Fabrication of three-dimensional kirigami structures with tunable properties
A method of forming a structural honeycomb includes cutting and folding a substrate sheet according to predetermined cutting and folding patterns and fold angles that cause the sheet to form a honeycomb having cells that each have at least one face abutting, or nearly abutting, the face of another cell. The honeycomb is then stabilized by joining abutting, or nearly abutting, faces to hold the honeycomb together. The honeycomb may have a prespecified three-dimensional shape. The folding pattern may include corrugation, canted corrugation, or zig-zag folds. Joining may employ fixed and/or reversible joinery, including slotted cross section, tabbed strip, angled strip, integral skin, sewn, or laced. At least some folds may be partially-closed to create bends and twists in the honeycomb structure. Some surfaces of the honeycomb may be covered with a skin or face sheet. The substrate sheet may have flexible electronic traces.
1 . A structure, comprising:
at least one three-dimensional structural honeycomb formed by a cut-and-folded substrate sheet that has been treated with a pattern of cut areas and folds according to predetermined cutting and folding patterns, the predetermined cutting and folding patterns being formulated according to parameters derived for creation of a three-dimensional honeycomb having specific properties, the folds having at least one predetermined folding angle that is sufficient to cause the cut-and-folded substrate sheet to form a honeycomb structure having a plurality of identical cells, each cell having at least one face abutting at least one face of another cell,
wherein at least some of the folds create contact between at least a first part of the substrate sheet and at least a second part of the substrate sheet that the first part of the substrate sheet did not abut before folding,
wherein the honeycomb structure has a shape having a height of the top and bottom, respectively, of the honeycomb structure at the honeycomb coordinates t(i, j) and u(i, j) that is, for each x and y value, t(x,y) and u(x,y), as established by the predetermined cutting pattern, the at least one predetermined fold angle, and the predetermined folding pattern, and wherein the cut-and-fold instructions parameters are A′(i, j), B′(i, j), C′(i,j), and D′(i,j), wherein:
A
i
,
j
=
A
i
,
j
′
+
W
i
B
i
,
j
=
B
i
,
j
′
+
W
i
C
i
,
j
=
C
i
,
j
′
+
V
i
D
i
,
j
=
D
i
,
j
′
+
V
i
such
that
;
A
i
,
j
′
=
{
u
(
4
i
,
j
)
if
j
≡
0
or
j
≡
1
u
(
4
i
+
1
,
j
)
if
j
≡
2
or
j
≡
3
(
mod
4
)
B
i
,
j
′
=
{
t
(
4
i
,
j
)
if
j
≡
0
or
j
≡
1
t
(
4
i
+
1
,
j
)
if
j
≡
2
or
j
≡
3
(
mod
4
)
wherein
C
′
(
i
,
j
)
and
D
′
(
i
,
j
)
are
determined
recursively
,
such
that
;
C
i
,
j
′
=
{
C
i
,
j
-
1
′
-
t
(
4
i
+
2
,
j
)
+
t
(
4
i
+
1
,
j
-
1
)
if
j
≡
0
C
i
,
j
-
1
′
-
t
(
4
i
+
2
,
j
)
+
t
(
4
i
+
2
,
j
-
1
)
if
j
≡
1
C
i
,
j
-
1
′
-
t
(
4
i
+
1
,
j
)
+
t
(
4
i
+
2
,
j
-
1
)
if
j
≡
2
C
i
,
j
-
1
′
-
t
(
4
i
+
1
,
j
)
+
t
(
4
i
+
1
,
j
-
1
)
if
j
≡
3
(
mod
4
)
with
D
′
(
i
,
0
)
=
C
′
(
i
,
0
)
+
t
(
4
i
+
2
,
0
)
-
u
(
4
i
+
2
,
0
)
,
then
;
D
i
,
j
′
=
{
D
i
,
j
-
1
′
-
u
(
4
i
+
2
,
j
)
+
u
(
4
i
+
1
,
j
-
1
)
)
if
j
≡
0
D
i
,
j
-
1
′
-
u
(
4
i
+
2
,
j
)
+
u
(
4
i
+
2
,
j
-
1
)
)
if
j
≡
1
D
i
,
j
-
1
′
-
u
(
4
i
+
1
,
j
)
+
u
(
4
i
+
2
,
j
-
1
)
)
if
j
≡
2
D
i
,
j
-
1
′
-
u
(
4
i
+
1
,
j
)
+
u
(
4
i
+
1
,
j
-
1
)
)
if
j
≡
3
(
mod
4
)
;
and
the offset parameters are
V
i
=
max
j
(
B
i
,
j
′
-
C
i
,
j
′
)
and
W
i
=
max
j
(
D
i
,
j
′
-
A
i
,
j
+
1
′
)
;
and
at least one joinery mechanism attached to or between at least some abutting faces in a manner that stabilizes the structural honeycomb into a fixed shape.
2 . The structure of claim 1 ,
wherein the folds are of one or more types selected from the group consisting of corrugation-type folds, which project vertically in the z-direction to the vertices of the cells, and cross-sectional folds, which project horizontally in the y-direction to the edges between adjacent cells in the x-direction.
3 . The structure of claim 2 , wherein the structural honeycomb is a volume-filling honeycomb.
4 . The structure of claim 1 , wherein at least one of the top or bottom of the structural honeycomb forms an arbitrary two-dimensional shape along its length.
5 . The structure of claim 1 , wherein the structural honeycomb has zig-zag corrugation folds in at least one direction.
6 . The structure of claim 1 , wherein the structural honeycomb has canted corrugation folds in at least one direction.
7 . The structure of claim 1 , wherein the joinery mechanism is glue or spot welding.
8 . The structure of claim 1 , wherein the joinery mechanism is reversible.
9 . The structure of claim 8 , wherein the reversible joinery mechanism is at least one of: slotted cross sections, tabbed strip joinery, angled strip joinery, an integral skin, sewn joinery, or laced joinery.
10 . The structure of claim 1 , wherein the joinery mechanism functions as a skin covering at least part of the structure.
11 . The structure of claim 1 , further comprising a skin or face sheet that at least partially covers at least one surface of the honeycomb.
12 . The structure of claim 1 , wherein at least some of the folds are only partially closed.
13 . The structure of claim 1 , further comprising at least one set of flexible electronic traces applied to the substrate sheet.
14 . The structure of claim 1 , wherein the structure is an aircraft wing.
15 . The structure of claim 1 , wherein the structure is a shoe sole.
16 . The structure of claim 1 , wherein the structural honeycomb is formed according to the method comprising:
cutting a substrate sheet according to a predetermined cutting pattern, the predetermined cutting pattern being formulated according to parameters derived for creation of a three-dimensional honeycomb having specific properties;
after the step of cutting, folding the cut substrate sheet according to a predetermined folding pattern and at least one predetermined fold angle, wherein folding comprises at least contacting at least a first part of the substrate sheet against at least a second part of the substrate sheet that the first part of the substrate sheet did not abut before folding, the predetermined folding pattern being formulated according to parameters derived for creation of a the three-dimensional honey comb having specific properties, and the at least one predetermined fold angle being sufficient to cause the cut and folded substrate sheet to form a structural honeycomb having a plurality of cells, each cell having at least one face abutting, or nearly abutting, at least one face of another cell, wherein the predetermined cutting pattern, the at least one predetermined fold angle, and the predetermined folding pattern together comprise cut-and-fold instructions for the structural honeycomb, wherein the parameters used in creating the cut-and-fold instructions for the structural honeycomb are calculated by:
using predetermined functions t(x,y) and u(x,y), giving the height of the top and bottom, respectively, of the desired honeycomb shape for each x and y value, to calculate the parameters in terms of the values of functions t(x,y) and u(x,y) at the honeycomb coordinates t(i, j) and u(i, j), wherein the cut-and-fold instructions parameters are calculated by:
calculating unshifted parameters A′(i, j), B′(i, j), C′(i,j), and D′(i,j), providing the cut-and-fold instructions parameters through the column shifts, according to the formulas:
A
i
,
j
=
A
i
,
j
′
+
W
i
B
i
,
j
=
B
i
,
j
′
+
W
i
C
i
,
j
=
C
i
,
j
′
+
V
i
D
i
,
j
=
D
i
,
j
′
+
V
i
calculating
:
A
i
,
j
′
=
{
u
(
4
i
,
j
)
if
j
≡
0
or
j
≡
1
u
(
4
i
+
1
,
j
)
if
j
≡
2
or
j
≡
3
(
mod
4
)
B
i
,
j
′
=
{
t
(
4
i
,
j
)
if
j
≡
0
or
j
≡
1
t
(
4
i
+
1
,
j
)
if
j
≡
2
or
j
≡
3
(
mod
4
)
writing
formulas
for
C
′
(
i
,
j
)
and
D
′
(
i
,
j
)
recursively
:
C
i
,
j
′
=
{
C
i
,
j
-
1
′
-
t
(
4
i
+
2
,
j
)
+
t
(
4
i
+
1
,
j
-
1
)
if
j
≡
0
C
i
,
j
-
1
′
-
t
(
4
i
+
2
,
j
)
+
t
(
4
i
+
2
,
j
-
1
)
if
j
≡
1
C
i
,
j
-
1
′
-
t
(
4
i
+
1
,
j
)
+
t
(
4
i
+
2
,
j
-
1
)
if
j
≡
2
C
i
,
j
-
1
′
-
t
(
4
i
+
1
,
j
)
+
t
(
4
i
+
1
,
j
-
1
)
if
j
≡
3
(
mod
4
)
setting
D
′
(
i
,
0
)
=
C
′
(
i
,
0
)
+
t
(
4
i
+
2
,
0
)
-
u
(
4
i
+
2
,
0
)
;
performing
recursion
according
to
the
formula
D
i
,
j
′
=
{
D
i
,
j
-
1
′
-
u
(
4
i
+
2
,
j
)
+
u
(
4
i
+
1
,
j
-
1
)
)
if
j
≡
0
D
i
,
j
-
1
′
-
u
(
4
i
+
2
,
j
)
+
u
(
4
i
+
2
,
j
-
1
)
)
if
j
≡
1
D
i
,
j
-
1
′
-
u
(
4
i
+
1
,
j
)
+
u
(
4
i
+
2
,
j
-
1
)
)
if
j
≡
2
D
i
,
j
-
1
′
-
u
(
4
i
+
1
,
j
)
+
u
(
4
i
+
1
,
j
-
1
)
)
if
j
≡
3
(
mod
4
)
;
and
calculating the offset parameters according to the formulas
V
i
=
max
j
(
B
i
,
j
′
-
C
i
,
j
′
)
and
W
i
=
max
j
(
D
i
,
j
′
-
A
i
,
j
+
1
′
)
.
and
stabilizing the structural honeycomb by joining one or more abutting, or nearly abutting, honeycomb faces in a manner that holds the structural honeycomb together.
17 . The structure of claim 16 , further comprising an outer skin or face sheet at least partially covering the structural honeycomb.
18 . The structure of claim 16 , wherein the cut-and-fold instructions include instructions to include or exclude cells in order that the structural honeycomb will have a prespecified outline.
19 . The structure of claim 16 , wherein the cut-and-fold instructions include instructions to create a honeycomb wherein at least one of the top or bottom of the honeycomb forms an arbitrary shape along its length.
20 . The structure of claim 16 , wherein the structural honeycomb is a volume-filling honeycomb.