Numerical Study on the Quantitative Structure-Property Relation of Lattice Truss Metals

Jiyeon Kim; Dongmyoung Jung; Yongwoo Kwon

doi:10.2320/matertrans.MT-MB2022015

Abstract

The structure-property relationship of lattice truss metals is investigated by correlating effective properties with quantitative structural parameters such as Maxwell stability parameter, truss thickness, and coordination number excluding porosity. Even at the same porosity, the effective properties can vary many folds due to the structural difference. The elastic modulus and the thermal conductivity of nine representative lattice trusses are calculated using finite element analysis and are correlated with the structural parameters. It is confirmed that the structural dependence of the elastic modulus is higher than that of the thermal conductivity, as stated by M. F. Ashby. The Maxwell stability parameter and the coordination number have more effects than the truss thickness. All structural parameters except the porosity show weak correlation with properties. In other words, each parameter makes a little contribution. Overall, it seems that their combination causes a significant difference in the physical properties.

1. Introduction

Porous metals have many advantages such as their sufficient strength at low density, high specific surface area, sound or vibration absorption, and so on. Therefore, they attract increasing attention in many industrial fields such as automobile or robot, building material, biomaterial, catalyst, battery electrode, and so on. The porous metals may have various pore structures such as open, closed, and lotus, according to the manufacturing method. In the open structure, the pores are connected to each other so that a fluid can flow in the material, in the closed structure, the pores are isolated from each other, and in the lotus structure, the cylindrical pores are aligned in a certain direction.¹^–⁵⁾ Most of the porous metal research has focused on “pores” such as porosity, size and shape of pores, size distribution, etc. Many studies on the physical properties of porous metals have been conducted in that point of view.⁶^–⁸⁾

A porous metal can be treated as a composite of air and metal, and therefore, its physical properties are inevitably different from those of the base metal. The concept of effective properties is used to distinguish the properties of porous metals from those of the base material. Although the effective properties are largely governed by the porosity, they can be doubled purely depending upon the metal structure even at the same porosity. The metal has both the role of load bearing and thermal and/or electrical conduction. Structural dependence does not significantly affect the conduction properties, but it does have a significant effect on the stiffness.⁹⁾ If the structural dependence is well understood, it will be beneficial to the structural design of the porous metal for the modulation of the physical properties.

In this study, we investigate the metallic lattice trusses in the viewpoint of the structural dependence of effective thermal conductivity and elastic modulus. The lattice truss is highly likely to be manufactured and applied in practice according to the development of 3D printing technology in the future, so the results of this study should be useful.¹⁰^–¹⁵⁾ We analyze nine lattice truss structures whose unit cells are shown in Fig. 1. Three lattices are selected for each of cube, octa, and octet families. Effective thermal conductivity and elastic modulus are calculated from heat conduction and compressive loading simulations, respectively. Maxwell stability parameter, truss radius, and coordination number are selected as three structural parameters besides the porosity because our interest is pure structural dependence excluding the porosity effect. As stated, the property can vary significantly even at the same porosity, implying that the structural parameters may play some important roles.

Fig. 1

Lattice trusses investigated in this work. All the lattices have the same porosity of 0.75.

2. Numerical Method

Finite element analysis (FEA) is performed for unit cells of the nine lattice trusses shown in Fig. 1 to calculate their effective properties.¹⁶⁾ All the boundary conditions are set for a plate where the unit cell is infinitely repeated in the x- and y-directions while finitely repeated in the z-direction. Both the compressive loading and the heat conduction are solely along the z-direction in the mechanical and thermal simulations. Only normal components of the effective properties are calculated. The base material is nickel whose properties are elastic modulus 200 GPa, yield strength 150 MPa, Poisson’s ratio 0.29, and thermal conductivity 90 W/m·K. It is also assumed that the base material is isotropic and homogeneous.

The effective elastic modulus of the lattice truss material, E_eff [Pa], is extracted based on Hooke’s law.

\begin{equation} E_{\textit{eff}} = \sigma_{\textit{app}}/\varepsilon_{\textit{res}} \end{equation}

(1)

where σ_app [Pa] and ε_res [dimensionless] are applied stress (input) and resulting strain (output), respectively. The deformation of the lattice truss is shown in Fig. 2(a) where color means local stress in the base metal. Purely elastic deformation is assumed. The boundary conditions are applied assuming that the unit cell is in a plate as mentioned. Lateral surfaces (zx- and yz-planes) freely displace in the tangential directions and are fixed in the normal directions. The bottom surface is fixed (constrained), and the top surface is free to move along the normal direction. A compressive load is applied to the top surface, and then local stress distribution and deformation are calculated.

Fig. 2

Finite element analysis for the octa-cross unit cell (porosity of 0.75). (a) deformation of the Octa-cross unit cell and local stress distribution [Pa] within the solid by compressive loading along z-direction, (b) local temperature distribution [K] within the solid for a temperature difference between top and bottom of the unit cell.

The effective thermal conductivity, k_eff [W/mK], is determined from the applied temperature gradient (input), |∇T|_app, and resulting heat flux (output), q_res [W/m²].

\begin{equation} k_{\textit{eff}} = q_{\textit{res}}/|\nabla T|_{\textit{app}} \end{equation}

(2)

The temperature distribution in the lattice truss is shown in Fig. 2(b) where color means local temperature in the base metal. The top and bottom xy-surfaces are Dirichlet boundaries having fixed temperatures, and the heat flux along the z-direction is calculated as a result. Then, the effective thermal conductivity is obtained. Lateral surfaces (zx- and yz-planes) are insulated (Neumann) boundaries, which is also for a plate condition where heat conduction is purely in the normal direction of the plate.

Three structural parameters are used to quantify a lattice truss as mentioned. The first one is truss radius. One may simply think that a thicker truss is more rigid, which is true for a specific lattice. However, a thicker truss inevitably has lower porosity for the lattice. We want to make a comparison between lattices with the same porosity. It is very difficult to observe the effect of the truss radius completely in isolation. For this reason, the correlation between the truss radius and the stiffness may be weak. The second one is the coordination number (CN), that is, the average number of struts per joint in a unit cell. One may think that higher CN yields higher rigidity. However, this is not simple for different structures with the same porosity, either. The third one is the Maxwell’s stability parameter (MSP), M, a topological measure for a structure consisting of struts and joints.¹⁷^–¹⁹⁾ Equation (3) and (4) are for 2D and 3D structures, respectively.

\begin{equation} M = b - 2j + 3 \end{equation}

(3)

\begin{equation} M = b - 3j + 6 \end{equation}

(4)

where b is the number of struts and j is the number of joints. Figure 3 demonstrates the MSP, M, for 2D structures. In Fig. 3(a), the structure (b = 4, j = 4) has a negative value of M = −1 from eq. (3). Such a structure is called bend dominated, implying that it cannot withstand the load and is prone to collapse because it has a minimum number of connections. In Fig. 3(b), the structure (b = 5, j = 4) has M = 0, called stretch dominated. It has strong connectivity. So, when a load is applied, the structure is stable because the load is distributed over the structure. In Fig. 3(c), the structure (b = 6, j = 4) has M > 0, called over-constrained. Since more struts are connected at a joint, they hold each other when a load is applied. So, it is the most stable structure that yields the least amount of deformation. One also may think that higher MSP may yield higher rigidity. However, both CN and MSP in these arguments for the situation increases by adding more struts, implying lower porosity. The comparison between the different lattices with the same porosity may not be simple. Therefore, we investigate the correlation between the effective properties and the quantitative structural parameters explained above for the 3D lattice trusses.

Fig. 3

The demonstration of Maxwell’s stability criterion: (a) bend-dominated, (b) stretch-dominated, and (c) over-constrained structures.

3. Data and Results

For the nine lattice trusses given in Fig. 1, the effective properties are analyzed for the porosity range of 0.6 to 0.9, and the results are shown in Fig. 4. For the lattices with the same porosity, the difference in thermal conductivity is small while that in the elastic modulus is as large as a few folds depending on the porosity. Lines for Maxwell-Eucken model, upper bound, and power law are added for comparison. All the data for the lattice trusses exist within the theoretical range. The Maxwell-Eucken model in eq. (5) describes the effective properties of a two-phase mixture composite in which two phases are randomly arranged.²⁰^–²²⁾

\begin{equation} k_{\textit{eff}} = k_{1}\frac{2k_{1} + k_{2} - 2(k_{1} - k_{2})\phi_{2}}{2k_{1} + k_{2} + (k_{1} - k_{2})\phi_{2}} \end{equation}

(5)

where k_eff is the effective property of the mixture, k₁ and k₂ are the properties of phase 1 and 2, respectively, and ϕ₂ is the volume fraction of the phase 2. For the thermal conductivity of a porous metal, if the phase 2 is air, ϕ₂ is the porosity and the conductivity k₂ is nearly zero. Then, the effective thermal conductivity of the porous metal is simplified to k_eff ≈ 2k₁(1 − ϕ₂)/(2 + ϕ₂). The upper and lower bounds models in eq. (6) and (7) describe the property of the lamellar structures whose normal directions are perpendicular and parallel to the energy flow, respectively.²³⁾ In other words, the upper and lower bounds correspond to resistors in-parallel and in-series, respectively.

\begin{equation} k_{\textit{eff}} = k_{1}\phi_{1} + k_{2}\phi_{2} \end{equation}

(6)

\begin{equation} k_{\textit{eff}} = \left(\frac{\phi_{1}}{k_{1}} + \frac{\phi_{2}}{k_{2}}\right)^{-1} \end{equation}

(7)

In case of the porous metal, the upper and lower bounds models, eq. (6) and (7), becomes k_eff ≈ k₁(1 − ϕ₂), k_eff ≈ 0, respectively. The power law in eq. (8) is a modification of the upper bound model for a porous material that deviates from the lamella structure such that solid phases are connected in both perpendicular and parallel ways, i.e., intricately.²⁴^,²⁵⁾

\begin{equation} k_{\textit{eff}} = k_{1}(1 - \phi_{2})^{n} \end{equation}

(8)

The exponent n is 3 for elastic modulus and 1.5 for thermal conductivity. Note that there is a limitation to the analytical treatment of real porous structures including the lattice trusses. So, the computational approach is necessary. In Fig. 4, the effective thermal conductivity data match well with the power law line while the effective elastic modulus data lie between the Maxwell-Eucken model and the power law lines. Most importantly, the structural dependence is much more significant in the elastic modulus than the thermal conductivity as M. F. Ashby stated.⁹⁾

Fig. 4

Porosity vs. effective properties. (a) Young’s modulus, (b) thermal conductivity.

From now on, let us investigate the purely structural effects by examining the lattice trusses with the same porosity of 0.75. Figure 5, 6, and 7 show the effects of the truss, the average coordination number at a joint, and the Maxwell stability parameter, respectively. In all these figures, the difference in the thermal conductivity according to structural parameters is very small while that in the elastic modulus is large. Although thermal conductivity plots show a relatively high R²-value, it has little physical significance. Figure 5 shows little dependence on the truss radius as indicated by low R²-values. As mentioned before, the effect of the truss radius is very difficult to separate. Even among the lattices in the same family, there is no consistent trend. On the other hand, Fig. 6 and 7 show some structural dependence on the CN and the MSP. Especially, the elastic modulus of the lattices in the cubic and octa families increases with the CN and the MSP. In case of the octet family, the Octet truss has a lower elastic modulus although it has higher CN and MSP values than the V-octet and the Octet-cross, which may be due to its far smaller truss radius as shown in Fig. 5.

Fig. 5

Effective properties vs. truss thickness for the lattice trusses with the same porosity of 0.75. (a) Young’s modulus, (b) thermal conductivity.

Fig. 6

Effective properties vs. average coordination number for the lattice trusses with the same porosity of 0.75. (a) Young’s modulus, (b) thermal conductivity.

Fig. 7

Effective properties vs. Maxwell stability parameter for the lattice trusses with the same porosity of 0.75. (a) Young’s modulus, (b) thermal conductivity.

Table 1 shows the evaluated MSP values that are in the decreasing order for each family of lattice trusses. For each family, the higher the M value, the larger the elastic modulus tends to be. However, the Octet-truss structure in the Octet family deviates from this trend as mentioned before. It can be said that this generally follows the Ashby’s theory well.⁹⁾ Over-constrained (M > 0) is indicated in red, stretch-dominated (M = 0) is indicated in green, and bend-dominated (M < 0) is indicated in blue in Fig. 5 to 7.

Table 1 Maxwell stability parameters of lattice trusses.

From the analysis results so far, it is possible to infer some design rules for the lattice truss material having desired properties. Consider making a plate consisting of metal lattice trusses that has sufficient rigidity without feeling cold to the touch. In this case, the plate needs to have a low thermal conductivity and sufficient at the same time. First, find the porosity with the desired thermal conductivity in Fig. 1. And, if the highest rigidity is required, select the Octa-cross structure. If the elastic deformation is required, select another structure having a lower elastic modulus. Figure 8 compares the physical properties of all 0.75 porosity lattices. As shown in Fig. 1, the elastic modulus order is mostly similar at different porosity.

Fig. 8

Young’s modulus vs. thermal conductivity for the lattice trusses with the same porosity of 0.75.

4. Conclusions

In this study, the structural dependence of lattice truss properties was quantitatively investigated. By analyzing structures with the same porosity, as stated by Ashby’s theory, the structural dependence is relatively small in the thermal conductivity, while it is quite large in the elastic modulus. On the elastic modulus vs. thermal conductivity plot, the structure with a large E/k value, that is, the structure that exists at the top, has a higher stiffness merit. In other words, a lattice truss with a larger E/k value can have the same stiffness with a smaller porosity, which is effective in lowering the thermal conductivity. Among the structures we calculated, the Octa-cross structure has the largest E/k value. The Maxwell stability parameters and the average coordination number generally represent the elastic modulus tendencies well although they are not perfect. We believe that the results of our research will serve as basic data for controlling the properties of lattice truss and designing new lattice structures.

Acknowledgements

This research was supported in part by the Creative Convergence Research Project (CAP-16-10-KIMS) funded by National Research Council of Science and Technology (NST), in part by 2021 Hongik University, Korea, through the University Research Fund, in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B05050256), and in part the Nano Material Technology Development Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2022M3H4A6A01018639).

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