1 Introduction and motivation

Lattice structures are a combination of unit cells arranged periodically in two or three dimensions to form a cell compound. Additive manufacturing (AM) is used to produce lattice structures, since conventional subtractive manufacturing processes are not able to realize such complex internal geometries. AM lattice structures are an interesting field of study because they show the possibility of high mass reduction, while stiffness requirements still get accomplished [1,2,3,4]. A special type of lattice unit cells are auxetic cells [5]. At the macroscopic structural level, auxetic unit cells have a negative Poisson ratio [6,7,8,9]. This property can be used in many technical applications, such as damping or compliant mechanisms [10,11,12] and optical applications [13]. So far, the properties of auxetic structures have not been used as mechanical compensators for aberrations in precision optical systems. This study provides the basis for the development of a novel auxetic mounting structure for optical lenses that compensates for thermal defocus in refractive optical systems. The focus of this work is the investigation of the mechanical behavior of two different lattice structures.

2 Physical description

Powder-bed based manufacturing techniques like laser beam fusion of metals (PBF-LB/M) are common techniques for manufacturing metallic lattice structures [14,15,16,17]. Aluminum alloys or titanium is commonly used due to their low density and high mechanical strength [2, 16, 18, 19]. A challenge of PBF-LB/M, especially in the manufacturing of thin lattice structures, is the adhesion of residual powder on manufactured surfaces [20, 21]. Residual powder affects the effective mechanical properties due to material agglomeration. The quality of numerical simulations on the mechanical behavior of PBF-LB/M manufactured lattice structures will be analyzed by finite-element method (FEM). The influence of different types and sizes of finite elements (FE) is characterized based on experimental load cases. Therefore, the deviations between experiment and FEM are determined by a body-centered cubic cell structure with additional struts in the vertical direction (BCCZ), see Fig. 1 for its computer-aided design (CAD) model. To avoid scaling effects, the samples are arranged with seven BCCZ unit cells in each Cartesian direction [22]. They are manufactured out of Al-40Si alloy (see Fig. 2) [23]. Each unit cell is designed with an edge length of L = 7 mm and a strut diameter of d = 1 mm, resulting in a relative density of ρ* = 15%. Experiments and numerical analyses are performed for three different load cases. Quasi-static tensile, compressive and shear loads are investigated on three samples each.

Fig. 1
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CAD model of the BCCZ unit cell

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Fig. 2
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PBF-LB/M manufactured BCCZ samples on build platform with support structure

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Based on these results of numerical modeling of thin lattice struts, the analytical–numerical investigation of two-dimensional auxetic reentrant cells is set up. Auxetic unit cells are used as passive defocus compensator in the mounting structure. The reentrant cell is a type of auxetic cell and is chosen due to its large possible range of achievable Poisson ratio (see Fig. 3) [8, 24]. This allows a wide range of defocus compensation for the intended application.

Fig. 3
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Auxetic reentrant cell

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By varying the three geometric parameters height H, length L, and angle Θ, it is possible to easily adjust the auxetic Poisson ratio according to Eq. (1) with a = H / L and ΔH > 0 [25]:

$$v_{Aux} \, = \, - \frac{{\cos \theta \left( {a\, - \,\cos \theta } \right)}}{{\left( {\sin \theta } \right)^{2} }}$$
(1)

For isotropic linear-elastic materials, the Poisson ratio is defined as the negative ratio of lateral contraction εy to longitudinal extension εx and can also be written as a function of the two Lamé constants λ and μ [26]:

$${\upnu }_{Iso}=-\frac{{\varepsilon }_{y}}{{\varepsilon }_{x}}= \frac{\lambda }{2(\lambda +\mu )}$$
(2)

The theoretical limits of υ for isotropic solid materials are -1 < υ < 0.5 [26]. For a solid cylinder with diameter D and cylinder height A (see Fig. 4), the Poisson ratio can be calculated with change in diameter ΔD and change in cylinder height ΔA as follows:

Fig. 4
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Poisson ratio of an isotropic linear-elastic cylinder

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$${\upsilon }_{Iso,cylinder}=-\frac{\frac{\Delta D}{D}}{\frac{\Delta A}{A}}$$
(3)

When the deformation of the initial cylinder is caused by a mechanically connected outer cylinder (made of a material with a higher coefficient of thermal expansion (CTE) than the inner cylinder) due to a change in temperature (ΔT), the lateral strain ΔD/D can be calculated from the difference in thermal expansion between the inner and outer cylinder (ΔCTE) as follows:

$$\frac{\mathrm{\Delta D}}{{\text{D}}}=\mathrm{\Delta CTE}\cdot \mathrm{ \Delta T}$$
(4)

Substituting Eq. (4) into Eq. (3) gives the following relationship:

$${\upsilon }_{Iso,cylinder}=-\mathrm{\Delta CTE}\cdot \mathrm{ \Delta T}\frac{A}{\Delta A}$$
(5)

If a solid hollow cylinder is replaced by an auxetic lattice structure consisting out of reentrant cells, a mounting structure for an optical element can be designed using Eqs. (1) and (5). This structure can compensate for a thermal defocus Δf corresponding to the change of cylinder height ΔA. Knowing ΔCTE, ΔT and adjusting the geometric parameters A, H, L, and Θ, a reentrant cell can be designed that fulfills the following condition:

$${\upupsilon }_{Aux}\triangleq {\upupsilon }_{Iso,cylinder}$$
(6)

To validate this concept, the auxetic reentrant cells are analyzed and evaluated in Chapters 3.4 and 3.5. Titanium is the material of choice for this application due to its lower CTE and higher yield strength compared to Al-40Si. These results provide the basis for the development process of an auxetic mounting structure for refractive optical elements consisting of two-dimensional reentrant cells.

3 Investigation of mechanical properties

For the investigation of the mechanical properties, the BCCZ samples are analyzed numerically and experimentally, and the deviations are quantified. Results on the optimal numerical modeling of thin lattice struts, as well as the knowledge of the source and quantity of deviations and manufacturing imperfections, are applied to numerical analyses of the reentrant cell. These analyses begin with the analytic-numerical evaluation of the effective Poisson ratio at the unit cell level. This concept is then extended to two-dimensional reentrant cell compounds consisting out of 3×3 = 9 unit cells and 7×7 = 49 unit cells.

3.1 Numerical analysis of BCCZ cell compounds

The numerical analyses are performed with different types and sizes of FEs. One-dimensional (1D) beam elements (ANSYS BEAM189) and three-dimensional (3D) tetrahedron elements (ANSYS SOLID186) are used. Fine and coarse meshing is applied to each element type. Thus, the BCCZ sample is meshed in four different ways in order to analyze the deviations between numerical and experimental data depending on the element type and size. Table 1 shows the element type, size, and the resulting node number NN.

Table 1 Finite element types and mesh settings for the BCCZ sample
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Both the element type and the element size strongly influence the node number of the FE model and thus the computation time is scaled. Due to the application-oriented goal of reducing computational effort and time, FE models with NN >> 1,000,000 were avoided. Therefore, further refinement of the element size with 3D elements << 0.25 mm was not performed. Simulations were carried out with the same mesh settings of Table 1 for all three load cases. One sample interface is fixed with all six degrees of freedom (DOF), while the other sample interface is loaded with a force F = 1 kN in the corresponding direction of each load case. The resulting equivalent stress is evaluated and scaled to the ultimate tensile strength Rm of the material used (Rm, Al40Si = 275 MPa [27]). This scaling factor is used to calculate the estimated failure force FF as a function of the applied load. All FEM results are shown in Table 2.

Table 2 FEM results for the tensile, compression, and shear load
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3.2 Experimental analysis of BCCZ cell compounds

Each of the three load cases (tensile, compression, shear) was performed with three samples. Quasi-static loading was applied by a moving crosshead at a velocity of v = 0.5 mm/min. Specially designed devices were manufactured to ensure that the samples are decoupled from other stress states (see Fig. 5 as an example of the shear test setup). All experiments were performed with a ZwickRoell Z020 tension–compression testing machine. The load was applied until the samples reached their failure point. Force–displacement diagrams were recorded using a crosshead (accuracy ± 0.1 µm) and a load cell (accuracy ± 200 N).

Fig. 5
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Experimental setup for the shear test

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Figure 6 shows the force–displacement diagram of the three BCCZ samples under tensile loading. All samples show almost perfect linear-elastic material behavior up to failure, as expected for a brittle material [28]. The force F increases almost linearly as the displacement S increases. For all samples, abrupt failure occurs at displacement S = 1.03 – 1.19 mm with maximum failure force FF = 11.90 kN – 13.93 kN.

Fig. 6
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Force–displacement diagram of the BCCZ samples under tensile load

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Figure 7 shows the force–displacement diagram of the BCCZ samples under compressive loading. Compared to the tensile load, the maximum forces are lower and a kind of plastic deformation occurs. After the almost linear-elastic increase, the force F stagnates while the displacement S continues to increase. This is contrary to the brittle material properties of Al-40Si [29]. By visual inspection of the crushed samples under tensile (see Fig. 8) and compression (see Fig. 9) loading, this contradiction is explained by detailed investigation of the different appearance of crack growth in the samples.

Fig. 7
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Force–displacement diagram of the BCCZ samples under compression load

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Fig. 8
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BCCZ sample after tensile load

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Fig. 9
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BCCZ sample after compression load

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While the tensile sample (Fig. 8) fails in the horizontal direction along the entire cross-section, the failure of the compression sample (Fig. 9) appears as a “shear band” under an angle of 45°. This effect is caused by buckling and thus, by piecewise failure of individual struts (initial buckling on the upper left unit cell row). Consequently, the effective stressed cross-section of the next lower horizontal plane is reduced by the left row of cells. As a result, the next row of failing struts moves down to the right. This continues until the failure region reaches the lower plane of unit cells, which finally leads to the observed “shear band” and is represented by the plastic deformation-like region in Fig. 7. This buckling-induced effect has also been observed by other researchers [15, 16, 30, 31].

Figure 10 shows the force–displacement diagram for all three BCCZ samples under shear loading. All samples show almost perfect brittle material behavior. There is no region of plastic deformation and failure occurs abruptly. The maximum failure forces range from FF = 4.90 kN – 6.24 kN, corresponding to a displacement of S = 4.39 mm – 4.73 mm. Table 3 shows the arithmetic means as well as the minimum and maximum deviations of the experimental results.

Fig. 10
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Force–displacement diagram of the BCCZ samples under shear load

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Table 3 Arithmetic mean values and deviations of experimental results
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The mean values are compared with the numerical results from Chapter 3.1. Therefore, the percentage deviations of the 1D FEM and 3D FEM results from the experimental data are calculated as follows:

$${\Delta }_{1D/3D}= \left|1-\frac{{F}_{1D/3D}}{{F}_{Exp}}\right|$$
(7)

Table 4 summarizes all the results and the corresponding deviations. The numerically predicted failure forces provide conservative results for all load cases. The results from 1D FE modeling (6% – 28%) show lower deviations than 3D FE modeling (21% – 38%), which is contrary to the basic understanding of FEM [32], but has also been observed by other researchers [33, 34]. The high deviations of 3D FE modeling can be explained by the fact that the solutions have not yet converged due to insufficiently fine meshing. Further mesh refinement could not be implemented, because the computer capacity is limited by about 6 million nodes. Another future option to overcome these limitations would be to develop bulk materials with effective material properties to replace complex lattice structures in FE analyses. Also noticeable are the smaller but still high deviations in the 1D FE modeling (except for Δ1D-Compression = 6%). This can be explained by simplified assumptions in the physical description of finite beam elements [35].

Table 4 Evaluation of experimental and numerical results
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3.3 Additive manufacturing of thin lattice structures for precision applications

It is shown that large-volume lattice structures with strut diameters of 1 mm can be manufactured by PBF-LB/M and the structural behavior can be predicted by FEM. Although the FEM results show partially large deviations, the experimental data exceed the numerically determined failure forces. One reason for the deviations is powder adhesion, which influences the structural behavior. Maximum agglomerations are observed from 250 µm to 500 µm (see Fig. 11). Consequently, d ≥ 1 mm is set as the minimum strut diameter for technical applications. This is mainly caused by overhanging struts since the lattice structures are manufactured without support structures (see Fig. 2). These material agglomerations overlay the general surface roughness of PBF-LB/M manufactured components, which was experimentally determined to be Sq = 4.3 µm for Al-40Si without any further surface treatment in a previous study [20]. For precision technical applications, agglomeration due to powder adhesion can be disadvantageous because the structural behavior cannot be precisely determined. Therefore, on the one hand strut diameters d < 1 mm are not considered in the development of the future auxetic optical lens mount. On the other hand, surface treatment techniques for thin AM lattices need to be investigated to reduce production-related agglomeration.

Fig. 11
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PBF-LB/M manufactured BCCZ unit cell with material agglomerations

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3.4 Numerical analysis of reentrant unit cells

Based on the previous results, the following simulations are carried out with sufficiently fine-meshed models to ensure numerically converged results. Ti6Al4V is selected as the material model. It has a low CTETi6Al4V = 9 ppm / K and a high yield strength RP, Ti6Al4V = 850 MPa [36]. The low CTETi6Al4V leads to a higher defocus compensation Δf and the high yield strength RP, Ti6Al4V allows more deformation and thus a higher compensatable temperature range. Numerical analyses of the reentrant unit cell are performed to investigate the load-implied deformation behavior and to compare it with the analytical description in Eq. (1). Therefore, two different CAD models of the reentrant cell are designed. Figure 12 shows on the left and center the CAD model designed for meshing with 1D beam elements. In Fig. 13, the CAD model designed for meshing with 3D tetrahedron elements is shown on the left and in the middle. Both FE models are loaded with a lateral displacement of 0.001 mm (B) and the resulting longitudinal deformation is used to calculate the numerical Poisson ratio.

Fig. 12
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CAD model with boundary conditions, FE mesh and deformation behavior of a reentrant cell modeled with 1D beam elements

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Fig. 13
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CAD model with boundary conditions, FE mesh and deformation behavior of a reentrant cell modeled with 3D tetrahedron elements

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The sensitivity of the three geometrical parameters H, L and Θ to the resulting Poisson ratio is investigated by parameter studies. Angles of Θ = 25° – 80°, lengths of L = 5 mm – 12 mm, heights of H = 10 mm – 25 mm, and a fixed strut thickness of t = 1 mm are defined as input. Fig. 14 summarizes the results of the sensitivity analysis of the parameter study performed with the 1D FE model.

Fig. 14
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Sensitivity analysis of Poisson ratio of the reentrant unit cell modeled with 1D beam elements

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Height H, length L and angle Θ are each shown with their corresponding percentage sensitivity to the result of the analytical and numerical Poisson ratio and the deviation between the analytical and numerical calculation. The differences in the sensitivities for all parameters H, L, and Θ between the analytical (blue) and the 1D numerical (orange) calculation of Poisson ratio are negligibly small. Θ has the highest influence, while L has the least influence on the Poisson ratio. The sensitivities to the deviation between the analytical and numerical calculations are shown in grey. It can be seen that Θ has the greatest influence on the deviation of the Poisson ratio from the analytical to the numerical calculation, while H has no influence. Therefore, Fig. 15 shows the relative deviation of the numerical Poisson ratio from the analytical as a function of Θ and L. The plot is a response surface from more than 100 single parameter samples. The blue area shows deviations below 10%, so most of the analyzed parameter space is in good correlation. Only for small length L and large angle Θ (upper left corner) or very large length L and very small angle Θ (lower right corner) deviations between 15 and 40% are observed.

Fig. 15
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Numerical deviation of Poisson ratio of the reentrant unit cell modeled with 1D beam elements

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The same conceptual analysis is performed for the reentrant unit cell modeled with 3D tetrahedron elements. Figure 16 shows the sensitivities of H, L, and Θ to the analytical and numerical Poisson ratios and to the deviation between them. The general behavior is similar to the 1D FE modeling, only the sensitivities of H and L changed in percentage. The differences in the sensitivities of H and L to the analytical (blue) and numerical (orange) Poisson ratios increased to 9%. Angle Θ is the most sensitive parameter to both, Poisson ratio and the analytical–numerical deviation. Again, height H has no effect on the analytical–numerical deviation.

Fig. 16
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Sensitivity analysis of Poisson ratio of the reentrant unit cell modeled with 3D tetrahedron elements

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The deviation of the numerical from the analytical Poisson ratio is shown in Fig. 17. Since H has no significant effect on the relative deviation, it is shown only as a function of the parameters L and Θ. All blue areas represent deviations ≤ 10%. Most of the analyzed parameter space shows a good correlation between analytical and numerical modeling. Only for small lengths and large angles (upper left corner) deviations from 20 to 50% and for small lengths and small angles (lower left corner) deviations from 15 to 30% occur. Sensitivities for both types of FE modeling, 1D beam and 3D tetrahedron, show generally good correlation for all geometric parameters. In addition, the distribution and number of analytical–numerical deviations (compare Figs. 15 and 17) show good agreement for both FE modeling variants.

Fig. 17
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Numerical deviation of Poisson ratio of the reentrant unit cell modeled with 3D tetrahedron elements

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For final validation, the analytical description of the reentrant unit cell is compared with numerical calculations of reentrant cell compounds. Therefore, cell compounds of 3 × 3 = 9 unit cells and 7 × 7 = 49 unit cells are designed and numerically investigated in the following section.

3.5 Numerical analysis of reentrant cell compounds

For the numerical analyses of reentrant cell compounds, only 3D FE modeling is considered. The investigated parameter space is changed to a fixed height H = 15 mm (due to negligible influence on the numerical deviation), the ranges of Θ and L are increased to Θ = 45° – 85° and L = 7.5 mm – 25 mm. According to Eq. (5), higher defocus compensation Δf = ΔA is obtained with lower negative Poisson ratio, and according to Eq. (1), lower negative Poisson ratio is obtained with higher Θ and smaller a = H / L.

As an example, the lateral deformation behavior of a 3–3 cell compound under longitudinal compression is shown in Fig. 18. The analysis setup and Poisson ratio evaluation are performed in the same way as described for the reentrant unit cell in Chapter 3.4. Sensitivities are not evaluated, only numerical deviations from the analytical description are examined.

Fig. 18
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Lateral deformation behavior of a 3–3 reentrant cell compound

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The results for 3–3 cell compound are shown as a response surface in Fig. 19. The numerical deviation of the Poisson ratio from the analytical description is plotted as a function of Θ and L. Over the entire parameter space, there are no deviations > 20%. Most of the investigated parameter space shows deviations ≤ 10%. The highest deviations occur only around smaller angles and larger lengths (lower right corner).

Fig. 19
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Numerical deviation of Poisson ratio of the 3–3 reentrant cell compound

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The response surface resulting from the 7–7 reentrant cell compound is shown in Fig. 20. Similar behavior is observed for the numerical deviations of the Poisson ratio. The overall maximum deviations are negligibly higher (≈ 25%), but they occur for the same set of parameters. The area with deviations ≤ 10% is covered by most of the analyzed parameter space.

Fig. 20
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Numerical deviation of Poisson ratio of the 7–7 reentrant cell compound

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Since, according to Gibson et al. [37], at least 7 unit cells in each spatial direction are necessary to account for size effects of lattice structures, it can be concluded that the analytical description of the reentrant unit cell can also be used to describe the linear-elastic mechanical behavior of cell compounds.

4 Conclusion and outlook

This research paper investigates the mechanical behavior of additively manufactured lattice structures to establish a development process for designing a novel auxetic mounting structure to compensate for thermal defocus in refractive optical systems.

For this purpose, BCCZ lattice structures were analyzed numerically and experimentally under tensile, compressive, and shear loads. In good agreement with the results of other researchers, deviations ranging from 6 to 38% occur due to simplifications in the FE modeling, limitations in computational capacity and manufacturing effects (powder adhesion) that were not considered in the FE analyses [33, 34]. These findings were applied to the analytical–numerical study of auxetic reentrant cells. Good correlation (deviations < 10%) of the numerical results with the analytical description of the effective Poisson ratio is observed over a large parameter space by analyzing unit cells, 3–3 cell compounds and 7–7 cell compounds. The results provide fundamentals for the further development and design process of an auxetic mounting structure. The next steps in this research include experimental investigations to validate the numerical results of the reentrant cells based on cyclic tension–compression tests. For this purpose, a cell geometry will be designed, which will be used as a unit cell for the mounting structure consisting of two-dimensional cell compounds in a radial-symmetrical arrangement.