# The Mechanical Behavior of Polyline Crease Origami Self-Locking Structure under Quasi-Static Compressive Loading

Authors: Peiwen Zhang; YuLong He; Huawei Yang; Xuxian Yan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The unique mechanical behavior of polyline crease (PLC) origami structures inspired by the Tachi-Miura polyhedron (TMP) is investigated. We explore the potential usage as mechanical metamaterials that exhibit negative Poisson’s ratio, self-locking mechanism, and load capacity simultaneously. Poisson’s ratio, locked folding ratio, dimensionless force and folding ratio relationship, and elastic-plastic response during its compressive process (rigid folding motion or plastic bending/buckling) are investigated analytically and experimentally, respectively. The effect of original folding angle and the length-to-height ratio on negative Poisson’s ratio, locked folding ratio, dimensionless force, deformation modes, and the mean compression force are also considered. Based on the experiments, an approximate theoretical study has also been conducted, which shows a reasonable correlation with the experimental results. The results are significant to guide the engineering applications of PLC self-locking origami structures.


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## Body

## 1. Introduction

Origami-inspired metamaterial has become more and more important in the manmade material field due to their macroscale mechanical properties, and this is primarily determined by designing microscale structures purposefully [1–7]. In recent years, straight crease origami structures have been investigated extensively due to the feature of lightweight, high compact ratio, and ease to fold and deploy. Different from straight crease (SC) origami, curve crease pattern (CC) origami can be designed to be curved shape to occupy specific space [8–10], which makes them desirable for engineering applications. Recently, origami has been identified as a platform to implement programmable mechanical metamaterials. Origami has received significant interest in diverse applications because of its excellent properties that have been verified, including tunable Poisson’s ratio and stiffness [2, 11–15], programmable collapse [5], self-locking [16–18], and multi-stability [19]. A metamaterial is generated by stacking of individual units, in which preserving the folding kinematics between the sheets was described by Schenk [11]. Because of the special geometries, the origami metamaterials usually exhibit a negative Poisson ratio under in-plane deformations [9]. There are many materials or structures that also have negative Poisson's ratio properties, such as kirigami [20, 21], origami [22], and other materials [23–25]. Some material properties can be enhanced as a result of having a negative Poisson ratio, for example, the indentation resistance or hardness [23]. Poisson’s ratio switching is another important property that has been found recently for the Morph pattern, in which its Poisson’s ratio can change from negative infinity to positive infinity theoretically [26]. The quasi-static and dynamic mechanical behaviors at large plastic strains of the metal-based cellular materials have been widely studied to bring insight into their energy absorption capacity [27–35]. Moreover, previous investigations demonstrated that the energy absorption capacity depends on the material topology [36]. The relationship between the geometrical configurations of the origami metamaterials and overall mechanical properties has been also examined [37–41].Polyline crease (PLC) origami differs from SC origami and CC origami, in that the folded surface of the pattern keeps rigid motion before it reaches locked strain. Then, the folded surface is bent during the folding process after it reached the locked folding angle (or locked strain). Limited studies on the mechanical performance of such geometries have been conducted, in part because of the difficulty in parameterizing and modeling the pattern geometry. This study presents a new nondevelopable degree 4 unit cell. It is developed with a two-stage method that converts straight crease Miura-derivative base geometries into polyline crease geometries. The first stage involves fitting a polyline to a prismatic origami surface, and the second stage involves subdividing this curve into prismatic rigid origami slices with an enforced common configuration angle at slice interfaces. A generic degree 4 vertex origami possesses one continuous degree of freedom for rigid folding, and this folding process can be stopped when two of its facets bind together. Such facet binding will induce self-locking so that the overall structure stays at a prespecified configuration without additional locking elements or actuators. Self-locking offers many promising properties, such as programmable deformation ranges and piecewise stiffness jumps, which could significantly advance many adaptive structural systems. However, despite its excellent potential, the origami self-locking features have not been well studied, understood, and used.In this study, one new PLC origami unit cell with 4 interlayers (PLC-IL4) was designed firstly and then Poisson’s ratio was investigated. By exploiting various geometrical configurations, the self-locking mechanism was investigated, which can effectively induce piecewise stiffness jumps. The polyline crease origami offers the origami structures with extraordinary attributes that the straight crease origami does not have, such as the programmable locked folding ratios. The specimens with different original folding angles of the designed PLC-IL4 origami unit cell are then manufactured using cardboard. The effects of original folding angle of PLC-IL4 origami structure on the quasi-static behavior were investigated experimentally. Finally, the experimental results of PLC-IL4 origami structure were compared with that of theoretical prediction. The results of this research provide new scientific knowledge for the design and utilization of PLC origami structures for many potential engineering applications.


## 2. Negative Poisson’s Ratio

To calculateW, B, and H under various folding configurations, a half model of the PLC-IL4 origami structure was considered, as shown in Figure 1(e), which corresponds to Figures 1(a)–1(d). The folding angles γ and β are functions of α (original folding angle) and θ (varies by the degree of folding). Based on the geometry described in Figure 1, W, B, and H are obtained as follows:(1)W=l+l⋅cosθ+D/4tanα1+D/4tanα2,B=lsinθ+D4sinβ1+D4sinβ2,H2=D4cosβ1+D4cosβ2,tanθ2=tanα1⋅sinβ1=tanα2⋅sinβ2,sinθ2=sinα1⋅cosγ12=sinα2⋅cosγ22.Figure 1
Illustrations of PLC-IL4 origami structure. (a) The PLC origami pattern in its planar state. (b) A Miura-ori in its folded state. (c) The fabrication process of a PLC unit cell. (d) A unit cell of a PLC origami fabricated from two thin cardboards bonded together. (e) The sketch of a half model of PLC origami unit cell. The dimension of the unit cell isw, b, and h in x-, y-, and z-directions, respectively.
(a)(b)(c)(d)(e)Poisson’s ratios of the PLC-IL4 are investigated by defining them as follows:(2)υZX=−dW/WdH/H,υZY=−dB/BdH/H.Differentiating equation (1) with respect to the folding angles and plugging them into equation (2), Poisson’s ratios were obtained as follows:(3)υZX=−2lsinθtanα1cosβ1cos2θ/2cosβ1+cosβ2l+lcosθ+D/4tanα1+D/4tanα2sinβ1+sinβ2tanα1cosβ1/tanα2cosβ2,υZY=8lcosθtanα1cosβ1cos2θ/2+Dcosβ1+Dtanα1/tanα2cosβ1cosβ1+cosβ24lsinθ+Dsinβ1+Dsinβ2sinβ1+sinβ2tanα1cosβ1/tanα2cosβ2.


### 2.1. The Effect of Original Folding Angle on Poisson's Ratio

Poisson’s ratio of Miura-ori has been verified to be negative as testified by some theoretical studies [11, 12, 19, 26]. A similar conclusion was obtained in this study, and the PLC-IL4 origami unit cells with different original folding angles exhibit negative Poisson’s ratio in the folding process. The effect of original folding angle on Poisson’s ratio was investigated using 9 different angle values such as α1 = 50°, 55°, 60°, 65°, 70°, 75°, 80°, and 85° and α2 = α1-5°, while the length and width were fixed as l = 10 mm and D = 40 mm. In Figure 2, the range of negative Poisson’s ratio is shown, and it can be seen that the value of υZX at the start of folding stage increases, but the value of υZY decreases with the decreasing original folding angle. The region of negative Poisson’s ratio is also decreasing with the decreasing original folding angle. The main reason for this phenomenon is that the reentrant extent of the PLC-IL4 origami structure with the small original folding angle α1 is more obvious than that with big angle values. So, the lateral displacement of origami cell with the small original folding angle is larger. It is noteworthy that the color range of Poisson’s ratio (υZX and υZY) occupied will decrease as the original folding angle α1 increases, which corresponded to the decreasing folding ratio. The main reason is that this PLC origami unit cell should satisfy the geometrical relations as shown in equation (1).(4)tanθ2=tanα1⋅sinβ1.=tanα2⋅sinβ2Figure 2
Contour plot of out-plane Poisson’s ratio (υZX, υZY) of PLC-IL4 origami unit cell with different original folding angles.Thus,(5)sinβ2=tanα1tanα2⋅sinβ1.It should be noted that whileβ2 ∈ [0; 90°], the range of β1 is limited to satisfy 0≤β1≤arcsintanα2/tanα1. This is to avoid the collision between two adjacent planes during the folding process. For example, while α1=85° and α2=α1−5°=80°, the range of sinβ1 is 0≤sinβ1≤0.4962, which corresponded to 0≤β1≤29.75°. Then, the range of folding ratio (β1/90°) is 0≤β1/90°≤0.331.A schematic of a special folding rate is shown in Figure3. The folding rate is defined as the ratio of the angle β1 to 90°. For a specimen, while its β1 = 45° in the folding process, the corresponding folding rate is 0.5.Figure 3
A schematic example of a specific folding ratio.The analytical contour plot of locked folding ratio as a function of continuousα1−α2 and the original folding angle α1 is shown in Figure 4. The analytical contour plot provides a quantitative tool for designing on request self-locking behavior and original folding angle, which can be used to optimize PLC-IL4 origami programmable structures. It is seen that the locked folding ratio decreases with the increasing original folding angle α1 while α1−α2 is fixed. At a certain locked folding ratio, the PLC-IL4 origami structure can be designed to different geometrical configurations. For example, while the engineering requirement of PLC-IL4 origami structure locked folding ratio is 0.4, we only need to draw a parallel line of x-axis that through point 0.4. Then, different geometrical configurations could be obtained (such as α1 = 74°, α2 = 64°; α1 = 81.1° and α2 = 75.1°). Similarly, if the original folding angle is fixed as α1 = 70°, we also can obtain different locked folding ratios by changing the value of α1−α2. It also can be found that the locked folding ratio will be equal to zero (which means the self-locking behavior eventually disappears), while the original folding angle α1 increases to 90°. This phenomenon indicates that the PLC-IL4 origami structure will degenerate to another new type of PLC structure in which the self-locked behavior does not exist, which can be exploited for the purpose of intelligent control and metamaterial design.Figure 4
Locked folding ratio change in PLC-IL4 origami structure with differentα1−α2 values.


### 2.2. The Effect of Length-to-Width Ratio on Poisson’s Ratio

The length-to-width ratio also can exert an influence on Poisson’s ratio of PLC origami unit cell, which has been reported in the investigation of Miura-ori [12]. By exploiting different length-to-width (l/D) ratios, the effect of it on Poisson’s ratio was investigated, as shown in Figure 5. Five unit cell widths (10 mm, 20 mm, 30 mm, 40 mm, and 50 mm) were used, while the length and the original folding angle were fixed as l = 10 mm, and α1 = 60° and α2 = 55°. Then, five different length-to-width (l/D) ratios were obtained through calculation. The folding ratio of this PLC-IL4 origami structure only reaches 0.617 due to the self-locking behavior. It is observed that the absolute values of Poisson’s ratio υZX increase with the increasing l/D ratio, but the area that the negative Poisson’s ratio occupied is decreasing. It means that the υZX of PLC-IL4 origami unit cell with different length-to-width ratios will decrease to remain close to zero when the folding ratio increases to 0.4. The υZY of PLC-IL4 origami unit cell with different length-to-width ratios is tending to zero early and does not exhibit negative values in the folding process. This phenomenon suggests that the length-to-width ratio has less impact on υZY than that on υZX.Figure 5
Contour plot of out-plane Poisson’s ratio (υZX, υZY) of PLC origami unit cell with different l/D ratios.


## 2.1. The Effect of Original Folding Angle on Poisson's Ratio

Poisson’s ratio of Miura-ori has been verified to be negative as testified by some theoretical studies [11, 12, 19, 26]. A similar conclusion was obtained in this study, and the PLC-IL4 origami unit cells with different original folding angles exhibit negative Poisson’s ratio in the folding process. The effect of original folding angle on Poisson’s ratio was investigated using 9 different angle values such as α1 = 50°, 55°, 60°, 65°, 70°, 75°, 80°, and 85° and α2 = α1-5°, while the length and width were fixed as l = 10 mm and D = 40 mm. In Figure 2, the range of negative Poisson’s ratio is shown, and it can be seen that the value of υZX at the start of folding stage increases, but the value of υZY decreases with the decreasing original folding angle. The region of negative Poisson’s ratio is also decreasing with the decreasing original folding angle. The main reason for this phenomenon is that the reentrant extent of the PLC-IL4 origami structure with the small original folding angle α1 is more obvious than that with big angle values. So, the lateral displacement of origami cell with the small original folding angle is larger. It is noteworthy that the color range of Poisson’s ratio (υZX and υZY) occupied will decrease as the original folding angle α1 increases, which corresponded to the decreasing folding ratio. The main reason is that this PLC origami unit cell should satisfy the geometrical relations as shown in equation (1).(4)tanθ2=tanα1⋅sinβ1.=tanα2⋅sinβ2Figure 2
Contour plot of out-plane Poisson’s ratio (υZX, υZY) of PLC-IL4 origami unit cell with different original folding angles.Thus,(5)sinβ2=tanα1tanα2⋅sinβ1.It should be noted that whileβ2 ∈ [0; 90°], the range of β1 is limited to satisfy 0≤β1≤arcsintanα2/tanα1. This is to avoid the collision between two adjacent planes during the folding process. For example, while α1=85° and α2=α1−5°=80°, the range of sinβ1 is 0≤sinβ1≤0.4962, which corresponded to 0≤β1≤29.75°. Then, the range of folding ratio (β1/90°) is 0≤β1/90°≤0.331.A schematic of a special folding rate is shown in Figure3. The folding rate is defined as the ratio of the angle β1 to 90°. For a specimen, while its β1 = 45° in the folding process, the corresponding folding rate is 0.5.Figure 3
A schematic example of a specific folding ratio.The analytical contour plot of locked folding ratio as a function of continuousα1−α2 and the original folding angle α1 is shown in Figure 4. The analytical contour plot provides a quantitative tool for designing on request self-locking behavior and original folding angle, which can be used to optimize PLC-IL4 origami programmable structures. It is seen that the locked folding ratio decreases with the increasing original folding angle α1 while α1−α2 is fixed. At a certain locked folding ratio, the PLC-IL4 origami structure can be designed to different geometrical configurations. For example, while the engineering requirement of PLC-IL4 origami structure locked folding ratio is 0.4, we only need to draw a parallel line of x-axis that through point 0.4. Then, different geometrical configurations could be obtained (such as α1 = 74°, α2 = 64°; α1 = 81.1° and α2 = 75.1°). Similarly, if the original folding angle is fixed as α1 = 70°, we also can obtain different locked folding ratios by changing the value of α1−α2. It also can be found that the locked folding ratio will be equal to zero (which means the self-locking behavior eventually disappears), while the original folding angle α1 increases to 90°. This phenomenon indicates that the PLC-IL4 origami structure will degenerate to another new type of PLC structure in which the self-locked behavior does not exist, which can be exploited for the purpose of intelligent control and metamaterial design.Figure 4
Locked folding ratio change in PLC-IL4 origami structure with differentα1−α2 values.


## 2.2. The Effect of Length-to-Width Ratio on Poisson’s Ratio

The length-to-width ratio also can exert an influence on Poisson’s ratio of PLC origami unit cell, which has been reported in the investigation of Miura-ori [12]. By exploiting different length-to-width (l/D) ratios, the effect of it on Poisson’s ratio was investigated, as shown in Figure 5. Five unit cell widths (10 mm, 20 mm, 30 mm, 40 mm, and 50 mm) were used, while the length and the original folding angle were fixed as l = 10 mm, and α1 = 60° and α2 = 55°. Then, five different length-to-width (l/D) ratios were obtained through calculation. The folding ratio of this PLC-IL4 origami structure only reaches 0.617 due to the self-locking behavior. It is observed that the absolute values of Poisson’s ratio υZX increase with the increasing l/D ratio, but the area that the negative Poisson’s ratio occupied is decreasing. It means that the υZX of PLC-IL4 origami unit cell with different length-to-width ratios will decrease to remain close to zero when the folding ratio increases to 0.4. The υZY of PLC-IL4 origami unit cell with different length-to-width ratios is tending to zero early and does not exhibit negative values in the folding process. This phenomenon suggests that the length-to-width ratio has less impact on υZY than that on υZX.Figure 5
Contour plot of out-plane Poisson’s ratio (υZX, υZY) of PLC origami unit cell with different l/D ratios.


## 3. The Dimensionless Force and Folding Ratio Relationship

The force and folding ratio relationship was investigated to validate the effect of original folding angle on the PLC-IL4 origami unit cell, as shown in Figure6. The PLC-IL4 origami unit cell calculation model was established by rigid plates connected by torsional spring along the crease lines. The folding behavior of the PLC-IL4 origami unit cell was considered under a uniaxial force (F) in the z-direction. By applying virtual displacement (δu) to the PLC-IL4 unit cell and using the principle of virtual work, the following equation was obtained.Figure 6
Sketch of PLC origami unit cell and calculation model with a torsional spring.(6)Fδu=2nh1Mh1δβ1+nh2Mh2δβ2+nb1Mb1δγ1+nb2Mb2δγ2.The bending momentsMh and Mb are along horizontal and inclined crease lines, respectively, and nh1,nh2,nb1,nb2 are the number of the horizontal and inclined crease lines corresponded to β1,β2,γ1, and γ2; the relationship between bending moment and angle is assumed as linear:(7)Mh=kθβ−β0;Mb=kθγ−γ0,where kθ is a spring constant, and β and γ are the original folding angles for horizontal and inclined crease lines, respectively.By applying variation to equation (1) with respect to the folding angles, the following relationships can be obtained:(8)δθ=2tanα1cos2θ2cosβ1δβ1=2tanα2cos2θ2cosβ2δβ2,(9)12cosθ2δθ=−12sinα1sinγ12δγ1=−12sinα2sinγ22δγ2.Therefore,(10a)δβ2=tanα1cosβ1tanα2cosβ2δβ1,(10b)δγ1=−2cos3θ/2cosβ1cosα1sinγ1/2δβ1,(10c)δγ2=−2cos3θ/2tanα1cosβ1sinα2sinγ2/2δβ1.The height of the PLC-IL4 is as follows:(11)H0−u=D2cosβ1+D2cosβ2.Then,(12)δu=12Dsinβ1δβ1+12Dsinβ2δβ2.Substituting equations (7)–(12) into equation (6), the relationship between force and displacement of PLC-IL4 can be obtained as follows:(13)Fkθ/D=2sinβ1+tanα1cosβ1/tanα2cosβ2sinβ22nh1β1−β10+nh2β2−β20tanα1cosβ1tanα2cosβ2−nb1γ1−γ102cos3θ/2cosβ1cosα1sinγ1/2−nb2γ2−γ202cos3θ/2tanα1cosβ1sinα2sinγ2/2.The numbers of the torsional spring along horizontal and inclined crease lines arenh1=4, nh2=8, nb1=8, and nb2=8, respectively. Then, the relationship between force and displacement of PLC-IL4 can be obtained finally as follows:(14)Fkθ/D=16sinβ1+tanα1cosβ1/tanα2cosβ2sinβ2β1−β10+β2−β20tanα1cosβ1tanα2cosβ2−γ1−γ102cos3θ/2cosβ1cosα1sinγ1/2−γ2−γ202cos3θ/2tanα1cosβ1sinα2sinγ2/2.The force and folding ratio relationships of PLC-IL4 origami unit cell with four different original folding angles (α1 = 55°, 65°, 75°, and 85° and α2 = α1−5°) based on equation (12) are presented in Figure 7. It is observed that the relationship between the normalized force and the folding ratio is not unique. The normalized force of PLC-IL4 unit cell with α1 = 55° and 65° increases gradually, while that of PLC-IL4 unit cell with α1 = 75° and 85° tends to develop a higher peak load before the load drops during the folding process. This indicated that the PLC-IL4 unit cell with α1 = 75° and 85° exhibits local structural softening, which is called “type II” structures, while the PLC-IL4 unit cell with α1 = 55° and 65° does not exhibit local structural softening, which is called “type I″ structures [42]. There is a local minimum point of normalized force corresponding to the folding ratio for the PLC-IL4 unit cell with α1 = 75° and 85°, while the normalized force increases in both directions. This indicates that the PLC-IL4 origami unit cell under α1 = 75° and α1 = 85° also exhibits two stable configurations, which are consistent with the results in [19].Figure 7
Effect of original folding angle on normalized force and folding ratio relationship.


## 4. Experimental Investigation

### 4.1. The Mechanical Properties of Cardboard

All the specimens were cut from the cardboard KZ-230G with 0.3 mm thickness. The density of the cardboard is 0.8016 g/cm3. Quasi-static tensile tests at 20 mm/min were conducted to determine the properties of the cardboard using an Instron testing system (Instron 5544, Instron, Norwood, MA, USA). The dimensions of samples are shown in Figure 8(a). Force and displacement were recorded by the Instron testing system. Figure 8(b) shows the force-displacement curves of the specimens. It can be found that the test results are very close to each other. Therefore, the average mechanical properties of all specimens are as follows: E (Young's modulus) = 4.02 GPa; σy (yield stress) = 12.81 MPa; σu (ultimate stress) = 34.19 MPa; and n (hardening exponent) = 0.5017.Figure 8
Quasi-static testing of cardboard KZ-230G: (a) sketch of sample dimensions; (b) force-displacement curves obtained from the tests of samples.
(a)(b)


### 4.2. Specimen Preparation

PLC-IL4 origami unit cell specimens, 60 mm long, with a width also of 60 mm, were fabricated from two thin cardboards bonded together. A photograph of samples with an original folding angle (65°) is shown in Figure9. A total of 40 samples were fabricated. To examine the effect of original folding angle on the force-displacement relationship, five nominally identical specimens were used for each test condition. Samples were identified by a unique label; e.g., PLC-IL4-50-1 indicates a sample with polyline crease pattern, divides height into 4 equal parts, an original folding angle α1 = 50°, and is the first specimen used to investigate the effect of folding angle value.Figure 9
PLC-IL4 samples: (a) sketch of sample dimensions; (b) a sample of PLC-IL4-70-4 with an original folding anglesα1 = 70° and α2 = 65°.
(a)(b)


### 4.3. Test Result Discussion

To gain insight into the effect of original folding angle, the PLC-IL4 samples of eight different angle values were tested at quasi-static compression of 20 mm/min, as summarized in Figure10(a). It is seen that the force-displacement curve exhibits initial peak, while the original folding angle increases over 65°. The load capacity of the samples decreases with the increasing original folding angle, while their displacement exceeds the rigid folding part. The possible reason is that the PLC-IL4 unit cell with small α1 exhibits local structural softening, while that with larger does not exist. Figure 10(b) presents the deformation process of the sample with an original folding angle (α1 = 70°, α2 = 65°). The compressive force of the sample is less than 1N until the displacement increases to 32 mm (corresponded to locked folding ratio of 0.615), which is mainly caused by the friction in the rigid folding stage, but the force increases sharply after the displacement exceeds the critical value of rigid folding motion due to the plastic deformation.Figure 10
Typical force-displacement history curves: (a) quasi-static compressive response of samples with five different original folding angles; (b) deformation process of the sample with an original folding angle (α1 = 60°,α2 = 55°).
(a)(b)The experimental locked folding ratio of specimens with different original folding angles and the theoretical prediction values is plotted in Figure11. As mentioned above, the locked folding ratio decreases with the increasing original folding angle α1, while α1-α2 is fixed. As shown in Figure 11, it is clear that all the experimental data points are concentrated basically around the theoretical prediction line, herein indicating that the experimental measurements can fit theoretical predictions well.Figure 11
Comparison of the theoretical prediction locked folding ratio and experimental results.Figure12 shows the typical deformation modes of PLC-IL4 origami structure with different original folding angles. It can be seen that the original folding angle value has less influence on the deformation modes. The forms of plastic hinge only occur in the adjacent inclined creases, while the sample compressive process exceeds the rigid folding motion (which corresponded to folding ratio that exceeds the locked folding ratio). The main reason is that the value of α1-α2 is relatively small, which will result in a little plastic area. Thus, the deformation modes of all the samples are small elastic-plastic deformations that are very similar to each other.Figure 12
Typical deformation/failure modes of the PLC-IL4 origami structure with different original folding angles.


## 4.1. The Mechanical Properties of Cardboard

All the specimens were cut from the cardboard KZ-230G with 0.3 mm thickness. The density of the cardboard is 0.8016 g/cm3. Quasi-static tensile tests at 20 mm/min were conducted to determine the properties of the cardboard using an Instron testing system (Instron 5544, Instron, Norwood, MA, USA). The dimensions of samples are shown in Figure 8(a). Force and displacement were recorded by the Instron testing system. Figure 8(b) shows the force-displacement curves of the specimens. It can be found that the test results are very close to each other. Therefore, the average mechanical properties of all specimens are as follows: E (Young's modulus) = 4.02 GPa; σy (yield stress) = 12.81 MPa; σu (ultimate stress) = 34.19 MPa; and n (hardening exponent) = 0.5017.Figure 8
Quasi-static testing of cardboard KZ-230G: (a) sketch of sample dimensions; (b) force-displacement curves obtained from the tests of samples.
(a)(b)


## 4.2. Specimen Preparation

PLC-IL4 origami unit cell specimens, 60 mm long, with a width also of 60 mm, were fabricated from two thin cardboards bonded together. A photograph of samples with an original folding angle (65°) is shown in Figure9. A total of 40 samples were fabricated. To examine the effect of original folding angle on the force-displacement relationship, five nominally identical specimens were used for each test condition. Samples were identified by a unique label; e.g., PLC-IL4-50-1 indicates a sample with polyline crease pattern, divides height into 4 equal parts, an original folding angle α1 = 50°, and is the first specimen used to investigate the effect of folding angle value.Figure 9
PLC-IL4 samples: (a) sketch of sample dimensions; (b) a sample of PLC-IL4-70-4 with an original folding anglesα1 = 70° and α2 = 65°.
(a)(b)


## 4.3. Test Result Discussion

To gain insight into the effect of original folding angle, the PLC-IL4 samples of eight different angle values were tested at quasi-static compression of 20 mm/min, as summarized in Figure10(a). It is seen that the force-displacement curve exhibits initial peak, while the original folding angle increases over 65°. The load capacity of the samples decreases with the increasing original folding angle, while their displacement exceeds the rigid folding part. The possible reason is that the PLC-IL4 unit cell with small α1 exhibits local structural softening, while that with larger does not exist. Figure 10(b) presents the deformation process of the sample with an original folding angle (α1 = 70°, α2 = 65°). The compressive force of the sample is less than 1N until the displacement increases to 32 mm (corresponded to locked folding ratio of 0.615), which is mainly caused by the friction in the rigid folding stage, but the force increases sharply after the displacement exceeds the critical value of rigid folding motion due to the plastic deformation.Figure 10
Typical force-displacement history curves: (a) quasi-static compressive response of samples with five different original folding angles; (b) deformation process of the sample with an original folding angle (α1 = 60°,α2 = 55°).
(a)(b)The experimental locked folding ratio of specimens with different original folding angles and the theoretical prediction values is plotted in Figure11. As mentioned above, the locked folding ratio decreases with the increasing original folding angle α1, while α1-α2 is fixed. As shown in Figure 11, it is clear that all the experimental data points are concentrated basically around the theoretical prediction line, herein indicating that the experimental measurements can fit theoretical predictions well.Figure 11
Comparison of the theoretical prediction locked folding ratio and experimental results.Figure12 shows the typical deformation modes of PLC-IL4 origami structure with different original folding angles. It can be seen that the original folding angle value has less influence on the deformation modes. The forms of plastic hinge only occur in the adjacent inclined creases, while the sample compressive process exceeds the rigid folding motion (which corresponded to folding ratio that exceeds the locked folding ratio). The main reason is that the value of α1-α2 is relatively small, which will result in a little plastic area. Thus, the deformation modes of all the samples are small elastic-plastic deformations that are very similar to each other.Figure 12
Typical deformation/failure modes of the PLC-IL4 origami structure with different original folding angles.


## 5. Theoretical Analysis

The typical deformation mode for theoretical prediction of PLC-IL4 structure with different original folding angles is proposed based on experiments, as shown in Figure13. Then, a simple mathematical formula was established to predict the energy absorption of PLC-IL4 samples with different original folding angles.Figure 13
Typical deformation/failure modes of the PLC-IL4 origami unit cell.Based on the deformation modes, the superfolding element theory [43], and the existing research [42, 44], the plastic energy dissipated by inclined plastic hinges, Et, can be respectively calculated as follows:(15)Et=4MPD/42r1tanγ10∫β10π/2cosβ1sinα1dβ1+4MPD/42r1tanγ20∫β20π/2cosβ2sinα2dβ2.The energy balance between external work done by the compressive force and energy absorption in PLC-IL4 samples is applied, and the mean compressive force can be obtained by dividing the total energy absorption by the final compression displacement.(16)Pmt⋅0.73h0=Et.The experimental quasi-static mean compressive force can be calculated as follows:(17)Pme=∫δ1δ2Pxδ2−δ1,where δ1 and δ2 are starting point and the destination of the plastic deformation process, and h0 = δ2-δ1.The theoretical predictions,Pmt, calculated from equation (16) and the experimental results, Pme, obtained from equation (17), of samples are listed in Table 1.Table 1
Comparison of theoretical prediction and experimental results of mean compressive force.
Specimen typeδ1 (mm)δ2 (mm)h0 (mm)Et (J)Pmt (N)Pme (N)PLC-IL4-55-133.775016.230.037423.1582.498PLC-IL4-55-231.8953.1621.270.037422.4103.311PLC-IL4-55-335.5652.1516.590.037423.0904.103PLC-IL4-55-434.4151.8317.420.037422.9433.820PLC-IL4-55-531.5051.1719.670.037422.6063.459PLC-IL4-65-135.0853.0317.950.10167.7542.979PLC-IL4-65-234.4053.9519.550.10167.1192.959PLC-IL4-65-336.1053.6519.550.10167.1192.293PLC-IL4-65-433.2953.6820.390.10166.8262.237PLC-IL4-65-532.4453.2620.820.10166.6852.416PLC-IL4-75-131.5854.1422.560.176710.7292.675PLC-IL4-75-231.8654.5522.690.176710.6681.798PLC-IL4-75-333.0354.5121.480.176711.2692.191PLC-IL4-75-433.2054.3921.190.176711.4232.644PLC-IL4-75-528.0455.0026.960.17678.9781.259PLC-IL4-85-114.3554.5940.240.561119.1010.681PLC-IL4-85-216.475538.530.561119.9490.446PLC-IL4-85-322.2154.7932.580.561123.5920.612PLC-IL4-85-414.9354.7239.790.561119.3170.779PLC-IL4-85-513.1954.5541.360.561118.5840.534The prediction mean compression forces are compared with the experimental results, as shown in Figure14. It can be seen that the data points of PLC-IL4-55 specimens are concentrated basically around a straight line with a slope of 1, herein indicating a reasonable correlation between the experimental compression force and the predicted value of mean compression forces. However, for the large angle case, there is a greater difference between the theoretical predictions and the experimental results. The difference between the theoretical predictions and the experimental results becomes larger with the increasing original folding angle. It is evident that the theoretical prediction of the large angle specimens tends to overestimate the mean compression force. The main reason is as follows: the first reason is that the effect of friction is not considered throughout the folding process, and it is impossible to overcome friction in the experiment. The second reason is the large original folding angle specimens undergo a large plastic deformation in the initial folding process, which is not taken into account in the approximate theoretical model of the elastic-plastic deformation stage, as shown in Figures 15 and 16. The third is the deformation pattern gradually changes from bending to buckling with the increasing original folding angle. For this reason, the theoretical model in this study is only suitable for designing origami structures with small angles (<60°), and structures with large angles require modification of the theoretical model due to the transformation of the deformation mode. The corresponding theoretical consideration, as a part of our current and future work, is worthy to be further conducted.Figure 14
Comparison of the theoretical predictions and experimental results.Figure 15
Quasi-static compressive curves of samples with different original folding angles.
(a)(b)(c)(d)Figure 16
Deformation process comparison of PLC-IL4-55 and PLC-IL4-85.


## 6. Conclusion

The quasi-static mechanical behavior of polyline crease origami self-locking structures with different original folding angles and length-to-height ratios under axial compression loading was investigated. The out-plane Poisson’s ratioυZX of PLC-IL4 origami structures exhibits a negative Poisson ratio under axial compression deformations, while the υZY exhibits a positive Poisson ratio. It is obtained that the original folding angle has a great impact on Poisson’s ratio than that of the length-to-height ratio. The PLC-IL4 origami structures can design on request self-locking behavior and original folding angles, which can be exploited for the purpose of intelligent control and metamaterial design.The different load capacity and failure modes of the PLC-IL4 with different original folding angles are discussed through the experimental results. It is found that the load capacity of samples is mainly controlled by the original folding angle values, while the deformation/failure modes are very similar to each other. Based on the experiments, an approximate mathematical formula was established to predict the mean compression force for every type of sample. A reasonable agreement between the theoretical predictions and experimental results is obtained.



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*Source: 2324167-2022-02-18.xml*