Swinburne Research Bank http://researchbank.swinburne.edu.au Author: Lv, Yang; Zhang, Ying; Gong, Neng; Li, Zhongxian; Lu, Guoxing; Xiang, Xinmei On the out-of-plane compression of a Miura-ori patterned sheet 105022 2019 International Journal of Mechanical Sciences 161-162 October http://hdl.handle.net/1959.3/450931 Title: Article number: Year: Journal: Volume: Issue: URL: Copyright: Copyright © 2019 Elsevier Ltd. All rights reserved. Author's Accepted version is licensed as follows: Creative Commons Attribution-NonCommercialNoDerivatives 4.0 International (CC BY-NC-ND 4.0) https://creativecommons.org/licenses/by-ncnd/4.0/. This is the author’s version of the work, posted here with the permission of the publisher for your personal use. No further distribution is permitted. You may also be able to access the published version from your library. The definitive version is available at: https://doi.org/10.1016/j.ijmecsci.2019.105022 Swinburne University of Technology | CRICOS Provider 00111D | swinburne.edu.au Powered by TCPDF (www.tcpdf.org) On the out-of-plane compression of a Miura-ori patterned sheet Yang Lv 1, Ying Zhang 2, Neng Gong 1, Zhong-xian Li 1,3, Guoxing Lu 4,*, Xinmei Xiang 5* 1. Tianjin Key Laboratory of Civil Structure Protection and Reinforcement, Tianjin Chengjian University, Tianjin 300384, China; 2. School of Materials Science and Engineering, Tianjin Chengjian University, Tianjin 300384, China; 3. Key Laboratory of Coast Civil Structure Safety of Ministry of Education, Tianjin University, Tianjin 300072, China; 4. Faculty of Science, Engineering & Technology , Swinburne University of Technology, Hawthorn, Vic 3122, Australia; 5. School of Civil Engineering, Guangzhou University, Guangzhou Panyu University City Outer Ring Road No. 230. Guangzhou 510006, China. Abstract: Miura-ori patterned sheets have been recognized as efficient energy absorption devices that can serve as the core of sandwich structures or a layer of metamaterials. In this paper, first, the geometric characteristics of a Miura-ori patterned sheet were examined. Second, a Miura-ori patterned sheet was fabricated using 0.3-mm thick aluminum sheets by three-step compression, and the patterned sheet was used as the core by bonding it between two aluminum skins to make a sandwich panel. The quasi-static out-of-plane compression behavior of both the Miura-ori patterned sheets and their corresponding sandwich panels was investigated experimentally and numerically by using finite element analysis (FEA). The simulation results show a reasonable agreement with the experimental results. A parametric analysis of the Miuraori patterned sheets was carried out using FEA to demonstrate the effect of cell wall thickness, side length, dihedral angle and sector angle. Empirical formulae were obtained for the peak crushing force and the mean force. Keywords: Miura-ori patterned sheet; sandwich panel; out-of-plane compression; quasi-static; energy absorption * Corresponding authors. E-mail address: glu@swin.edu.au (Guoxing Lu); xiangxm@gzhu.edu.cn (Xinmei Xiang) 1 1. Introduction Origami is an ancient Japanese method of paper folding. Many complex 3D shapes can be fabricated easily by simply folding a flat paper along predesigned creases. Because of the potential in a wide range of applications, origami has attracted research interests in both mathematics and engineering fields. In rigid origami, the surface surrounded by crease lines is not allowed to stretch or bend during folding. Origami has been introduced in energy-absorbing thin-walled structures [1-4] , programmable robots with precise actuation control [5] and the planar fabrication of 3D mechanisms [6]. One of the essential rigid origami patterns is the Miura-ori [7] , the unique geometric structure of which has attracted attention from scientists, mathematicians, engineers, and artists. A Miura-ori patterned folding sheet can be regarded as a tessellation of multiple unit cells, each made up of four identical facets in the shape of parallelograms, and the facets are assumed to be infinitely thin and rigid. Miura-ori patterned sheets can potentially be used as the core of sandwich structures [8-10] because of the high stiffness-to-weight ratio, open ventilation channels, continuous manufacturing process, higher energy absorbing ability and better impact properties when compared with conventional honeycomb cores. In recent years, Miura-ori patterned sheets have gained much attention. Fischer et al. [11] produced and tested various folded cores with different unit cell geometries and base materials. Heimbs et al. [12-13] studied the mechanical behavior of folded core structures for sandwich composites under compression using a virtual testing approach. They studied the constitutive modeling of the cell wall material, imperfections and the representation of cell wall buckling, folding or crushing phenomena. Schenk and Guest [14-15] proposed a novel manufacturing process, which uses cold gas-pressure to fold sheets into Miura-ori. Wei et al. [16] analyzed the Poisson’s ratio of the Miura-ori metamaterial and the effective bending stiffness of the unit cell and solved the inverse design problem for the optimal geometric and mechanical response. Zhou et al. [17] presented a parametric study on the Miura-ori folded core models using finite element analysis (FEA). Pydah and Batra [18] carried out numerical studies on the response of elastic-plastic Miura-ori core sandwich plates subject to a range of loading rates. Xiang et al. [19] presented a parametric study of the three-point bending tests on sandwich panels with Miura-ori folded cores using the FEA method, and the yielding moment, 2 fully plastic bending moment and elastic buckling moment of the incipience of core buckling were obtained. In addition, Xiang et al. [20] also performed a parametric study of the quasi-static out-of-plane compression tests on the arc-Miura arches and showed that their energy absorption was two to four times that of the corresponding monolithic arch. Zhang et al. [21] carried out quasi-static in-plane compression tests on Miura-oribased metamaterials and demonstrated that the origami-based metamaterials with the selected geometric parameters could absorb more energy than the corresponding honeycombs. Dynamic compression tests on similar specimens were also performed by the same research group [22] , and they found that the energy absorption capacity increased with the loading rate. Heimbs et al. [23] concluded that under quasi-static compression, the load-displacement property mainly depended on the base material of the Miura-ori patterned sheet; however, few studies of Miura-ori patterned sheets made of aluminum have been carried out. In this paper, the geometric characteristics of a Miura-ori patterned sheet are examined first. Experimental and FEA results are reported for the quasi-static compressive behavior of both the Miura-ori patterned sheets and their sandwich panels. A parametric analysis of the Miura-ori patterned sheets with different cell wall thicknesses, side lengths, section angles and dihedral angles is carried out. Empirical formulae are proposed for the peak crushing force and the mean force. 2. Geometry and Fabrication of the Miura-ori Patterned Sheet 2.1 Geometry of Miura-ori The Miura-ori pattern can be constructed by the tessellation of multiple units. A unit consists of four identical parallelograms, as shown in Figure 1(a). The mountain and valley crease lines are marked as solid and dotted lines, respectively. Once the crease lengths a and b and the sector angle α are given, the motion of the unit is dependent on the dihedral angle θ, as shown in Figure 1 (b). The height h, dihedral angle γ, and dimensions u, l, and w can be expressed in terms of the dihedral angle θ. The relations between these variables are given as follows: h = a sin α sin θ (1) 3 γ = 2 arcsin cos θ (2) 1 − sin 2 θ sin 2 α γ  = w 2= b sin α sin   2 2b sin α cos θ (3) 1 − sin 2 α sin 2 θ l= 2 a 2 − h 2= 2a 1 − sin 2 α sin 2 θ (4)  w  2ab cos α u =− b2   = l 2 (5) 2 For the case of a=b=20 mm and the dihedral angle α=60°, the dimensions of the patterned sheet u, l, and w are plotted against the dihedral angle θ, as shown in Figure 2. It can be seen that because of the cosine function relationship, the width w decreases slightly when the dihedral angle increases from 0° to 40°, and when the dihedral angle θ increases from 60° to 90°, the width w decreases significantly with θ. The relationship between the height h and the dihedral angle θ is a sine function. (a) (b) γ h α a α a α w b Z Y Y X θ b u l X Figure 1. (a) One unit of the Miura-ori pattern in the original flat position [10] and (b) definition of parameters [10]. 4 Figure 2. Rigid folding motion of the patterned sheet with dihedral angle θ. 2.2 Fabrication of the Miura-ori Patterned Sheet Thin 1060 aluminum sheets with a thickness of 0.3 mm were used to fabricate the Miura-ori patterned sheets. The density was 2700 kg/m3. The dimensions of the Miuraori patterned sheets are a=b=20 mm, α=60° and θ=50°, with 5 units in each direction. The theoretical dimensions of the sheets after rigid folding are h=13.27 mm, w=29.76 mm, l=29.93 mm, and γ=118.42°. Molds of ABS resin were first made by using 3D printing technology to facilitate the fabrication of the Miura-ori patterned sheets. Young’s modulus of the ABS resin was 2.4 GPa, the density was 1040 kg/m3 and the tensile strength was 36 MPa. To avoid possible breakage of the aluminum sheets at the mountain and valley crease lines when compressed directly on the mold with the dimensions of the target sheet (Figure 3), three pairs of molds with gradually changing dihedral angles were employed. The dihedral angles (θ) of these three pairs of molds are 25°, 38° and 50°. The mold with a dihedral angle θ of 50° is shown in Figure 4 as an example. The corresponding dimensions of the patterned sheet after being compressed on two pairs of molds are as follows. When the dihedral angle θ equals 25°, h=7.32 mm, w=33.74 mm, l=37.22 mm, and γ=153.76°; when the dihedral angle θ equals 38°, h=10.66 mm, w=32.27 mm, l=33.84 mm, and γ=137.33°. The sizes of the 5 three pairs of molds are 196.8×168.6×7.3 mm, 181.0×161.3×10.66 mm and 163.0×148.8×13.27 mm. Figure 3. Two photographs showing breakage at the mountain and valley crease lines during the compression process. Figure 4. Three-dimensional-printed ABS resin molds with the dihedral angle θ equals to 50°. With the three pairs of molds, the Miura-ori patterned sheets were obtained by compressing the 0.3-mm thick aluminum sheets between the male and the female mold step by step. Typical images of the patterned sheet after each step of compression are shown in Figure 5. It should be noted that the compression process should be conducted with great care to avoid damage of the mountain and valley crease lines. A 5.0-kN load was applied in the first compression step, and loads of 7.0 kN and 10.0 kN were exerted during the second and third steps. The displacement of the crosshead and resistance force is shown in Figure 6 for the three steps of the compression processes for both 6 correct operation (without fracture) and incorrect operation (with fracture). The maximum compression force was 21 kN, and the maximum strain of the ABS mold was 0.04%, which was very small and within the elastic strain. Therefore, ABS molds could be assumed undeformed. The Miura-ori patterned sheets were then placed into an oven to heat to 350 ℃ and incubated for 5 minutes to release any residual stress. Figure 5. Miura-ori patterned sheets after the first, second and third step of compression 21 Resistance force (kN) 18 third step right operation wrong operation 15 second step 12 9 first step 6 3 0 0 2 4 6 8 10 12 14 Displacement of the actuator (mm) Figure 6 Plot of force against the crosshead displacement 3. Experiments 3.1 Tensile Stress-Strain Curve for the Parent Aluminum Sheet Tensile tests of the aluminum specimen were conducted to obtain the mechanical property of the aluminum used in this study. Universal testing machine Autograph AGS-X was used to conduct the tensile test and the maximum load was 100kN. The loading speed of the tensile test was 3mm/min. The dimensions of the dog-bone shaped 7 specimen are shown in Figure 7(a), and the thickness of the sample is 2.0 mm. Figure 7(b) is a true stress-strain curve obtained by converting the engineering stress-strain relationship. From the true stress-strain curve, Young’s modulus was 10 GPa, and the yield stress was 108 MPa. (a) 120 Stress (MPa) 100 80 60 40 20 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Strain (b) (c) Figure 7. (a) Dimensions of the dog-bone shaped specimen (in millimeters); (b) a photography of the tensile test and (c) the true stress-strain curve. 8 3.2 Compressive Tests for Miura-ori Patterned Sheets To evaluate the characteristics of the Miura-ori patterned sheets in the out-of-plane direction, compressive tests were conducted as shown in Figure 8(a) with a cross-head speed of 5 mm/min by placing, loosely initially, the patterned sheet between two thick flat plates. The load-displacement curves are shown in Figure 8(b) for three nominally identical specimens. The curves approximately show three stages of the deformation. In the beginning part of the first stage, corresponding to a displacement ( δ ) between 0 and 0.5 mm, the upper compression plate was not in close contact with the specimen, so the force gradually increased with the displacement. For the displacement ( δ ) between 0.5 and 2.5 mm, the deformation can be considered elastic, and the load increased almost linearly with the displacement. The maximum force in the first stage was approximately 4.5 kN. In the second stage, with a displacement ( δ ) from 2.5 mm to 9.0 mm, the cell walls of the patterned sheet structure began to buckle and collapse, and the force of the specimen decreased slightly with the increase in the displacement. This stage is critical to the energy absorbing capacity of the Miura-ori patterned structures because of its large range of displacement. From Figure 8(b), it can be seen that most deformation developed during the second stage when a significant amount of energy can be absorbed. In the third stage, i.e., the compacting stage, the load increased rapidly with the displacement, and the patterned sheet further collapsed with some facets touching the upper and bottom loading plates. With the number of touching facets increasing, the Miura-ori patterned sheet was compacted densely between the upper and bottom loading plates, and the specimen eventually lost the energy absorbing capacity. (a) 9 10 9 Specimen 1 Specimen 2 Specimen 3 8 Force (kN) 7 6 5 Elastic stage Plastic buckling stage 4 3 Compacted stage 2 1 0 0 (b) 1 2 3 4 5 6 7 8 9 10 11 δ (mm) Figure 8. (a) Setup of the out-of-plane compressive test and (b) load-displacement curves of three nominally identical Miura-ori patterned sheets. a=b=20 mm, t=0.3 mm, the dihedral angle θ equals 50°, h=13.27 mm, w=29.76 mm, l=29.93 mm, and γ=118.42° 3.3 Compressive Tests of Sandwich Panels with Miura-ori Patterned Sheet as Core In this case, the crease lines of the Miura-ori patterned sheet and the top and bottom thick aluminum skins were first polished by the cross intersect method, and then the patterned sheet was glued to the skins with an epoxy adhesive to form a sandwich panel. After some very small initial loading on both the upper and bottom skins for five minutes, the precured epoxy adhesive was further cured for 24 hours at room temperature. The size of the skins is 200×200×2 mm to avoid any local buckling and noticeable deformation. Tests with the sandwich panels were conducted in the same way as for the patterned sheet. The test setup and the load-displacement curves are shown in Figure 9 (a) and (b), respectively. Figure 9(b) indicates that the load-displacement curves are similar to those of the Miura-ori patterned sheets, with the exception of a much higher load. The load-displacement curves show basically three stages as well: an elastic stage, a plastic buckling stage and a compacting stage. Again, in the beginning of the first stage for the displacement of 0 to 0.5 mm, the crosshead was not in full contact with the specimen and the resistance force gradually increases with the displacement. For a displacement between 0.5 and 2.3 mm, the load-displacement curves are nearly linearly elastic, and 10 the slope is much higher than that of the patterned sheet because of the bonding of the creases to the upper and bottom aluminum skins. At the end of the first stage, the stiffness gradually decreases with the displacement probably because of slight local buckling of the patterned sheet. The maximum loads of specimen 1, specimen 2 and specimen 3 are 8.65 kN at 1.76 mm, 9.51 kN at 2.26 mm and 9.94 kN at 2.61 mm, respectively. Specimen 1 has the lowest load-carrying capacity in the first stage because part of the epoxy adhesive cracked during the loading process. In the second stage, with a displacement from 2.3 to 8.0 mm, the load-carrying capacity was stable and later slowly increased with the displacement, which is different from the case of the patterned sheet for which the force drops all the time. The cell walls of the patterned sheet structure began to buckle and collapse. The load-carrying capacities for specimen 1, specimen 2 and specimen 3 are between 6.13 kN and 11.57 kN, 7.31 kN and 11.87 kN, and 7.69 kN and 11.26 kN, respectively, which are much higher than those of the patterned sheet core only. In the third stage, the patterned sheet was further compacted, and the facets of the patterned sheet began to contact each other. Finally, the specimen lost the energy absorbing capacity. For the compression of the origami core, the plastic deformation was mostly located in the creases of the core. When the origami core was glued with the top and bottom skins, the core could undergo much larger plastic deformation in the face of the cell wall because the creases could not slide freely due to the constraint from the skins, which significantly increased the load-carrying capacity and hence the energy absorption. (a) 11 35 30 Specimen 1 Specimen 2 Specimen 3 Force (kN) 25 20 15 Plastic buckling stage Elastic stage 10 Compacted stage 5 0 0 1 2 3 4 (b) 5 6 7 8 9 10 δ (mm) Figure 9. (a) Setup of the out-of-plane compressive test and (b) the load-displacement curves of sandwich panels. 4. Finite Element Simulation and Analysis 4.1 Finite Element Simulation Finite element simulation was carried out by using ABAQUS/Explicit (see Figure 10 (a) and (b)). The model was created by using Solidworks software. The parameters of the model were the same as those of the specimens in the experiment, which were a=b=20 mm, α=60° and θ=50°, h=13.27 mm, w=29.76 mm, l=29.93 mm, and γ=118.42°, with a total of 5 units in each direction. The elements of the Miura-ori cores were linear quad elements, type S4R. The data of the true stress and true plastic strain used in ABAQUS were obtained from the engineering stress-strain curve of the 1060 aluminum alloy (see Figure 7(b)). Its density was 2700 kg/m3, Young’s modulus was 10 GPa, and the yield stress was taken as 108 MPa. The upper and base plates were modeled as rigid bodies, and the friction coefficient was set as 0.3. Corresponding to the two types of specimens and tests in the experiments, the Miura-ori core was placed between the upper and base plates either freely or with the top and bottom creases being tied with the upper and base plates. Different mesh sizes were tested, and the simulation results showed a tendency to converge to a stable value when the element size was smaller than 1.0 mm. Thus, an element size of 1.0 mm was selected as the default element size for all the simulations in this study. 12 (a) Specimen (b) compressive test Figure 10. FEA: (a) meshed specimen; (b) compressive test. 4.2 Comparison between FEA and Experimental Results The load-displacement curves from Specimen 1 of the Miura-ori patterned sheet and Specimen 3 of the sandwich panel are shown in Figure 11. In the FEA, the results from the tests with compression speeds of 0.04 m/s and 0.4 m/s for the sandwich panels were compared (see Figure 11). The difference of the peak force was within 12%, showing that under quasi-static compression, the loading speed did not greatly affect the peak force. The peak force was caused by the interaction between the core and the skins. The approximate plateau stage and the densification (compacting) stage of the forcedisplacement curves for both the free and tied cases matched well with the experimental results. However, the elastic stages and the initial peak force did not match well. The experimental results show a much smaller value of initial stiffness and peak force because in the experiments, the dimensions or the Miura-ori units were not exactly the same as the ideal ones. Their folding angle θ has a larger value, for example. Additionally, at some places the creases of the Miura-ori core were not in full contact with the upper and base plates due to imperfections in the specimens. These two factors led to the observed less stiff curve with a lower peak force. However, the later part of the curves and the compacting stage do agree well. For energy absorption analysis, the FEA could be an effective method to conduct the parametric studies in the next Section. 13 Figure 11. Comparison of the results from the FEA and experiment for both the free and tied (sandwich) cases. The deformation of both the Miura-ori patterned sheet and the sandwich panel is shown in Figure 12 and Figure 13, respectively. From Figure 12(b) and Figure 13(b), it appears that the load reached its maximum value (Fmax) when elastic buckling occurred. Comparing Figure 12(c) with Figure 13(c), it is seen that for the free case (see Figure 12(c)), the patterned sheet slid on the supporting plate during the compression process, and local buckling was less obvious than that of the sandwich panel core. For the sandwich panel, because of the constraint from the upper and bottom skins, plastic hinges existed along the edges of the core cell (see Figure 13(c)), which significantly increased the load-carrying capacity and hence the energy absorption. (a) δ=0 (b) at displacement F = Fmax (c) δ=5 mm Figure 12. Deformation of the Miura-ori patterned sheet: (a) δ=0; (b) at the displacement when F = Fmax and (c) δ=5 mm. 14 (a) δ=0 (b) F = Fmax , elastic buckling starts (c) δ=5 mm Figure 13. Deformation of the sandwich panel core: (a) δ=0; (b) the moment when F = Fmax and (c) δ=5 mm. 4.3 Parametric Study Parametric studies were conducted, and the material properties used were the same as those described in Section 4.1. The friction between the Miura-ori core, and the upper/base plates was neglected. The influence on the load-carrying capacity is studied for cell wall thickness t, side length a, dihedral angle γ , and section angle α; the remaining parameters are given a fixed value. Four groups of cores are investigated: Group 1 - cores with different thickness t (a=20 mm, γ = 75o and α = 60o ); Group 2 - cores with different side lengths a (t=0.3 mm, γ = 45o and α = 60o ); Group 3 cores with different dihedral angle γ (a=20 mm, t=0.3 mm, and α = 60o ); and Group 4 - cores with different sector angles α (a=20 mm, t=0.3 mm, and γ = 75o ). 4.3.1 Load-displacement curves The flat Miura-ori sheet was compressed by two rigid plates in the out-of-plane direction. The load-displacement curves of the out-of-plane compression of the patterned sheet were obtained for different groups. Group 1: The load-displacement curves of the sheet with cell wall thicknesses of t=0.2 mm, 0.3 mm, 0.4 mm, 0.5 mm and 0.6 mm are shown in Figure 14(a). It is seen that a larger t led to a larger value for stiffness and load-carrying capacity, indicating a stronger structure under out-of-plane compression. Group 2: The load-displacement curves of the sheet with side lengths of a=3 mm, 10 15 mm, 20 mm, 25 mm and 30 mm are shown in Figure 14(b), and similar stiffnesses and plateau forces were obtained. It can be concluded that the mechanical properties were not significantly influenced by the side length of the cell unit. One exception is that for a=3 mm, the nondimensional displacement at the compacting stage seems less. Group 3: The load-displacement curves of the patterned sheet with dihedral angles γ = 45o , γ = 75o , γ = 105o , γ = 120o and γ = 135o are shown in Figure 14(c). This figure shows that a smaller γ led to a higher value for both the stiffness and the initial peak force because the Miura-ori core structure was more condensed with a smaller γ . However, it seems that a small γ led to an unstable structure in the out-of-plane compression process because the force deceased rapidly after sudden buckling of the cell walls. Group 4: The load-displacement curves of the sheet with different section angles α are shown in Figure 14(d). It is found that, compared with those with α=60° and 75°, the structures with α=30° and 45° had a lower peak force but a higher “plateau” force, which is favorable in the design of protective structures. 16 Figure 14. Fore-displacement curves: (a) Group 1 - different thicknesses (t); (b) Group 2 different side lengths (a); (c) Group 3 - different dihedral angles ( γ ); (d) Group 4 - different sector angles (α). 4.3.2 Analysis of the Peak Force Based on observations from the FEA, the maximum load occurred when the cell wall started to buckle. If one piece of the cell wall is treated as a regular column (see Figure 15), according to Euler's theory of elastic buckling, the maximum load which a column can bear is Fmax ∝ EI a2 (6) where E is the elastic modulus of the material, and I is the minimum area moment of inertia of the cross section ( I = at 3 /12 ). Therefore, t Fmax ∝ Et 2 ⋅   a (7) 17 Figure 15. A plate (approximated as a column) under axial loading The relationships between the peak force (Fmax) and the geometrical parameters of the Miura-ori core, i.e., the cell wall thickness (t) and the side length (a), were obtained from the FEA. Figure 16 shows the relationship between Fmax and t, and from the double logarithmic plot, it is seen that Fmax is approximately linearly proportional to t2 for the Miura-ori cores with different side lengths and dihedral angles. The relationship between Fmax and side length (a) for the Miura-ori cores with different cell wall thicknesses and dihedral angles is shown in Figure 17. It is found that when a is small, Fmax is almost constant; however, when a ≥ 10 mm , Fmax decreases with a, and the slope was approximately -0.5 in the logarithmic plot, which suggests Fmax ∝ 1 . a 0.5 18 Figure 16. The relationship between the peak force (Fmax) and cell wall thickness (t) Figure 17. The relationships between the peak force (Fmax) and side length (a) t According to Equation (7), Figure 18 plots Fmax/ t2 against ( ), taking into account the a effect of the dihedral angle (θ). When t/a is small, Fmax cos θ / Et 2 increases with t/a, while when t/a is large, it is almost a constant. Using least squares fitting and including 19 Young’s modulus E, the following formulae are obtained for the peak force: when t/a<0.033 (i.e., a/h > 30), Fmax cos θ t = 3.16   2 Et a 0.34 (8) when t/a≥0.033 (i.e., a/h < 30), Fmax cos θ = 0.98 Et 2 (9) Figure 18. Relationship between the peak force and geometrical parameters. 4.3.3 Mean Force After the peak force is reached, the force decreases with displacement. To analyze the energy absorption performance during this stage, the mean force of a load-displacement curve is used, which is defined as δ2 Fmean ∫δ = Fd δ 1 δ 2 − δ1 (10) 20 where δ1 is the displacement when the force reaches the peak value, and δ 2 is the displacement when the Miura-ori core is densified and the force starts to increase dramatically. The maximum bending moment that a section (the wall of a Miura-ori unit) can carry is defined as the fully plastic bending moment M p ( M p = σ y at 2 4 ), at which the entire cross section has reached yielding with stress equal to the yield stress ( σ y ) for a perfectly plastic material. When this point is reached, a plastic hinge is formed (see Figure 19). According to moment equilibrium, we have Fmean ⋅ a ∝ M p (11) Fmean ∝ σ y t 2 (12) Therefore, Figure 19. Sketch of the force and fully plastic bending moment when a plastic hinge is formed Figure 20 shows the relationships between the mean force (Fmean) and the cell wall thickness (t). From the logarithmic plot, it is seen that Fmean is approximately linearly proportional to t2 for the Miura-ori cores with different dihedral angles. From Figure 14(b), it is seen that the plateau force was almost the same for the Miura-ori cores with 21 different side lengths. Taking into account the effect of the dihedral angle (θ), Fmean /t2 is plotted against t/a based on the data from the FEA (see Figure 21). Fmean /t2 slightly increases with t/a, and the value of γ has some effect; however, the trend is unclear. As a first approximation, Fm ean ( cos θ ) 0.44 / σ y t 2 is regarded as a constant, and an empirical formula is given, as follows, Fm ean ( cos θ ) σ yt 2 0.44 = 87.98 （13） The above equation gives a predication for the mean force with a maximum error of less than 10%. Figure 20. The relationship between the mean force and the cell wall thickness. 22 Figure 21. Relationship between the mean force and the geometrical parameters. 5. Conclusions This paper has investigated the out-of-plane compressive properties of Miura-ori patterned sheets and sandwich panels with the patterned sheet as the core. The patterned sheets were fabricated by three-step compression, and in each step, a pair of molds with increasing section angles was used. The out-of-plane compression tests of the Miuraori patterned sheets and the sandwich panels were carried out to examine their loadcarrying capacity and the energy absorption capacity. Finite element analysis (FEA) using ABAQUS/Explicit was performed, and a reasonably good agreement with the experimental data was gained. A parametric analysis of the Miura-ori patterned sheets with different cell wall thicknesses t, side lengths a, dihedral angles γ and sector angles α was carried out using the FEA method. The results show that a larger t or a smaller γ led to a higher stiffness and loadcarrying capacity in the out-of-plane compression process. However, a small value of γ led to the force decreasing rapidly after the initial buckling of the cell walls. It was found that when α was small, the structures had a lower peak force and a higher mean force, which is favorable in the design of protective structures. 23 Both the maximum force (Fmax) and the mean force (Fmean) are linearly proportional to t2. When the side length (a) is small, Fmax is almost constant. When a is large, Fmax ∝ 1 . For all the Miura-ori cores with different side lengths, Fmean is almost a 0.5 constant. These findings are useful in the analysis of the energy absorption of structures and materials made of Miura-ori patterns. Acknowledgements The authors gratefully acknowledge the partial support of this research by the National Key Research and Development Program of China under grant number 2016YFC0701100, the National Natural Science Foundation of China under grant numbers 51708390 and 51578361. G Lu and X Xiang wish to thank the financial support from the Australian Research Council (Grant Nos. DP160102612 and DP180102661). 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