit to the right, and with the same structure shifted one unit to the left. It can be seen that the forces in the diagonal members cancel each other out with the result that in Fig. 2(a) F = 0.

Therefore, the restoring moment results in an axial restoring force on the bistable actuator of Ps=Mc/Yc (10) in which the lever arm is y3=acosq52 (11) It should be noted that there is an important issue concerning the evaluation of the plate bending stiffness of the plate D5. It can be seen that this depends on the Young's modulu5 of the plate. It it is made of a linear elastic material, no problems are caused. If, however, the plate exhibits viscoelastic relaxation properties, as is likely if polymers are used, great care must be taken in the design.

Viscoelastic materials tend to exhibit an initial high unrelaxed modulus, Eu, which quickly degrades to a much lower relaxed value, ER. The value that should be taken as indicative of the material is strongly dependent on the rate of relaxation of the material and on the proposed rate of transition between the straight and folded configuration.

For the moment, however, the transition is assumed to occur instantaneously and therefore the unrelaxed modulus Eu iS used.

We next turn to the restoring force on the bistable actuator due to the rotation in hinge 3.

This is an example of the commonly-encountered compliant mechanism component known as a living hinge' and is a compliant equivalent to a conventional pin-joint and consists of a short flexible beam component connecting two significantly stiffer regions of structure. sketch of the deflection path of the end of a compliant cantilever, consisting of a living hinge 20 at the built-in end and a substantially stiffer section 22, subject to an increasing moment at the free end 24 is shown in Fig. 3(a). The equivalent structure is shown in Fig. 3(b). The living hinge is replaced by a pin joint at its geometric centre which connects two rigid elements. A spring is included with a stiffness that matches the flexural stiffness of the living hinge. It can be seen that the approximate deflection path that is followed by the pseudo-rigid-body model provides a good estimation of the displacement behaviour of the compliant mechanism.

Adopting a similar approximation for hinge 3, the free body diagram in Fig. 2(b) is used. Assuming that this hinge is realized with a small finite length Lh, and given the rotation of the hinge due to actuation of the bistable element is equal to P2-P1, the curvature of the compliant hinge is (12) The living hinge therefore exerts a restoring moment Mh = DhKh in which Dh is the plate bending stiffness per unit width multiplied by the total hinge width. Once again care must be taken with the value of Young's modulus in the hinge bending stiffness if viscoelastic material properties are expected.

This restoring moment in the compliant hinge 3 is assumed to act at a point half-way along the length of the hinge, and, referring to Fig. 1(c), results in an axial restoring force on the actuator of Ph=2Mh/yh (13) in which the hinge lever arm y_(aLh/2)c0sc2 (14) Therefore for a given initial (straight) geometry and a bistable actuating element with stroke AL, an initial estimate of the restoring force on the bistable actuator in the folded configuration can be obtained by superposing the results of the two separate analyses gogaI = 2(M5 /y5 + Mh lYb) (15) The bistable actuator should be designed with an actuation force of at least the value predicted by Eqn. 15 if the backing structure is to be multistable, i.e. if it is to be able to remain in the folded configuration.

The above analysis will now be used to compare different designs and evaluate the effects of varying certain parameters. In the following comparison the unrelaxed material properties of a typical rapid-prototype polymer (Si50) (purchased in 2005 under the trade name Accura 50 Plastic by 3D Systems Corp., 26081 Avenue Hall, Valencia, CA 91355 U.S.A., used in solid state stereolithography systems) having a Young's modulus of 2500 MPa and a Poisson's ratio of 0.3 were assigned to the multistable backing structure, and the flexible plate was assumed to have a Young's modulus of 1000 MPa and a Poisson's ratio of 03. The material properties are not of great importance, however, as this analysis is primarily intended to provide a comparison of different geometrical configurations.

Two different values of plate thickness were considered (corresponding to available film thicknesses) of 0.25 mm and 0.65 mm respectively. Two equilateral structures (a = L) were chosen with characteristic lengths of 10 mm and 15 mm. Characteristic lengths of greater than about 15 nun were felt to be undesirable as they would restrict the radius of curvature that the plate could be folded to.

Characteristic lengths of less than about 10 mm make it difficult to envisage that a bistable snap-through truss of sufficiently high actuation force could be incorporated at least at the prototype stage. A range of typical actuation strokes for the bistable element were taken, using LL= 4, 5, 6 mm. A compliant hinge thickness of 0.2 mm was used as the lowest achievable thickness using the SLA rapid-prototype method, and a hinge length Lb of 2 mm was chosen.

Results for the local plate radius of curvature under actuation and the corresponding required bistable actuator resisting force are shown in Table 1 for t5 = 0.25 mm, and Table 2 for t = 0.65 mm.

L (mm) AL (mm) R5 (mm) P (N) Ph (N) Ptotal (N) 4 20 1.0 2.4 3.4 5 16 1.4 3.4 4.8 6 13 1.9 4.6 6.5 4 46 0.3 0.9 1.2 5 37 0.4 1.2 1.6 6 30 0.5 1.5 2.0 Table 1. Radius of curvature (R5) (approximate) and required bistable actuation force (Ptotai) for various multistable plate configurations, t = 0.25 mm. All lengths are in mm, and all forces are in N. L (mm) AL (mm) R5 (mm) P5 (N) Ph (N) Ptotai (N) 4 20 18 2.4 20.4 5 16 24 3.4 27.4 6 13 33 4.6 37.6 4 46 4.7 0.9 5.6 5 37 6 1.2 7.2 6 30 8 1.5 9.5 Table 2. Radius of curvature (R5) (approximate) and required bistable actuation force (Ptotai) for various multistable plate configurations, t = 0.65 mm. All lengths are in mm, and all forces are in N. From this simple analysis it is possible to make a number of remarks relating to the design of the multistable support (backing) structure. Most clearly, it can be seen that increasing the plate thickness can have a significant effect on the restoring force (which is proportional to the thickness cubed) . It can also be seen that increasing the characteristic length of the support structure by 5 mm is sufficient to lower the estimated forces in the actuator by a factor of up to four times. These two conclusions are important when such a multistable structure is being designed, as very small changes in plate thickness and characteristic length can make the difference between a structure that locks by a large margin, and a structure that is not rnultistable at all. Simple Euler buckling calculations reveal that for a single snap-through truss fabricated from Si50, a peak snap-through force of about 2 N is achievable. Reference to Table 1 shows that there are feasible configurations for which it would be possible to have a single snap-through truss per actuating stage thus reducing the model complexity and hence aiding fabrication.

On the basis of this analysis it was decided to take an equilateral structure with a characteristic length of about 15 mm, an actuation stroke of about 4 mm, and a plate that is as thin as possible, in order to maximize the likelihood of achieving a working prototype. The width of the structure was fixed at about 40 rrun as this was considered large enough to accommodate a bistable snap-through truss.

The multistable structure typically requires a bistable structure that results in a change in length when transitioned between its stable states. The bistable structure chosen is the well-known snap-through truss (see for example Baant, Z., and Cedolin, L., 1991. Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories, 1st ed. Oxford University Press) . It is implemented as a compliant structure with living hinges, fabricated from Si50 polymer which exhibits viscoelastic properties. In this section, viscoelasticity will be briefly discussed and then the effects of viscoelasticity on the bistable behaviour of a snap-through truss are investigated.

Viscoelastic materials creep (i.e. exhibit a time dependent increasing strain when subject to a constant stress) and relax (i.e exhibit a time-dependent decaying stress when subject to a constant strain) . A viscoelastic material may be represented in terms of a time-dependent creep compliance (16) a0 and a time-dependent relaxation modulus (17) Co in which the subscript 0 denotes a constant applied value.

Functional representations of Eqns. 16 and 17 may be obtained from material test data and used to determine the response of a viscoelastic material to an arbitrary stress and strain input. We will focus on the case of stress relaxation as this is the situation that is observed when a viscoelastic structure is subject to displacement-controlled changes in configuration.

Time-dependence of viscoelastic materials means that there is no longer a unique correspondence between stress and strain. For a linear viscoelastic material the response may be predicted by means of the Boltzman Superposition Principle (BSP) . The principle states that if a series of strain steps have been applied to a linear viscoelastic material, the stress at any given time will be a function of all the previous load steps. In other words the material exhibits a history-dependent response. This is illustrated by considering the discrete strain load history depicted in Fig. 4.

Referring to Eqn. 17, each strain increment occurring at time t results in a stress at time t of E(t -t1) Therefore, for an arbitrary load history, the stress in a linear viscoelastic material at time t subject to a discrete load history may be written as (18) For a continuously varying strain history the BSP has the integral form a(t)= jE(t-u)-du (19) If a functional approximation to a linear viscoelastic material is available, Eqn. 19 may be used to determine the stress due to an arbitrary strain loading history. The functional representation may be in any form, but commonly either an exponential decay form or a power law form is used. For nonlinear viscoelastic materials, the BSP is no longer valid and must be modified accordingly.

The effect of viscoelastic material behaviour on the stability and behaviour of a snap-through truss structure is now presented. In this case the truss elements are considered to be viscoelastic springs 30, 32 that are constrained to deform axially. This structure is illustrated in Fig. 5.

The single parameter which is chosen to describe the configuration of the system is the deflection 6. This means that the system is displacement-controlled, and that a relaxation modulus approach may be used. For any given 6 the strain in each individual element may be evaluated according to 6=[(L+o2_2LO8cose)2_LO]/LO (20) in which L0 = L/cosa is the initial member length. The stress at any given time may be evaluated using the superposition principle equation, as explained above.

First, we consider the displacement-controlled case. The material properties assumed are those of rapid-prototype polymer Si50 which were determined from experimental relaxation tests. This material is chosen because it is used to fabricate the model described below. Details are not provided here, as they required a nonlinear viscoelastic material model and a corresponding modified superposition principle. The form of the results shown below are, however, generic for all viscoelastic materials which exhibit time-dependent relaxation between unrelaxed and relaxed material properties.

The first load rate -the fast transition -consists of an almost instantaneous snap-through transition (0.2 s), where the structure is then held for 1000 s. This hold is to permit the material to recover if necessary, although this will be seen to be unnecessary as the material does not have time to relax over a rapid transition. The structure is then rapidly displaced back to the original configuration where it is held for a further 1000 s. The second load rate -the slow transition -consists of the structure being snapped-through over 40 s and then being held for 960 s. The structure is then displaced back to its original configuration over 40 s where it is held for a further 960 s. The calculated force profiles for these two load cases are shown in Fig. 6a and 6b.

The fast load rate (Fig. 6a) produces a symmetrical force response which is indicative of elastic behaviour. If the load rate is fast compared to the viscoelastic relaxation time of the material, then it may in fact be considered as linear elastic with a Young's modulus equal to the unrelaxed modulus E. The slow load rate (Fig. 6b) produces an asymmetrical force response in which the material appears to lose stiffness over the transition.

This reduction in stiffness will continue down to the point at which the material has fully relaxed and may be considered as linear elastic with a Young's modulus equal to the relaxed modulus ER. As the structure is held in the displaced configuration, however, it may be seen that the material recovers. In the case of the slow transition the time is sufficient for the material to recover fully and the return transition is the same as the initial transition.

As the data plotted in Figs. 6a and b were obtained using displacement-controlled loading, they represent history dependent equilibrium paths. It is not possible at this stage to reach any firm conclusions about the stability of these paths. Because a structure fabricated from a viscoelastic material is non-conservative, any evaluation of strain energy is valid only for a particular time and load history. As the concept of an energy well has lost meaning for such a dissipative system, it is necessary to seek alternative definitions of stability.

Although it is no longer possible to define the structural stability by means of a potential, we can still however use Liapunov's definition of a stable state as one in which the application of an infinitesimally small force does not result in large deflections. The extension of Liapunov's definition introduced here follows Baant and Cedolin (Baant, Z., and Cedolin, L., 1991. Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories, 1st ed. Oxford University Press) in which it is stated that for some applications it is useful to consider stability from the viewpoint of infinitesimal loading cycles. We note that under a cyclic load, an inelastic non-conservative structure will behave in one of two ways.

Either the structure will settle down into elastic-like behaviour within a few cycles in a phenomenon known as shakedown', or increasingly divergent behaviour will be observed, known as incremental collapse' . The latter case may be considered a suitable definition of instability.

This is applied to the stability analysis of viscoelastic structures as follows. If a viscoelastic structure is released in an equilibrium configuration, its stability may be assessed by the application of a small oscillating load which has a period substantially less than the relaxation time of the viscoelastic material. If the effect of consecutive force oscillations is to produce divergent displacement behaviour then the structure may be considered to be unstable. If the displacement response to consecutive force oscillations is bound then the structure may be considered stable.

This is illustrated by reference to the same viscoelastic bistable truss analysed with respect to Fig. 5. In this case, in a finite element analysis was carried out to simulate the truss being subjected to a displacement-controlled transition. The transition time was chosen to ensure that the material relaxed over the transition. At this point, the loading regime is changed to force-control and the structure is released. The structure immediately moves to a zero-force configuration.

Following release, small oscillations of �0.1 N are applied as the structure is allowed to relax. These oscillations are applied (a) immediately following the transition, (b) after 10 s, (C) after a further 100 s and then again (d) after an additional 1000 s. The resulting displacements are plotted against the applied force in Fig. 7 (shown as lines A, B, C and D) and in more detail in Figs. 8a-d, corresponding to A, B, C and D, respectively. The response of the structure to the oscillating load at all times following the snap-through quickly settles into a cyclic response. Therefore the structure may be considered stable at all times.

This is an effective, but computationally expensive technique, so it is not considered necessary to carry out this process for the majority of analyses, as convergence of the finite element method under force control is sufficiently indicative of stability. If, however, there is any doubt, the oscillating force method may be used to assess the instantaneous stability of any given equilibrium configuration.

For the loading regime described above, the viscoelastic truss was demonstrated to be instantaneously stable at all times. It is of interest to show how the same structure may become unstable when it is subjected to a constant force over a long duration. The effect of such a case is sketched for the viscoelastic snap-through truss in Fig. 9.

It has already been noted that a viscoelastic material behaves instantaneously as linearly elastic with Young's modulus E before relaxing. Under an applied force, it has also been noted that the material relaxes to a point at which it may be considered linear elastic with the reduced Young's modulus E0.

In Fig. 9 consider the instantaneous force-controlled displacement of the truss from its original configuration 0, to the points on the equilibrium curve labelled A and B. We consider first the case when the truss is loaded to point A. If the structure is held at this displacement for a period of time, the effective modulus of the material decreases, and so the location of the limit point corresponding to the onset of negative stiffness and structural instability moves with time as the material relaxes. The locus of this limit point between the unrelaxed and the relaxed material states is sketched in the figure. As the material relaxes, it can be seen that the force corresponding to this limit point moves below the force corresponding to equilibrium point A. The initially stable structure suddenly becomes unstable and a dynamic jump will ensue. This phenomenon is known as creep buckling.

When the structure is instantaneously loaded to point B, the configuration of the structure changes as the material relaxes, but the structure never becomes unstable, because the force magnitude at point B is below the peak of the long-term response of the structure. This indicates that if a viscoelastic structure is designed with no load-rate information, it can be important to analyse it in its relaxed' material state to be certain that it is stable for all times following the loading.

Prior to the detailed design and analysis of a polymeric folding structure, the two dimensional concept illustrated in Fig. 1 was realized as a three-dimensional structure using a simple cardboard model. A structure consisting of three adjacent units based around equilateral triangles of side about 100 mm was fabricated.

Each bistable unit consisted of a pair of snap-through truss structures. All compliant hinges were realized with thin sheets of paper. It was found that the structure behaved as expected and remained in a curved configuration after actuating a central bistable element, even when the actuation force was removed and that 15° of fold (or curvature) in the plate is achieved by actuation of a single bistable unit. It was clear at this stage that the structure was viable, so a more detailed design was commenced, and this is discussed below.

The next stage in the design process was to produce a detailed three-dimensional concept for the multistable structure. This is shown in Fig. lOa, in which the support structure 40 is indicated attached to a flexible screen 42 or, more generally, flexible plate. The support structure is shown in greater detail in Fig. lOb which also includes the nomenclature chosen for the different parts of the structure: connecting hinges 1,2 spacing structure 44 snap-through truss support structure 46 snap-through truss hinges 48 webs 50 screen (or plate) attachment points 52 web hinges 1 support structure reinforcement 54 bistable snap-through truss 56 It was decided that the compliant hinges be made as short as possible for two reasons. First, a short hinge would more closely capture the behaviour of the pseudorigid model.

Second, as the hinges are subject to compressive forces, if they are too slender there is the undesirable possibility that they will buckle.

It can be seen that hinges 1, 2 and 3 in Fig. 1 have been replaced in the embodiment shown in Fig. lOa and Fig. lOb with thin flexural components (living hinges) . The bistable actuator connecting hinges 1 and 2, is implemented as a snapthrough truss 56 of the type described above. The snap-through truss support structure 46 is required to be as rigid as possible to prevent the peak resisting force of the snap-through truss 56 being reduced. The purpose of the spacing structure 44 is to move the snap-through truss 56 down so that the line of action of the snap-through truss 56 is in the correct location.

Although only three actuating units are shown in Figs. lOa and lOb, and only the actuation of the middle unit will impart curvature to the flexible plate 42, it can be seen that the structure may be extended to have as many actuation units as may feasibly be manufactured.

This compliant design was then implemented in a finite element analysis in order to carry out several design iterations to enable the final configuration and dimensions to be determined.

A number of design iterations were carried out starting from the compliant mechanism concept illustrated in Figs. lOa and lOb. The primary goals of the design were to achieve a structure that locked in the folded configuration at all times and which was as lightweight as possible. The analysis was carried out using the finite element model shown in Fig. 11.

The finite element model will now be described. The nonlinear finite element solver used was ABAQUS v6.5. The webs, compliant hinges, snap-through truss members, support structure reinforcement, and the flexible plate were modelled with quadrilateral shell (S4R) elements. The spacing structure 44 and the snap-through truss support structure 46 were modelled using solid (C3D8) elements.

Design variables included: the depth of the multistable (support) structure 40; the geometry of the snap-through truss support structure 46; and the lengths and thicknesses of the compliant hinges 1,2, 3 and webs 50. All actuation forces and displacements were applied to the centre actuation unit 60 labelled in Fig. 11.

80 g/m2 paper which has an average thickness of 0.1 mm was used to represent the flexible plate 42. It was assumed to behave as a linear elastic material with Young's modulus 2000 N/mm2 and Poisson's ratio 0.3. It was also discovered that it was not possible to achieve the required peak snap-through force in the bistable truss element using Si50 whilst keeping the truss geometry within manufacturing limits. As an interim measure, 0.5 mm thick steel strips were incorporated in the design in the locations labelled support structure reinforcement 54 in Fig. 10. It is shown below, that the use of an alternative rapid-prototyping process means that such reinforcement is unnecessary and permits the design of truly monolithic structures.

The final design which was shown to lock into a folded configuration and be stable in this state over long time scales had the following dimensions: a depth of 35 mm; a snapthrough truss 56 element length of 10 mm; a maximum height above the flexible plate 42 of 15 mm; web hinge length 3 mm; snap-through truss hinge length 2 mm; and all compliant hinges had thickness 0.2 mm. The flexible plate was 100 mm long by 70 mm wide.

Three separate analyses were carried out on this final design: first assuming the Si50 was unrelaxed to predict the instantaneous snap-through performance; second assuming the Si50 was fully relaxed to determine the long-term response; and third a fully viscoelastic analysis which incorporated a slow transition to illustrate the snap-through behaviour when the Si50 was partially relaxed, and then oscillating loading over time to assess the stability of the structure as the material fully relaxes. In all cases, displacement boundary controls were applied to the central actuating unit.

By inspection of the results from the unrelaxed analysis, plotted in Fig. 12, it can be seen that the peak force required to actuate a transition from the straight to the folded configuration is 1.13 N. Perhaps unexpectedly, the snap-through equilibrium path is not rotationally symmetric about the half way point, as was observed for the simple viscoelastic truss. The main reason for this is that the backing (support) structure was fabricated in the straight configuration. Because of the use of compliant hinges, when the backing structure is folded (or curved) the restoring moments in these compliant hinges reduce the stability of the truss in the folded configuration resulting in the observed asymmetrically-bistable behaviour.

When the external force is removed, the structure moves by a very small amount to the configuration indicated by the dot in Fig. 12.

By inspection of the results from the relaxed analysis plotted in Fig. 13 it can be seen that there is a substantial reduction in required snap-through force down to a value of approximately 0.16 N representing a reduction of 85%. It can be seen, however, that the backing (support) structure still locks in the folded configuration.

This means that the design places no restrictions on the rate of transition and removes the possibility of creep-buckling such as illustrated in Fig. 9. The relaxed material property represents the worst possible case. The dot indicates the point to which the structure moves when the external actuation force is removed. It can be seen that the transition has been foreshortened by 0.5 mm in addition to having the peak forces reduced by the relaxation of the material.

The above two analyses represent an upper-and a lower-bound respectively on the performance of the backing structure. A full viscoelastic analysis may be used to determine the performance for a given rate of loading. The response shown in Fig. 14 is for a transition taking place over 40 s, which is sufficient for the material to relax partially. The effect of this relaxation is apparent as the response (which is bounded by the responses from the relaxed and unrelaxed analyses) is closer to the unrelaxed response at the beginning of the transition, but tends increasingly towards the relaxed response over the duration of the transition as the material becomes more relaxed.

The results of the application of oscillating loads to assess stability may be seen on the right hand side of the plot following the transition in Fig. 14. These loads were applied immediately following the transition, after 10 S had elapsed, after a further 100 s had elapsed, and finally after an additional 1000 s had passed. Although these responses are not shown here in detail, it was observed under close inspection that, at instantaneous points throughout the relaxation following transition, the structure behaved in a cyclic non-divergent fashion. This indicates that the backing structure is following a stable equilibrium path. The third and fourth set of oscillations produced coincident responses as the structure had reached its final configuration.

It can be seen that the structure does not recover fully to the displacement predicted by the unrelaxed analysis. This is due to the history-dependent nature of the response. If a final configuration is exactly required for a design, the viscoelastic nature of the material should be taken into consideration.

A demonstration model of the multistable backing structure was fabricated from Si50 polymer using the SLA rapid-prototype technique for the design whose performance is set out above. The model was manufactured using a Viper SLA Machine manufactured by 3D Systems Inc (see www.3dsystems.com, accessed 13 August 2007) . This was then integrated with the steel reinforcement strips and the paper sheet representing the flexible plate described above.

The structure behaved as predicted. Schematic views of the assembled structure in both its straight and folded configurations are shown in Figs. 15(a) and (b) (these views are reproduced from photographs) . The predicted response is shown in Figs. 15(c) and (d) . It can be seen that there is a close agreement between the predicted and the achieved curvature in the compliant plate due to actuation of a single element (corresponding to the central actuation unit shown in Fig. 11) A closer view of the actuated bistable element in both closed and open configurations is shown in Fig. 16.

Salient features are marked, which may be compared with Fig. 10.

Further work was carried out using an alternative rapid-prototypical technology enabling improvements to the multistable structure design. This technology is PolyJet (trade name) 3 Dimensional Printing, developed by Objet Geometries Ltd, 2 Floltzman St. Science Park, P.O. Box 2496, Rehovot 76124, Israel, www.2object.com, accessed 13 August 2007. The FuliCure (trade name) 720 material was used.

In this process, a printer jets photopolymeric material in layers which are immediately cured with UV light. If material is not required a gel-like support structure is deposited instead which may be removed following the build.

This use of removable support structure means that encapsulated structure may be manufactured. This means that the snap-through support structure, as shown in Fig. 10, may be restrained with transverse bars above and below the snap-through truss. This significantly stiffer configuration removes the need for the additional reinforcement strip, resulting in a structure which is truly monolithic.

Without further detailed analysis, the multistable folding structure based around equilateral triangles of side 10 mm and consisting of 5 bistable units was redesigned to be manufactured using the PolyJet (trade name) process. The design was modified to make the support gel easier to remove. A 0.15 mm thick flexible plate was incorporated.

A detailed sketch of the new design is shown in Fig. 17 and sketches of the resulting structure, showing the effect of sequential actuation of the bistable units, are provided in Fig. 18.

The structure has flexible plate 42 and similar snap-through truss elements 56 as in the embodiment shown in Figs. lOa and lOb. However, in addition, the structure has top and bottom transverse bars 70 in order to provide additional support to the bistable truss elements 56. In addition, support material removal holes 72 are shown in the drawing. These are present as a result of the PolyJet process, in which the monolithic structure is built with a soluble support material in intervening spaces, this support material being dissolved away in a later stage of the process to leave the desired monolithic structure.

These results illustrate that it is possible to construct the multistable folding structure and the flexible plate as a single monolithic entity, and that increasing the number of bistable units in the chain means that large folding angles may be achieved. Although not shown in Fig. 18, the effect was fully reversible, and each individual unit could be actuated independently, resulting in 2 = 32 possible discrete configurations.

It has been shown that the multistable structure can be produced and that it operates as predicted by the inventors.

In particular it has been shown that the actuation of a single bistable element imparts a smooth local curvature to an attached flexible plate when it is offset from the plate by means of suitable support structure. The bistable elements also provide structural rigidity to the flexible plate in all configurations.

Injection molding is expensive and not viable for the construction of one-off prototype models, and hence was not used in the present work. However, the present inventors consider that all the designs described above could be manufactured by injection molding to make large numbers of devices. The use of this process could be beneficial for the fatigue life of the final structure. By requiring the molten polymer to flow through the living hinges during molding, the molecules become aligned along the hinge, which consequently becomes much stronger. Provided a satisfactory viscoelastic material model for the desired polymer is available, or can be determined, it has been shown that the behaviour of a multistable structure fabricated from this material can be well predicted.

It has previously been considered that the p'resence of viscoelastic effects in multistable structures is undesirable and should be alleviated or avoided, either by finding less susceptible materials or ensuring that the material does not have the opportunity to become relaxed.

The latter is constraining as it limits designs to those which require instantaneous transitions between stable states. The former is possible, but it is the case that polymers are much more suitable than many other materials for the construction of compliant structures. The present inventors have surprisingly demonstrated that if the presence of viscoelasticity is accommodated into the design process, its effects need not be considered detrimental.

The preferred embodiments have been described by way of example. On reading this disclosure, modifications of these embodiments, further embodiments and modifications thereof will be apparent to the skilled person, and as such are within the scope of the present invention.

Claims (19)

1. A reversibly deformable structure including a flexible substrate and at least one multistable element connected with and spaced apart from the substrate via a support arrangement, the multistable element having at least a first passively stable configuration and a second passively stable configuration different to the first configuration, wherein arranging the multistable element from the first configuration towards the second configuration causes a corresponding change in shape of the substrate, the multistable element and the substrate together providing a first stable structure configuration and at least a second stable structure configuration.

2. A structure according to claim 1 wherein a difference in length of the at least one multistable element between the first passively stable configuration and the second passively stable configuration is LL.

3. A structure according to claim 2 wherein the at least one multistable element is spaced apart from the substrate, via the support arrangement, by a distance xL in at least one of the first and second configurations, x being at least 1.

4. A structure according to any one of claims 1 to 3 wherein the support arrangement provides at least three connection locations to the substrate for the multistable element.

5. A structure according to claim 4 wherein a length change of the multistable element causes movement of at least one of the at least three connection locations relative to the others, thereby causing a change in curvature of the substrate.

6. A structure according to any one of claims 1 to 5 wherein hinge connections are provided between the support arrangement and the substrate.

7. A structure according to any one of claims 1 to 6 wherein hinge connections are provided between the support arrangement and the at least one multistable element.

8. A structure according to claim 6 or claim 7 wherein the hinge connections are living hinges.

9. A structure according to any one of claims 1 to B wherein the support arrangement includes trusses defining a substantially triangular connection of the at least one multistable element, the apex of the triangle being located at the substrate.

10. A structure according to claim 9 wherein the trusses are flexible and flex in use.

11. A structure according to any one of claims 1 to 10 wherein the substrate is a strip or wire.

12. A structure according to any one of claims 1 to 10 wherein the substrate is a plate.

13. A structure according to any one of claims 1 to 12 wherein further rnultistable elements are provided, the multistable elements forming a chain of multistable elements spaced apart from and connected to the substrate by a corresponding support arrangement.

14. A method of forming a structure according to any one of claims 1 to 13 including the step of injection moulding the at least one multistable element.

15. A configurable apparatus including a plurality of reversibly deformable structures according to any one of claims 1 to 13, sharing a common substrate.

16. A configurable apparatus including a plurality of reversibly deformable structures according to any one of claims 1 to 13, further including a flexible plate to which each substrate of the reversibly deformable structures is attached.

17. A configurable apparatus according to claim 15 or claim 16 including at least 5 multistable elements.

18. A configurable apparatus according to any one of claims 15 to 17 wherein the overall shape of the apparatus depends on the configuration of each multistable element, and the number of permutations of configurations of all of the multistable elements is at least 10.

19. Use of a configurable apparatus according to any one of claims 15 to 18 in which the shape of the configurable apparatus is stably altered, the use including the step of applying force to the substrate in order to reconfigure at least one of the multistable elements in at least one of the reversibly deforrnable structures.