JOM

DOI: 10.1007/s11837-015-1322-y Ó 2015 The Minerals, Metals & Materials Society

Additive Manufacturing of Metal Cellular Structures: Design and Fabrication LI YANG,1,4 OLA HARRYSSON,2 DENIS CORMIER,3 HARVEY WEST,2 HAIJUN GONG,1 and BRENT STUCKER1 1.—Department of Industrial Engineering, University of Louisville, Louisville, KY 40292, USA. 2.—Department of Industrial & System Engineering, North Carolina State University, Raleigh, NC 27695, USA. 3.—Department of Industrial & System Engineering, Rochester Institute of Technology, Rochester, NY 14623, USA. 4.—e-mail: [email protected]

With the rapid development of additive manufacturing (AM), high-quality fabrication of lightweight design-efficient structures no longer poses an insurmountable challenge. On the other hand, much of the current research and development with AM technologies still focuses on material and process development. With the design for additive manufacturing in mind, this article explores the design issue for lightweight cellular structures that could be efficiently realized via AM processes. A unit-cell-based modeling approach that combines experimentation and limited-scale simulation was demonstrated, and it was suggested that this approach could potentially lead to computationally efficient design optimizations with the lightweight structures in future applications.

INTRODUCTION In recent years, additive manufacturing (AM) has received considerable attention for its potential in transforming global manufacturing industries. In some of the leading application areas such as aerospace, biomedical, and automotive, AM has demonstrated unprecedented flexibility for part consolidation, function integration, and lightweighting of structure and component designs.1–3 Lightweight design is one area that AM addresses well where other traditional manufacturing technologies become largely impractical.4–6 Because of the requirements for design optimization and the complex resulting geometries, full-freedom lightweight design often involves multiscale analysis, which makes pure finiteelement based design computationally demanding. Topology optimization methods have been investigated as a feasible solution.7–9 However, currently topology optimization has been used only for maximum stiffness design to date, and it lacks sufficient capabilities for hierarchical design integrations at multiple scales. Another common approach that attempts to achieve the blalance between design convenience/efficiency and design accuracy is the unit cell design approach, which uses representative geometries to efficiently perform design tasks.

Cellular structures are characterized by the large percentage of porosity in the solids10 and can often be treated as assemblies of cells with solid edges or faces.11 In past decades, cellular structures have received a lot of interest due to their promising potential in a wide range of engineering applications.10–13 It is well known that cellular structures exhibit a combination of mechanical and thermal properties such as high stiffness-to-weight ratio,11,13–15 high energy absorption,16–18 and low heat conductivity,19 which are highly desired in applications such as aerospace structures, automobiles components, stiffening spatial fillers, impact cushions, thermal insulations, sandwich cores, vibration dampers, and civil structures.13 In addition, due to their large surface areas, cellular structures are also extensively used as catalysts and filters 10,14,20,21 and biological interfaces.22–24 The unique advantage of adopting analytical modeling-based cellular design is that it becomes possible to integrate multiple design objectives and to achieve a combination of mechanical properties through design optimization. This is enabled by the explicit geometry–property relationships of the representative unit cells, which have simplified forms and could be readily manipulated via optimization methods. Traditionally, due to manufacturability

Yang, Harrysson, Cormier, West, Gong, and Stucker

issues, limited theories exist for the design methodology of three-dimensional (3D) cellular structures that can be verified experimentally. In addition, the use of AM processes also brings about manufacturing-related issues such as staircase effects, surface finish, and defects, which also need to be considered in the design. This article demonstrates a feasible cellular design approach based on analytical modeling of unit cell geometries and experimental verification using powder-bed fusion AM processes. Mechanical properties of a re-entrant auxetic cellular geometry including Poisson’s ratio, elastic modulus, and yield strength were derived based on the solid material and the geometric design of the unit cell structure. The design tool also attempted to incorporate the design factors related to multiple manufacturing issues with the AM processes. The results were verified by experiments and simulations to verify the accuracy of the model and the boundary effect of the actual structures.

Fig. 1. Unit cell designs.

DESIGN Unit cells have been widely employed for the design of cellular structures due to their ability to simplify analysis and represent characteristics of the geometries.11,13 The unit cell structure could be effectively treated as a network of beams or trusses connected by rotational or rigid joints. Both rotational hinge joints and rigid joint assumptions have been adopted in previous modeling analyses with cellular structures.25,26 Preliminary study with metal cellular parts fabricated by selective laser melting (SLM) or electron beam melting (EBM) processes exhibit bending dominated strut behavior. Therefore, in this article the rigid joint assumption was adopted for metal cellular structures. Figure 1 shows the cellular structure investigated in this study. Figure 1a shows a re-entrant auxetic cellular structure, which is expected to exhibit negative Poisson’s ratios in all three principal directions. The auxetic structure was of interest due to its exceptional shear performance and energy absorption.27,28 Figure 1b shows the geometric design parameters of this re-entrant auxetic structure, which includes the length of the vertical strut H, the length of the re-entrant strut L, re-entrant angle h, and the dimensions of the cross sections of the strut (not shown in Fig. 1b). In this article, some modeling details of the auxetic cellular design are presented to demonstrate the design methodology. Additional details about the modeling are presented elsewhere.29–31 Due to structural symmetry, the re-entrant auxetic structure is represented by two characteristic directions, which are the z direction and x or y directions as shown in Fig. 1a. The modeling was established for an ideal structure that has infinite numbers of unit cells in each of the three principal directions, which minimizes boundary effects. Consider a remote compressive stress that is applied on the re-entrant

Fig. 2. Uniaxial loading in z direction.

auxetic structure from either z or x directions. The equivalent force F applied to each unit cell could be readily obtained via force equilibrium, which is not shown here. Depending on the loading condition, different types of structural symmetry could be further applied to the unit cell structure and simplify it. Figures 2 and 3 show the simplified structures under z-directional and x-directional compressive loading. Note that due to the lower degree of symmetry, the structural analysis for the x direction is more complex, and the re-entrant struts at two directions are subject to different loading cases. Using the similar approach that was developed by Onck et al.32 and further incorporates shearinduced deformations, a set of equations was obtained for the elastic modulus E and Poisson’s ratios m of the re-entrant auxetic structure as:29–31 2

mzx ¼ 

L 6 ðEt 2 þ 5GÞ cos hða  cos hÞ L2 sin2 h Et2

mxz ¼ 

2

h 4a þ 6 sin 5G þ E

sin2 h cos hða  cos hÞ

(1)

(2)

Additive Manufacturing of Metal Cellular Structures: Design and Fabrication

Fig. 3. Uniaxial loading in x (equivalent to y) direction.

the structure was considered to fully yield when the entire cross section of an arbitrary strut in the unit cell achieves the yield strength of the material. For a relatively ductile material with low-strain-rate sensitivity, this approach could provide a rough estimation of the structural yield strength. The stress distribution of a beam under the combination of normal stress, shear stress, bending moment, and the maximum allowable stress (i.e., the yield strength) of the re-entrant auxetic structure under x-directional and z-directional loading were obtained by solving for Eqs. 5 and 6: ðr2Y 

9r2z L4 sin6 h 3 Þt 16t4

4rY

4r2 L4 sin4 h cos2 hrY 2L3 sin3 h rz  z  ¼ 36r2 L4 sin6 h 8 t 64r2Y  1 t4

(5)

ðr2Y 

9L4 ðacos hÞ2 sin2 h cos2 hr2x 3 Þt 16t4



 2 2 h 2 16L4 cos 2 þ 4 ða  cos hÞ sin hrY rx sin h    144L4 ðacos hÞ2 sin2 h cos2 hr2x 256r2Y  t t4

Fig. 4. Modulus of the structure.

¼

Ez ¼

Ex ¼

ða  cos hÞ 2aL2 sin2 h Et2

L2

cos2

þ

4rY

L4 ð2Et 4

þ

3L2 Þ sin4 5Gt2

h

1 1  L2 3 hða  cos hÞ ð2Et4 þ 5Gt 2Þ

(3)

(4)

where a = H/L and G is the shear modulus of the solid material. Shear was included as it was found that for medium relative density designs, the shearinduced deflection of the struts could not be neglected.30 In the calculation of yield strength of the structures, the von Mises criterion was used, and

4

(6)

L3 ða  cos hÞ sin h cos hrx 4

With the analytical equations (Eqs. 1–6), the mechanical properties of the re-entrant auxetic structure could be readily designed. Figures 4 and 5 show the design maps of the modulus and strength of the structure in z and x directions, respectively. The mechanical properties of the re-entrant auxetic cellular structure in two representative directions exhibit opposite trends, which coincide with the Poisson’s ratio trends in these directions. With larger negative Poisson’s ratio, the mechanical properties of the structure also tend to increase. This makes it possible to use the Poisson’s ratio as a design indicator for quick design screening. It is worth noting that there exist two possible failure modes when the

Yang, Harrysson, Cormier, West, Gong, and Stucker

Fig. 5. Strength of the structure.

Table I. Geometrical design for experiments

A1/B1 A2/B2 A3/B3 A4/B4

h (°)

H (mm)

L (mm)

Relative density (%)

mxz

mzx

70 70 45 45

5.5 4.14 7.74/ 7.8 6.9/ 6.46

4.25 3.2 3.77 3.3

13.41 22.06 22.45 27.18

–2.71 –2.71 –0.52 –0.51

–0.37 –0.37 –1.90 –1.90

structure was loaded in the z direction, which are the yield failure and the elastic failure. The elastic failure mode is common for cellular structures with struts oriented along the loading directions, and it could be catastrophic to the structures. Therefore, by obtaining design maps as shown in Figs. 4 and 5, designers can choose combinations of mechanical properties in the two principal directions and avoid undesirable mechanical behavior. Another interesting observation is that for cellular structures, the Poisson’s ratios of the structures are also controllable. For solid materials, the Poisson’s ratio is usually considered a constant,

which determines the bulk deformation behavior of the structures. With the control of Poisson’s ratio, it becomes possible now to carefully tune the structures to respond to multiaxis stress in specific ways, which might possess practical values in applications such as energy absorption and assembly fitting. EXPERIMENTS When the designed cellular structures are converted into real structures, some of the assumptions made during modeling are no longer applicable and

Additive Manufacturing of Metal Cellular Structures: Design and Fabrication

therefore need to be addressed by experiments and simulations. This information is then incorporated into the model designs as corrective factors or used for further analysis and more accurate first-principle-based modeling. Experiments could provide accurate information about the properties of the actual structures. However, care must be taken to properly decouple factors that could introduce errors into the analysis. Due to the computational difficulty, the simulation of the cellular structure could only be performed on limited scales. On the other hand, the simulation results usually allow for detailed investigations of the boundary effects without complicating the discussions with manufacturing-related issues. The proper combination of both methods facilitates rapid verification of the cellular models and can further provide insights for model improvement. In this study, re-entrant auxetic cellular samples made from Ti6Al4 V using EBM was used for the experimental evaluations. Four auxetic designs were evaluated as listed in Table I. The cross-sectional geometry of all the struts was designed as a square. Design groups A and B had slight differences mainly due to strut thickness. Group A has a strut thickness of 0.8 mm and group B has a strut thickness of 0.9 mm. From Table I, it is apparent that for both groups, design pairs 1,2 and 3,4 enables evaluations of the effect of both relative

density and Poisson’s ratio on the mechanical properties of the samples. All the samples were designed to have 4 9 4 9 4 number of unit cell repetitions. The samples were fabricated successfully with minimum dimensional variations upon inspection. Figure 6 shows some of the fabricated samples. It was also revealed upon closer inspection that there exist multiple issues that are closely related to the EBM process and the realization of cellular structures. As shown in Fig. 7a, due to the thermal dissipation during the fabrication process, the surface of the parts fabricated via EBM and other powder-bed fusion AM processes suffer from significant surface sintering, which reduces the geometrical accuracy of the structures and creates crack-initiation sites. The surface roughness also affects the accurate determination of effective strut dimensions. To obtain a ‘‘mechanically equivalent dimension,’’ which could be used in the mechanical property predictions in the analytical model, the minimum fully solid-sectional dimension was used as the dimension of the struts as shown in Fig. 7. Another issue was related to the realization of the cellular designs. When the conceptual cellular geometry as shown in Fig. 1b was realized by fabrication, the thickness of the struts with the actual structure resulted in an enlarged joint area and shorter effective strut length as shown in Fig. 8. To compensate for this, a corrective factor was

Fig. 6. Ti6Al4 V samples via EBM.

Fig. 8. Effective strut length.

Fig. 7. Surface sintering.

Yang, Harrysson, Cormier, West, Gong, and Stucker

Fig. 9. Comparison of modulus. Fig. 10. Comparison of strength.

introduced to account for the reduction of effective strut length DL as: DL ¼

t 2 sin h

(7)

Also, as all the cellular geometries were modeled in SolidWorks (Dassault Syste`mes, Waltham, MA), finite-element simulations with the modeled structures were also carried out in COMSOL (COMSOL Inc., Burlington, MA). Quasi-static elastic analyses were performed; therefore, no special setting for the meshing was used except to define the mesh that was smaller than the thickness of the struts. Yield was predicted by checking for the threshold stress values, which spread through the entire thickness of an arbitrary strut. After all the correction factors were incorporated, the comparison between the analytical model predictions, the finite-element simulation, and the experimental results are shown in Figs. 9 and 10. In general, the simulation results agree very well with the experimental results, which implies that the material properties of the cellular structure are consistent and predictable. The theory agrees very well with the experiments in the z direction, while significant discrepancies were observed in the x direction. Further simulation-based analysis revealed that the size effect of the re-entrant auxetic cellular

structure is minimum, and therefore it did not contribute to the significant error in the theoretical prediction. The primary cause of inaccuracy was speculated to be contributed by the higher-order stress coupling effects due to the reduced degree of symmetry in the structure when it was loaded in the x direction. It was found via simulation that when the re-entrant auxetic structure was loaded in the x direction, the re-entrant struts were also subject to torsional stress as a result of the compatibility constraint between struts. As shown in Fig. 11, the additional torsional stress tends to introduce additional deformation in the struts, and when the deformation was restricted by the continuity requirement of material at the joints, it could potentially result in significant change of stress distribution within the struts, thus contributing to the deviation of the modulus and strength of the structures compared to the model. Higher-order coupling effects could complicate the unit cell modeling and reduce the efficiency of this approach, and further work is needed to investigate this issue. DISCUSSION In the geometric modeling of this work, the material properties of the cellular structures were assumed to be homogeneous, which is a very rough approximation. In fact, it has been known that AM

Additive Manufacturing of Metal Cellular Structures: Design and Fabrication

processes introduce intrinsic anisotropy into the structure due to the layered process. In addition, the energy input pattern during the fabrication process could result in different microstructure across the samples and thus potentially contribute to the variation of material properties. Through preliminary work with the selective laser melting (SLM) process, it was found that the process control could significantly affect the quality of the solid struts.32 Overheating becomes more significant as shown in Fig. 12, which could result in complete failure of the features with certain sizes. In Fig. 12, all three struts were fabricated with the same process parameters and settings, and the scanning path was designed to include a contour and a singleline hatching. However, as the diameter of the laser beam was around 0.1 mm, significant overheating occurred with the 0.2 mm feature under the designed scanning strategy, which resulted in the deformation of the resulting feature as shown in Fig. 13. As a conclusion, certain thin features might

Fig. 11. Torsional warping.

Fig. 12. Scanning path analysis for small feature sizes 0.1–0.3 mm.

Fig. 13. Thin features generated by SLM process.

not be readily manufacturable under certain process settings, and a complete thermal-mechanical analysis is needed to fully characterize the potential effect of dimensions and scanning strategies on the quality of the parts. In the design, it is often desired to treat the cellular structures as a homogeneous continuum, which could then be treated with traditional design analysis techniques. The material matrix of a solid material also includes shear modulus values; therefore, to apply homogenization, the shear modulus also needs to be derived for the cellular structures using a similar approach as demonstrated in this article. However, this approach faces significant challenges in accounting for the boundary effects. First, actual structures are often geometrically complex, which would result in incomplete unit cells if conformal modeling was not used. Incomplete unit cells do not contribute to the structure in the same manner, and therefore they could potentially affect the overall performance of the structures. Second, at the boundaries, the loss of structural symmetry also results in additional force and moment components, which create a ‘‘weak spot’’ for the cellular structures. Third, the semidiscontinuous solid phases in the cellular structures could also create localized effects under certain loading conditions. Therefore, it is still impractical to apply the cellular unit modeling approach to the lightweight design of actual structures, and more work is needed in this area to understand the boundary issue. On the other hand, due to the mathematical simplicity as demonstrated in this work, the unit cell design could be easily incorporated into other larger scale design models and optimization

Yang, Harrysson, Cormier, West, Gong, and Stucker

methods, therefore opening up opportunities for advanced multiobjective structural design for lightweight structures. CONCLUSION In this article, a hybrid approach that combines analytical modeling, experimentation and simulation was demonstrated for the design of 3D cellular structures. A re-entrant auxetic cellular structure was modeled, and the results showed good agreement between the theories and the experiments when structural symmetry is maximized. Additional modeling work is needed when the symmetry breaks down. In addition, manufacturing factors such as energy input, scanning strategy, and defects are closely coupled with the actual cellular design, which will need to be incorporated into the geometric models to enable more accurate designs. The results obtained from this work still lack sufficient applicability with actual structures; however, the demonstrated methodology could be further developed and expanded, which has the mathematical advantage of being readily incorporated into other design tools. REFERENCES 1. T. Catts, GE printing engine fuel nozzles propels $6 billion market, Bloomberg, http://www.bloomberg.com/news/201311-12/ge-printing-engine-fuel-nozzles-propels-6-billion-mar ket.html. Accessed 12 Nov 2013. 2. GE Reports, This electron gun builds jet engines, http:// www.gereports.com/post/94658699280/this-electron-gun-buildsjet-engines. Accessed 17 Aug 2014. 3. B. Coxworth, World’s first 3d-printed titanium bicycle frame could lead to cheaper, lighter bikes, Gizmag, http://www.giz mag.com/3d-printed-titanium-bicycle-frame/30760/. Accessed 8 Feb 2014. 4. Y.-H. Lee and K.-J. Kang, Mater. Des. 30, 4434 (2009). 5. A.-J. Wang, R.S. Kumar, and D.L. McDowell, Mechan. Adv. Mater. Struct. 12, 185 (2005). 6. V.S. Deshpande, N.A. Fleck, M.F. Ashby, and J. Mechan, Phys. Solids 49, 1747 (2001).

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