Design method of density gradient minimal surface structure

Abstract

This invention discloses a design method for a density gradient minimized surface structure, aiming to solve the problem of poor mechanical property balance in existing uniform minimized surface structures and obtain a minimized surface structure with balanced performance. The method comprises the following steps: 1. Using the RegionPlot3D command in Mathematica software, implicit function plotting is performed to obtain a Gyroid-type uniform minimized surface, where parameter l is the cell size and C is an isoparameter; 2. Determining l and C, C is adjusted according to an nth-order curve (n≥1) to change the relative density along the Z-axis from ρ... 匀 The gradient becomes 2ρ 匀 A three-dimensional model is generated, where L=ml and m is a natural number; 3. Samples are prepared by selective laser melting additive manufacturing. Experiments have shown that the secondary gradient structure can achieve a balance between static and dynamic mechanical properties, and is suitable for energy absorption, protection and other fields.

Description

Technical Field

[0001] This solution relates to the field of minimal surface structure design, specifically to a design method for a surface structure with minimal density gradient. Background Technology

[0002] Energy-absorbing materials possess a unique property during their deformation process: the ability to effectively dissipate impact kinetic energy from external sources. In this process, the material converts impact energy into deformation energy through changes in its internal microstructure, thereby reducing or preventing damage to surrounding structures. This ability to dissipate impact energy makes energy-absorbing materials particularly important in various protective and cushioning applications.

[0003] Minimal surface structures are an emerging type of porous material, characterized by a unique surface with uniform thickness and continuous curvature in three-dimensional space. A key feature of this structure is that every point on its surface is at the same distance from its surroundings, resulting in a smooth and continuous surface. Uniform minimal surface structures have wide applications in engineering and architectural design, particularly in the design of porous materials, such as lightweight, high-strength aircraft and buildings.

[0004] In the comprehensive evaluation system of mechanical metamaterials, commonly used evaluation dimensions include: relative density, strength, plateau stress, load efficiency, and specific energy absorption. In current designs of minimal surface structures, only one dimension of the material's properties are typically outstanding, while other dimensions are relatively poor, resulting in generally poor overall balance across dimensions. This limits the material's use in final applications. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to provide a design method for a surface structure with minimal density gradient.

[0006] The purpose of this invention is to provide a design method for a surface structure with minimal density gradient, so as to obtain a minimal surface structure with relatively balanced performance.

[0007] The specific technical solution of the present invention to solve the above-mentioned technical problems is as follows: Step 1: Use the implicit function plotting command of the mathematical software Mathematica RegionPlot3D to obtain a uniform minimum surface.

[0008] Step 2: Determine the cell size that makes up the uniform minimal surface. l The isoparameter C is adjusted to form a functional expression. The isoparameter C changes according to the nth curve (n≥1), generating a three-dimensional model with a minimal surface structure with gradient density distribution.

[0009] Step 3: Prepare gradient structure specimens using additive manufacturing technology; Step 4: Perform a static compression test on the above-mentioned structural specimen to obtain the stress-strain curve of the reinforced structure and calculate the energy absorption of the structure.

[0010] Step 5: Conduct a dynamic impact test on the above-mentioned structural specimen to test the stress-strain curve of the above-mentioned structural specimen under dynamic impact load.

[0011] Furthermore, the uniform structure minimum surface is a Gyroid-type minimum surface, and the formula for a Gyroid-type minimum surface is as follows:

[0012] In the formula, l Let C be the length of the cell, and C be the equivalent parameter.

[0013] Furthermore, the density of the minimum surface of the density gradient structure varies along the Z-axis.

[0014] Furthermore, the isoparameter C is adjusted so that it varies according to an nth-order curve, as follows: The uniform minimal surface structure obtained in step one has a relative density of... (0< <1), the parameters corresponding to the formula for the uniform minimum surface are: The parameters corresponding to the formula for the surface with minimum density gradient are: With relative density Let Z be the dependent variable and its spatial location be the independent variable. The initial value of the independent variable Z is 0, and the final value of the independent variable Z is 0. L， And L = ml (m is a natural number), constructing an nth-degree function, the relative density at Z = 0 is... , so that at Z=L =2 This yields a minimal surface structure in which the relative density varies with the spatial position of the sample.

[0015] Furthermore, the density gradient structure has a minimum curved surface cell size. l The diameter is 4mm, and the length (L) is 20mm. Attached Figure Description

[0016] Figure 1 This is a table of linear density gradient changes and a corresponding 3D model diagram; Figure 2 This is a table of secondary density gradient changes and a corresponding 3D model diagram; Figure 3 This is a table of three density gradient changes and a corresponding 3D model diagram; Figure 4 These are actual images of uniform density structure samples and linear density gradient samples. Figure 5 These are actual images of secondary and tertiary density gradient samples. Figure 6 This is a schematic diagram of a quasi-static test setup; Figure 7 This is a schematic diagram of the dynamic testing device; Figure 8 It is the quasi-static nominal stress-strain curve of the G-type minimal surface mechanical metamaterial; Figure 9 This is a deformation diagram of the quasi-static deformation mode of a G-type minimal surface mechanical metamaterial; Figure 10 It is the nominal stress-strain curve of a surface mechanical metamaterial with minimal density gradient; Figure 11 This is a static deformation diagram of a surface-mechanical metamaterial with a minimum density gradient; Figure 12 It is the dynamic nominal stress-strain curve of a uniform G-type minimal surface mechanical metamaterial; Figure 13 This is a dynamic deformation diagram of a uniform G-shaped minimal surface mechanical metamaterial; Figure 14 It is the dynamic stress-strain curve of a surface mechanical metamaterial with a minimum linear gradient; Figure 15 This is a dynamic deformation diagram of a linear density gradient G-type minimum surface mechanical metamaterial; Figure 16 It is the dynamic stress-strain curve of a quadratic density gradient G-type minimal surface mechanical metamaterial; Figure 17 This is a dynamic deformation diagram of a G-type minimum surface mechanical metamaterial with a secondary density gradient; Figure 18 It is the dynamic stress-strain curve of a triple density gradient G-type minimal surface mechanical metamaterial; Figure 19 This is a dynamic deformation diagram of a cubic gradient G-type minimal surface mechanical metamaterial; Figure 20 This is the result of the static mechanical property evaluation of the metamaterial; Figure 21 It refers to the dynamic energy absorption characteristics of mechanical metamaterials. Detailed Implementation

[0017] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0018] Step 1: Use the implicit function plotting command of Mathematica RegionPlot3D to obtain the minimum surface of the Gyroid-type uniform structure; the formula for the Gyroid-type minimum surface is as follows:

[0019] In the formula, l Let C be the length of the cell, and C be the equivalent parameter.

[0020] Step 2: Determine the cell size that makes up the uniform minimal surface. l And the isoparameter C, the isoparameter C is adjusted to vary according to an nth-order curve (n≥1), generating a 3D model of a minimal surface structure with a gradient density distribution. Specifically, the isoparameter C is adjusted to vary according to an nth-order curve as follows: The uniform minimal surface structure obtained in step one has a relative density of (0< <1), the parameters corresponding to the formula for the uniform minimum surface are: The parameters corresponding to the formula for the surface with minimum density gradient are: With relative density Let Z be the dependent variable and its spatial location be the independent variable. The initial value of the independent variable Z is 0, and the final value of the independent variable Z is 0. L Construct an nth-degree function such that Z = L place =2 This yields a minimal surface structure in which the relative density varies with the spatial position of the sample.

[0021] The density of the minimum surface of the density gradient structure varies along the Z-axis; Step 3: Prepare gradient structure specimens using additive manufacturing technology; Specifically, in this example, based on the implicit functional equations of the Gyroid minimal surface structure, the following three density gradient structures were designed, including: Figure 1 The linear density gradient structure (FG-L) shown is as follows: Figure 2 The secondary density gradient structure (FG-Q) shown and as Figure 3 The three density gradient structures shown (FG-T) exhibit different density distributions along the Z-axis, and the cell sizes vary. l The diameter is 4mm, and the length (L) is 20mm.

[0022] Note Figures 1 to 3 In the figure, the horizontal axis represents the side length of the sample, in mm.

[0023] The relative density variations of the three density gradient structures are shown in Table 1: Table 1. Design of minimum surface structures with different gradients in Gyroid

[0024] Uniform surface-based mechanical metamaterial specimens with minimal density gradients were prepared using selective laser melting (SLM). Eight specimens of each type were printed. The mass of all additively manufactured mechanical metamaterials was measured using an electronic balance with a measurement accuracy of 0.01g, and the three-dimensional geometric dimensions of the mechanical metamaterials were measured using a digital vernier caliper with an accuracy of 0.01mm. The measurement results are shown in Table 2.

[0025] Table 2 Geometric Information of Additive Manufacturing Minimal Surface Mechanics Metamaterials

[0026] Step 4: Perform a static compression test on the above-mentioned structural specimen to obtain the stress-strain curve of the density gradient structure and calculate the energy absorption of the structure.

[0027] In this project, an electronic universal testing machine was used to conduct quasi-static compression tests on a minimal curved surface lattice structure with a nominal strain rate of 0.001 / s. The quasi-static test setup is shown in Figure 6.

[0028] After the test, the force-displacement curve is exported using the universal testing machine's testing software. The stress is obtained by dividing the force by the cross-sectional area of ​​the specimen. (1) The strain is obtained by removing the space from the initial length of the specimen, i.e. (2) The test results are as follows: Figure 8 shows the quasi-static nominal stress-strain curve of a G-type minimal surface mechanical metamaterial with a relative density of 30%. The horizontal axis represents nominal strain, and the vertical axis represents nominal stress, both in MPa. The curve can be clearly divided into three stages: the linear elastic stage, the plateau segment, and the dense segment.

[0029] Figure 9 shows the quasi-static deformation modes of a uniform G-type minimal surface mechanical metamaterial with a relative density of 30%. When the strain is 0, the sample is in an unloaded state. When the strain reaches 0.2, it can be seen from the figure that only slight deformation occurs in the upper part of the sample, while the lower end of the sample has developed an inclined deformation band from left to right. When the strain reaches 0.4, it can be seen from the figure that the deformation band that appeared earlier has developed into a denser state, while the rest of the sample maintains its initial state relatively well. Subsequent deformation extends outward from the shear deformation area. When the strain is 0.6, the pores that exist in the mechanical metamaterial itself have disappeared, and the thin walls are in contact and compressing each other. This corresponds to the stage of rapid stress growth in the stress-strain curve. At this time, the sample has gradually entered the compaction stage.

[0030] Figure 10 shows the nominal stress-strain curves of G-type minimal surface mechanical metamaterials with different density gradient forms under quasi-static loading. The figure shows that the stress-strain curves of the three types of mechanical metamaterials with different density gradient forms all exhibit continuous strengthening characteristics. The initial peak stresses of the three density gradients are different, with the linear density gradient structure having the highest stress and the cubic density gradient structure having the lowest. The nominal stress-strain curves of the quadratic and cubic density gradient mechanical metamaterials have similar geometric characteristics, both showing a significant stress drop after exceeding the initial peak stress, followed by continuous strengthening. These results indicate that the macroscopic mechanical response of minimal surface mechanical metamaterials is related to the form of the density gradient, and the mechanical properties can be controlled by changing the density gradient.

[0031] Figure 11 shows the deformation modes of G-type minimal surface mechanical metamaterials with different density gradients under quasi-static loading. As can be seen from the figure, the initial deformation of the three different density gradient minimal surface mechanical metamaterials all first appears in the low-density region, and then the deformation gradually extends to the high-density region. Compared with the deformation mode of homogeneous mechanical metamaterials, which is mainly characterized by tilted deformation bands, the layer-by-layer collapse and crushing deformation mode exhibited by the density gradient mechanical metamaterials is more stable. As the deformation gradually extends to the high-density region, the stress amplitude in the stress-strain curve continuously increases.

[0032] Step 5: Conduct a dynamic impact test on the above-mentioned structural specimen to test the stress-strain curve of the above-mentioned structural specimen under dynamic impact load.

[0033] To obtain the complete response process of the specimen from initial to compaction under dynamic loading, a dynamic impact test was conducted based on a traditional split Hopkinson bar device. The test loading is shown in Figure 7, with a loading strain rate of approximately 1500 s⁻¹. -1 In the experiment, the specimen was attached to a bullet (impact rod). An acceleration device propelled the bullet to impact the incident rod at a specific velocity. Strain gauges attached to the surface of the incident rod output electrical signals, which were then converted into strain signals by a signal acquisition system and stored in a computer. The specimen was susceptible to failure during the dynamic impact, and the high-speed flying debris posed a certain danger. To protect personnel and the experimental equipment, protective devices were installed at the impact point to prevent injury from flying debris. A high-speed camera was placed at the impact point to record the entire dynamic deformation process of the specimen. After the experiment, the stress-time history curve of the specimen was obtained. The strain-time history curve was obtained by processing the high-speed photographic images, and the dynamic stress-strain curve of the specimen was obtained by combining the two.

[0034] The test results are as follows: Figure 12 It is the stress-strain curve of a uniform G-type minimal surface mechanical metamaterial under dynamic impact load, compared to the quasi-static nominal stress-strain curve ( Figure 8 The dynamic stress fluctuates continuously during the loading process, and its initial dynamic peak stress is higher than that of the static initial peak stress, indicating that increasing the loading speed can improve the mechanical properties of mechanical metamaterials. This is also the reason for studying the dynamic mechanical properties of mechanical metamaterials.

[0035] Figure 13 shows the deformation evolution of a uniform G-type minimal surface mechanical metamaterial under dynamic impact loading. As can be seen from the figure, the tilting deformation band that appears under quasi-static loading is not obvious at this point. The deformation is initially concentrated at the impact end, and then the deformation begins to concentrate in the area where the specimen contacts the impact rod.

[0036] Figure 14 shows the stress-strain curves obtained by loading along the negative and positive gradient directions of the linear density gradient metamaterial, respectively. From the figure, it can be observed that the geometric characteristics of the curves are strongly correlated with the loading direction. When loading along the positive gradient direction, the curve shows a continuous upward trend; when loading along the negative gradient direction, the curve initially oscillates due to the higher strength in the high-density region at the impact end. After a period of time, it also shows a continuous upward trend.

[0037] Figure 15 illustrates the deformation evolution along the negative and positive gradient directions of the linear density gradient metamaterial, respectively. As can be seen from the figure, when loaded along the negative gradient direction, no deformation occurs at the impact end; instead, deformation begins on the side furthest from the impact end, creating a deformation band perpendicular to the loading direction, which then develops into layer-by-layer crushing. When loaded along the positive gradient direction, deformation begins first at the impact end, creating a deformation band perpendicular to the loading direction, followed by a layer-by-layer crushing deformation process. These two deformation modes explain the continuous strengthening segment of the curve. Simultaneously, because the impact strength does not reach the yield strength of the high-density region when loaded along the negative gradient direction, the specimen does not deform significantly and maintains a high stress amplitude. This explains the initial high stress amplitude followed by a stable stress strengthening phenomenon in the stress-strain curve.

[0038] Figure 16 shows the stress-time history curves for loading along the negative and positive gradient directions of the quadratic density gradient minimum surface mechanical metamaterial, respectively. It can be seen from the figure that the geometry of the curves is similar to that of the linear density gradient structure, but the stress amplitude is slightly lower.

[0039] Figure 17 illustrates the deformation evolution under loading along the negative and positive gradient directions of the surface mechanical metamaterial with minimal secondary density gradient. As can be seen from the figure, when loading along the negative gradient direction, deformation does not occur at the impact end but rather at the contact point between the specimen and the impactor, exhibiting a layer-by-layer crushing deformation process. When loading along the positive gradient direction, deformation initially occurs at the impact end, and then gradually extends to the right from the impact point. These two deformation modes are the reason for the formation of its continuously reinforced curve segment.

[0040] Figure 18 shows the stress-strain curves of the metamaterial with a minimum cubic density gradient under loading along the negative and positive gradient directions. Similar to the stress-strain curves of the linear and quadratic gradient structures, when loading along the negative gradient direction, stress fluctuations occur initially, followed by a stable increase in stress. When loading along the positive gradient direction, only continuous strengthening is observed.

[0041] Figure 19 illustrates the deformation evolution of the cubic density gradient minimal surface mechanical metamaterial under loading along the negative and positive gradient directions, respectively. As can be seen from the figure, the dynamic deformation mode of the cubic gradient minimal surface mechanical metamaterial is similar to that of the previous two density gradient structures. However, under loading along the negative gradient direction, the sample exhibits significant lateral deformation, which is due to the longer low-density region.

[0042] Establishing a comprehensive evaluation system is fundamental to studying the static and dynamic energy absorption characteristics of minimal surface mechanical metamaterials. Static and dynamic tests are conducted on homogeneous and non-homogeneous minimal surface mechanical metamaterials to obtain quasi-static and dynamic stress-strain curves. Numerous indicators characterize the mechanical properties of mechanical metamaterials, such as strength and energy absorption. Considering only a single indicator cannot reasonably and comprehensively evaluate the performance of mechanical metamaterials, nor can it provide effective references for configuration design and optimization. Therefore, to comprehensively consider various mechanical properties, an evaluation method that comprehensively considers static and dynamic mechanical properties is needed. This method integrates indicators related to strength and energy absorption characteristics, mainly including relative density, strength, plateau stress, specific energy absorption, and load efficiency.

[0043] Relative density is the primary factor influencing the mechanical properties of mechanical metamaterials, carrying more weight than all other influencing factors. In lightweight porous materials, relative density is defined as the ratio of the density of the mechanical metamaterial to the density of its matrix material. Strength represents the near-linear relationship of mechanical properties up to a certain strength, after which a plateau is reached. Strength can be defined as the stress corresponding to 0.2% plastic strain.

[0044] Plateau stress is the average stress of a mechanical metamaterial throughout the entire compression process. It is an indicator of the overall structural load-bearing capacity and can be expressed as: (3) Energy absorption: This refers to the total energy absorption of a lattice material during the entire compression process due to plastic deformation, and can be expressed as: (4) Specific energy absorption is the energy absorbed by a mechanical metamaterial under unit mass during plastic deformation, and can be expressed as: in, It is the relative density of mechanical metamaterials. It is the density of the matrix material (5) Load efficiency is the stress efficiency of a lattice material during deformation, and can be expressed as: (6) After introducing the above indicators, a comprehensive evaluation system for evaluating the mechanical properties of mechanical metamaterials is constructed, including five dimensions: relative density, strength, plateau stress, load efficiency, and specific energy absorption.

[0045] Based on the above comprehensive evaluation system, the evaluation results of the static mechanical properties of homogeneous and three density gradient minimum surface mechanical metamaterials are as follows: Figure 20 As shown in the figure, the uniform density structure has the best strength, the linear density gradient structure has the best specific energy absorption, the cubic density gradient structure has the best load efficiency, and the quadratic gradient structure achieves a good balance among various mechanical properties.

[0046] The evaluation results of the dynamic mechanical properties of homogeneous and three density gradient minimum surface metamaterials are as follows: Figure 21 As shown in the figure, for a uniform density gradient, increasing the loading speed only improves the strength, while the increase in specific energy absorption and plateau stress is relatively small, resulting in a significant decrease in load efficiency. For density gradient structures, loading along the negative gradient direction yields better strength, plateau stress, and specific energy absorption compared to loading along the positive gradient direction. Loading along the linear density gradient direction, although the strength decreases compared to the uniform structure, all other properties are significantly improved. Loading along the positive gradient direction effectively protects the impacted object, and the secondary density gradient achieves a balance among various mechanical properties without significantly reducing specific energy absorption (-0.33%).

[0047] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A design method for a surface structure with minimal density gradient, characterized in that, Includes the following steps: Step 1: Use the implicit function plotting command of the Mathematica RegionPlot3D software to obtain the minimum surface with uniform structure; Step 2: Determine the cell size l and isoparameter C of the uniform minimal surface, and adjust the isoparameter C to change according to the nth curve (n≥1) to generate a three-dimensional model of the minimal surface structure with gradient density distribution. Step 3: Prepare gradient structure samples using additive manufacturing technology.

2. The design method for a surface structure with minimal density gradient according to claim 1, characterized in that, The uniform structure minimal surface is a Gyroid-type minimal surface, as shown in the following formula: In the formula, l Let C be the length of the cell, and C be the equivalent parameter.

3. The design method for a surface structure with minimal density gradient according to claim 1 or 2, characterized in that, The density of the minimum surface of the density gradient structure varies along the Z-axis.

4. The method of designing a density gradient minimal surface structure according to claim 3, wherein The isoparameter C is adjusted to vary according to an nth-order curve, as follows: The relative density of the uniform minimum surface structure obtained in step one is... ρ 匀 (0<ρ) 匀 <1） The parameters corresponding to the formula for a uniform minimum surface are: C 匀 The parameters corresponding to the formula for the surface with minimum density gradient are: With relative density Let Z be the dependent variable and its spatial location be the independent variable. The initial value of the independent variable Z is 0, and the final value of the independent variable Z is 0. L， And L = ml (m is a natural number), construct an nth-degree function such that at Z = L =2 This yields a minimal surface structure in which the relative density varies with the spatial position of the sample.

5. The method of designing a density gradient minimal surface structure according to claim 4, wherein The density gradient structure extremely small surface structure cell size l L is 20 mm.

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