Stacked-origami mechanical metamaterial with tailored multistage stiffness

Origami-baed metamaterial has shown remarkable mechanical properties rarely found in natural materials, but achieving tailored multistage stiffness is still a challenge. This study proposes a novel zigzag-base stacked-origami (ZBSO) metamaterial with tailored multistage stiffness property based on crease customization and stacking strategies. A high precision finite element (FE) model to identify the stiffness characteristics of the ZBSO metamaterial has been established, and its accuracy is validated by quasi-static compression experiments. Using the verified FE model, we demonstrate that the multistage stiffness of the ZBSO metamaterial can be effectively tailored through two manners, i.e. varying the microstructures (through introducing new creases to the classical Miura origami unit cell) and altering the stacking way. Three strategies are utilized to vary the microstructure, i.e. adding new creases to the right, left, or both sides of the unit cell. We further reveal that the proposed ZBSO metamaterial has several outstanding advantages compared with traditional mechanical metamaterials, e.g. material independent, scale-invariant, lightweight, and excellent energy absorption capacity. The unravelled superior mechanical properties of the ZBSO metamaterials pave the way for the design of the next-generation cellular metamaterials with tailored stiffness properties.


Introduction
Mechanical metamaterials are unique structures whose properties are determined not by their material composition as conventional structures but by the geometric configuration of the microstructure units [1][2][3]. Due to its unconventional mechanical properties, for instance, negative Poisson's ratio, negative effective mass, negative modulus, chirality, et al., it has broad application prospects [4][5][6][7][8]. The empirical method is the mainstream method of designing such mechanical metamaterials. Although this method has demonstrated powerful capabilities, it is difficult to exploit the superior properties fully. As a remedy, topology optimization methods have been further developed to design mechanical metamaterials with optimal (or locally optimal) mechanical properties [9][10][11]. Nevertheless, topology optimization may not always be possible obtaining the desired optimal solutions, thereby leading to the imperious demand for new design ideas for the mechanical metamaterial.
The origami technique may be one answer to the above question. Due to its simple concept, fascinating characteristic, and wide application prospects, origami has aroused great interest of mathematicians, scientists and engineers in recent year [12], leading to a series of innovative designs , for instance, foldable lithium-ion battery, origami robot, foldscope, bioinspired spring, active structures, energy absorbing-structures/materials sandwich panels. Furthermore, the recent inspiration for the design of mechanical metamaterials has made it even more compelling; especially recent studies reveal that origami-based mechanical metamaterials have unique properties that traditional mechanical materials do not have. Mechanical metamaterials based on Miura origami are the most widely studied among them. Miura origami itself is a mechanical metamaterial with two distinct Poisson's ratio properties, i.e. a negative one for in-plane deformations and a positive one for out-of-plane bending, which is dominated by the kinematics of the folding [36]. Through expanding on the design space of Miura origami, its variants can achieve adjusting Poisson's ratio between positive and negative [37]. Tunable negative Poisson's ratio can be realized by the cylindrical derivative of Miura origami, e.g. the TachiMiura polyhedron [38]. Stiffness represents the ability to resist deformation, which is another key performance index of mechanical metamaterials. The operating environment of engineering equipment is generally complex and changeable, resulting in structures with different stiffness characteristics for various load conditions that are significantly desired. Thus, the tailored stiffness characteristics of the mechanical metamaterial are essential for the engineering equipment, which is an uneasy task for traditional design methods. Cylindrical origami-inspired mechanical metamaterials were designed and analyzed via energy landscapes and strain variations; by controlling the stiffness, the required deployability and collapsibility can be realized [39]. By considering the mixed mode of deformations involving both rigid origami motion and facet bending, the deformation stiffness of mechanical metamaterials with the waterbomb bases can be tailored, forming a potential long-distance actuation mechanism with a single far-field force [40].
To further expand the design freedom of Miura origami, mechanical metamaterials based on curved crease origami can accomplish in situ stiffness manipulation by alteration of the curvature of the creases [41]. Although these studies have successfully achieved tailored stiffness characteristics, there is still room for further expansion of its design freedom.
Stacking origami mechanical metamaterials to structure novel mechanical metamaterials (SOMM) can gain more fruitful tailored stiffness characteristics due to introducing extra design freedoms, e.g. stacking order and sacking way [42]. For instance, the high-static-low-dynamic stiffness property can be harnessed by SOMM, which is extremely important in the field of low-frequency vibration isolation [43][44][45][46]. Our previous work revealed that stacking two Miura sheets to form a SOMM can achieve the realization of tailoring the dynamic stiffness [47]. By non-uniformly stacking Miura sheets, Chen and coworkers have designed a series of novel mechanical metamaterials, whose graded stiffness characteristics were investigated by combining kinematic analysis, numerical simulation and experimental method [48,49]. Interestingly, Li and coworkers combined the stacked Miura origami and rhombic honeycomb structure to develop new mechanical metamaterials with a two-stage programmable compressive strength [50]. The concept of this combination of different configurations to gain incredible performances has received considerable attention in many fields [51]. Alternatively, introducing additional microstructures to the mechanical metamaterial is another promising way to significantly improve the performance [52,53]. Nevertheless, studies on SOMM based on this idea are limited. In this study, we will show how to use the stacking and introducing additional microstructures strategies to tailor the stiffness of the mechanical metamaterial. The microstructures can be readily introduced by adding extra creases [54]. To this end, we have proposed a novel zigzag-base stacked-origami (ZBSO) metamaterial based on the unique geometric construction and architecture. By adding new creases to the classical Miura sheet to introduce microstructures, the proposed ZBSO metamaterial can manifest beneficial tailored stiffness properties, systematically investigated by numerical and experimental methods.

Geometric design
We then repeat the DMO unit cell in the x-direction and y-direction, resulting in nx and ny unit cells in the x-direction and y-direction, respectively.

Tensile tests
Brass (H62) is utilized for constituting ZBSO metamaterial mainly considering its good ductility.
To obtain the mechanical property of the brass, we use a machine CMT5105 (Type: SUST) with an electronic extensometer (Type: YYU-12.5/25) to conduct the tensile tests, finding that the brass with a thickness of 0.2 mm can be characterized by the parameters summarized in Table 1. These parameters will be utilized in the succeeding finite element (FE) simulations.

FE modelling
To capture the tailored stiffness property of ZBSO metamaterial, numerical simulations are performed utilizing nonlinear finite element code Abaqus/Explicit. Four-node shell elements (S4) are employed to mesh the ZBSO metamaterial. Global element size is chosen as 1.  Table 2.  It should be noted that, to ensure the accuracy of the simulations, the following two situations should be carefully considered [26]: (1) The ratio of artificial energy to internal energy is below 5% to make sure that the hour-glassing effect would not significantly affect the results； (2) The ratio of kinetic energy to internal energy is below 5% during most crushing processes to ensure that dynamic effects can be considered insignificant.

Fabrication of ZBSO
There are many manufacturing methods for making origami structure prototypes, e.g.
To ensure the accuracy of the sample, two pairs of male and female moulds for fabricating DMOS are processed by a vertical machining center (Type: HS-1066H), and the material used is 45 steel; the surface of the mold is further plated with a layer of chromium metal to increase the hardness and wear resistance of the mold. A hydraulic stamping press (type: LY-WDQ20A4) is employed to conduct the compression moulding of DMOS. A commercial glue ergo 1690 is used to form the ZESO metamaterial samples.

Experimental setup
The identical tensile machine used in subsection 2.1 is employed to experimentally investigate the quasi-static compression process and to further reveal the tailored stiffness property of the ZBSO metamaterial, as shown in FIG 3C. The ZBSO metamaterial sample is placed between the loading plant and the fixed plant. The lower end of the ZBSO metamaterial sample is fixed, and the upper end moves slowly down with the loading plant until the whole sample collapses. The data for the quasi-static compression process, i.e. the displacement and the load, is gathered by a data collector and then processed by a personal computer, finally obtaining the relationship between the displacement and the force.

Mesh sensitivity analysis
Mesh sensitivity analysis is performed to determine the mesh size to weigh the simulation  3A).
Thus, the mesh size will be employed in the FE models through all the simulations in this work.

FE modelling verification
To validate the effectiveness of the FE modelling, two sets of experiments are conducted using two different ZBSO prototypes, respectively. One ZBSO prototype is fabricated according to section 2.2 (corresponding to Test 1), and another is based on the parameters summarized in Table   3 (corresponding to Test 2). To make a full comparison, a comparison of the experiment and simulation results are conducted in terms of force-displacement curves and deformed shapes.  3C), there are two apparent stages for both experiment and simulation, namely, the compression stage and the densification stage, in which the former tends to be our concern. In the compression stage, again, for both experiment and simulation, four prominent stiffness areas correspond to the four layers of the ZBSO metamaterial can be found. A closer observation of the force-displacement curves it can also be found that the trends of the two curves are basically the same. Overall, the experimental results are a bit larger than those from the numerical simulations, except that when the displacement is smaller than approximately 10 mm. This may be attributed to the use of the glue introducing extra damping, resulting in greater resistance. Comparing FIGs 3B-C, it can be found that the latter exhibits obvious gradient characteristics as presented in [48].
This is because the four layers of the former are uniformly stacked, while the latter is gradient stacking. Moreover, the ZBSO metamaterial of Test 2 outperforms that of Test 1 in the energy absorption capacity, proving from the side that the stacking can tailored the mechanical properties of the ZBSO metamaterial.

The tailored multistage stiffness characteristic of the ZBSO metamaterial
In this section, we will show how the stiffness of the ZBSO metamaterial can be tailored by varying the microstructures (three strategies to add new creases) and altering the stacked way by utilizing the validated FE model.

Varying the microstructures
We introduce three strategies to add new creases to the ZBSO metamaterial and investigate how they tailor the stiffness. The related geometric parameters are identical to that used in the FE modelling of Test 1, as shown in Table 2. the ZBSO metamaterial is constructed by traditional Miura unit cells, which is geometrically similar to [48]. Look close to the force-displacement curve, it first slowly increases from zero, then enters the platform area, and finally densification occurs, which is also consistent with that observed in [48]. It can be found that, in this case, multistage stiffness is not realized since identical DMOS tubes for four stacks are used and no additional microstructures are introduced.
Now we keep the DMOS tubes identical but introduce new creases, e.g. n=1:2, n=1:3, n=1:4, and n=1:5. It can be found that multistage stiffness characteristics can be clearly observed. Moreover, the number of the creases introduced significantly influences the multistage stiffness characteristic.
Roughly speaking, as decrease the value of n (with more microstructures), the multistage stiffness characteristic becomes increasingly apparent. Specifically, for the ZBSO metamaterial with n=1:2, the first trough appears when the displacement is about 27mm, which lags other cases, e.g. it is roundly 15mm for the ZBSO metamaterial with n=1:3. It can also be found that the more creases are introduced, the earlier the first trough will emerge, and the more obvious the multi-level stiffness characteristic will be. To further demonstrate the phenomena mentioned above, several snapshots are extracted from the deformation shapes of the ZBSO metamaterials of the representative moments, i.e. the displacement equals 15mm and 27mm. When disp=15mm, the ZBSO metamaterials with n=1:1 and n=1:2 are in the platform area, and the deformation is mainly contributed by the rotation of the creases (served as plastic hinges); when n=1:3, the first layer of the ZBSO metamaterial happens to be self-locking with the emergence of the first trough of the force-displacement curve [48]; for the cases n=1:4 and n=1:5, the first layer of the ZBSO metamaterial has been self-locked and the facets begin to deform, leading to the deformation of the ZBSO metamaterial is dominated by both the rotation of the creases and the bending of the facets. Let's turn our eyes back to the case when disp=27mm, the deformation of the ZBSO metamaterial with n=1:1 is also caused by the rotation of the creases; while the ZBSO metamaterial with n=1:2 just enters the self-locking state for its first layer; for the ZBSO metamaterials with more microstructures, for instance, the ZBSO metamaterials with n=1:3 and n=1:4 undergo deformations contributed by layer 1 and layer 4, and layer 4 has not appeared self-locking; however, when n=1:5, self-locking has occurred in two layers of the ZBSO metamaterial, i.e. layer 1 and layer 2, which deformation mode is different from the previous two.
Therefore, it can be found that simply adding microstructures can significantly tailor the multistage stiffness of the ZBSO metamaterial.  Comprehensively analyzing all the three strategies, it is interesting to find that when fewer creases are introduced, each layer of the ZBSO metamaterial will be compressed before the self-locking of the first layer occurs, i.e. simultaneous deformation of the four layers is more obvious. In comparison, sequence deformation of each layer of the ZBSO metamaterial is clearly observed when more creases are added. Besides, from the perspective of energy absorption, the ZBSO metamaterial with more microstructures introduced, the better the energy absorption performance will be. The peak force obtained by the first two strategies is not much different and larger than that of the third strategy. Analyzing the peak force and the area contained in the force and displacement curve before the densification area, the ZBSO metamaterials obtained by these three strategies all have good application potential in the field of energy absorption.   Table 4.

Altering the stacked order
Adjusting the stacked order of these DMOS tubes leads to various ZBSO metamaterials; strictly ZBSO metamaterial (66°,68°)-(62°,64°)-(58°,60°)-(70°,72°) appears local instability in the middle layers earlier, leading to a considerable peak force occurs at roundly 70mm, which is often desired to be avoided in the field of energy absorption. However, its total energy absorption capacity is prominently superior to the other three. Furthermore, it is interesting to find that in the range of about 33-50mm of the force-displacement curve, the stiffness of this ZBSO metamaterial is almost opposite to that of other ZBSO metamaterials. Therefore, through the above discussion, we have verified that by changing the stacking order of the DMOS tubes, one can tailor the stiffness characteristics of the ZBSO metamaterial to a large extent.  changes in stiffness can be achieved in specific deformation regions. We also show that the proposed ZBSO metamaterial exhibits excellent energy absorption ability through force-displacement curves and deformation modes. It is worth emphasizing that although brass is used in this study, the proposed ZBSO metamaterial is inherently material independent, scale-invariant, and lightweight. In the future, the application of the proposed ZBSO metamaterial in other fields will be further explored, e.g. low-frequency vibration isolation load-bearing structures [43].