Foldcores made of thermoplastic materials: Experimental study and finite element analysis

Thin-Walled Structures 100 (2016) 170–179

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Foldcores made of thermoplastic materials: Experimental study and finite element analysis Shixi Zang a, Xiang Zhou a,n, Hai Wang a, Zhong You b a b

School of Aeronautics and Astronautics, Shanghai Jiao Tong University, China Department of Engineering Science, University of Oxford, UK

art ic l e i nf o

a b s t r a c t

Article history: Received 13 October 2015 Received in revised form 1 December 2015 Accepted 15 December 2015 Available online 22 December 2015

Foldcore sandwich structures have aroused considerable research interests in recent years as a promising alternative to honeycomb sandwich structures. While metallic, aramid and CFRP foldcores have been extensively studied, foldcores made of thermoplastic materials have not been well investigated. This paper presents an experimental and numerical study on thermoplastic foldcores. A manufacturing process involving folding with dies and shape-setting by heat-treatment was established. Both PET and PEEK foldcore specimens were successfully manufactured using this process and tested in compression. An explicit FE modeling protocol of PEEK foldcores were developed and validated with the test results. Furthermore, a parametric study on the weight-specific mechanical properties of various PEEK foldcore models subjected to virtual compression and in-plane shears was performed, through which the relationships between the mechanical properties and the geometric parameters were established. Finally, the mechanical properties of a PEEK model and its aramid counterpart with the same geometry were compared. It was shown that the PEEK foldcore has comparable or even better energy absorption performances than the aramid one. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Foldcore Thermoplastic material Manufacturing Finite element method

1. Introduction Composite sandwich structures, typically consisting of two thin and stiff faces bonded to a thick light-weight core, have received credit for engineering applications where weight-reduction is of high importance. Therein, aluminum or aramid honeycomb structures have acted as the most widely used core type in sandwich structures nowadays owing to their superior weight-specific mechanical properties. However, these cores are known to suffer from an undesirable problem in the moist environment where the closed hexagonal cells tend to trap condensed water, leading to serious deterioration of the mechanical performance over time. In the aerospace industry, for instance, this problem contributes an important reason for the field of applications of honeycomb sandwich structures being restricted to secondary components such as control surfaces or cabin panels of an aircraft [1]. During the search for advanced cellular cores, foldcores, made by folding sheet material into a three-dimensional structure using the principle of origami, emerged as a promising alternative to conventional honeycomb cores. This origami-like core family is open to a variety of base materials and a wide range of unit cell n

Corresponding author. E-mail address: [email protected] (X. Zhou).

http://dx.doi.org/10.1016/j.tws.2015.12.017 0263-8231/& 2015 Elsevier Ltd. All rights reserved.

geometries, thus allowing for tailored mechanical properties to achieve various design goals. Moreover, they do not have the moisture accumulation problem due to the existence of open channels. As a result, a considerable amount of work on foldcore sandwich structures is available in the literature. Only a few selected references are given here. Heimbs et al. [2] compared the performances of aramid and CFRP foldcores in compression, shear and impact experiments. Kintscher et al. [3] developed a test facility to measure the stiffness and strength of foldcores under combined compressive and shear loads. Klaus and Reimderdes [4] employed four-point bending test to investigate the residual strength of foldcore sandwich panels after impact. The mechanical performances of wedge-shaped aramid foldcores under compression and shear have also been studied experimentally [5]. Moreover, Sturm et al. [6] investigated the deformation of triggered and untriggered foldcore sandwich panels under crash relevant bending-compression loads by means of 3D CT scanning. In addition to experimental studies, endeavors were also focused on the numerical simulations of foldcore sandwich structures using the finite element (FE) method, which has been widely adopted in the development of new composite structures as a time- and costefficient research tool. For instance, Heimbs et al. [7] investigated flatwise compression properties of aramid and CFRP foldcores using the FE package LS-DYNA. Heimbs et al. [8] proposed a virtual testing method and compared flat-wise compressive properties of

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171

250 PET-test 1 PET-test 2 PET-test 3

200

PEEK-test 1 PEEK-test 2

compressive force (N)

PEEK-test 3 PEEK-FE

150

100

50

0 0

5

10

15

20

25

30

40

35

45

50

compressive strain (%)

Fig. 3. The weight-specific compressive force versus strain curves of PET foldcore specimens (black lines), PEEK foldcore specimens (blue lines) and the PEEK FE model (red line with triangular marks). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 90 80 70

true stress (MPa)

60 50 40 30 20

Fig. 1. (a) The PET foldcore specimen; (b) the PEEK foldcore specimen.

specimen-1

10

specimen-2

0 0

1

2

4

3

5

6

7

8

9

true strain (%)

Fig. 4. The true stress versus strain curves from the material tests of PEEK.

Table 1 The material properties of the PEEK sheet.

Fig. 2. The test setup for flat-wise compression.

Kevlar and CFRP foldcores. Fischer [9] developed a FE modeling procedure to study the mechanical properties of aluminum foldcores. Zhou et al. [10] performed a parametric virtual study on

ρ [kg/m3]

E [GPa]

ν [-]

σy [MPa]

σ U [MPa]

εf [-]

1055

3.1

0.4

45.5

86.0

0.08

Miura-based foldcores subjected to quasi-static loads. Gattas and You [11] investigated the behavior of curved-crease foldcores under low-velocity impact loads. Sturm et al. [12] presented a FE study on the failure mode of foldcore sandwich panels subjected to combined compression and bending. Furthermore, the mechanical performances of cylindrical foldcore sandwich structures under radial crush have been virtually studied [13]. Current research on foldcores mainly focuses on aluminum, aramid and CFRP ones. So far, the literatures on foldcores made of thermoplastic materials are limited, among which Grzeschik [14] presented an experimental study on foldcores made of different base materials including aluminum, aramid paper and PEEK polymer, and Klett [15] compared compression test results of

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Fig. 5. (a) Virtual flat-wise compression test; (b) virtual shear tests in the x–z (dashed arrow line) and y–z (solid arrow line) planes.

Nomex honeycomb cores and foldcores made out of PEEK and PC. In this paper, we present a study on the mechanical properties of foldcores made of two typical thermoplastic materials, i.e. polyethylene terephthalate (PET) and polyetheretherketone (PEEK). First, PET and PEEK foldcore prototypes based on a certain crease pattern were manufactured using a pair of metallic dies. Their flatwise compression behaviors were mechanically tested and compared. A FE modeling protocol of PEEK foldcores was then developed and validated with the test results. A parametric study on PEEK foldcore models with various unit cell geometries subjected to flat-wise compression and in-plane shear was performed using the validated FE modeling protocol. Moreover, the weight-specific mechanical properties of the PEEK foldcore were compared to those of an aramid counterpart with the same geometry. Finally, a brief discussion concludes the paper.

2. Experimental study of thermoplastic foldcores 2.1. Manufacturing of thermoplastic foldcores Current manufacturing methods for foldcores can be generally categorized into continuous and discontinuous approaches [16]. The typical discontinuous approach is based on transformable dies, which can fold simultaneously to deform the sheet material laid in between them [17]. The common way to produce foldcore continuously is to bend the sheet material along one axis in a first step to a certain wave pattern and then bend it along the second axis [18]. Current methods are mainly designed for the manufacturing of metallic or epoxy-based composite (e.g. CFRP) foldcores. They are associated with a high manufacturing cost. On the other hand, thermoplastic foldcores can be manufactured with relative ease due to the good processing property

Fig. 6. (a) The wave patterns of the input point sets in the x–z plane of models UM1 (solid lines), UM2 (dashed lines), UM3 (dash-dotted lines) and UM4 (dotted lines); (b) the wave pattern of the input point set in the y–z plane of all models.

possessed by thermoplastic materials. The process we developed to produce thermoplastic foldcores contains three major steps. First, the folding pattern is scored onto the sheet material using an electronic cutting device (Silhouette America Inc., USA). Second, the sheet material that is pre-folded along the creases is compressed between a pair of metallic dies whose corrugate surfaces are machined to be identical to the target foldcore geometry. Finally, the assembly is heat-treated in a furnace to set the base material into the desired shape. With this process, PET and PEEK foldcore specimens having a nominal size of 171  155  15 mm3 were manufactured, as shown in Fig. 1, where the PET foldcores were made from 0.23 mm thick PET sheet treated at 100 °C for 1.5 h for shape-setting and the PEEK group were made from 0.32 mm thick PEEK sheet treated at 200 °C for 4.5 h. The weights of the PET and PEEK foldcore specimens are measured at 9.51 g and 12.87 g, respectively. 2.2. Compression test of PET and PEEK foldcores Flat-wise compression tests with the PET and PEEK foldcore specimens described above were conducted. In the test, two 1 mm thick aluminum plates were bonded respectively to the two flatwise ends of each foldcore specimen through epoxy adhesive. The sandwich-like assembly was then placed between the flat load platens mounted on a E45.105 materials testing machine (MTS Systems, USA), as shown in Fig. 2 and compressed by 8 mm in the thickness direction, resulting in a maximum loaded effective strain of over 50%.

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173

Fig. 7. Unit cell models: (a) UM1, (b) UM2, (c) UM3 and (d) UM4.

Table 2 The geometric properties of models M1–M4. Model

hx [mm]

Tx [mm]

Ty [mm]

hy [mm]

Sc [mm2]

mc [kg/m3]

M1 M2 M3 M4

12 12 12 12

60 60 60 60

50 50 50 50

15 15 15 15

2.70E4 2.70E4 2.70E4 2.70E4

28.15 29.88 28.64 30.12

The compressive force normalized by the weight of the specimen versus strain curves of the PET and PEEK groups are compared in Fig. 3. It is noted that the PEEK group exhibits higher weight-specific stiffness and strength than the PET group. As a result, the numerical study in the sequel is carried out on PEEK models.

Besides, in order for quasi-static procedures to be simulated within a realistic computational time, convergence studies of various loading rates were also performed. The material parameters of PEEK used in the numerical models were obtained through material tests on specimens that were cut out of the same 0.32 mm thick PEEK sheet and subjected to the same heat treatment condition for the manufacturing of the PEEK foldcore speciments. The tensile tests were performed on a Z100 materials testing machine (Zwick Roell Group, Germany). The stress–strain curves gained from the tensile tests are shown in Fig. 4. According to the test results, an elastoplastic isotropic material model with the ductile failure criterion was employed to simulate the base material. The material properties are summarized in Table 1, where the strain hardening data is not shown for simplicity. 3.2. Validation

3. FE modeling of PEEK foldcores 3.1. FE modeling protocol In this paper, the FE analysis was performed in the FE package ABAQUS/Explicit (SIMULIA Inc., USA) due to its good capability to deal with large nonlinear deformation and complex contact conditions. All foldcore models were meshed with S4R elements, the four-node quadrilateral shell elements with reduced integration and hourglass control. With this type of element, the mesh density has a strong correlation with the accuracy of simulation results. A coarser mesh cannot truly represent the post-buckling behavior of the facets while an over-crowded mesh significantly augments computational cost due to the combined effect of the increased number of elements and reduced time increment of explicit FE analysis that is proportional to the element length. As a result, convergence study of different element sizes was conducted as a first step to find an appropriate element size for each model.

To validate the FE modeling protocol described above, a virtual compression test of the PEEK foldcore specimen in Fig. 1(b) was performed. The geometry of the specimen was acquired through an optical 3D measurement facility (Xjtop 3D Technologies, China) so that realistic geometrical imperfections resulting from the manufacturing process were included in the FE model. The compressive load was applied via two rigid panels RP1 and RP2 attached respectively to the flat-wise ends of the foldcore model with tie constraints, as shown in Fig. 5(a). During the loading, RP1 was completely fixed and RP2 translated half of the thickness towards RP1, resulting in a maximum loaded compressive strain up to 50%. According to the mesh density and loading rate convergence tests, an element size of 0.5 mm and a loading rate of 150 mm/s were chosen. The weight-specific compressive force versus strain curve from the virtual compression test is plotted as the triangle-marked red line in Fig. 3. A good agreement between the FE and experimental results can be observed.

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(a)

absorbed energy (kJ/m 2 )/density (kg/m 3 )

0.014 M1 M2 M3 M4

0.012

0.01

0.008

0.006

0.004

0.002

0 0

10

20

30

40

50

60

compressive strain (%)

(b) Fig. 8. (a) The weight-specific compressive stress versus strain curves and (b) the weight-specific energy absorption versus strain curves of models M1–M4 in the virtual compression test.

4. Parametric studies of PEEK foldcores 4.1. Parametric models In this section, we used the FE analysis to investigate the influences of different geometric parameters on the mechanical properties of PEEK foldcores subjected to three quasi-static loading conditions, i.e. flat-wise compression, shear in the x–z plane and shear in the y–z planes. The virtual compression test has been described in Section 3.2. The virtual x–z and y–z plane shear tests are illustrated by Fig. 5(b), where RP1 was fixed and RP2 is translated respectively in the x- and y-directions by a distance equal to half of the core thickness, resulting in a maximum effective shear strain of 50%. The geometric models of the foldcores were created with a new three-dimensional (3D) origami design method [19], known as the vertex method, which determines the vertex set V of a 3D origami structure out of two 2D input point sets V x and V y defined in the x–z and y–z planes of a Cartesian coordinate system, respectively. Using this method, a unit cell of the standard Miura-ori tessellation UM1, as shown in Fig. 7(a), can be represented by a single

Fig. 9. The deformed shapes of model M4 in the virtual compression test (a) at 1.6% strain with stress contour and (b) at 4.5% strain with strain contour.

period of a triangular wave pattern of period Tx and height hx in the x–z plane, as shown by the solid lines in Fig. 6(a), and a single period of a triangular wave pattern of period Ty and height hy in the y–z plane, as shown in Fig. 6(b). By changing the triangular wave in Fig. 6(a) to trapezoidal (dashed lines), sinusoidal (dashdotted lines) and piecewise elliptical (dotted lines) waves of the same period and height, respectively, three different types of unit cells UM2–UM4 can be obtained, as shown in Fig. 7(b)–(d). The details of the parametric models are described in Appendix A. In the sequel, we will first compare the mechanical behaviors of foldcores with different wave patterns in the x–z plane input. Then, the influences of wave parameters hx , Tx and Ty will be discussed. Throughout the section, hy is taken as 15 mm so that a constant core height equal to 15 mm is maintained.

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175

Fig. 10. (a) The weight-specific shear stress versus strain curves in the x–z plane and (b) the weight-specific shear stress versus strain curves in the y–z plane of models M1–M4 in the virtual shear tests.

4.2. Influence of wave pattern Consider first four foldcore models M1 to M4 consisting of 3  3 unit cell models UM1 to UM4, respectively. The geometric parameters of these models are summarized in Table 2, where Sc is the base area, i.e. the projected area of the foldcore in the x–y plane and ρc the density of the foldcore, i.e. the weight of the foldcore divided by the volume it takes. Hence, the weight-specific stress σ ̅ is defined as

σ̅=

Fr , (Sc ρc )

(1)

where Fr is the compressive or shear force acting on the rigid plate. Fig. 8(a) shows the weight-specific compressive stress versus strain curves of the four models, where a logarithmic scale is used to the abscissa for a better illustration of the regions of small strains. Three stages can be identified for the mechanical behaviors of these models under compression. In the first stage, the

Fig. 11. (a) The deformed shape of model M4 in the virtual shear test in the x–z plane at 35% strain with strain contour; (b) the deformed shape of model M4 in the virtual shear test in the y–z plane at 12% strain with strain contour.

compressive stress increases linearly with the strain until yield of the base material begins to propagate. Fig. 9(a) shows the moment corresponding to point A on the stress–strain curve of model M4 at which the model reaches the material yield point. In the second stage, the stress value decreases rapidly because of the combined effects of facet buckling and folding and crack initiation and propagation. The snapshot at point B on the stress–strain curve of model M4 is shown as an example in Fig. 9(b). In the third stage, the contact between the facets and the rigid plates occurs, resulting in an increase in the stress level. Note that model M4 has the highest compressive strength and absorbs the most compressive energy, i.e. the area under the stress–strain curve, as shown in Fig. 8(b). The weight-specific shear stress versus strain curves in the x–z

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model group M4HX

UM4HX1

UM4HX2

UM4HX3

UM4HX4

model group M4TX

UM4TX1

UM4TX2

UM4TX3

UM4TX4

model group M4TY

UM4TY1

UM4TY2

UM4TY3

UM4TY4

Fig. 12. Unit cell models UM4HX1–4, UM4TX1–4 and UM4TY1–4. Table 3 The geometric properties of models M4HX1–4, M4TX1–4 and MTY1–4. Model

hx [mm]

Tx [mm]

Ty [mm]

hy [mm]

Sc [mm2]

mc [kg]

M4HX1 M4HX2 M4HX3 M4HX4 M4TX1 M4TX2 M4TX3 M4TX4 M4TY1 M4TY2 M4TY3 M4TY4

6 9 15 18 18 18 18 18 18 18 18 18

60 60 60 60 90 75 45 30 30 30 30 30

50 50 50 50 50 50 50 50 75 62.5 37.5 25

15 15 15 15 15 15 15 15 15 15 15 15

2.70E4 2.70E4 2.70E4 2.70E4 4.05E4 3.38E4 2.02E4 1.35E4 2.00E4 1.69E4 9.98E3 6.75E3

1.11E 1.16E 1.28E 1.36E 1.83E 1.59E 1.13E 9.20E 1.27E 1.09E 7.57E 6.16E

2 2 2 2 2 2 2 3 2 2 3 3

and y–z planes are shown in Fig. 10(a) and (b), respectively. Similarly, three stages can be recognized for both x–z and y–z plane shear behaviors of these models. The first two stages are similar to those of the compression case discussed above whereas no densification occurs in the third stage for shear cases. Instead, it is characterized by severe damage propagation, leading to global failure of the model, as shown in Fig. 11. Again, model M4 exhibits the highest weight-specific shear strengths in both x–z and y–z planes. 4.3. Influences of wave parameters Consider now the influences of the wave parameters hx , Tx and Ty on the mechanical properties of PEEK foldcores. Since model M4 shows the highest weight-specific compressive and shear strengths, twelve foldcore models each consisting of 3  3 unit cell models with the piecewise elliptical pattern are considered. As shown in Fig. 12, the unit cells of the twelve models are categorized into three groups, where model groups M4HX, M4TX and M4TY have different hx , Tx and Ty , respectively. The detailed geometric parameters of these models are summarized in Table 3. The weight-specific compressive stress versus strain curves and

shear stress versus strain curves in the x–z and y–z planes of the twelve models are shown in Fig. 13(a)–(c), where the magenta, blue and green lines correspond to the data of groups M4HX, M4TX and M4TY, respectively. According to the results, the weight-specific compressive strength σu̅ ts increases with the increase in hx but decreases as Tx or Ty increases. A linear best-fit correlation between σu̅ ts and hx or Tx and a nonlinear best-fit correlation between σu̅ ts and Ty can be found, as shown in Fig. 14 (a). It is shown that Ty is the most influential parameter to the compressive strength of the foldcore. For the x–z plane shear case, the weight-specific shear strength τ u̅ xzts decreases with the increase in Ty based on a nonlinear correlation similar to that between σu̅ ts and Ty whereas hx and Tx have virtually no influence on τ u̅ xzts , as shown in Fig. 14(b). Finally, the weight-specific shear strength τ u̅ yzts in the y–z plane increases with the increase in hx and decreases with the increase in Tx or Ty . Linear best-fit correlations between τ u̅ yzts and all three parameters are applicable, as shown in Fig. 14(c), where Tx appears as the most influential parameter to the y–z plane shear strength of the foldcore.

5. Comparison with aramid foldcore In this section, the mechanical properties of model M4TY4 are compared to those of an aramid counterpart whose geometry is identical to model M4TY4. Aramid is chosen as it is a common base material of foldcores as well as honeycomb cores and has comparable mechanical properties to those of PEEK. The aramid foldcore model considered here is assumed to be made from the 0.3-mm thick aramid paper tested in Ref. [20]. The aramid paper is modeled with the same material model for PEEK described above. The detailed material properties of the aramid paper are listed in Table 4. The weight-specific stress versus strain curves of model M4TY4 and the aramid model are plotted in Fig. 15. Due to the higher modulus and strength and the lower density of the aramid paper, the aramid foldcore model has higher weight-specific compressive and shear stiffnesses and strengths than the PEEK model does.

S. Zang et al. / Thin-Walled Structures 100 (2016) 170–179

Fig. 13. (a) The weight-specific compressive stress versus strain curves, (b) the weight-specific shear stress versus strain curves in the x–z plane and (c) the weight-specific shear stress versus strain curves in the y–z plane of models M4, M4HX1–4, M4TX1–4 and MTY1–4. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

177

Fig. 14. (a) Changes of σu̅ ts with respect to hx , Tx and Ty ; (b) changes of τ u̅ xzts with respect to hx , Tx and Ty ; (c) changes of τ u̅ yz ts with respect to hx , Tx and Ty .

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Table 4 The material properties of the aramid paper.

ρ [kg/m3]

E [GPa]

ν [-]

σy [MPa]

σ U [MPa]

εf [-]

800

11.6

0.39

29.2

96.7

0.02

Fig. 17. An illustration of the longitudinal and transverse waves in a foldcore. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

6. Discussions and conclusions

Fig. 15. The weight-specific compressive stress versus strain curves and shear stress versus strain curves in the x–z and y–z planes of models M4TY4 and the aramid foldcore model.

Fig. 16. The weight-specific energy absorption versus strain curves of models M4TY4 and the aramid foldcore model in the virtual compression and shear tests.

Nevertheless, when one compares the weight-specific energy absorption curves, as shown in Fig. 16, the PEEK foldcore model assumes overall better energy absorption performance than the aramid model, with the absorbed energy of the aramid model quickly exceeded by the PEEK model at point C1 in the compression case, point C2 in the x–z plane shear case as well as point C3 in the y–z plane shear case due to the greater ductility (i.e. higher strain at which the material fails) that the PEEK sheet has. Incidentally, in the compression case, the absorbed energy of the aramid model surpasses that of the PEEK model later on by a relatively small proportion due to the densification of the foldcore model.

This paper has presented an experimental and numerical study on foldcores made of thermoplastic material. It was successfully demonstrated that the thermoplastic foldcores such as PET and PEEK specimens can be readily manufactured through a three-step process involving scoring, folding with dies and shape-setting by heat-treatment. It was also shown that an explicit FE modeling protocol of the PEEK specimen that involves an elastoplastic isotropic material model with the ductile failure criterion, mesh and loading rate convergence studies and real geometric imperfections from 3D-scanning gave generally fine prediction of the experimental results. Based on the validated FE modeling protocol and by virtue of the vertex method, a parametric study on the weight-specific mechanical properties of various PEEK foldcore models subjected to virtual flat-wise compression and in-plane shear tests was performed. First, it was shown that the wave pattern is correlated with the mechanical properties and among the four wave patterns considered, the piecewise elliptical pattern outperforms the others. According to the mathematical meaning of the vertex method [19], the longitudinal wave (i.e. the blue line in Fig. 17) in the foldcore model is identical to the y–z plane input whereas the period and the height of the transverse wave (i.e. the red line in Fig. 17) are determined by those of the x–z plane input. According to the simulation results, the larger the height of the transverse wave or the denser the foldcore in the transverse or longitudinal direction, the higher the weight-specific compressive strength and the shear strength in the y–z plane are, where the compressive strength is the most sensitive to the denseness in the longitudinal direction and the shear strength in the y–z plane to the denseness in the transverse direction; the weight-specific shear strength in the x–z plane increases as the foldcore becomes denser in the longitudinal direction and is generally independent of the transverse wave. Furthermore, a direct comparison on the weight-specific mechanical properties of the PEEK and aramid foldcore models of the same geometry reveals that despite the lower stiffness and strength, the PEEK foldcore exhibits comparable or even better energy absorption performance than the aramid foldcore owing to the better ductility that the PEEK sheet possesses. The work reported in this paper provides an insight into the mechanical properties of thermoplastic foldcores. Yet it has several limitations. Further investigations are recommended as follow-ups to the present study of thermoplastic foldcores. Foremost, the mechanical properties under dynamic loading conditions such as impact and other ambient conditions such as cold and elevated temperatures should be evaluated. Future work will consider thermoplastic materials containing carbon or aramid fibers, which

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is a promising way to increase the strength and energy absorption capacity of the thermoplastic foldcore.

Acknowledgments

Appendix A In the vertex method, the vertex set V of a 3D origami structure is generated through the following equation

Vi, j =V jy+⎡⎣ A j ⎤⎦ Vix,

(A.1)

V jy

⎤ ⎡1 0 0 ⎥ ⎢ cos cos + θ θ j−1 j⎥ ⎢ 0 0 ( − 1) j ⎡⎣ A j ⎤⎦=⎢ sin (θj − 1 − θj ) ⎥, ⎥ ⎢ sin θj − 1 + sin θj ⎥ ⎢ 0 0 ( − 1) j ⎢⎣ sin (θj − 1 − θj ) ⎥⎦

(A.2)

where the angular variable θj can be determined by

⎡ 0 ⎤ y y ⎢ ⎥ V j +1 − V j . ⎢ cos θj ⎥= y y ⎢⎣ sin θj ⎥⎦ V j +1 − V j

(A.3)

The solid, dashed, dash-dotted and dotted lines in Fig. 6(a) can be expressed by T ⎡ i −1 1+( − 1)i ⎤ Vix |UM1=⎢ Tx 0 hx ⎥ , i=1, 2, 3, ⎣ 2 ⎦ 2

⎤T ∑ Pl ⎥⎥ , i=1, …, 9, l= 1 ⎦

⎡ i−1 Vix |UM2 =⎢ Tx 0 ⎢⎣ 8

(A.4)

i

T ⎡ i−1 h ⎛ 2π (i−1) ⎞⎤ ⎟⎥ , i=1, …,N +1, Vix |UM3=⎢ Tx 0 x ⎜ 1−cos ⎠⎦ 2⎝ N ⎣ N

0

(A.5)

(A.6)

⎤T ⎛ i −1 hx s −1 ⎞2 ⎥ ⎟ + ( − 1) s 1 − 16 ⎜ − ⎝ N 2 2 ⎠ ⎥⎦

, i=1, …,N +1,

(A.7)

where Pl is the l-th element of an array P given by

⎡ h P =⎢ 0 0 x ⎣ 2

T ⎡ j−1 1+( − 1) j ⎤ V jy=⎢ Ty 0 hy ⎥ , j=1, 2, 3. ⎣ 2 ⎦ 2

(A.10)

⎡ ⎤ Vix |1UMk ⎢ ⎥ 2 ⎢ ⎥ ⎛ ⎞ T ⎢ j−1Ty+V x |UMk ⎜ y ⎟ +1 ⎥ i 3 Vi, j |UMk =⎢ 2 ⎥, k=1, 2, 3, 4. ⎝ 2h y ⎠ ⎢ ⎥ ⎢ ⎥ 1+(−1) j hy ⎢⎣ ⎥⎦ 2

(A.11)

Note that models UM1 to UM4 have the same x-directional width and z-directional height, equal to Tx and hy , respectively.

Vx

where and are the i-th point in and the j-th point in V y , respectively, and [Aj ] is a 3  3 matrix given by

⎡ i −1 Vix |UM4 =⎢ T ⎢⎣ N x

The solid line in Fig. 6(b) can be expressed by

Substituting Eqs. (A.4)–(A.7) and Eq. (A.10) into Eq. (A.1) yields

The financial support from National Natural Science Foundation of China (No. 51408357) is gratefully acknowledged.

Vix

179

⎤ hx h h 0 0 − x − x 0 ⎥, ⎦ 2 2 2

(A.8)

N is a large integer and s is an index given by

⎧ N 1,i≤ +1 ⎪ 4 ⎪ ⎪ N 3N s=⎨ 2, +1 ⎩ 4

(A.9)

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