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Article published in
Thin-Walled Structures, Vol. 192, 2023, 111114
https://doi.org/10.1016/j.tws.2023.111114
Design and mechanical performance of stretchable sandwich
metamaterials with auxetic panel and lattice core
Xing Chi Teng a, Xi Hai Ni a, Xue Gang Zhang a, Wei Jiang a, Yi Zhang a, Hang Hang Xu a, Jian Hao a, Yi Min Xie b, Xin
Ren a, *
a Centre for Innovative Structures, College of Civil Engineering, Nanjing Tech University, Nanjing, Jiangsu, 211816, PR China
b Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne, 3001, Australia
* Corresponding author: Xin Ren. Email: xin.ren@njtech.edu.cn
Abstract
Sandwich structures are widely used in engineering as lightweight and high-strength materials. In recent
years, there has been increasing attention toward new design ideas aimed at improving the adaptability and
variability of sandwich structures for broader applications. This paper proposes a new stretchable sandwich
structure. Experiments and finite element methods are conducted to provide the influence of stretchable
design on the mechanical properties of traditional sandwich structures. Then, three novel stretchable
sandwich structures with 3D lattice core layers are also designed to explore the applicability of this method.
The results demonstrate that the stress and energy absorption capacity of stretchable lattice sandwich
structures remains essentially constant. In addition, stretchable sandwich structures possess the capability
of in-plane deformation and active bending. By combining the functional characteristics of sandwich
structures and rotating polygons, stretchable sandwich structures can adjust their surface area, porosity, and
in-plane negative Poisson's ratio effect while maintaining bearing capacity. This study suggests
improvements in the utilization and adaptive capacity of sandwich structures in defense engineering and
other applications by overcoming the problem of limited deformation faced by traditional sandwich
structures.
Keywords: Negative Poisson’s ratio; Auxetic; Sandwich structure; Mechanical metamaterials; Lattice
structure.
1. Introduction
The sandwich structure is a widely used material in engineering that consists of two layers of plates
and an intermediate core layer [1, 2]. Sandwich structures can provide better strength and stiffness compared
to traditional monolithic materials due to their lightweight and high-strength characteristics, making them
suitable for a wide range of applications in aviation, automotive, construction, and other fields [3-5]. In the
aviation industry, aircraft and satellites use sandwich structures to reduce weight while maintaining strength
and stiffness [6]. Similarly, sandwich structures are used to manufacture vehicle components such as body
panels, doors, and engine hoods in the automotive industry [7]. The construction industry commonly utilizes
sandwich structures for exterior wall panels and roofs.
The mechanical properties of sandwich structures are predominantly influenced by the core layer,
which can be made of various shapes and materials such as honeycomb [8], foam [9], and fiber-reinforced
composite materials [10]. Depending on the shape and material of the core layer, sandwich structures can
be classified into different types such as metal and composite sandwich structures [11, 12]. Designs for
sandwich structures must consider their mechanical properties, including strength, stiffness, and bending
strength, among others, which vary depending on different application scenarios. Targeted designs can be
carried out accordingly. For instance, sandwich beam structures are mainly used for bending resistance [13],
foam aluminum sandwich panels are primarily used for energy absorption [14], and foam sandwich
structures are generally used for thermal or acoustic insulation [15].
Honeycomb sandwich structures were initially introduced in the early 20th century and are considered
a type of mechanical metamaterial that provides high specific strength and modulus due to their evenly
distributed structure under load [16-20]. Because honeycomb sandwich structures are relatively complex,
some scholars initially examined their mechanical properties by conducting tests on either the core layer or
the entire sandwich structure [21]. During the study of honeycomb sandwich structures, some scholars
established various equivalent stiffness theories for the honeycomb sandwich face and the whole sandwich
structure [22, 23]. They also investigated the correlation between the in-plane elastic modulus of the
honeycomb and the geometric configuration of the sandwich [24, 25]. Apart from the employed numerical
simulations, more novel and strong numerical methods have been recently proposed for the stress analysis
of composites. Among them, the "Differential Quadrature" and "Bezier" methods proved to have higher
stability and accuracy than other numerical methods [26, 27]. Based on numerical tests conducted to analyze
the bending and buckling of sandwich structures under various loading and boundary conditions, the current
solutions demonstrate a satisfactory level of accuracy when compared to Navier's solution, differential
quadrature method, and commercial finite element analysis models.
With the development of the field of metamaterials [28-32], lattice structures and negative Poisson's
ratio structures have gradually emerged [33-38]. Lattice structures are space cells composed of repetitive
and periodic 3D arrangements, including cubic lattice, orthorhombic lattice, monoclinic lattice,
rhombohedral lattice, and triclinic lattice. Negative Poisson's ratio structures, also known as "auxetic"
structures, expand outward horizontally when stretched longitudinally [39]. In contrast, traditional structures
shrink inward horizontally when vertically stretched. Negative Poisson's ratio structures can include various
forms such as re-entrant structures [40-44], chiral structures [45], rotating polygon structures [46], and
perforated plate structures [47, 48], among others. Of these, re-entrant structures are the most extensively
applied. When a horizontal tension is exerted on a re-entrant honeycomb structure, the longitudinal rod
moves outward, the inclined rod rotates, and the re-entrant angle increases, achieving the negative Poisson's
ratio effect [49-52]. Similarly, the rotating polygon structure is based on rigid polygons and rotatable
connecting nodes that cause the entire system to expand when subjected to tensile stress, thereby achieving
the negative Poisson's ratio effect [53, 54]. Since then, many excellent properties of re-entrant honeycomb
structures have been discovered, such as anti-dimpling, anti-indentation, and synclastic curvature behavior
[55, 56].
Then, some scholars have begun utilizing re-entrant honeycomb structures as cores for sandwich panels
due to their excellent properties. Yan et al. [57] demonstrated that auxetic honeycomb sandwich panels
(AHSPs) made of carbon fiber reinforced polymer (CFRP) have good ballistic impact performance via
numerical and experimental research. Pelinski et al. [58] conducted three-point bending tests on beams made
of different materials to determine the influence of thickness on the stiffness, strength, energy absorption,
and dissipation of sandwich beams with increased diameter cores. To reduce the mass and volume of modern
aerospace vehicles while maintaining their original load-carrying capacity, material structures need to be
designed to minimize weight while maintaining their original strength. Some scholars have utilized the
lightweight and high-strength characteristics of sandwich structures to meet this demand [59]. Usta et al.
[60] described the low-velocity impact behavior of composite sandwich panels with various types of noise
reduction structures and non-noise reduction prism cores, while Juliette et al. [61] developed a sound
insulation sandwich panel with a broad sound absorption performance. Both designs aim to improve
multifunctional structures used for aerospace applications. Li et al. [62] developed an ultra-lightweight
comprehensive thermal protection system (ITPS) structure using thin ceramic-based composite plates
combined with thick insulation layers to protect the substructures of hypersonic aircraft.
While extensive research has been conducted on the performance of various sandwich structures,
including their core types, research on surface panels is scarce. Most studies only focus on the thickness of
the panel. New design ideas and methods for sandwich structures have attracted attention in recent years,
particularly those aimed at improving the adaptability and variability of sandwich structures for broader
applications. This paper proposes a new type of deformable sandwich structure and stretchable design
method that can vary in-plane based on the deformation mechanism of rotating squares. Using experiments
and finite element methods, the influence of stretchable design on the mechanical properties of traditional
honeycomb sandwich structures was tested and a novel stretchable sandwich structure with a 3D lattice core
layer was designed. The results showed that the novel lattice sandwich structures do not reduce out-of-plane
compression performance compared to traditional sandwich panels, and they can deform in-plane and bend
to fit objects. Furthermore, the in-plane negative Poisson's ratio effect of the lattice sandwich structure was
analyzed. The novel sandwich structures have stable deformation during in-plane compression and tension,
with a negative Poisson's ratio effect and a Poisson's ratio of -1.
2. Design methodology and material properties
This section introduces the design method of stretchable sandwich structures, including structural
design and fabrication of the novel sandwich structure. Afterward, material properties are obtained by tensile
testing of splines.
2.1 Structural design of the novel sandwich structures
Based on the traditional sandwich panel structure, some novel sandwich structures were designed,
namely a novel honeycomb sandwich structure (HSS), a novel re-entrant honeycomb sandwich structure
(RHSS) and the novel lattice sandwich structures (LSS) which means the sandwich structures with a lattice
as the core. The novel stretchable sandwich panel structure is composed of sandwich structural units
connected by hinges. Fig. 1 shows the overall design of the novel sandwich structures.
Fig. 1. The design parameters and deformation mechanism of the stretchable sandwich structure. (a) The
design parameters of re-entrant honeycomb and regular hexagonal honeycomb. (b) The design parameters
of the elastic soft hinge (The soft hinges and re-entrant honeycombs are integrated designs.). (b, c, d)
Schematic of a stretchable sandwich panel after assembly with 3 × 3 unit cells and its top view.
The honeycomb sandwich unit is uniquely determined by the sandwich structure wall thickness t, side
length l, panel thickness T, and the unit length L. The connecting pieces are elastic hinges, which can rotate
freely after being connected. The connector is determined by the length and width , diameter d and
thickness . The rotation angle of the panel after stretching is α. Then, the total height H of the stretchable
sandwich panel structure and the side length S of the square area covered by the structure after rotation can
be expressed by the following two formulas:
(1)
(2)
where , n is the cell number of each side.
The relative density of the structure (3 × 3) is defined as the ratio of the cellular volume to the total
volume of the space where it is located. So, the relative density of regular hexagonal honeycombs () and
re-entrant honeycombs () can be expressed separately as:
(3)
(4)
After design, the novel sandwich structure is endowed with in-plane stretchability. The projection
above the structure is a typical rotating square structure, and its deformation mechanism is the same as that
of a 2D rotating square. After the sandwich panel structure is subjected to tension in the plane, the
honeycomb sandwich unit rotates and unfolds around the connector, causing the overall structure to expand.
Assuming that the square is completely rigid and does not deform, the in-plane equivalent Poisson's ratio of
the structure is: v12=v21=1.
2.2 Fabrication of novel sandwich structures
As shown in Fig. 2, the novel sandwich structures were fabricated. The HSS and RHSS are assembled
structures connected by hinges. The geometric parameters of the specimens are in Table 1. The parameters
of their connecting elements are = d = 2 mm, = 4 mm.
The stretchable sandwich structures in this study were produced using additive manufacturing technology.
The specimens consist of thermoplastic polyurethane (TPU) with a density of g/cm3 and were
manufactured using the HS403P printer by Hunan Farsoon High-Technology Ltd. in China. This high-
resolution 3D printer has a resolution ranging from 0.06 to 0.3 mm and uses selective laser sintering (SLS)
technology to fabricate these structures.
Although the four different forms of the structure have the same sandwich layer, their mechanical
properties have the following differences in structure and mechanism. (1) The boundary conditions are
different. As shown in Fig. 2, cells, traditional panels, and stretchable design structures have different
boundary conditions. (2) The interactions between cells are different. The stretchable structure has less
interaction between each cell than the traditional structure due to the existence of a void. And with the
increase of rotation angle, the contact will gradually decrease. (3) The traditional honeycomb forms are
different. The hexagonal honeycomb structure has a positive Poisson’s ratio, while the re-entrant honeycomb
structure has a negative Poisson’s ratio. Different sandwich layers have different transverse strains, so there
are different interactions.
Fig. 2. The 3D printed models of the honeycomb sandwich structure and re-entrant honeycomb sandwich
structure.
Table 1.
Geometric parameters of the cell of sandwich structures.
Models
Direct design parameters (mm)
Calculated parameters
t
l
T
L
H
S2
β
HSS
1
5
3
25
625
53.2%
RHSS
1
5
3
25
625
60.9%
2.3 Material properties of TPU
The material properties of the experimental panels were obtained through uniaxial tensile tests of
standard dumbbell-shaped specimens, as illustrated in Fig. 3. Four TPU standard dumbbell-shaped tensile
test specimens were fabricated and tested at a speed of 100 mm/min, according to the ISO 37:2017 standard.
The universal testing machine used for the uniaxial tension test was a WANCE ETM-104 B from China,
which had a measuring range of 20 kN. The data collected from the specimen tests could be employed to
verify and analyze the finite element model parameters.
Fig. 3. (a) Experimental setup for testing the 3D-printed specimen. (b) Stress-strain curves of the TPU
standard dumbbell specimens.
3. Experimental apparatus and finite element models
The experimental set-up of the novel sandwich structures can be seen in Fig. 4(a). The test of uniaxial
compression was performed using a universal testing machine (WANCE ETM-104 B, China) with a
measuring range of 20 kN and a maximum error of the test force indication value less than 0.5. In
uniaxial compression experiments, the compression rate of sandwich structures was set to 2 mm/min which
is corresponding to the quasi-static compression process.
The modeling software Rhino was used for the creation of the model, and the commercial finite element
software Abaqus/Explicit was used for the numerical simulation. Fig. 4(c) is the FE model of the HSS with
a rotation angle of 0°. For the mesh properties, hexahedron linear solid elements composed of 8 nodes with
reduced integration (C3D8R) were adopted. The “Linear” and “Reduced integration” were chosen in
geometric order and element controls, respectively. As shown in Fig. 4(b), after convergence analysis of
meshes, the size of 0.5 mm is selected, which can take into account the accuracy and efficiency of simulation
calculations. The average Young's modulus, ultimate strength and ultimate strain of the material are 18.7
MPa, 6.6 MPa and 0.69 respectively. The material model is a hyperelastic model, and the strain energy
potential is “Mooney-Rivlin” with the input source of the test data of uniaxial compression. Finally, to
simulate a more accurate stress-strain curve, the strain energy potential order was set to 2. The sandwich
panels are compressed between two discrete rigid bodies. The bottom discrete rigid body is fixed, while the
top rigid body moves down with a smooth analysis step of 100 mm/min. The target time increment in quality
scaling was set to 1×10-5 to improve the efficiency of computer calculation.
Fig. 4. Experimental apparatus and finite element models of the specimen. (a) The Experimental
arrangement. (b) The convergence analysis of the structure. (c) The finite element model of HHS with a
rotation angle of 0°.
4. Results analysis and discussion of the HSS and RHSS
This section presents a comparison between the results obtained from experiments and finite element
simulations, which exhibits a strong correlation between the two methods. Furthermore, the validity of the
finite element model was confirmed. Additionally, a thorough analysis of mechanical behavior and specific
energy absorption was conducted. Finally, the impact of various tensile angles on HSS and RHSS was
explored through finite element simulations.
4.1 Energy-absorbing properties and dense point
Energy-absorbing materials, commonly abbreviated as EA, possess excellent buffer and energy
absorption properties. These properties allow them to absorb a portion of the energy and reduce the impact
force during high-speed collisions or impacts, thereby mitigating potential harm to equipment or personnel.
The widespread use of EA materials in various fields, including automobiles, aircraft, and buildings, is
primarily driven by their ability to enhance safety measures. Sandwich structures, which are lightweight
structures with load-bearing and energy-absorbing properties, are often employed in this regard. Energy
absorption (EA) serves as a crucial indicator of a sandwich structural energy-dissipating capabilities and
can be defined as follows:
(5)
the displacement during compression is represented by , and P denotes the load applied under compression.
Energy efficiency is typically defined as the ratio of energy expended to the outcome of a specific task.
When less energy is used to achieve the same result, energy efficiency is higher, while it is lower when more
energy is required. Thus, high energy efficiency leads to greater energy conservation and more
environmentally friendly use of energy. The displacement of refers to the effective EA phase, which
precedes the densification phase. The displacement of the densification point is determined by the shift
in the energy efficiency peak point, as shown in the following formula:
, (6)
the stress value corresponding to instantaneous strain is denoted as . The densification strain, , can
be determined by maximizing energy absorption efficiency.
When comparing structures of different weights, it is unfair to only consider their total energy
absorption. The energy absorption capacity is also influenced by relative density. Hence, specific energy
absorption (SEA) is calculated and defined as the ratio of energy absorbed per unit mass m:
(7)
4.2 FE model validation and experimental results
The experimental results are verified by simulation, as demonstrated in Fig. 5 and Fig. 6. The
comparisons of the stress-strain curves and deformation modal results from both the experiments and finite
element simulations indicate a good agreement. However, variations in contact characteristics, material
parameters, and boundary conditions between the 3D printed samples and those of the finite element model
may result in experimental errors and unstable data due to unavoidable errors and defects of additive
manufacturing technology. The SLS method used for 3D printing is a common technique that uses lasers to
fuse powder materials into three-dimensional objects with desired shapes. Although it is widely used to
manufacture complex parts, prototypes, and small-batch productions, it can cause some uncertain cracks in
the specimen or powder that are difficult to remove, leading to higher or lower stresses and faster
compaction. In general, such errors fall within an acceptable range, and finite element simulations can be
used to study the designed structure.
The study indicates that the implementation of a stretchable sandwich structure can decrease stress
compared to a traditional single sandwich structure, particularly for HSS. With the stretchable design, the
stress at the point where the curves differ the most is reduced by nearly half due to a difference in the
boundary effect. When compared with THSS and TRHSS, each cell of the stretchable sandwich structure is
isolated, leading to less interaction between individual cells. Therefore, the stress of THSS and TRHSS
tends to be higher. During the compression process, THSS buckles gradually and uniformly, with an
observable stress plateau stage in which the stress remains relatively stable. Overall buckling form consists
of waveform distortion of the elastic body of the outer bar and lateral structure expansion. The TRHSS
shrinks inward in the compression process, displaying a negative Poisson's ratio effect. The internal bar is
the primary stress position, and the structure deformation remains stable. The compaction point of TRHSS
is later than that of THSS and has a higher effective strain. Stress increases throughout the process,
exceeding that of THSS. For HSS 0° and RHSS 0°, each cell operates as a unit cell, so the deformation
mode is similar to THSS and TRHSS. As demonstrated in Fig. 6(d), when the strain reaches 0.24, RHSS
experiences overall right-side buckling, causing the large gap between the experimental load and the finite
element curve load at the beginning. The buckling of a single cell is relatively random, as shown in the finite
element deformation diagram of RHSS at 0°. In the diagram of strain 0.24, the right-most cell is buckling
to the right. The energy absorption histogram and specific energy absorption line diagram of the four
structures are displayed in Fig. 5(c). RHSS has better energy absorption and specific energy absorption
structures because its stress and strain range is higher than that of HSS.
However, for the sandwich structure with a stretchable design, multiple sandwich cells are connected
into a single piece of stretchable sandwich structure through hinges, rather than cutting the whole into
multiple pieces and then connecting. Therefore, it is necessary to compare the relationship between the
stretchable sandwich structure and a single cell, so as to study the influence of stretchable design on the
sandwich cells.
Fig. 5. Some mechanical properties of the four structures. (a) Experimental and finite element stress-strain
curves. (b) Energy efficiency curves of four structures. (c) Energy absorption (EA) and specific energy
absorption (SEA) properties.
Fig. 6. Comparison of deformation modes between experiment and finite element. (a) THSS. (b) TRHSS.
(c) HSS with a rotation angle of 0°. (d) RHSS with a rotation angle of 0°.
4.3 The simulation of unit cells and novel sandwich structures
To investigate the mechanical properties of novel stretchable sandwich structures, which use traditional
hexagonal honeycomb and re-entrant honeycomb as core materials, a series of sandwich structures with
different parameters were designed. Models included single-block HSS (RHSS), HSS (RHSS) with rotation
angles of 0° and 60° (3 × 3), and THSS (TRHSS). Their mechanical properties were analyzed using the
same methods and settings as before. Fig. 7(a) displays a model diagram of the simulated structure, while
Fig. 7(b, c) shows stress-strain curve results obtained by finite element simulation. The energy absorption
histogram and specific energy absorption line chart are presented in Fig. 8.
Fig. 7. (a) Models for finite element simulation. (b) Stress-strain curves of sandwich structures with
hexagonal honeycomb core. (c) Stress-strain curves of sandwich structures with re-entrant honeycomb core.
Fig. 8. The EA histogram and SEA line chart of a sandwich structure. (a) Sandwich structures with
hexagonal honeycomb core. (b) Sandwich structures with re-entrant honeycomb core.
As shown in Fig. 7, the load-bearing capacity of the stretchable sandwich structure is lower compared
to that of the traditional sandwich structure, further confirming the experimental results. Of note, the stress-
strain curves of stretchable sandwich structures with varying rotation angles and sandwich structure cells
are almost identical, indicating that the stretching angle hardly affects the load-bearing capacity of the
structure. This means that the area of the stretchable sandwich structure can be freely adjusted without any
impact on the load-bearing capacity. In various protective structure applications, stretchable sandwich
structures can be utilized as thermal protection material design since the rotating angle can be adjusted to
regulate the hole size in the panel, which then regulates the transfer and distribution of heat flow, thus
enhancing the effectivity and lifespan of the thermal protection material. RHSS has twice the energy
absorption of HSS, which suggests that the re-entrant hexagonal core structure can absorb more energy and
provide better protection when used as a protective component. Due to the similar stress-strain curve, the
energy absorption capabilities of stretchable sandwich structures and sandwich structure cells with different
angles are also close. The load-bearing capacity and energy absorption effect of the stretchable sandwich
structure is equivalent to those of the sandwich structure cells. Compared to the traditional sandwich
structure, the stretchable sandwich structure can be adjusted within the plane, while the design of the soft
hinge allows the structure to bend out of the plane while maintaining panel rigidity. This makes it ideal for
fitting non-flat surfaces during protection. However, the load-bearing capacity and energy absorption effect
of the stretchable sandwich structure is lower than those of the traditional sandwich structure of the same
type. This is because a single block of sandwich structure has stronger interaction and mutual constraint
between each cell, resulting in coordinated deformation. A lattice structure consists of a series of ordered
points that can be any material or structure. When subjected to force, each point of a lattice structure deforms
independently. Therefore, the stretchable sandwich structure designed with a lattice structure as the core
may have similar mechanical properties to the traditional sandwich structure when subjected to out-of-plane
compression due to the characteristics of the lattice structures.
5. Experimental testing and finite element simulation of novel lattice sandwich structures
This section primarily investigates the design parameters and mechanical properties of LSS.
Experiments and finite element analysis were conducted to study the changes in mechanical properties of
the traditional sandwich structure after implementing the stretchable design. The tests on LSS and finite
element analysis were carried out under the same conditions as HSS. Fig. 9 and Table 2 present the models
and dimensions of the three kinds of LSS.
5.1 Structural design of the lattice sandwich structures
Fig. 9 illustrates the design of several lattice sandwich structures (LSS) which consist of classic lattice
structures, namely, square lattice, regular hexagonal lattice, and re-entrant hexagonal lattice. These three
LSS types are respectively referred to as Sq-LSS, He-LSS, and Re-LSS. Fig. 9(b) presents the parameters
of these three LSS. As shown in Fig. 10, the novel lattice sandwich structures were fabricated with the
technology of 3D printing. The LSSs are the structure of the whole. The LSS core is a 3D structure that
possesses a much lower relative density than traditional honeycomb sandwich structures, making it
lightweight, highly porous, and capable of significant deformation. The parameters of the connecting
element are 2 mm cubes. To compare their mechanical properties, uniform parameters were adopted to
ensure that the relative density of the three structures is equal. The calculation formula for the relative
density of the LSS is provided below:
(8)
Fig. 9. The design parameters of the stretchable lattice sandwich structure (LSS). (a) Schematic diagrams
of lattice sandwich structures. (b) The unit cell and parameter design correspond to the three LSS.
Fig. 10. The 3D printed models of the lattice sandwich structures.
Table 2.
Geometric parameters of the cell of sandwich structures.
Models
Direct design parameters (mm)
t
l
T
L
H
Sq-LSS
2
2
20
20
He-LSS
2
2
20
20
120
Re-LSS
2
2
20
20
60
5.2 Experimental results and discussion
Fig. 11 presents the stress-strain curves of lattice sandwich structures (LSS) with three classic lattice
cores, He-LSS, Re-LSS, and Sq-LSS, as well as EA and SEA. These LSSs are composed of rods, which
have simple structures and are easy to design and construct. He-LSS features two orthogonal regular
hexagonal honeycombs, and its stress-strain curve exhibits three stages. In the first stage, the overall
structure mainly experiences continuously increasing elastic reaction force. After reaching the initial peak
stress, the structure enters the stress platform stage during which strain continues to increase but stress
remains constant. The structure in this stage demonstrates uniform reaction force change and stable and
predictable deformation, which is an essential factor for good energy absorption. When the strain reaches
0.55, the He-LSS structure begins to compact, leading to a sharp rise in stress. Re-LSS, on the other hand,
consists of two re-entrant honeycombs, and its stress-strain curve shows a staged linear increase without
obvious differentiation between stages. The curve does not have a clear yield stage and can be considered
as an elastic stress-strain curve similar to a spring. Sq-LSS is formed by two rectangles and is one of the
most classic 3D lattice structures. Its stress-strain curve is similar to that of He-LSS and is also divided into
three stages. However, due to the straight shape of its rods, after entering the plastic stage, straight rods will
cause stress to decrease when they bend, unlike honeycomb structures.
The energy absorption effect of LSS is closely related to the magnitude of stress. Through observing
the stress-strain curves, energy absorption, and specific energy absorption of the same core structure, the
color area in Fig. 11 shows the difference between stretchable sandwich structures and traditional structures.
It indicates that designing stretchable sandwich structures for lattice cores is feasible, and the stress of the
structure will not decrease after the design is implemented. The stretchable design even increases their
bearing capacity and energy absorption.
Fig. 11. Stress-strain curves and EA, SEA properties of the three LSS. (a) He-LSS; (b) Re-LSS; (c) Sq-
LSS.
5.3 Experimental and FE modeling of Re-LSS
The study produced Re-LSS specimens and finite element models stretched to different angles. Fig. 12
and Fig. 13 present the stress-strain curves and deformation mode of Re-LSS with various rotation angles.
Among them, Re-LSS (P) had the highest stress, followed closely by Re-LSS 0°, while Re-LSS 30°
exhibited a lower stress. The experimental data matched well with the finite element curves, but small
defects in the specimens led to slightly lower experiment data compared to the finite element results due to
the limitations of additive manufacturing technology.
Fig. 13 displays the deformation comparison of experiments and finite element results for Re-LSS (3
× 3) at three different angles. The deformation modes of the three structures were identical, and the stress
clouds under the same strain were similar. The deformation revealed that the individual cells of Re-LSS did
not significantly affect each other. The impact of stretchable design on lattice structures was relatively small.
The finite element outcomes corresponded one by one with the experimental data, and thus the established
Re-LSS model can be utilized for further finite element analysis.
Table 3 presents the relevant calculation parameters and Young's modulus of the three LSS cells.
Young's modulus of the structural element is calculated based on the stress-strain curves of these structures.
Sq-LSS cells exhibit the highest Young's modulus, which is 5.3 times that of He-LSS and 9.4 times that of
Re-LSS.
Fig. 12. The stress-strain curves of Re-LSS with different rotation angles.
Fig. 13. The deformation mode of Re-LSS with different rotation angles.
Table 3.
Calculated parameters of the cell of sandwich structures.
Models
(cell)
Calculated parameters
Young’s modulus
S2 (2)
β
E (kPa)
Sq-LSS
400
23.2%
1255
He-LSS
400
23.2%
239
Re-LSS
400
23.2%
133
5.4 Analysis of in-plane auxetic behavior of Re-LSS
The Re-LSS (6 × 6) was established to investigate the in-plane negative Poisson’s ratio performance
and deformation stability of stretchable sandwich structures. The dimensional parameters used in this study
were derived from the initial design data. As illustrated in Fig. 14, cyclic loading was applied to the Re-LSS
30°, which was placed between two discrete rigid bodies. The lower rigid body was fixed, while the upper
rigid body moved downward with a smooth analysis step of 45 mm in one second and then moved vertically
upward after one second, forming a hysteresis curve of one cycle. Fig. 14(a) demonstrates the deformation
modes at each stage of loading, whereas Fig. 14(b) shows the load-displacement curve applied to the
structure, and Fig. 14(c) illustrates the surface area and in-plane Poisson's ratio at different rotation angles
during the entire process. Due to the unique deformation mechanism of rotating squares, the lateral direction
contracts inward during compression. Re-LSS exhibits an obvious negative Poisson's ratio effect, and the
Poisson's ratio remains stable at -1 throughout the process. When the structure is stretched, the surface area
of the structure reaches almost twice its initial state when the rotation angle reaches 90°. On in-plane
compression, the deformation occurs layer by layer, gradually from the bottom to the top, resulting in
multiple peaks and steep drops in the curve (Fig. 14(b)). At a displacement of 40 mm, some rotating squares
come into contact with each other, causing the panel to show an alternating pattern and leading to significant
fluctuations in the curve. Overall, Re-LSS can be stably compressed and stretched in-plane while exhibiting
some bending out-of-plane. During the deformation process, hinge connections experience the main stress
location. The load remains relatively constant during in-plane compression of the structure. The strength of
the connecting rod determines the force required for deformation. The planar tension and compression of a
sandwich plate can be easily achieved using a flexible hinge or another freely rotating connecting rod.
For stretchable design structures, the structure can be regulated for large-scale stretching and
compression. The hinge can also be bent, allowing the original rigid plate structure to achieve a certain
degree of bending. When used as protective components, stretchable structures can regulate the area and fit
surfaces with a certain curvature. When subjected to local loads, stretchable structures exhibit synclastic
curvature. Traditional sandwich structures with a single-panel construction are an approximate overall
deformation mode. In contrast, stretchable structures have better flexibility, and only part of the structure is
deformed. However, the bearing capacity of novel stretchable structures under the same displacement is
lower. The synclastic curvature and local low reaction force of stretchable structures may have potential
applications, such as in wearable devices and soft robots. By utilizing this property, the structure can fit
joints better and move with less binding force. Novel sandwich structures may have broad application
prospects in the field of soft robots that require flexible activities and protection engineering that requires
anti-collision and energy absorption.
Fig. 14. The in-plane auxetic behavior of Re-LSS (6 × 6). (a) The deformation mode under compression
and tension of Re-LSS with the rotation angle of 30°. (b) The force-displacement curves. (c) The surface
area and in-plane Poisson’s ratio of the Re-LSS with different rotation angles.
6. Conclusion
This study proposed a novel concept for designing stretchable sandwich structures that enable high
strain and flexibility, making them applicable in complex, dynamic environments.
Inspired by 2D rotating polygons with negative Poisson's ratio effects, a stretchable honeycomb
sandwich structure was designed using traditional regular hexagonal and re-entrant cores connected by
freely rotating soft hinges. Uniaxial compression experiments showed that the load-bearing capacity of
stretchable sandwich structures decreased compared to traditional ones. Eight different specimens were
designed to investigate their load-bearing capacity and energy absorption performance further. The results
indicated that the rotation angle did not significantly affect mechanical properties, whereas sandwich
structures with re-entrant cores exhibited higher stress and energy absorption ratios than those with regular
hexagonal cores, with more stable deformation behavior. Nevertheless, the stress of the stretchable design
remained lower than that of the traditional structure. To explore the applicability of this method, three lattice
sandwich structures with lattice cores were designed, indicating minimal changes in stress and energy
absorption after stretchable design and not always decreasing. Finally, Re-LSS was selected to analyze its
mechanical properties and deformation mode through experiments and finite element analysis. The pressure-
tension hysteresis loop of Re-LSS within one cycle was studied for the in-plane deformation performance,
which revealed a significant auxetic effect and a Poisson's ratio of -1. By integrating the functional features
of sandwich structures and rotating polygons, it becomes feasible to devise stretchable sandwich structures.
These structures offer adjustable area, porosity, and in-plane auxetic effect while ensuring a certain load-
bearing capacity. In defense engineering and other applications, this study provides suggestions for
improving the utilization and survivability of sandwich structures and overcoming the problem of the limited
deformation of traditional sandwich structures.
It is worth noting that this work is a preliminary exploration and more research should be conducted in
the future to investigate the performance of the proposed sandwich panel. The NSP structure can be further
optimized in materials, hinges, and other areas. Soft materials, plastic materials, or rotating shaft designs
can be used as hinges to make the structure deform more smoothly without rebound or instability. Harder
materials can be used for the panel and core structures to increase protection. A reasonable selection of
surface and core materials can effectively meet the performance requirements of pre-designed panels.
However, sandwich panels with rotating blocks have some limitations. The connecting hinge is the main
part under force and deformation during in-plane tension, so a unique design is required. Meanwhile, the
bending load-bearing capacity of the structure is lower at each connection point of the rotating panel. When
subjected to local loads, the position of the action point needs to be considered. With more researchers
delving into this topic, a more comprehensive understanding of its complexities and challenges can be
expected to be gained.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (grant number
51978330); Qing Lan Project of Jiangsu Province; Natural Science Foundation of Jiangsu Province (grant
number BK20220103).
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