Totimorphic assemblies from neutrally stable units

Significance

Existing shape-morphing materials and structures are limited in their ability to transition between a few stable configurations. Here we show how to create structural materials that have an arbitrary range of shape-morphing capabilities using the idea of neutral stability, as embedded in a unit cell with a range of energetically equivalent configurations. These structures allow for a decoupling of the local and global geometry from the local and global mechanical response and thus allow for independent control of the structure and mechanics, laying the foundation for engineering functional shapes using a new type of morphable unit cell.

Abstract

Inspired by the quest for shape-shifting structures in a range of applications, we show how to create morphable structural materials using a neutrally stable unit cell as a building block. This unit cell is a self-stressed hinged structure with a one-parameter family of morphing motions that are all energetically equivalent. However, unlike kinematic mechanisms, the unit cell is not infinitely floppy and instead exhibits a tunable mechanical response akin to that of an ideal rigid-plastic material. Theory and simulations allow us to explore the properties of planar and spatial assemblies of neutrally stable elements, and solve the inverse problem of designing assemblies that can morph from one given shape into another. Simple experimental prototypes of these assemblies corroborate our theoretical results and show that the addition of switchable hinges allows us to create load-bearing structures. Altogether, totimorphs pave the way for structural materials whose geometry and deformation response can be controlled independently and at multiple scales.

A range of mechanical systems that arise in soft robotics, flexible antennae, prosthetics, etc. require transitions between multiple relatively stable conformational states that are all easily accessible without large energetic barriers between them (1, 2). Recent advances in manufacturing technologies are a step in this direction and have enabled the fabrication of shape-morphing structures that respond to a range of external stimuli such as heat, light, humidity, pH, magnetic or electric fields, and so forth (3, 4). However, these structures typically have only a finite number of metastable equilibrium states separated from each other by energetic barriers that can be overcome only in the presence of a guided external energy input. This raises a natural question: Can one design a material structure that is so malleable that it has an infinite number of energetically equivalent states? Also, can one control the possibly heterogeneous response of these structures on multiple scales? A possible set of candidates are mechanisms, floppy structures that are akin to a set of underconstrained linkages that have a number of zero-energy internal modes. However, such systems are unstable to small perturbations and difficult to control. To create mechanism-like infinitely malleable (totimorphic) structures that are also stable and controllable, we need to have a finite resistance to deformation that is independent of the magnitude of deformation. This study provides one possible answer suggested by the use of neutrally stable structural assemblies that are totimorphic, stable, and controllable.

Our starting point is the observation of Clerk Maxwell (5), who defined the criteria associated with stable mechanical structures in terms of the difference between the number of degrees of freedom and the number of constraints in the system. When the difference is exactly zero, the structure is isostatic and stable, while when the difference is either positive or negative we get either underconstrained structures with zero-energy floppy modes or overconstrained systems with states of self-stress (6). An unusual class of structures that sits precariously at the boundary between isostatic and floppy structures are neutrally stable (NS) structures (NSSs) (7), marginally stable objects typically made of a combination of very stiff and very soft elements that are internally strained but globally equilibrated. A key feature of these structures is the presence of geometric constraints that allow for a one (or more) parameter family of large deformations that redistribute the internal stresses while keeping the total strain energy invariant. Thus, no additional external work is required to change or maintain their configurations, making these objects infinitely malleable. This makes NSSs different from typical monostable or even multistable unstrained or prestrained [e.g., tensegrity (8)] structures which require external work to deform them to switch between states. Instead, NSSs exhibit an infinity of equivalent equilibrium states that are locally stabilized by internal friction and require minimal external work to transition between them.

An everyday example of an NSS is the Anglepoise lamp (9, 10) designed almost a century ago, with a lamp head that is infinitely morphable by virtue of its having a set of opposing springs in tension that change their lengths while the total energy remains constant. The presence of finite frictional or other dissipative forces provides local stability to the local conformation of the NSS, but this is not a robust approach for assemblies of NSSs. To use these as the basis for creating more complex shapes, we need to 1) develop a generalized approach for the design and assembly of NSS and 2) develop methods to “lock” and “unlock” the NSS once it has reached a prespecified conformation.

Neutrally Stable Unit Cells and Their Assembly.

Our structures use a minimal NS unit cell as the basis for our assembly shown schematically in Fig. 1A (see physically in Fig. 3A). It is constructed using two “zero-length” springs (11) (i.e., springs that are stretched substantially relative to their rest length), a lever of length $L$, and a rigid link of length $2L$ (springs between lever to strut are connected through point attachments). The freely rotating lever is attached to the link via a planar joint (with 1 degree of freedom [DoF]) and allows for a range of configurations characterized by a single internal angle $𝜗$. For each value of this angle, the individual springs with stiffness $k^{\left(1\right)},k^{\left(2\right)}$ are stretched differently, and the total energy of this system is given by

$$\begin{array}{ll}\hfill {\mathcal{E}}_{\text{el}}& =\hspace{0.17em}\frac{1}{2}{k}^{\left(1\right)}{\left(l^{\left(1\right)}\right)}^2+\frac{1}{2}{k}^{\left(2\right)}{\left(l^{\left(2\right)}\right)}^2\hfill \\ \hfill & =\hspace{0.17em}2L^2\left[k^{\left(1\right)}{\cos }^2\left(\frac{𝜗}{2}\right)+k^{\left(2\right)}{\sin }^2\left(\frac{𝜗}{2}\right)\right]\hfill \end{array}$$

.For the special case $k^{\left(1\right)}=k^{\left(2\right)}$ (which we will restrict ourselves to, but see SI Appendix for details of how this is not as restrictive as it seems), this energy is independent of $𝜗$, and the structure has a constant total energy for all orientations of the lever; this can be viewed as a mechanical realization of Thales’ theorem.

Fig. 1.

NS unit cells and their assembly. (A) Schematic of an NS unit cell constructed from two rigid elements, a strut and a lever, and two elastic elements associated with zero natural-length springs of stiffness $k^{\left(1\right)}=k^{\left(2\right)}$. (B) A simple planar assembly of two unit cells allows for three deformation modes (shear, splay, and compression). A combination of such modes enables the transformation of NS assemblies (with multiple unit cells) into various configurations. (C–E) Assembly of NS structures can have a Poisson’s ratio ($\nu =-{\epsilon }_{\text{trans}}/{\epsilon }_{\text{axial}}$), that is (C) positive with $\nu ={\cot }^2\phi$, (D) zero, and (E) negative (auxetic) with $\nu =-\cot \phi /\cot \left(\phi /2\right)$. Multiple struts are rigidly connected (in D and E) or through a pin joint (in C). In all diagrams, a yellow-gray disk indicates the attachment of a spring to a rigid element, a black disk denotes connections between rigid elements via pin joints, a yellow-gray disk surrounded by a black circle indicates a pin joint between a pair of rigid elements, and arcs on levers represent independent rotational degrees of freedom.

Fig. 3.

Experimental models of totimorphic NS units and assemblies. (A) An experimental realization of a two-dimensional NS unit cell with the lever and strut assembled using laser-cut plastic sheets. The composite image shows different equipotential configurations of the lever with respect to the base link. The black circle at the bottom of the lever indicates a pin joint and the blue circle indicates the location where the elastic strings are attached (see SI Appendix, Fig. S3 for further details). (Inset) Schematic of the pin joint showing the internal frictional force that leads to a frictional torque. Although the configurations shown are equipotential, i.e., their elastic energies are equivalent, a finite torque ($\tau$) is required to overcome internal frictional resistance. (B) The torque ($\tau$) as a function of angle $𝜗$ for unit cells constructed with different spring constants $k$ and the geometric scale $L$ is nearly constant (shown on a log-linear scale). (Inset) Constant torque is required to perturb the configuration (near $𝜗=\pi /2$). (C) An experimentally realized assembly of two-dimensional NSS with Poisson’s ratio that is zero (Left) and negative (Right), just as theoretically predicted (Fig. 1 D and E). (D) Experimental realization of a 3D NS-net that morphs into surfaces with zero, positive, and negative Gaussian curvature $\mathcal{K}$. (E) A stimuli-responsive NS-net with gallium-lubricated joints. When the gallium is in its solid state there is an increase in the internal friction in unit cells enabling load-carrying (ratio of load carried to NS-net weight shown on top). (F) Exposure to a higher temperature ($\mathrm{35}°$C) melts the gallium in the joints, reducing the internal friction, and hence the threshold torque, so that the structure collapses under the load.

To extend the capability of the minimal unit cell shown in Fig. 1A, we construct a simple planar NSS by pinning the levers of two NS unit cells as shown in Fig. 1B. Using the Tchebychev–Grübler–Kutzback (CGK) criteria in two dimensions (12), we can calculate the number of DoF of such a structure with $n$ rigid bodies connected at $g$ joints with $f_i$ DoF per joint using the formula $\#\mathrm{D}\mathrm{o}\mathrm{F}=3\left(n-1\right)-3g+{\sum }_{i=1}^g{f}_i$. In Fig. 1B, by connecting two NSSs we have $n=4$, $g=3$, and $f_i=1$ (corresponding to each pin joint), leading to a structure with 3 internal DoF. We visualize these DoF in Fig. 1B as the shear, splay, and compression modes of the planar NSS. These local modes of deformation of the planar unit cell when combined with the global assembly of NSS gives us the flexibility to create a new class of metamaterials. Although these structures can be deformed without an energetic cost they are not traditional mechanisms with floppy (zero-energy) modes. Instead, as we will see, they have a different type of response close to that of ideal rigid-plastic solid, with one tunable parameter that is experimentally accessible.

As first examples of NSS assemblies, we show that we can design planar equipotential NSSs with varying Poisson’s ratio (the negative ratio of the transverse strain to the longitudinal strain) by simply varying the connectivity and orientation of the unit cells. In Fig. 1C the structure has 1 DoF and a positive Poisson’s ratio, in Fig. 1D the structure has 8 DoF and zero Poisson’s ratio, and in Fig. 1E the structure has 5 DoF and a negative Poisson ratio. We note that the multiple internal DoF in Fig. 1 D and E are due the unconstrained internal modes of a subset of the assembly; later we will show how to arrest/control these. We can also assemble NS units into two-dimensional structures that have the ability to shift between different shapes (see SI Appendix, Figs. S2 and S3).

To allow for the morphability of NSSs into three-dimensional (3D) structures starting from planar two-dimensional neutrally stable nets (NS-nets), we replace the planar pin joint between the lever and the strut in the unit cell (Fig. 1A) by a spherical joint, leading to a modified unit cell shown in Fig. 2A (see also SI Appendix, Fig. S4). Furthermore, to ensure that the deformation of the NS-net only involves redistribution of the elastic energy stored in the prestretched elastic springs, we connect neighboring struts (or levers) using NS joints (see SI Appendix, Fig. S7A) that allow for a single rotational DoF between two struts (or levers). Then, at each NS joint, neighboring levers/struts can move in the local tangent plane or normal to it. NS-nets thus represent a natural generalization of Tchebychev nets (13) by including an additional local translational DoF at the scale of the unit cell (see SI Appendix, section S6), in addition to the single orientational (shear) DoF at every joint,.

Fig. 2.

Simulations of totimorphic assemblies using NS units. (A) An NS-net assembled from NS unit cells can also undergo out-of-plane deformation. The lever and the strut in the unit cell are connected via a spherical joint, and the connections between the neighboring cells are made via regular NS joints (same DoF as a pin joint; see SI Appendix, Fig. S7 for details). Zoomed-in views on the right show the joints and their motions. (B) Deployment of a catenoid, tessellated using 342 NS unit cells (connections highlighted in the zoomed image) into a helicoid (see SI Appendix, sec. S5). (Inset) The large displacement of the highlighted section of the catenoid. (C) Eversion of an open cylinder tessellated from 420 unit cells. The top of the cylinder is fixed, while the bottom end is everted and then displaced vertically in small steps until the entire structure is turned inside-out. All the intermediate states during the eversion are mechanically stable. (Inset) The folding of an axial strip during eversion.

Fig. 2A shows an example of a patch of an NS-net consisting of six unit cells, each with a spherical joint between the lever and the strut; NS rotation at the spherical joints allows for local out-of-plane deformation of the patch. Just as in the planar case, the DoF of such a NS-net can be evaluated using the CGK equation (12) that yields the expression DoF = $6\left(n-1\right)-6g+{\sum }_{i=1}^g{f}_i$; thus, for the assembly shown in Fig. 2A, $n=14$, $g=15$, $f_i$ for each spherical joint is 3, and for each NS pin joint is 1, yielding a total of 19 DoF.

Inverse Design of NS-Nets.

To solve the inverse problem of designing NS structural materials that can morph from a planar periodic structure to a given complex 3D shape we use an optimization approach similar to that recently deployed in other geometric optimization problems in origami and kirigami design (14–16). Algorithmically, this corresponds to minimizing the shape mismatch error between the points on the target surface and equivalent points on the reference structures while satisfying the geometric constraints of neutral stability, implemented using the optimization routine fmincon in MATLAB. The cost function given an initial and target shape then takes the form

$$\mathcal{E}=\hspace{0.17em}\sum _{i,j}\Vert {\mathbf{a}}^{\left(i,j\right)}-{\tilde{\mathbf{a}}\left(i,j\right)\Vert }^2+{\Vert {\mathbf{b}}^{\left(i,j\right)}-{\tilde{\mathbf{b}}}^{\left(i,j\right)}\Vert }^2,$$ [1]

where $\Vert {\mathbb{A}\Vert }^2={\text{tr}}^2\left(\mathbb{A}\right)+\text{tr}\left({\mathbb{A}}^2\right)$ and ${\mathbf{a}}^{\left(i,j\right)},{\tilde{\mathbf{a}}}^{\left(i,j\right)}$ and ${\mathbf{b}}^{\left(i,j\right)},{\tilde{\mathbf{b}}}^{\left(i,j\right)}$ are the first and second fundamental forms (17) of the initial and target shapes evaluated at coordinate $\mathbf{x}\left(u_j,v_j\right)$ (see SI Appendix, sectin S4 for details). Since all the deformations of NS-net are equipotential, one can realize multiple shapes with same net subjected to the local planarity constraints imposed by the NS pin joints connecting the neighboring unit cells.

SI Appendix, Fig. S4D shows the ability of an NS-net to switch between two human faces with different geometric features. This is enabled by the capability of any patch of unit cells within an array to locally deform without affecting the neutral (mechanical) stability of the neighboring cells. The decoupling of the geometry from the mechanical response in NSS is in contrast with classical spring or elastomeric networks whose response to local perturbations is nonlocal and dimension-dependent. We characterize the ability of an NS-net to morph into simple objects such as a cylinder, a sphere, and a hyperboloid and show the effect of varying the unit cell size on the shape morphing ability in SI Appendix, Fig. S4 A–C.

While NS-nets with periodic patterns of unit cells are capable of a range of shape-transforming capabilities, their ability to morph into shapes with large variations in curvature is limited by the size of the unit cell. To further the capability of NS-nets to morph into complex topographies with multiscale features accurately we need to construct nonuniform NS-nets with varying unit cell size. The problem of tessellating the global minimization problem of solving for the entire 3D surface with NS cells in a single step (via energy minimization) is a nonconvex (and elliptic) problem that arises from the constraints of neutral stability, and computationally expensive. An alternate approach of working with this constrained optimization problem is to convert the problem into a parabolic one using a marching method which tiles the surface in a stepwise manner, analogous to a recent additive approach to origami (18).

The tiling process starts by discretizing a selected contour on the target surface into a set of edges of equal length (see SI Appendix, Fig. S5A), each corresponding to the strut of a unit cell. We then construct the unit NS cells corresponding to these struts by finding the appropriate location of the levers. One end of the lever is at the midpoint of the strut and the other end is found by minimizing the vertex distance from the surface of the target shape while satisfying the constraints of neutral stability. This allows a selected contour on the surface of the target object to become isomorphic to a connected layer of NS cells. We repeat this procedure with the optimized locations of the lever ends as the starting location to find the next connected layer and repeat the process until the entire surface is covered with NS units as shown in SI Appendix, Fig. S5A. All the connections between individual unit cells are treated as planar NS joints whose plane of rotation is defined by the orientation of struts and levers on the tiled surface.

Fig. 2B shows the NS-net tessellation for a catenoid, a surface with uniform negative Gauss curvature and zero mean curvature, using such a procedure. When the axis of tiling is along the direction of rapid change in the principal curvature, such as in the catenoid (see SI Appendix, Fig. S5B), our algorithm results in large number of units in the high curvature region. Beginning with a catenoid, a continuous deformation applied to all NS components morphs it to a helicoid with intermediate target shapes that are all minimal surfaces (SI Appendix, Eq. S7). Such a geometric deployment is possible without instabilities such as wrinkles on the NS-net because of the local in-plane shear DoF provided by the individual NS units as quantified in SI Appendix, Fig. S5D. We note that the planar constraints between the neighboring unit cells imposed by the NS pin joints restrict arbitrarily large deformations of the NS-net on the global scale and hence the final achievable shape of the helicoid has an expected roughness, as evident in Fig. 2B. Using the same algorithm we also tessellate an open cylinder, a surface with zero Gauss curvature and uniform mean curvature, and demonstrate its eversion in Fig. 2C. We carry out this eversion by fixing the top of the cylinder while deforming the other end axially (details in SI Appendix, section S5). Note that all the partially everted states are energetically equivalent and stable. Similar to the catenoid transition, the local in-plane energy-free shear modes enable large deformations (shown in SI Appendix, Fig. S5E).

Experimental Realization of Neutrally Stable Assemblies/Structures.

To complement our computational results with physical realizations of NS-nets we created neutrally stable unit cells using laser-cut plastic sheets with the levers attached to the link through freely rotating pin joints and elastic cords to mimic zero-length springs (see SI Appendix, Fig. S3). NS unit cells/nets in our experiments differ from ideal floppy structures as they are internally stressed. Furthermore, the finite size of the spherical and pin joints along with the presence of friction implies the need for a finite torque $\tau$ to rotate the struts at any joint by an angle $𝜗$ in each unit cell (Fig. 3A, Inset). The value of this torque can be controlled by changing either the spring stiffness or the lever length within a unit cell, since this changes the frictional forces at the joints and thence the torque. In Fig. 3B we see a proportional increase in the torque with increasing spring stiffness $k$ and lever length $L$; furthermore, we see that the torque is relatively independent of $𝜗$ over a range of angles (see Fig. 3B, Inset) (see also SI Appendix, Figs. S8 and S9). This response is reminiscent of a rigid-plastic material (19), for long just a mathematical idealization of plastic behavior in solids, but one that arises here physically as a consequence of the internal friction at the joints.

In Fig. 3C we show physical realizations of planar NS-nets with both zero and negative Poisson’s ratio that follow their theoretical idealizations in Fig. 1 D and E (see SI Appendix, Fig. S3 D and E for details of their mechanical response). To show how to achieve a shape-morphing NS-net in three dimensions, we use ball and socket joints to connect the levers and struts within a unit cell and also to connect neighboring unit cells (see SI Appendix, Fig. S3F). This allows us to fabricate NS-nets that can form surfaces of zero Gauss curvature (e.g., a cylinder), positive Gauss curvature (e.g., a sphere), and negative Gauss curvature (e.g., a hyperboloid) as shown in Fig. 3D. In all these cases the deformations of the NS-net are primarily in the joints connecting nearest neighbor NS units, just as shown in SI Appendix, Fig. S4B.

Although friction in the NS-net enables a rigid-plastic response, the structural materials made from NS unit cells do not have the ability to support substantial loads owing to the large number of internal DoF. To overcome this limitation and to move toward an NS-net which can be rigidified on-demand while retaining the flexibility that arises from neutral stability, we use the phase-changing property of gallium at near room temperatures (melting point of $30°$C) and fill all the ball joints with gallium (20). Gallium can then act either as a viscous lubricant in the ball joints in its liquid state (at room temperature) or as a solid that impedes rotation (at low temperatures). Thus, by controlling the local joint temperature, we can make the NS-net multimorphable when the gallium is in liquid state or stiff when the gallium is in its solid state. This ability to rigidify the NS-net allows it to support external loads as shown in Fig. 3E, where a cylindrical NS-net can carry a load until it is heated to about $\sim 35°$C when the structure buckles and collapses (Fig. 3F).

Our approach of using neutrally stable unit cell-based assemblies offers a simple approach to totimorphic assemblies by separating the geometry of the assembly from its mechanical response at both the individual and collective level. The local geometry of the unit cell can be varied by changing both its overall size and the length of the single movable strut, while its plastic response can be changed by varying either the stiffness of the springs within the structure or the length of the struts and links. This allows for individual neutrally stable structures to be built on any scale and then assembled into structures by modulating the size and stress within a unit cell to create spatially heterogeneous material structures. At a practical level, the mechanical response of individual cells can be further controlled by tuning the rheological response of the pin/spherical joints by using a phase-changing material such as gallium. Altogether, this allows for totimorphic NS assemblies to simultaneously have both local structural flexibility and a heterogeneous mechanical response.

Data Availability

All study data are included in the article and/or supporting information.

Change History

April 05, 2022: Movies S1–S5 have been updated.