Shapeable sheet without plastic deformation

Randomly crumpled sheets have shape memory. In order to understand the basis of this form of memory, we simulate triangular lattices of springs whose lengths are altered to create a topography with multiple potential energy minima. We then deform these lattices into different shapes and investigate their ability to retain the imposed shape when the energy is relaxed. The lattices are able to retain a range of curvatures. Under moderate forcing from a state of local equilibrium, the lattices deform by several percent but return to their retained shape when the forces are removed. By increasing the forcing until an irreversible motion occurs, we find that the transitions between remembered shapes show cooperativity among several springs. For fixed lattice structures, the shape memory tends to decrease as the lattice is enlarged; we propose ways to counter this decrease by modifying the lattice geometry. We survey the energy landscape by displacing individual nodes. An extensive fraction of these nodes proves to be bistable; they retain their displaced position when the energy is relaxed. Bending the lattice to a stable curved state alters the pattern of bistable nodes. We discuss this shapeability in the context of other forms of material memory and contrast it with the shapeability of plastic deformation. We outline the prospects for making real materials based on these principles.

Figure 1

Visualization of the floppy modes in a $7\times 7$ puckered lattice of Fig. 2. Each box shows the three values of floppiness corresponding to the three Cartesian displacements of one node. Thus, the height of the left, rear box (which is 0.98) shows the floppiness $F_i$ corresponding to the free vertical displacement of that corner node. Likewise, the width and depth of this box are proportional to the floppiness $F_i$ for $x$ and $y$ horizontal displacements. Evidently, displacements normal to the lattice have relatively large floppy content.

Figure 2

(a) Zero-energy shape of a random lattice of springs with a rest length given by $L=a(1+0.1R[-0.5,0.5\left]\right)$, where $a$ is the length of a spring in the equilateral case and $R[x,y]$ is a random uniform distribution given values between $x$ and $y$. (b) Top, symmetric case, hexagonal pyramid with equal springs in the middle; bottom, nonsymmetric case, hexagonal pyramid with two different lengths in the middle $a_m=1.05a$ and $a_l=1.15a$. Notice that it is no longer symmetric. (c) Zero-energy surface of the puckered lattice with unit cells of (b) (bottom). Two dimensionless parameters could be defined for these systems: (1) the ratio of height and system size, $\alpha =h/(\sqrt{N}a\sim 0.06)$; (2) the ratio of the unequal pyramid strut lengths, $\beta =a_l/a_m\sim 1.1$. This ratio controls the nonplanarity of the hexagons.

Figure 3

Lattice configurations resulting from the shaping procedure described in the first paragraph of Sec. 3, using a random lattice of size $8\times 8$. (a) Starting from a goal configuration, all the springs are sitting in a smooth surface but are frustrated; (b) relaxed shapes for a random lattice with bending energy; (c) relaxed shapes for a random lattice with extra springs at the edges; (d) averaged two-dimensional projection of the shapes as described in the text and a comparison to the original “flat” sheet: The dotted (blue) line is the desired shape, the dashed (green) line is a sheet with extra springs at the edges, the dash-dotted (pink) line is a sheet with bending energy, and the yellow line is the original flat random sheet. All shapes were translated and rotated so as to get the best fit.

Figure 4

Lattice configurations resulting from shaping procedure described in the first paragraph of Sec. 3, using a $7\times 7$ puckered lattice. Floppy modes were eliminated with extra springs. (a) The desired shape (cylinder with subtending angle of $3\pi /2$); (b) relaxed springs in a puckered lattice with three rest lengths ($a_m=1.05a,a_l=1.15a$); (c) lattice with two rest lengths ($a_m=a_l=1.1a$); and (d) lattice with all springs of equal rest length ($a_m=a_l=a$).

Figure 5

Fifteen random sheets of size $8\times 8$ shaped as half a cylinder. The result presented here is averaged over the $x$ direction. The error measured by Eq. (1) gives $\eta =0.06\pm 0.04$.

Figure 6

Representation of the sheet with the center of mass fixed and a force applied on one of the middle nodes: The left front node is completely fixed, the right front node is restricted to a line, the left back node is restricted to a plane, and the remaining one is completely free. The force applied on the middle point is causing curvature to the sheet.

Figure 7

Hysteresis in a random array of springs of size $8\times 8$. (a) Unstabilized floppy structure; (b) stabilized by additional edge springs as in Fig. 15. Notice how in (a) the transitions between configurations are continuous; each increase in the height of the midpoint, however small, results in a new surface. In (b) forcing the midpoint up does not always result in a new surface. There is an energetic barrier to transform to a new configuration. The plot shows average values. Black dots result from forcing the midpoint up; red squares result from forcing it down. Some examples of the relaxed sheet at various points along the loop are presented; color is darker at the front of the sheet.

Figure 8

Effect of global forcing on a shaped sheet. A puckered $7\times 7$ sheet with stiff edges was shaped as an almost closed cylinder (represented by red dots in both figures). We applied an outer potential, forcing it more inwards (left figure), resulting in 6% deformation. After removing the force, the sheet returns to the original shape (right figure).

Figure 9

Perspective view of the simplified lattice used for the calculation of bulk modulus in Sec. 4. Lattice vectors: ${\mathbf{a}}_{\mathbf{1}}=(l_1,l_2,0),{\mathbf{a}}_{\mathbf{2}}=(l_3,0,0),\mathbf{b}=l_4{\mathbf{a}}_{\mathbf{1}}+l_5{\mathbf{a}}_{\mathbf{2}}+(0,0,l_6),\mathbf{c}=l_7{\mathbf{a}}_{\mathbf{1}}+l_8{\mathbf{a}}_{\mathbf{2}}+(0,0,l_9)$.

Figure 10

Bistability in a random lattice with stiff edges of size $8\times 8$. (a) Bistable nodes (horizontal lines) and nodes of positive angular deficit (vertical lines) shown on top of the actual nodes. (b) Histogram for the distribution of angular deficit for 252 nodes (in seven different realizations of globally flat random sheets of size $8\times 8$: blue, positive angular deficit and bistable; yellow, positive angular deficit and monostable. Only when the angular deficit is close to zero do we get a behavior that deviates from expectation. For large angular deficit, the proportion of monostable nodes falls to zero. We do not present nodes of negative angular deficit since, as mentioned in the text, those were almost always monostable.

Figure 11

Shaping the sheet changes some of the bistable nodes. An example for an $8\times 8$ random lattice with stiff edges. Bistable nodes are marked with a blue circle for the flat sheet and with a red dot for the cylindrically curved one.

Figure 12

Puckered lattice as in Fig. 15 with all bistable nodes pointing down (in gray), and the same lattice with one bistable node (marked by a dot) flipped up (in black). (a) Sheet with spring lengths as in Fig. 2; (b) sheet with longer springs and bigger mismatch ($a_l=1.4a$ and $a_m=1.2a$), resulting in more curvature. The bottom figure presents density plots for the change of local height along the sheet after flipping. The perturbed region extends along a line away from the flipped node.

Figure 13

Puckered lattice used in Fig. 12. It is either forced to curve up (a) or down (b), as described in the text. If the force is removed, the shape retains some curvature when bistable nodes point outward on the curved surface (c), but not when they point inwards (d).

Figure 14

(a) Puckered lattice used in Fig. 8. It is then forced beyond its elastic limit and, when relaxed, finds a new configuration (b). The normalized change in dihedral angles between each pair of triangles (controlling bending energy) is represented by the width of the lines in (c); the displacement of a node is given by the size of the dots.

Figure 15

An example of a random triangular lattice with nodes $x_{\alpha }$ and $x_{\beta }$ connected by springs of rest length $l_{\alpha \beta }$. Red lines represent extra springs at the edges.