Adaptive enhancement design of triply periodic minimal surface lattice structure based on non-uniform stress distribution

The Schwarz primitive triply periodic minimal surface (P-type TPMS) lattice structures are widely used. However, these lattice structures have weak load-bearing capacity compared with other cellular structures. In this paper, an adaptive enhancement design method based on the non-uniform stress distribution in structures with uniform thickness is proposed to design the P-type TPMS lattice structures with higher mechanical properties. Two types of structures are designed by adjusting the adaptive thickness distribution in the TPMS. One keeps the same relative density, and the other keeps the same of non-enhanced region thickness. Compared with the uniform lattice structure, the elastic modulus for the structure with the same relative density increases by more than 17%, and the yield strength increases by more than 10.2%. Three kinds of TPMS lattice structures are fabricated by laser powder bed fusion (L-PBF) with 316L stainless steel to verify the proposed enhanced design. The manufacture-induced geometric deviation between the as-design and as-printed models is measured by micro X-ray computed tomography (µ-CT) scans. The quasi-static compression experimental results of P-type TPMS lattice structures show that the reinforced structures have stronger elastic moduli, ultimate strengths, and energy absorption capabilities than the homogeneous P-TPMS lattice structure.


Introduction
Inspired by the nature of porous structures such as bone and wood, more and more researchers are trying to produce cellular metals. Cellular metals replace solid material through the void to act as the filling material to meet the load-bearing and lightweight requirements of structures and products in engineering. They have been widely used in many fields such as aerospace, electronics and communications, atomic energy, electrochemistry, petrochemicals, separation, filtration, noise reduction, shock absorption, heat exchange, and biological transplantation [1] . The major advantage is their inherently lightweight property and their ability to effectively absorb compression energy [2] with high area-to-volume ratio characteristics, which can be used as heat exchangers and provide acoustic and vibration damping. In addition, revolutionary additive manufacturing (AM) techniques offer unique manufacturing solutions compared with traditional methods.
Combined with the AM process, cellular metals with complex structures could be fabricated. The design methods of cellular metals can be summarized into three kinds [3] . Owing to their manufacturing technology, two-dimensional (2D) honeycombs and three-dimensional (3D) foams commonly used as core layers are generally random and irregular. In fact, the mechanical and physical properties of AM are unstable. The repetition rate of controlling the internal shape of complex structures is low [4] . Second, the developed lattice structures are prone to stress concentration. The bandgap characteristics of a missing rib lattice structure composed of beam elements [5] have been investigated. The third type of porous structure is the triply-period minimal surface (TPMS) with good mechanical properties and heat conductivity, which has been widely used. The TPMS is a kind of periodic implicit surface with zero mean curvature [6] . Compared with other structures, it has a remarkable feature that not only can be precisely expressed by mathematical functions but also changes its surface structure, porosity, and volume specific surface area to adjust the mechanical and physical properties of lattice materials, giving full play to the advantages of architecture and topology. Moreover, TPMS structures have a smoother transition of the connection points of the multiple struts within the structure, which may reduce stress concentrations, thereby improving the load carrying capacity [3] .
TPMS structures have several unit types including gyroid, diamond, and primitive. Among the performances of these structures, the mechanical properties have been studied in detail, including the elastic behavior [7] , Poisson's ratio [8][9] , the anisotropy [10][11][12] , the fatigue behavior [13] , and the vibration and buckling properties [14][15] . In addition to the mechanical property indexes, other properties of the TPMS have also been explored. Examples include compression properties [16][17][18][19][20][21] , heat exchangers [22] , crashworthiness [23] , phase change energy storage [24] , and bone scaffold design [25] . However, for most TPMS structures, the mechanical properties are different in the direction, and it is possible to control the anisotropic properties or generate isotropic structures by adjusting the TPMS parameters [10] . The mechanical properties of the TPMS are controlled by the solid material, the relative density, and the thickness [26] . A phase field approach was proposed to study the damage caused by cracks on the TPMS and compare the crack damage of structures with different shapes and parameters [27] . Lee et al. [28] found a linear relationship between the shear modulus or the bulk modulus and the relative density by varying the relative density of the P-type structure, indicating a tensile-dominated behavior. However, under the macroscopic uniaxial load, the deformation mechanism is mainly composed of tension and bending. The strength of the bending-dominated deformation mode is quadratically proportional to the relative density. The strength of the tension-dominated deformation mode is higher than that of the bending-dominated deformation mode, but the toughness is lower than that of the bending-dominated deformation mode. By changing the relative density of the TPMS structure, a more efficient structure is obtained from the point of view of strength and weight.
Many attempts have been made to design the TPMS structures by controlling parameters. By studying the mechanical properties and deformation modes of the lattice structures of different polycrystal-inspired square-cells, it is concluded that the properties of symmetric bicrystals and quad-crystals are better than those of polycrystals. The structure of high-quality energy absorption could be designed from the microscopic perspective [29] . The TPMS functionally graded porous scaffolds with varying partial unit cell thickness can be used to select appropriate structural design variables for specific applications. Define different values for the period parameter. Then, the curvature parameter could produce a gradient or non-uniform TPMS porous structure [6] . By adjusting the parameters, the TPMS can not only achieve drastic changes in the surface but also maintain continuity and smoothness. The adjustment of different parameters could also directly control the relative density [30] . Moreover, the gradient TPMS structure was obtained by offsetting different wall thicknesses [31] . To enhance the comprehensive optimization of overall performance of structures, different TPMS gradient models were designed to control parameters such as the thickness, the porosity, and the pore size [16] . By mixing the gradient relative density and different structures [32] , a structure with an adjustable mechanical response was designed. However, the gradient scaffold relies on the regulation of the overall structure. Bonatti and Mohr [11] proposed a bias-function-based method for generating new open-pore shell lattice unit cells. Callens et al. [33] proposed a parametric metamaterial design method for the shell mesh by hyperbolic stitching of multiple materials. Through the optimization of variable shell thickness and shell shape, more design space was provided for the structure design of elastic isotropic porous structures [34] . Designed elastic isotropic structures with anisotropic material constitutive conditions were proposed by a gradient-based optimization method [35] based on the Zener anisotropy ratio. Therefore, an adaptive reinforcement design method was proposed to analyze the quantitative mechanical properties of a unit cell with variable thickness.
Defined by mathematical expressions, the TPMS structures consist of smooth surfaces. However, stress distributes unevenly on the surface during deformation, which undermines the performance of the TPMS unit cell. In the present work, an adaptive reinforcement design based on non-uniform stress distribution in structures with uniform thickness is proposed. Taking the yield strength of materials as the threshold, the structure is divided into enhanced and non-enhanced regions according to the local stress distribution. After the offset of different thicknesses, the two parts are smoothly connected. This enhancement mode preserves the cubic symmetry of the lattice structure. Experimental and numerical analyses are performed to study the mechanical properties. Finite element simulations are carried out to characterize the effects of different reinforcement design gradients on the mechanical properties, and the effectiveness of the reinforcement design is obtained by comparing the mechanical properties of homogeneous structures. The TPMS structures are fabricated by laser powder bed fusion (L-PBF) with 316L stainless steel to verify the proposed enhanced design. The manufacture-induced geometric deviation is quantified by the micro X-ray computed tomography (µ-CT) scanning and modeling comparison. The elastic modulus, the bulk modulus, the shear modulus, the ultimate strength, and the specific energy absorption (SEA) of these enhanced structures under compression are compared with those of the uniform TPMS structures.
2 Architecture of the novel TPMS lattice structure

Mathematical description and modeling
Additive manufactured TMPS lattice structures have received tremendous attention due to their porous architectures and flexible design space in topology, mechanical properties, and mass transport behavior [22][23]30,[36][37][38][39][40][41][42][43] . In this work, the Schwarz primitive triply periodic minimal surface (P-type TPMS) is investigated. The TPMS lattice structures are described by the following implicit notation of trigonometric functions: where ω is the frequency, which can be calculated from the period length L of the structure L = 1/ω, and the constant C controls the shape of the surface.
In this study, all the P-type TPMS lattice structures are modeled by using the isosurface extraction technique and exported as .stl files, which were introduced in detail in our previous study [44] . The thickness of uniform TPMS shells is defined as t. Different TPMS models with various thicknesses t and constants C were reconstructed and discussed. The obtained mechanical properties are compared with those in our previous work [44] . The structure with C = 0 and t = 1.0 is named as C0T10, and the relative density of the structure is about 23.3%. Considering the high mechanical properties and porosity of C0T10, the structures studied in this paper are based on C0T10.

Non-uniform enhanced design method
A novel adaptive thickness enhancement design method is proposed to improve the mechanical properties of TPMS lattice structures, as shown in Fig. 1. By considering the non-uniform stress distribution in TPMS structures with uniform thickness under compression, the local stress concentration could be reduced and the strength could be improved by the thickness distribution design of TPMS lattice structures. The C0T10 TPMS architecture is used to confirm the effectiveness of the proposed enhanced method. The unreinforced primitive solid unit cell was named as U1010 (uniform structure with a thickness of 10 mm). Two enhanced methods are proposed to improve the mechanical properties of U1010. The design structure with the same relative density is named as RXXYY, and that with the same thickness of non-enhanced regions is named as EXXYY. YY is the thickness value of the enhanced region in the TPMS surface, and XX is the thickness value of the other region.  Figure 1 illustrates the workflow of the non-uniform enhancement design method. First, a uniform thickness P-type unit cell finite element model for shell elements is established. Then, uniaxial compressive loads are applied to the unit cell model until the structure reaches its ultimate strength. Second, we use an in-house Python script to extract the Mises stress from the one-eighth unit cell model, and use the yield strength of the parent material as a threshold to classify the elements into two groups. The area to be strengthened is mainly located near the axis of symmetry, as shown in Fig. 1. Then, the elements of two different groups are offset with different thicknesses. In the RXXYY-type design, the thickness of elements with stresses less than the strength of the parent material is reduced to maintain the relative density of the lattice structure, while in the EXXYY-type design, the thickness remains unchanged. Finally, the transition area is smoothly connected, and the enhanced unit cell structure can be obtained by the symmetry of the one-eighth model. The compressive stress-strain curves of TPMS lattice structures contain three stages, i.e., the elastic-plastic stage, the plateau stage, and the densification stage. The P-type TPMS unit cell is modeled by using the isotropic surface extraction technique, and the equivalent force-strain curve of the structure is extracted under uniaxial compression and periodic boundary conditions (PBCs). The effects of the enhanced method and thickness on the mechanical response of P-type TPMS are studied, as shown in Fig. 2. The unit cell length is 10 mm for all numerical models. They are meshed by using tetrahedral elements (C3D4). The tetrahedral mesh is 1.0 mm, which is determined by analyzing mesh convergence. An isotropic elastoplastic constitutive model is adopted for the parent material. The multi-cell is placed between two rigid platens, and the imposed loading is quasi-static displacement. The friction coefficient is 0.2. The elastic-plastic constitutive relation of the parent material is used in the simulations, and the parameters are determined by material testing under tension. Numerical simulations are performed to analyze the mechanical responses of different TPMS lattice structures under compression.

Determination of elastic properties
The representative single-cell approach computes the effective elastic mechanical properties of the lattice structure. The elastic mechanical properties include the uniaxial modulus E, the shear modulus G, the bulk modulus K, and the Zener anisotropy index A. Different P-type TPMS lattice structures are simulated. P-type TPMS lattice structures have cubic symmetry which has three independent elastic constants. The equivalent constitutive equations of lattice structures are shown as follows:  where σ ij denote the stress, ε ij denote the strain, and C 1 , C 2 , and C 3 are three independent components that fully represent the elastic modulus tensor, in which ν is Poisson's ratio. The bulk modulus K is a measure of how much material deforms in response to the pressure around the surface, The elastic anisotropy of cubic crystals can be evaluated by using the Zener index, .
The P-type TPMS lattice structures are strictly cubically symmetric, which also simplifies the analysis of the relationship between the uniaxial stiffness or the yield strength and the loading direction. If a direction is treated as a unit vector, it is computed on one triangle on the unit sphere. The vertices of this triangle are located on one side of the cell (shorthand

Experimental verification 4.1 Materials and manufacturing
Three samples (R0912, U1010, and E1015) are chosen from the designed TPMS lattice structures. An electro optical system (EOS) M290 selective laser melting printer is used to produce 316L stainless steel TPMS lattice structures. The 316L stainless steel powder used is with a particle size generally ranging between 20 µm and 50 µm. The laser power is set to 200 W, and the scanning speed is 1 000 mm/s. The fabricated samples are shown in Fig. 4, and the fabricated weights are shown in Table 1.
The 225 kV µ-CT in Beijing Institute of Technology, China is used to evaluate the geometric deviation and porosity of the as-printed samples. The 3D image-based models are reconstructed and compared to evaluate the surface deviation of CAD design samples and fabricated samples.
The source voltage and current of the X-ray source are set to 140 kV and 71 µA, respectively, and the effective voxel is 15.15 µm.

Quasi-static compressive experiments
Dog bone specimens prepared in the same furnace as the structure are tested under tension, as shown in Fig. 4. The WDW-200 universal testing machine is used to test the samples with a loading speed of 1.0 mm/min at the compressive testing. The displacement and actual experimental process recorded by the video extensometer (RTTS-100) use the displacement difference to estimate the engineering strain.

Manufacture-induced geometric deviation
The enhanced design TPMS lattice structures are fabricated by the L-PBF, as shown in Fig. 4. The weights of the as-design and as-printed structures are shown in Table 1. Due to the manufacturing accuracy of the L-PBF, the enhanced structure R0912 with a non-uniform thickness is 6.3% heavier than the uniform structure U1010.
To quantify the geometric deviation of TPMS lattice structures, µ-CT is used to scan the samples and to reconstruct the 3D models. These as-printed models are compared with the asdesign models. Figures 5(a)-5(c) show the geometric deviation of three TPMS lattice structures. The deviation between the as-design and as-printed models of three TPMS lattice structures in most areas is less than 0.3 mm, and the structural deviation in the thickened region is within 0.15 mm. Thus, the lattice structures designed by the proposed method could be successfully manufactured by the L-PBF. However, the deviation increases at places where the slope of the surface is large, and the largest deviation occurs in the transition region as a result of low manufacturing precision (see Fig. 5(c)). It is worth noting that the deviation in the middle part of the inner surface is significantly large. This indicates that the prepared specimens are actually thinner than the designed specimens, which would result in lower mechanical properties. According to the statistical error curve, the design and manufacturing errors are within 3%. Figure 5(d) shows the pore distribution of the uniform sample prepared by the L-PBF calculated based on the CT scanning section, and the porosity is 0.55%.

Elastic properties
To determine the elastic properties of TPMS lattice structures under small deformation, the directionality of the structures is determined by Young's modulus, which is calculated by the finite element calculation. A common measure of cubic symmetry anisotropy is the Zener ratio [45] . As shown in Fig. 1, after designing a one-eighth non-uniformly thickened Ptype TPMS lattice structure, the unit cell structure is obtained with cubic symmetry through mirror symmetry. Young's modulus surfaces of different structures are shown in Fig. 6, which intuitively shows the anisotropy of uniform structure and reinforced structure based on the P-type surface. If the structure is isotropic, Young's modulus surface is a sphere and A = 1.
The Zener ratio is 2.02 in the uniform TPMS lattice structure U1010, as shown in Fig. 6(a). The modulus along the [100] main direction is smaller than that along the [111] main direction (E100<E110<E111). The modulus of the cell varies with the spatial orientation. Figures 6(b) and 6(c) show the two enhanced structures R0912 and R0813 with the same relative density. The anisotropy indices are A = 1.76 and A = 1.91, respectively. Figures 6(e) and 6(f) show two structures of E1013 and E1015 with different relative densities. The anisotropy indices are A = 1.81 and A = 1.68, respectively. All Young's modulus surfaces of TPMS lattice structures are consistent with the same shape. However, compared with the uniform structure, the anisotropy indexes of these structures with thickness adaptive design are reduced. Young's modulus surface tends to be spherical, and the structures tend to be isotropic.
For more comparability, the normalized Young's modulus E, the normalized bulk modulus K, and the normalized shear modulus G can be derived by  where ρ * represents the relative density of the unit cell, and E b , K b , and G b are the elastic modulus, the bulk modulus, and the shear modulus of the parent material, respectively. Five types of TPMS lattice structures are simulated. The elastic modulus E is the elastic modulus along the [100] principal direction, and the bulk modulus K and the shear modulus G can be calculated from the components of the effective elastic modulus tensor. Figure 7 presents a histogram of the normalized elastic modulus, shear modulus, and bulk modulus of different TPMS lattice structures. It can be seen that the normalized elastic modulus E of these designed structures is significantly improved compared with the uniform structure. In the R group with the same relative density, the elastic modulus E of R0912 is 17.0% higher than that of the uniform structure U1010, and the normalized elastic modulus E is 9.2% higher than that of U1010. In the E group, the normalized elastic modulus of E1015 with a larger weight is 26.5% higher than that of U1010. The normalized shear modulus and bulk modulus of E1013 and E1015 are higher than those of U1010. The normalized bulk moduli of the R group structures are also slightly higher than those of U1010. G of R0813 in the R group increases by 3%, and G in the E group increases by 7.5%. However, the normalized bulk modulus in the R group structures decreases compared with the uniform structure.  The experimental and simulation measured elastic modulus and yield strength of different lattice structures are shown in Figs. 9(a) and 9(b). The stress corresponding to 0.2% plastic strain is taken as the yield strength. In numerical simulations and experimental results, the improvement trends of the enhanced design method are the same. The elastic modulus and the yield strength of E1015 are the highest as a result of higher weight and enhanced design. With the same relative density, the elastic modulus and the yield strength of structures are improved by the adaptive thickness design. The elastic modulus of R0912 is 23.5% higher than that of the uniform structure U1010, and the yield strength is 10.2% higher. The proposed design method is indeed effective to improve the mechanical properties of TPMS lattice structures. Figure 10 compares the compressive deformation process of TPMS lattice structures. The simulation results show the equivalent plastic strainε p of the structures during compression. The failure modes of the three structures are all layer-to-layer modes. The experimental results agree well with the simulation results. The deformation modes of P-type structures are stable. The stress distributions of structures are adjusted by using the adaptive thickness design. Thus, the strength of E1015 and R0912 is improved.

SEA
The SEA is defined as the absorbed energy per unit mass of a structure. The energy absorption capacity of a lattice structure is characterized by calculating the area under the stress-strain curve. The expression of SEA is The SEA of different TPMS lattice structures is shown in Table 2. The SEA of the R0912 structure, which is calculated from the experimental results and numerical simulations, is lower by less than 2% over the uniform structure. For the E1015 structure with high weight, the SEA of the structure is improved by 8.7% in the experiment and 21.3% in the numerical simulation. The results show that the enhanced design method could improve the SEA of TPMS lattice structures effectively.

Conclusions
In this study, an adaptive thickness design method is proposed to improve the mechanical properties of TPMS lattice structures. Two types of lattice structures are designed. One keeps the same relative density, and the other keeps the thickness of non-enhanced regions the same. These lattice structures are fabricated by the L-PBF with 316L stainless steel. The manufacture-induced geometric deviation is quantified by using µ-CT scanning. The elastic modulus, the bulk modulus, the shear modulus, the ultimate strength, and the SEA of these enhanced lattice structures under compression are compared with those of uniform TPMS lattice structures. The adaptive enhancement design can be applied to various TPMS structures with different relative densities, and can be easily combined with other reinforcement design methods. The following conclusions can be drawn.
(i) The as-printed models of enhanced TPMS lattice structures are reconstructed from CT images and compared with the as-designed models. The geometric deviation is within 3%, and the porosity of TPMS lattice structures is lower than 0.55%. The proposed adaptive designed lattice structures can be fabricated by the L-PBF.
(ii) The effective Young's modulus surfaces and Zener ratio of different TPMS lattice structures are calculated. The anisotropy indexes of these enhanced design lattice structures are reduced, and the mechanical responses are closer to isotropic indexes.
(iii) Compared with the uniform lattice structure U1010, the normalized elastic modulus of the R0912 lattice structure increases by more than 17%, and the yield strength of R0912 increases by more than 10.2%.

Conflict of interest The authors declare no conflict of interest.
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