1 Introduction

Cellular materials with different topologies are commonly used as energy absorbers due to their light weight, relatively low collapse strength and long plateau stress, which develop large plastic strains leading to high energy absorption efficiency. Sandwich structures with various core materials have found wide applications utilizing topologies with random and periodic microstructures [1]. A comprehensive overview of the recent advances in the development of corrugated structures for energy absorption applications can be found in [2].

In addition to the established core materials, innovative core materials, such as three-dimensional structures created by folding flat sheets [3] that resemble origami folding patterns, have been gaining increasing interest. It was shown that the Miura-ori foldcore possesses acceptable weight-specific mechanical properties compared to monolithic structures and conventional cores such as in-plane loaded honeycombs, corrugated cores, lattices and trusses [4, 5]. A parametrization of Miura-ori pattern was formulated by Gattas et al. [6] for crease orientation, crease alignment, developability, flat foldability and rectilinearity. Schenk and Guest [3] explored the out-of-plane kinematics of single planar Miura-ori sheet, while Gattas et al. [6] formulated five Miura base plate geometries: the arc pattern, the arc Miura pattern, the non-developable Miura pattern, the non-flat foldable Miura pattern and the tapered Miura-ori pattern.

Together with the geometric characteristics, the mechanical properties of origami patterns have been extensively investigated. Their elastic properties were explored, e.g., stiffness [7], shear modulus and buckling resistance [4], Poisson’s ratio [8] and compressive modulus [9]. Various folded cores made of different base materials were analyzed using mechanical testing and simulations by Fisher et al. [10]. The set of the geometric parameters of the Miura-ori materials allow numerous combinations between them, thus forming materials with different mechanical properties. It was established that cores made of a partially folded Miura origami pattern have many favorable characteristics for impact-resistant applications [11, 12].

Miura-ori patterned sheets have been extensively analyzed predominantly to obtain their mechanical characteristics under out-of-plane compression, as the high strength in this loading direction makes the structure suitable for use as core material in sandwich structures. A parametric analysis of Miura-ori patterned sheets using finite element (FE) analysis to demonstrate the effects of cell geometry and wall thickness was carried out by Lv et al. [13]. The compressive behavior of several nylon Miura-ori sheets was investigated experimentally and numerically by Xiang et al. [14], where the non-unique dependence of strength on the sector angle \(\alpha \) was observed. The out-of-plane compressive properties of Miura-ori-based material were examined by Ma et al. [15] for graded stiffness to show that the structural properties can be tuned to achieve superior energy absorption capability to uniform tessellating repeat units.

A parametric study on four basic dimensions of Miura-ori core sandwich panels made by 3D printing polymer under static loading conditions was virtually and experimentally carried out by Xiang et al. [16] using origami with uniform and graded properties. Analytical modeling was recently conducted by Qiang et al. [17] to examine the quasi-static response of Miura-ori-based metamaterials under compression in the in-plane and out-of-plane loading directions.

The shear properties of Miura-ori-based materials have been less comprehensively studied compared to the out-of-plane compression. The mechanical behavior under compressive, shear and impact loads of folded cores made of prepreg sheets of carbon fiber/aramid fiber was examined only by Heimbs et al. [18, 19]. For corrugated cores, the out-of-plane compressive, transverse shear and longitudinal shear responses have been measured at three relative densities and compared with analytical and FE predictions to assess the properties of pyramidal and square honeycomb, prismatic corrugated and diamond cores [20]. Numerical studies have assessed Miura-based folded core models under quasi-static compression, shear and bending as reported by Zhou et al. [21].

Being a structure with a periodic topology, the Miura-ori core materials can be considered as a type of cellular materials, with their mechanical properties defined as functions of the relative density in search of weight-optimized structures. In the classical theory of cellular materials [22], the relative density is considered as one of the major material characteristics, and the relationship between material strength and relative density is sought to construct weight-efficient absorbers. The mechanical properties of many cellular materials with periodic topology depend on the loading direction, and for most cellular topologies, the strength is a monotonic function of the relative density. It was shown by Zhang et al. [23] that the relative density of Miura-ori metamaterials is not a monotonic function of the geometric parameters of the unit cells, i.e., the same relative density can be achieved for different combinations of these parameters. Besides, it was established that the compressive strength in the weakest in-plane direction of loading is not a unique function of the relative density [23]. The relationship between relative density of the origami-based materials and their mechanical characteristics in the different loading directions would provide valuable design information when considering the numerous combinations between the cell parameters.

The aim of the present study is to explore the relationships between the strength and relative density of Miura-ori materials, which can be used as core of sandwich structures, when considering the out-of-plane compressive and shear loading. A parametric analysis based on FE simulations is used to establish analytical relationships between the strength and density. The layout of the paper is arranged as follows. The basic geometric characteristics defining the relative density are summarized and the analyzed models are specified; the FE models are described. The mechanical characteristics of the generated models, namely the elastic moduli and characteristic stresses in the three loading directions, are obtained via FE simulations. Next, using these data, analytical approximations are defined and summarized while comparing the models’ response to the loading in different directions. A brief comparison between the collapse response of Miura-ori materials and hexagonal prismatic honeycomb with equal relative densities subject to shear and out-of-plane compression is presented. Conclusions of this study are provided at the end of the paper.

2 Miura-ori Core Models

2.1 Geometric Characteristics of the Miura-ori Unit Cell

The geometry of a Miura-ori sheet is a herringbone pattern that emerges from a series of convex mountain and concave valley creases. Vertices are formed when four creases intersect, and four adjacent vertices bound equal-area facets arranged with inversion symmetry (Fig. 1). This fold pattern is defined by two stationary crease lengths a, b, sector angle \(\alpha \) and vertex angle \(\gamma \), which quantify the extent of folding. \({X}_{1}\) and \({X}_{2}\) are the in-plane coordinates, while \({X}_{3}\) is the out-of-plane coordinate. The deformation process of the Miura-ori material is strongly influenced by the direction of loading. The compression of the origami in the \({X}_{2}\)-direction producing in-plane folding is characterized by the rotation of the rigid facets forming “mountains” and “valleys.” In this case, deformations are localized along the plastic hinge lines, and origami motion can be described as a 1DOF system [23] resulting in the lowest strength. The compression in the \({X}_{1}\)- and \({X}_{3}\)-directions causes buckling of the cell walls, and the deformation is governed by the buckling modes depending on the loading direction, and in general, the Miura-ori material exhibits higher strength when compressed in these directions with the highest strength associated with the loading in the reentrant orientation of the cells (\({X}_{1}\)-direction in Fig. 1) [17].

Fig. 1
figure 1

Miura-ori unit cell

Full size image

The geometric relationships between the parameters of Miura-ori cell presented in Fig. 1 are

$$ H = a\sin \theta \sin \alpha $$
(1)
$$ S = b\frac{\cos \theta \tan \alpha }{{(1 + \cos^{2} \theta \tan^{2} \alpha )^{1/2} }} $$
(2)
$$ L = a\left( {1 - \sin^{2} \theta \sin^{2} \alpha } \right)^{1/2} $$
(3)
$$ V = \frac{ab\cos \alpha }{L} $$
(4)
$$ \sin \left( {\frac{\gamma }{2}} \right) = \frac{\cos \theta }{{\cos \alpha \left( {1 + \cos^{2} \theta \tan^{2} \alpha } \right)^{1/2} }} $$
(5)

where \(L, S\) and \(H\) are the cell dimensions in the \(X_{1}\)-, \(X_{2}\)- and \(X_{3}\)-directions.

2.2 Relative Density of Miura-ori Materials

An assembly of unit Miura-ori cells can be considered as a cellular material, and similarly to the other types of cellular materials, the mechanical properties can be sought as functions of the relative density and the properties of the base material [23].

The initial density of Miura-ori-based metamaterial is defined as

$$ \rho_{0} = \rho_{s} \rho_{0}^{*} ,\quad \rho_{0}^{*} = \frac{tab\sin \alpha }{{H\left( {\theta_{0} } \right)L\left( {\theta_{0} } \right)S\left( {\theta_{0} } \right)}} $$
(6a, b)

where \({\rho }_{s}\) and \({\rho }_{0}^{*}\) are the density of the base material and relative density of the metamaterial, respectively; \({\theta }_{0}\) defines the initial folding angle; and \(t\) is the thickness of the cell wall. It can be shown that, the relative density is proportional to the ratio \(t/a\) for a given folding angle \({\gamma }_{0}\) and it varies with the variation of the sector angle \(\alpha \).

The variations of the relative density per unit wall thickness of the cells, \({\rho }_{0}^{*}/t\) , with the dihedral angle \({\gamma }_{0}\) for several fixed values of angle \(\alpha \) are shown in Fig. 2. It is observed that the density is not a unique function of the cell geometry, and it varies in a non-monotonic way with angle \({\gamma }_{0}\); for any given angle \(\alpha \), a minimum density occurs corresponding to a particular value of dihedral angle \({\gamma }_{0}\). Consequently, the same relative density can be achieved for different combinations of the cell characteristics, including geometry and wall thickness.

Fig. 2
figure 2

Relative density per unit cell wall thickness depending on the initial folding angle \({\gamma }_{0}\), \(a=b\): a \(a=\) 20 mm and different sectional angles \(\alpha \); b equal cell height—M1: \(\alpha = 31.3^\circ\), \(a \)= 20 mm and M4: \(\alpha = 60^\circ\), \(a \)= 12 mm

Full size image

The present study focuses on the single-layer sandwich core when four Miura-ori types of foldcores are considered. In the subsequent analysis, the initial dihedral angle \(\theta_{0}\) (Fig. 1) is fixed and the side lengths \(a\) and \(b\) are assumed equal. Three values of the sector angle \(\alpha\) and two values of the side lengths are used to obtain densities depending on the cell geometry. According to Eq. (1), the height of the core varies with \(\alpha\) and \(\theta_{0}\), and the folding angle is \(\gamma_0=f(\alpha,\theta_0)\) (Eq. 5). The characteristic parameters of the analyzed models M1–M4 are given in Table 1, which presents materials with different initial relative densities depending on the cell geometry and cell wall thickness. Figure 2a shows the density variation of models with cell sizes \(a = b =\) 20 mm to illustrate the influence of the sector angle \(\alpha\) (models M1, M2 and M3). Figure 2b presents the effect of the cell size on the density variation (M1 and M4). Note that models M1 and M4 are characterized by equal cell size \(H\) and their relative densities are equal for equal cell wall thicknesses. Therefore, the mechanical characteristics of these two models can be directly compared.

Table 1 Relative density of the Miura-ori cores depending on the cell geometry and cell wall thickness
Full size table

2.3 Geometric Characteristics of the Miura-ori Foldcores and the FE Models

The geometries of the analyzed Miura-ori foldcores are presented in Fig. 3a–d. The corresponding FE models were developed by the commercial software ABAQUS/Explicit as schematically shown in Fig. 4. The core is sandwiched between two “rigid” plates. The “rigid” plates are modeled as thick elastic plates (3 mm thick) with elastic modulus ten times the elastic modulus of the core material. The plates are tied along the top and bottom fold lines of the core during the simulation. Shell elements S4 are used for the core with an average size of 1 mm following a convergency analysis. Shell elements S4R are used for plates with an average size of 3 mm. General contact algorithm available in ABAQUS is used to model the interaction between the walls of the Miura-ori cells and rigid plates, assuming a friction coefficient of 0.05.

Fig. 3
figure 3

Geometry of the Miura-ori foldcores: a model M1; b model M2; c model M3; d model M4; e the model for a honeycomb core with equivalent density to model M3

Full size image
Fig. 4
figure 4

Virtual tests (FE model)

Full size image

The mechanical properties of the origami aluminum foil material of all thicknesses are assumed the same and are described by an elastic–perfectly plastic material model with Young’s modulus \(E_{s}\) = 71.52 GPa, yield stress \(\sigma_{s}\)= 116.3 MPa and density \(\rho_{s}\)= 2700 kg m-3.

Two types of virtual tests, namely out-of-plane compression and shear in the \(X_{1} - X_{3}\) and \(X_{2} - X_{3}\) planes, are considered. A node set is defined on the entire plate RPT to facilitate the displacement-controlled displacements in the corresponding loading direction. The shear and compression are applied at speed 10 mm/s in the corresponding direction. In the compression test, all degrees of freedom of the bottom plate (RPB) were constrained while the out-of-plane displacement of the top plate (RPT) was unconstrained. In the shear tests, RPB was fixed, but the in-plane displacements in the \(X_{1}\)- and \(X_{2}\)-directions of RPT were prescribed to calculate the corresponding shear in the \(X_{1} - X_{3}\) and \(X_{2} - X_{3}\) planes.

3 Mechanical Characteristics

The force–displacement characteristics obtained from the FE simulations are used to calculate the stress–strain characteristics of the core materials. The strains and stresses are obtained as

$$ \varepsilon_{3} = u_{3} /H, \gamma_{13} = u_{1} /H, \gamma_{23} = u_{2} /H $$
(7a-c)
$$ \sigma_{3} = F_{3} /A, \tau_{13} = F_{13} /A, \tau_{23} = F_{23} /A $$
(8a-c)

where \(u_{1} , u_{2}\) and \(u_{3}\) are the displacements in the corresponding directions; \(A\) is the loaded area of the core; and \(F_{13}\), \(F_{23}\) and \(F_{3}\) are the resulting forces in the three loading directions. Five cell wall thicknesses are used to obtain the analytical approximations of the cores’ mechanical properties as functions of the relative density. Typical stress–strain characteristics in the three loading directions are shown in Fig. 4 for models M1 (\(\alpha = 31.3^\circ , t\)= 0.2 mm) and M3 (\(\alpha = 60^\circ , t\)= 0.33 mm). Despite the equal relative densities, model M3 is characterized by a higher strength in comparison to model M1. Comparable compression and shear strength \(\tau_{13}\) characterize model M3, while the strength of model M1 is much lower in all loading directions. Besides, the out-of-plane compressive strength of model M1 is considerably lower than the shear strength \(\tau_{13}\). The latter behavior is related to the smaller cell wall thickness and smaller angle \(\alpha {\text{of model M}}1\). The higher sector angle of model M3 results in a higher initial folding angle (\(\gamma_{0} = 127^\circ )\), which defines a higher upright position of the cell walls, thus increasing the initial collapse load.

The elastic moduli are obtained at 0.1% strain for all analyzed geometries. It is found that the elastic moduli are linear functions of the relative density, \(E_{3} = C_{3} \rho_{0}^{*} E_{s}\), but with different proportionality coefficients \(C_{3}\) depending on the cell topology (see Table 2). The largest elastic moduli correspond to the out-of-plane compression, followed by the shear moduli \(G_{13}\) and \(G_{23}\).

Table 2 Mechanical properties of the Miura-ori cores—elastic moduli
Full size table

The maximum compressive stress \(\sigma_{3} = \sigma_{{3,{\text{Max}}}}\) corresponding to the collapse load and \(\tau_{13} = \tau_{{13,{\text{Max}}}}\) were recorded while \(\tau_{23} = \tau_{23,0.5\% }\) was used to obtain the analytical approximations of the characteristic stresses defined in a general form: \(\sigma_{3} = C_{3} \left( {\rho_{0}^{*} } \right)^{{\beta_{3} }}\), \(\tau_{13} = C_{1} \left( {\rho_{0}^{*} } \right)^{{\beta_{1} }}\) and \(\tau_{23} = C_{2} \left( {\rho_{0}^{*} } \right)^{{\beta_{2} }}\). The analytical expressions are summarized in Table 3. The stresses in the three loading directions are represented by different power-law functions of the relative densities. The maximum collapse stresses in the out-of-plane direction hold the largest power coefficients \(\beta_{3}\), indicating the strongest dependence on the relative density. A strong dependence of the steer stress \(\tau_{13}\) on the relative density is also observed, as identified by the relatively large coefficients \(\beta_{1}\), while the shear strength \(\tau_{23}\) depends almost linearly on the relative density with smaller coefficients \(\beta_{2}\). The obtained relationships between the elastic moduli and relative densities as well as between the compressive and shear strengths and relative density have been verified for \(t/a\) = 0.0375, which results in \(t\) = 0.45 mm for \(a\) = 12 mm and 0.75 mm for \(a\) = 20 mm.

Table 3 Mechanical properties of the Miura-ori cores—maximum strength in the three loading directions
Full size table

Figure 6a–e shows graphic illustrations of the calculated stresses and their variations for each model. Note that the same relative density is achieved when using larger cell wall thicknesses for the models with higher initial folding angle \(\gamma_{0}\) (Table 1). In general, the examined models possess the higher strength when loaded in shear in the \(X_{1}\)-direction, which corresponds to the reentrant orientation of the cells. Figure 6a–c explains the influence of the sector angle \(\alpha\) on the initial strength of models with side length \(a = b\) = 20 mm. The dotted lines in Fig. 6a (model M2) present the analytical approximation of the stresses according to the relationships in Table 3. It is shown that the increase of \(\alpha\) leads to a rapid increase of the out-of-plane stress \(\sigma_{3}\) when increasing the relative density. The reason for the increase in strength is described in relation to Fig. 5. Besides, the values of \(\sigma_{3}\) become comparable with the shear stresses \(\tau_{13}\) for model M3 (Fig. 6b and c).

Fig. 5
figure 5

Typical stress–strain characteristics of two models with equal relative density \(\rho_{0}^{*}\) = 0.04: a model M1, \(t\) = 0.2 mm; b model M3, \(t\) = 0.33 mm

Full size image
Fig. 6
figure 6

Comparison between the mechanical characteristics of the Miura-ori models: a M1 and M2; b M2 and M3; c M1 and M3; d M1 and M4, e M3 and M4

Full size image

Figure 6d directly compares models M1 and M4, as these models have the same densities for equal cell wall thicknesses and equal heights \(H\), and their stress variation is related to the folding pattern. It is revealed that the higher values of all stresses of model M4 are related to the higher initial folding angle \(\gamma_{0}\). Finally, Fig. 6e shows that the characteristic stresses of models M3 and M4 are nearly equal, regardless of the cell wall thicknesses for relatively low relative densities \(\rho_{0}^{* } < 0.05\) (\(\gamma_{0}\)=135 kg.m3) when the sector angles are equal. The out-of-plane stress \(\sigma_{3}\) and shear stress \(\tau_{13}\) related to model M3 increase more rapidly than those of model M4 for \(\rho_{0}^{* } > 0.05\).

The results in Fig. 6 show that, unlike most cellular materials, the relative density of Miura-ori materials cannot be considered as the only parameter defining their strength properties. The folding angles play the major role when defining the initial strength in the out-of-plane and shear loading directions.

In general, the initial strengths of Miura-ori materials under compression and shear increase with the increase in the initial folding angle \(\gamma_{0}\). However, it is important to point out the strong dependence on the sector angle \(\alpha\). The Miura-ori materials possess the lowest shear strength in the \(X_{2}\)-direction for all analyzed angles \(\alpha\). On the other hand, the shear strength in the \(X_{1}\)-direction increases when \(\alpha\) increases, becoming comparable to and even higher than the out-of-plane compressive strength for \(\alpha = 60^\circ\) (models M3 and M4).

4 Comparison between Core Materials with Periodic Structures and Equal Density

It was shown by numerous experimental tests and numerical simulations that the initial folding angle \(\gamma_{0}\) determines the initial peak force under out-of-plane and in-plane compression. Therefore, the initial high collapse strength in compression and shear does not necessarily define a material with high energy absorbing capacity as a significant force drop can develop after the initial peak. According to Fig. 6 and Tables 2 and 3, models M3 and M4 are characterized by the highest initial strength. Here, the full collapse responses of these models are compared, assuming the same relative density \(\rho_{0}^{*}\) = 0.0404. This density is achieved for cell wall thickness \(t\) = 0.33 mm in model M3 and for \(t\) = 0.2 mm in model M4. As the prismatic honeycombs have found a wide application as core materials, this material is selected as a typical core material for a comparison with the Miura-ori cores. The honeycomb relative density is

$$ \rho_{0}^{*} = 2t/\left( {l\sqrt 3 } \right) $$
(9)

where \(l\) is the side length. Therefore, a honeycomb (HC) model comprising 8 \(\times\) 8 hexagonal cells with a side length of 9.44 mm and height of 14.14 mm (equal to the height of model M3) is constructed to compare the two models with \(\rho_{0}^{*}\) = 0.0404 and wall thickness \(t\) = 0.33 mm. This model (Fig. 3e) has the same relative density as model M3 for equal cell wall thicknesses.

The responses to out-of-plane loading and shear loading in the \(X_{1}\)-direction of the Miura-ori models are compared with the responses to compressive and shear loading of the honeycombs applied in the \(X_{1}\)- and \(X_{2}\)-directions. To ensure a full collapse of the models under shear loading, the out-of-plane degree of freedom of the upper plate (RPT) remains unconstrained. In this case, the shear load produces a displacement in both the shear and compressive directions, resulting in total displacements \(\delta_{13} = \left( {u_{1}^{2} + u_{3}^{2} } \right)^{1/2}\) (Fig. 7) and \(\delta_{23} = \left( {u_{2}^{2} + u_{3}^{2} } \right)^{1/2}\). The displacement in the out-of-plane direction is \(\delta_{3} \equiv u_{3}\). To enable the comparison between the different models, the following variables are defined

$$ \overline{{F_{3} }} = F_{3} /A, \overline{{F_{13} }} = F_{13} /A, \overline{{F_{23} }} = F_{23} /A $$
(10a,b)
$$ \overline{{\delta_{3} }} = \delta_{3} /H, \overline{{\delta_{13} }} = \delta_{13} /H, \overline{{\delta_{23} }} = \delta_{23} /H $$
(11a,b)
Fig. 7
figure 7

Initial and deformed shapes of model M4 under shear loading

Full size image

The force–displacement characteristics are presented as \(\overline{{F_{3} }} = f\left( {\overline{{\delta_{3} }} } \right)\), \(\overline{{F_{13} }} = f\left( {\overline{{\delta_{13} }} } \right)\) and \(\overline{{F_{23} }} = f\left( {\overline{{\delta_{23} }} } \right)\).

The compressive and shear loads are applied at speed of 100 mm/s in the FE simulations. A deformed shape of the Miura-ori core under shear load is shown in Fig. 7, where the out-of-plane and shear displacements are marked.

A comparison between the force–displacement curves \(\overline{{F_{3} }} \left( {\overline{{\delta_{3} }} } \right)\) and \(\overline{{F_{13} }} \left( {\overline{{\delta_{13} }} } \right)\) for the origami models M3 and M4 is shown in Fig. 8a. Although the initial strengths of these models are nearly equal, the compressive forces of both models decrease rapidly after the peak forces are reached. The shear forces also decrease, but to a higher level, while the force–displacement curve of model M3 marks a stronger material.

Fig. 8
figure 8

Comparison between the force–displacement characteristics of models with equal relative density \(\rho_{0}^{*}\) = 0.0404: a Miura-ori models M3(\(t\) = 0.33 mm) and M4(\(t\) = 0.2 mm); b Miura-ori model M3 (\(t\) = 0.33 mm) and the HC model (\(t\) = 0.33 mm)

Full size image

Figure 8b presents the out-of-plane compression and shear responses of model M3 and the honeycomb model. Similar to the Miura-ori core, the compressive force \(\overline{{F_{3} }} \left( {\overline{{\delta_{3} }} } \right)\) of the honeycomb model decreases rapidly after the initial peak load, but it reaches a high plateau level, which is significantly higher than the compressive force of model M3. The shear force–displacement curves in the two loading directions, \(\overline{{F_{13} }} \left( {\overline{{\delta_{13} }} } \right)\) and \(\overline{{F_{23} }} \left( {\overline{{\delta_{23} }} } \right)\) (see Fig. 3d for the axis orientation), are compared with the force–displacement characteristic of the origami model \(\overline{{F_{13} }} \left( {\overline{{\delta_{13} }} } \right)\) in its stronger direction. The shear forces of the HC model, however, do not exhibit high peak loads, but are characterized by nearly constant plateau stresses. While the plateau region of force \(\overline{{F_{13} }} \left( {\overline{{\delta_{13} }} } \right)\) of the Miura-ori model is comparable to force \(\overline{{F_{23} }} \left( {\overline{{\delta_{23} }} } \right)\) of the honeycomb model, the latter model exhibits higher shear strength in the \(X_{1} - X_{3}\) plane represented by \(\overline{{F_{13} }} \left( {\overline{{\delta_{13} }} } \right)\).

The shear response of the honeycomb model is dominated by the in-plane shear of the cell walls oriented parallel to the \(X_{3}\)-axis. According to the geometry of the cells, they represent stocky plates defined by a relatively small ratio \(t/H\) = 43.67. According to Gheitasi and Alinia [24], the stocky plates under shear load yield before buckling. They have some post-yield capacity, buckle plastically, and their ultimate and critical loads under shear are equivalent. This response type corresponds to the relatively flat shear force–displacement curves of the HC model in Fig. 8b. The presence of a large number of cell walls placed in the \(X_{1}\)–\(X_{3}\) plane governs the higher force \(\overline{{F_{13} }}\) compared to \(\overline{{F_{23} }}\).

The walls of the Miura-ori cells are inclined with respect to both shear loading directions (\(X_{1}\)- and \(X_{2}\)-axes), resulting in a low buckling capacity, followed by a significant post-buckling reserve (Fig. 8a) and the ultimate load coincides with the formation of inclined yield bands. The high initial peak followed by a plateau stress of the Miura-ori material reflects the described response.

5 Conclusions

Virtual tests of shear and out-of-plane compression of Miura-ori foldcores were carried out to reveal the material characteristics as functions of the relative density. Empirical approximation of the elastic modulus and collapse strength as functions of the relative density were defined for origami with several topologies. The non-unique relationships between the core density and material strength in the examined loading directions were revealed, pointing out the strong influence of the Miura-ori topology. Linear relationships were established between the elastic moduli and relative densities of the Miura-ori metamaterials, while power-law functions of the relative density with different exponent constants were established for the strength in different loading directions. As expected, the material strength increases in all directions with the increase in the cell wall thickness.

It was shown that the Miura-ori materials possess higher shear strength when loaded in the \(X_{1} - X_{3}\) plane, and it increases with the increase in the relative density. However, this characteristic is strongly influenced by the sector angle \(\alpha\). It is noted that the strength of the Miura-ori materials as a function of the relative density is nearly invariant with respect to cell dimensions if the folding angle \(\gamma_{0}\) and sector angle \(\alpha\) are equal.

The comparison between the collapse responses of Miura-ori materials with equal densities, equal folding and sector angles, but different side lengths, shows the superiority of the material with the lower relative density per unit wall thickness (model M3). The comparison between the latter material and a honeycomb with the same core height, wall thickness and relative density shows that the prismatic honeycomb exhibits higher strength in all loading directions.

It can be concluded that the non-unique relationship between the mechanical characteristics of the Miura-ori foldcores and their relative density provides the possibility for multi-parameter optimization when compromising between the peak load and energy absorption, particularly when the core is expected to function under shear loading.