The sandwich structure of cushioning packaging has an important influence on the cushioning performance. Mathematical fractal theory is an important graphic expression. Based on Hilbert fractal theory, a new sandwich structure was designed. The generation mechanism and recurrence formula of the Hilbert fractal were expressed by Lin’s language, and the second-order Hilbert sandwich structure was constructed from thermoplastic polyurethane. The constitutive model of the hyperelastic body was established by using the finite element method. With the unit mass energy absorption as the optimization goal, the fractal sandwich structure was optimized, and the best result was obtained when the order was 2.5 and the unit layer thickness was 0.75 mm. The Hilbert sandwich structure was compared with the rice-shaped sandwich structure commonly used in industry, and the Hilbert fractal structure had better energy absorption. This has practical significance for the development and application of new cushioning packaging structures.
Fractal structures, originally referred to as broken and irregular fragments, were later used to describe special geometric figures. Compared with simple geometric shapes, fractal structures have various advantages, including self-similarity, multi-scale symmetry, and compact structures. Fractal structures can fill the plane by recursion, increase the number of effective structures, realize the transfer of mechanical loads more efficiently, effectively suppress global failure, and improve the recoverability of structures [
In 1891, the German mathematician Hilbert constructed a curve that could pass through all the points in the square lattice [
Recently, as a powerful mathematical tool, two-scale fractal theory has appeared in the recurrence of fractal theory [
Many scholars at home and abroad have made corresponding research on Hilbert fractal curve. He [
Since a fractal structure has strong space filling ability and recurrence relation, it could achieve a distinct effect when it is used in a buffer structure. In the field of porous concrete, He et al. [
Meza et al. [
At present, the research on the cushioning performances of fractal structures around the world is focused on automobile energy absorbers, but sandwich structures have been seldom considered. The research on Hilbert fractals is rarely used in the field of mechanics. Therefore, in this study, the main characteristic structure of Hilbert fractal is extracted, and the recursive formula is constructed by using logo language, and the three-dimensional model of sandwich panel is constructed. As shown in
A Hilbert curve has three characteristics: ① it can fill the whole plane image, ② it is highly tortuous and continuous but non-derivable, and ③ it has self-similarity [
The essence of the Lin’s system language is the string rewriting technology based on the LOGO language. The Lin’s system is a triple < V, ω, P > whose elements are defined as follows:
The drawing of a Hilbert curve is a process of constantly applying rules to symbols, and each order of the Hilbert curve can be expressed by
These three levels represent Hilbert curves of orders 1, 2, and 3, respectively, as shown in
In this study, TPU was used to prepare the fractal structure by a light-cured 3D printing method, and the room temperature and relative humidity were set to 25°C and 40%, respectively. The printing parameters of all the models are shown in
Printing temperature (°C) | Print layer thickness (mm) | Single-layer exposure time (s) | Exposure intensity (mW/cm^{2}) |
---|---|---|---|
25 | 0.1 | 10 | 9 |
The compression behaviors were tested using an American Instron 3369 Universal Material Testing Machine (American Instron Company, USA) with a load of 50 kN.
ANSYS, a nonlinear finite element software, has high efficiency and accuracy in modeling nonlinear and large-deformation materials such as rubber [
Based on the above mathematical expressions, a three-dimensional model was established. Overall, the model was 100 mm long, 100 mm wide, and 18 mm high. At the same time, in ANSYS, the explicit dynamics module was selected, and a 3D model was entered. Specifically, it consisted of a Hilbert fractal structure, mass block, and support plate, as shown in
Different hyperelastic models have different fitting effects on the complex deformation behaviors of rubber materials [
The linear least squares method [
In ANSYS, the explicit dynamics module was selected. The first step was to establish a finite element model. This consisted of the Hilbert fractal structure, mass block, and support plate. The second step was to set the material properties. The Hilbert fractal structure materials used a Yeoh second-order constitutive model. Rigid blocks and rigid support plates were set according to the material properties of structural steel. The third step is to mesh the model. Firstly, the material attributes are given to the calculation model, and then the mesh is divided in an appropriate way. The entity uses hexahedral element mapping grid, the size of Hilbert fractal structure unit is 1.5 mm, and the total grid unit is 109,600, as shown in
Dumbbell specimens were designed according to the ISO527–2 standard “Testing Methods for Tensile Properties of Plastics” [
According to ANSYS, the contact force between the upper late and the specimen was extracted as a reaction stress. The forward displacement of the upper discrete rigid body was extracted as the compression displacement, and the XY data were output. The XY data were imported, converted into stress–strain values by using software ‘Origin’, and the corresponding curve was drawn. The tested results showed that the finite element model could well reflect the deformation characteristics under different compression displacements, and the results were in good agreement with the compression experiments, as shown in
The stress–strain curves of the TPU were obtained by extracting the XY data from ANSYS. The stress–strain curves of the finite element model and TPU model were plotted, and the curves were in good agreement at each stage, as shown in
In the plateau stress stage, the TPU model was similar to the finite element model, but due to mesh collisions, some of the finite element models showed stress reduction. However, the overall trend was similar, and the errors of key numerical points were in the range of 5%–10%. Therefore, the accuracy of the finite element model was verified, and it could be used for subsequent research.
Photographs of the quasi-static compression test of the Hilbert fractal structure are shown in
According to the quasi-static compression tested results, the compression process of the Hilbert fractal structure could be divided into two parts: A platform area and a densification area. According to the analysis of the quasi-static compression tested data of the Hilbert fractal structure, when the maximum compressive stress of 4.17 MPa was applied, the deformation of the structure was 13.5 mm, and the compressive strain was 0.71. The energy absorption efficiency reflects the energy absorption in the densification stage. When the compressive strain was less than 0.5, the energy efficiency gradually increased, reaching 15% at the highest, and 0.5 was taken as ε_{d} (densification strain), as shown in
The order of the Hilbert fractal structure was varied, and its expression forms were different. In this study, the static compression performance of the Hilbert fractal structures of orders 2, 2.5, and 3 with the same layer thickness and wall thickness was mainly studied. Each structure is shown in
With the increase in the order, the energy-absorbing effect of the fractal structure was also greatly improved. ANSYS finite element simulations showed that the higher the order was, the greater the load under the same deformation was, as shown in
With the increase in the order, the mass and volume of the structure also increased. To obtain the best energy-absorbing structure, it is necessary to integrate factors such as the mass, volume, and energy-absorbing effect. The total energy absorption, energy absorption per unit mass, and energy absorption per unit volume of the different orders are shown in
Quality/g | Volume/cm^{3} | Total energy absorption (EA)/J | Energy absorption per unit mass (SEA_{m}) /J⋅g^{−1} | Energy absorption per unit volume (SEA_{v}) /kJ⋅m^{−3} | |
---|---|---|---|---|---|
2 order | 102.04 | 0.0876 | 0.3081 | 0.0030 | 3517.64 |
2.5 order | 132.64 | 0.1250 | 0.8295 | 0.0063 | 6636.00 |
3 order | 240.70 | 0.1936 | 1.3853 | 0.0058 | 7155.48 |
The unit layer thickness of the Hilbert fractal structure played an extremely important role in the process of the structure bearing pressure. Through the parametric setting of the Hilbert fractal structure in SolidWorks, structures with different unit layer thicknesses were generated. The thicknesses of the upper and lower panels remained unchanged at 2 mm, and the layer thicknesses of all the units in the middle were changed to 0.5, 0.75, and 1 mm. As shown in
A downward velocity load was applied to the parameterized Hilbert fractal structure with a velocity of 100 mm/min and a fixed lower surface, the displacement and equivalent effects in ANSYS were extracted, and the energy absorption per unit mass was manually calculated. The stress–strain energy efficiencies of Hilbert fractal structures with different cell wall thicknesses are shown in
According to the finite element analysis, the Hilbert fractal structure with different wall thicknesses had different buffering effects, and the greater the wall thickness was, the better the buffering effect was. However, with the increase in the wall thickness, the element mass also increased. With the energy absorption per unit mass as the index, the Hilbert response surface analysis can be improved.
The response surface optimization method can optimize multiple targets at the same time, and the best design point can be obtained from the sample points generated from the set parameter values for the optimal design [
In parameter response surface analysis, the response results or test values of the sample points are selected within parameter ranges, and a fitted functional relationship is established by means of regression analysis. For a Hilbert fractal structure, the order (n) and unit layer thickness (t) are taken as input parameters, and the maximum equivalent stress, strain, and mass are taken as output parameters. The results of the response surface analysis are shown in
Different orders (n) and unit layer thicknesses (t) | Quality/g | Total energy absorption (EA)/J | Energy absorption per unit mass (SEA_{m})/J⋅g^{−1} |
---|---|---|---|
2 order (1.00 mm) | 102.04 | 0.3081 | 0.0030 |
2 order (0.75 mm) | 83.23 | 0.2091 | 0.0025 |
2 order (0.50 mm) | 64.06 | 0.1198 | 0.0019 |
2.5 order (1.00 mm) | 132.64 | 0.8295 | 0.0063 |
2.5 order (0.75 mm) | 94.84 | 0.7198 | 0.0076 |
2.5 order (0.50 mm) | 78.57 | 0.4091 | 0.0052 |
3 order (1.00 mm) | 240.70 | 1.3835 | 0.0010 |
3 order (0.75 mm) | 185.95 | 0.9814 | 0.0057 |
3 order (0.50 mm) | 110.27 | 0.6988 | 0.0063 |
Based on the response surface analysis, the system provided the greatest advantages when the order (n) was 2.5 and the unit layer thickness (t) was 0.75 mm, as highlighted in
The “M-shaped” core structure is a kind of special sandwich structure in which members of “M-shaped” structures are added between adjacent vertical studs. This structure is commonly used in polypropylene sandwich panels. In this study, we also used the stereolithography 3D printing method to prepare a rice-shaped sandwich structure, with dimensions of 100 mm × 100 mm × 20 mm to ensure that the size was the same as the Hilbert fractal structure, as shown in
The GB-T 8168–2008 Static Compression Test Method for Packaging Buffer Materials 13 [
The stress–strain curves of the M-shaped and Hilbert fractal structures were examined. The total energy absorption of the Hilbert fractal structure was 120% greater, and the energy absorption per unit mass was 72% greater, as shown in
Through the string rewriting technology of L (Lin’s) language, the recursive formula of the Hilbert fractal structure was constructed using three variables. By using the Yoah second-order hyperelastic constitutive model, experiments and simulations of the Hilbert fractal structure were compared, and the maximum error was 8.21%. Thus, the simulations were basically considered to be reliable.
With the energy absorption per unit mass as the index, the order (n) and the unit layer thickness (t) were optimized. When the order (n) was 2.5 and the unit layer thickness (t) was 0.75, the energy absorption per unit mass increased from 0.0030 to 0.0076 J⋅g^{−1}.
Compared with the commonly used rice-shaped sandwich structure, the Hilbert fractal structure was superior in terms of the total energy absorption and unit mass energy absorption, with values that were 120% and 72% greater, respectively. Thus, this structure has good market application prospects and provides new ideas for the design of sandwich structures.
Thermoplastic polyurethane
Three-dimensional
Total energy absorption
Energy absorption per unit mass
Energy absorption per unit volume
Order
Unit layer thickness