Supplementary Information. for. Origami based Mechanical Metamaterials

Transcription

1 Supplementary Information for Origami based Mechanical Metamaterials By Cheng Lv, Deepakshyam Krishnaraju, Goran Konjevod, Hongyu Yu, and Hanqing Jiang* [*] Prof. H. Jiang, C. Lv, D. Krishnaraju, Dr. G. Konjevod School for ngineering of Matter, Transport and nergy Arizona State University, Tempe, AZ 8587 (USA) -mail: [*] Prof. H. Yu School of arth and Space xploration School of lectrical, Computer and nergy ngineering Arizona State University, Tempe, AZ 8587 (USA)

2 Change of Length W The length W is given by q. () of the main text. Figure Sa shows the derivatives of W s two terms, i.e., ( n ) cos β η φ cos ( φ / ) and cos / φ, along with W, as a function of φ, for n = =, η = / and β = 78.5, the same parameters used in Fig. b. It can be seem that these n 5 two derivatives work against each other with ( n ) cos β η > 0 φ cos ( φ / ) to decrease W, while with cos / < 0 φ to increase W, from a planar state to a collapsed state. Therefore, the one among these two derivatives with larger absolute value dominates the change of W. It is apparent that when one folds a Miura-ori from its planar state to a collapsed state, ( n ) cos β η > 0 φ cos ( φ / ) dominates firstly to decrease W. Once the stationary point is reached, increase W. cos / < 0 φ starts to dominate and Out-of-plane Poisson s Ratio Using q. () in the main text, the out-of-plane Poisson s ratios, related to in-plane strains upon height change, are given by β ε cot sin sin 3 = = ε cos β ε 0 ν ν ( ) ( n ) ( ) ( n ) + ( ) β φ η β φ ε sin sin cos cos 3 = = ε cos β ηcos β cos φ ε 0 (S) dh where ε = is the strain in x 3 -direction. Figure Sb shows that ν 3 is positive across the entire H range of φ, due to φ 0,β, and monotonically decreases from to 0 as the Miura-ori varies from

3 its completely collapsed state ( φ = 0 ) to the planar state ( φ β = ), i.e., [ ] ν. For an extreme 3 0, case when β 90 or φ 0, ν3. Figure Sc shows ν 3 for n = 3 and η = 0.5. It is observed that ν 3 can be both negative and positive, separated by a boundary defined by ( n ) η β cos = cos which is shown as the white dashed line. Figure Sc also shows that ν 3 [, ]. Nonlinearity of the lastic nergy with respect to the Coordinates of Vertices Depending on the type of rigid origami and the number of dihedral angles per unit cell, the elastic energy can be always expressed by U T ( α α ), n = k, (S) total i i i eq i= α i where k i are the stiffness constants of the dihedral angles α i with i, eq α as the equilibrium angle, and T n is the number of types of dihedral angles. To obtain the stiffness matrix K and non-equilibrium force P, we need to express U total in terms of the coordinates of vertices. Considering the case of Miura-ori where T α n =, the two dihedral angles are given as cos sin β sin = α = β φ ( ) cos cot tan, (S3) where φ is the projection angle between two ridges. β 0,90 and φ 0,β. Using the distances between the three vertices,, 3 (Fig. c) that form this angle, φ can be determined by using the cosine rule, R R3 R3 φ = cos +, (S4) RR3 3

4 where R ij is the distance between vertices i and j. By combining qs. (S) to (S4), a relationship between the elastic energy and the coordinates of vertices can be obtained. Clearly, this relationship is nonlinear, i.e., U total is nonlinear with respect to the coordinates of vertices. Bulk dulus of Miura-ori The bulk modulus K of Miura-ori can be defined by K θ =, (S5) p p= 0 where p is the hydrostatic pressure. Using the principle of superposition, when only p( = σ ) is applied and σ σ 0 = =, we have ( ) θ = ν ν ε, where σ, σ σ are normal stresses in 3 x, x, and x 3 directions, respectively. Using a similar approach for σ and σ, the bulk modulus K K ε ε ε = ν ν + ν ν + ν ν, where σ = σ = σ = p. is given by ( ) ( ) ( ) σ σ σ For vanishing nominal stress σ = σ = σ 0, the tensile moduli are given by ε ε ε σ σ σ =, =, =. (S6) σ = 0 σ = 0 σ = 0 Thus the bulk modulus K is given by ν ν ν ν ν ν K = + +. (S7) Here the tensile moduli are the tangential moduli of the stress-strain curve, i.e., dσ dσ dσ =, =, =. dε dε d 0 dε ε = dε = 0 dε = 0 4

5 Work Conjugate Relation Stress and duli for Miura-ori The elastic energy density W tot for a (n, n ) Miura-ori is given by W tot = V ( n )( n ) k ( α α, eq ) + ( n )( n ) k ( α α, eq ), (S8) where ( ) ( ) ( ) β φ β φ sin V = ( n ) ab n acos + bcos sin sin cos (S9) is the volume of this (n, n ) Miura-ori. The work conjugate relation provides stress by taking derivatives of W tot with respect to strains, i.e., W W W σ =, σ =, σ =. tot tot tot ε ε ε (S0) Since the Miura-ori is a periodic structure and for a given Miura-ori (i.e., fixed (n, n ), a, b, and β), the deformation can be solely determined by a single parameter φ, these derivatives can be implemented by taking derivatives with respect to φ, i.e., W / φ W / φ W / φ σ =, σ =, σ =. ε / φ ε / φ ε / φ tot tot tot (S) The strains are explicitly given by dl ε = = cot dφ L dw tan cos cos ε = = dφ. (S) W n a b ( n ) a β b ( ) cos β + cos tan cos ( β) dφ ( β) dh ε = = H sin sin Thus the stresses are obtained as 5

6 tan / σ = κ V σ σ ( n ) a β + b V ( n ) acos β bcos ( φ / ) β cot / cos cos / = κ, (S3) cot / sin sin / = cos β κ where cos / κ = ( n )( n ) k ( α α, eq ) sin β sin /. (S4) cos β dv + ( n )( n ) k ( α α, eq ) Wtot cos / sin sin / d φ β The moduli are given by dσ dσ / dφ = = dε dε / dφ ε = 0 dφ= 0 dσ dσ / dφ = = dε dε / dφ ε = 0 dφ= 0 dσ dσ / dφ = = dε dε / dφ ε = 0 dφ= 0. (S5) Implementation of q. (S5) leads to k = ab ζ ξ tan k ζ = ξ η β φ ( ) ( ) n ( ) ηcos β + cos φ φ ( n ) cos cos ( ) ab cot k ζ = ab ξ β cot sin sin 4 cos β (S6) where = ( n )( n ) ( ) + ( n )( n ) ζ 4 cos φ cos β 4 and ( n ) sin ( ) ( n ) cos cos ( ) sin sin ( ) 3 ξ = φ η β + φ β φ. Here k is the spring constant of the hinges for dihedral angles for Miura-ori. 6

7 Range of Tensile and Bulk dulus Given that n [ 3, ], [ 3, ] n, and φ 0,β collapsed state ( φ = 0 ) and for the planar state ( φ β, [ ] 0, = ), [ ] 0, with 0 for the completely with 0 for the planar state ( φ = β ) and for the completely collapsed state ( φ = 0 ). varies from a finite positive value (depending on n, n, and φ) to at both the planar and completely collapsed states. Figures Sa-c show the tensile moduli,, and normalized by k / ab as a function of φ for a few representative n and n, and η = /, β = 45. Above discussed trends are observed. Now we study the bulk modulus using q. (S7). Since some extreme values (e.g., 0,, and ) present in either tensile moduli or Poisson's ratios, it is interesting to study the extreme values of K. At ν ν3 φ 0, ν ν, 3 ν3 ν3 0, and 0, thus the bulk modulus K 0. ν ν3 ν ν3 At φ β, 0, ν3 ν3 0, and, thus the bulk modulus K 0. Another interesting point is at a particular state (φ) for the prescribed n, n and β, the right hand of q. (S7) vanishes, which provides an infinity bulk modulus. The vanishing of the right hand of q. (S7) is determined by the numerators, specifically ν ν3, ν ν3, and ν3 ν3, and again the condition of vanishing these three terms is given by their numerators. Based on q. (4) in the main text and q. (S), the numerators of ν ν3, ν ν3, and ν3 ν3 happen to be identical, 6 4 ( + β) ( n ) ( )( ) ( n ) cos cos cos. (S7) + ηcos φ cos β + cos β ηcos β 3 3 The vanishing of the expression (S7) provides a particular state of folding characterized by the angle φ and dependent on n to reach an infinity bulk modulus. Figure Sd shows the bulk modulus K 7

8 normalized by k / ab as a function of φ for a few representative n and n, and / η =, β = 45, where the signature of changing from 0 to and then 0 is represented. Non-local Interactions in the Ron Resch Pattern Figure S3a shows the planar state of a Ron Resch pattern, which features some equilateral triangles connected by some right triangles. The insets of Fig. S3a show two different folded states of Ron Resch patterns with the upper left one for a dome shape and the upper right one for a completely collapsed state or namely, a Ron Resch plate. Three dihedral angles, β, β, and β 3 are required to describe this rigid origami folding (Fig. S3b). When β = β = β3 = 80, i.e., all triangles are in the same plane, it represents a planar state (e.g., Fig. S3a). When β 0,80, β 0,80, and β 3 0,80, it corresponds to a curved state, illustrated by the upper left inset of Fig. S3a as an example. When β =, β = 0, β 3 = 90, it describes another planar but more compact state (illustrated by the upper 0 right inset of Fig. S3a), by the name of a Ron Resch plate. It is noticed that there are two types of vertices in a Ron Resch pattern, specifically, the centroids of the equilateral triangles (e.g., the vertex marked by a solid blue dot in Fig. S3a) and vertices between the right and equilateral triangles (e.g., the vertex marked by an open blue dot in Fig. S3a). The non-local feature can be similarly observed from these two vertices. For example for the solid blue vertex, it is seen that its motion influences its nearest-neighbor vertices (i.e., the ones marked by solid red dots) through dihedral angles β and β, and its second-neighbor vertices (i.e., the ones marked by solid green dots) through dihedral angles β 3. Buckling Analysis of a Ron Resch Plate and a Six-Fold Supporting Structure These two structures are all periodic so that only the unit cells are utilized to conduct the buckling analysis. Figure S6 shows the unit cell of these two structures. Same thickness, height of supporting, elastic modulus and Poisson s ratio are assigned to the two models. The finite element package 8

9 ABAQUS is used, where the eigenvalues of the different modes can be calculated by using the built-in buckling module. Critical load then can be obtained by using th i cr P th i = Pλ (S8) where th i cr P is the critical load for the i th mode, P is the infinitesimal load applied in the simulation, th i λ is the i th eigenvalue. For the Ron Resch plate, 84,58 R3 (3-node triangular shell) elements are used, with the fixed displacement boundary conditions along the in-plane directions of the plate and the spike. Contact at the spike between the ground plane and the Ron Resch plate is considered. A very small concentrated load is applied at the centroid of the plate. For the six-fold supporting structure, 49,68 S4 (4-node doubly curved shell) elements are used. Same boundary conditions and loads are applied. The cross-sectional properties and material properties of these two structures are the same. Here only the first buckling mode is concerned. The buckling analysis shows that the critical load for the Ron Resch plate is 57% higher than that for the six-fold supporting structure. 9

10 Figures (a) (b) (c) Supplementary Figure S. Geometry characteristics of Miura-ori. (a) Change of size W. Here the change rate of W s two terms with respective to φ is also shown. (b) Contour plot of out-of-plane Poisson's ratio ν 3 as a function of φ and β. (c) Contour plot of out-of-plane Poisson's ratio ν 3 as a function of φ and β. 0

11 (a) (b) (c) (d) Supplementary Figure S. Tensile and bulk moduli for the Miura-ori. (a) Tensile modulus as a function of φ for (3,3) and (3,3) Miura-ori. (b) Tensile modulus as a function of φ for (3,3) and (3,3) Miura-ori. (c) Tensile modulus as a function of φ for (3,3) and (3,3) Miura-ori. (d) Bulk modulus K as a function of φ for (3,3) and (3,3) Miura-oris. Here η = /, β = 45 and all moduli are normalized by k / ab.

12 (a) (b) (c) (d) Supplementary Figure S3. Ron Resch pattern. (a) The planar state of a Ron Resch pattern, where the solid lines are for mountain creases and the dashed lines are for valley creases. Insets are two different folded states. On the upper left is a dome shape and the upper right is a completely collapsed state. (b) Three dihedral angles β, β, and β 3 are needed to describe a Ron Resch pattern. (c) One type of non-local element for the Ron Resch pattern with the centroid of the equilateral triangle as the central vertex. (d) Another type of non-local element for the Ron Resch pattern with the intersections between pleated triangles as the central vertex.

13 Supplementary Figure S4. Shear deformation of a Miura-ori. Deformation of a (3,3) Miura-ori under shear loading along the negative x direction. It is observed that the opposite relationship between shear loading and shear deformation. 3

14 (a) (b) (c) Supplementary Figure S5. Histograms of the three dihedral angles (a) β, (b) β, and (c) β 3 for three Ron Resch patterns, namely a Ron Resch dome, a tube and a stingray. 4

15 (a) (b) Supplementary Figure S6. Unit cell for buckling analysis. (a) Completed collapsed Ron Resch plate. (b) Six-fold supporting structure. 5

16 References. B. Liu, Y. Huang, H. Jiang, S. Qu, K. C. Hwang, Computer Methods in Applied Mechanics and ngineering 93, 849 (004).. B. Liu et al., Physical Review B 7, 8 (Jul, 005). 6