Residual Stress Generation in Additive Manufacturing of Complex Lattice Geometries

Abstract

Residual stresses developed during additive manufacturing (AM) can influence the mechanical performance of structural components in their intended applications. In this study, thermomechanical residual stress simulations of the laser powder bed fusion (LPBF) process are conducted for both simplified (plate and cube-shaped) geometries as well as five complex lattice geometries fabricated with Inconel 718. These simulations are conducted with the commercial software package Simufact Additive^©, which uses a nonlinear finite element analysis and layer-by-layer averaging approach in determining residual stresses. To verify the efficacy of the Simufact Additive^© simulations, numerical results for the plate and cube-shape geometries are analyzed for convergence and compared to experimental residual stress results available in the literature. Numerical residual stress results are subsequently compared for five complex lattice geometries. Results suggest that lattice geometry can play a significant role in the distribution and magnitude of residual stresses, which are significant in some applications.

1 Introduction

The presence of residual stress in additively manufactured structures is being considered for structures that have unique features in which the residual stress variations would be complex. The structures chosen are referred to as triply periodic minimal surface (TPMS) lattice structures. A characteristic for these designs is material surrounding uniquely defined voids. A primary advantage lattice structures present is the ability to dampen the effect of impact. A study has been carried out by researchers that relate the use of lattice design as internal structures within a projectile. There are other specific applications where lattice structures can be applied, such as within a turbine engine of an aircraft (Ref 1). Reference 2 discusses potential application to debris collision with an aircraft (Ref 2). The current study provides a numeric evaluation of TPMS lattice structures to determine the resulting residual stress produced through the laser powder bed fusion (LPBF) additive manufacturing process.

Additive manufacturing (AM) processes can create complex parts characterized with fine features throughout. In comparison with well-known manufacturing techniques, parts created with AM are built with material being added through the process of model construction, whereas the well-known manufacturing technique removes, casts, or forges material to create parts. The LPBF process is also known as selective laser melting (SLM), which is an AM process used to create builds by sintering a metal powder via energy from a laser beam (Ref 3). To begin printing using the LPBF process, a file of a solid model is uploaded to a software package that is used with the 3D printer. The printer reads the file to ‘slice’ the model into layers, where the build will ultimately be printed using a layer-by-layer approach. Process parameters such as laser power, laser speed, hatch spacing, and layer thickness define the printing process. Metal powder is added to a chamber on one side of the build plate as the next step for preparation of the printing process.

An example of an LPBF printing process and printer layout is presented in Fig. 1. When the printing process begins, the first chamber moves slightly upward until enough material can be rolled across the printer to uniformly coat the build plate by the recoater roller. The laser is aimed through focal lenses and is reflected from an angled mirror to the powder bed. The laser sinters a select area of the material as determined by the slicing of the model file.

Fig. 1

Reproduced with permission from EMPA under the CC BY 3.0 license

Example of the laser powder bed fusion process (Ref 4).

This creates a melt pool of material for each pass of the laser. The material affected begins to cool and solidify with surrounding material. This also includes the next layer of material to be sintered on top of the previously sintered material. The rapid cooling rate of the material during the LPBF process is beneficial in creating parts with high resolution, but it also creates a significant thermal gradient resulting in residual stress creation in the build (Ref 5,6,7). As material layers are processed, cooling rates and thermal gradients within any given layer will change and vary between those of surrounding layers. This variation of cooling rates and thermal gradients leads to the buildup of residual stress.

The residual stress, especially in smaller models with fine geometry, can ultimately lead to detrimental effects of fatigue such as cracking and warping (Ref 5). The printer repeats sintering layers of material until all layers of the model have been printed. Afterward, the build powder bed chamber is removed from the printer for post-processing. The excess powder is removed from around the build, followed by the removal of the build plate. The build plate is removed from the printer by unscrewing four bolts from the top surface of the build plate with an Allen wrench. Post-processing heat treatment can be implemented after removal from the printer as well. (Ref 8)

A prior study by Spear served as primary motivation for the present work, wherein multiple TPMS lattice structures were created and exposed to a high impact to evaluate their strain rate sensitivity (Ref 6). The study by Spear did not include any characteristics of residual stress, although results found in the study coordinate quite well with impact experiments. The scope of the current study is evaluating any effects that residual stress may have on the overall environment produced in lattice structure. To do so, the same lattice designs considered by Spear are also considered in the current work. Three of the five lattice structures considered herein were closely analyzed in the motivating study: Diamond, I-WP, and Primitive. The two other lattice structures considered in this paper are gyroid and lidinoid structures.

The complex lattice geometries observed in both the motivating and current study are periodic and open cell architectures, selected for analysis because of their energy absorption and dispersion characteristics (Ref 6). The difference in architecture is displayed in previous work from Gibson et al. and Kennedy (Ref 9, 10). The lattice geometries are created by implementing a Matrix Laboratory (MATLAB) script provided by Spear (Ref 6, 11). The MATLAB script defines the geometric structures from user defined characteristics such as cell width, cell height, build width, build height, surface thickness, and cell density. Shape functions also define the unit cell geometry for each type of lattice. Presented in Fig. 2 is an example of the complexity of the lattice structures considered; a unit cell for the primitive lattice structure is displayed. The unit cells of all lattices are modeled in 3D. The surface thickness of the unit cell is very thin. The geometry is not only complex by the shape of the unit cell but by the void area of the cell as well. The trigonometric representations utilized to create the shape of the structures are given by Spear (Ref 6).

Fig. 2

Primitive surface cell (Ref 6)

After fabrication, the mechanical testing of the previous study consisted of compression testing using MTS Systems Corporation (MTS) Universal Testing Machine (UTM). Additional testing was also conducted to observe deformation behavior and mechanical response of the lattice structures (Ref 6).

The goal of this study is to provide insight into the residual stresses developed during LPBF and their potential role in mechanical performance of complex lattice structures. In so doing, a preliminary goal is to validate the use of Simufact Additive^© Modeling Software for basic plate and cube geometries used in a previous experimental study of residual stress generation in LPBF. The resulting comparison between simulations and experimental results provides the confidence needed to move forward with residual stress analysis of complex lattice structures, which is the remaining focus of this work.

2 Theories and Analysis Methods

2.1 Eigenstrain Theory and Measurement Methods

The Eigenstrain Reconstruction Method (ERM), also known as the inherent strain method, has been shown to be useful to determine the residual stress of various objects, from welded joints to cross-sectional areas of superalloy compressor blades (Ref 12,13,14). Eigenstrain is often used to refer to permanent inelastic strain of a specimen. Figure 3 is a one-dimensional diagram representing the yield stress (σ_Y), yield strain (ε_Y), plastic strain (ε_pl), elastic strain (ε_el), and total strain (ε_total) at a point in the specimen (Ref 15). The inelastic process begins after the material has reached its yield point; stress or strain being applied to make the specimen permanently deform. The total strain of the model is also represented by equation (1), the sum of the elastic strain (e_ij), and inelastic strain (ε_ij*) (Ref 15). Residual stress occurs in the inelastic region of the stress-strain relationship—where compatibility is satisfied (Ref 15). Another visual representation of this theory is shown in Fig. 4.

$$\varepsilon_{ij} = e_{ij} + \varepsilon_{ij}^{*}$$

(1)

Fig. 3

Inelastic stretching of a tensile specimen

Fig. 4

Residual stress-strain diagram

Each portion of a geometry is evaluated for its specific set of residual stresses. When a geometry is in global equilibrium, the distribution of the residual stress field is dictated by the remaining variation in the plastic strain (Eigenstrain). The computation of the residual stress is carried out by determining the remaining elastic strains at a coordinate point in the geometry and is used for calculation in Hooke’s law constitutive equations.

Experimental methods used to find the residual stress in parts created using laser powder bed fusion include x-ray diffraction (XRD), hole drilling, and the contour method, among others. Bartlett et al. described the experimental methods for residual stress setermination into two main categories: destructive distortion-based measurements and diffraction techniques (Ref 8). Hole drilling and contour method are two experimental methods that are a part of the destructive distortion-based category, as the lattice structures are, as mentioned, distorted and deformed in the experimental process. XRD is a diffraction technique where the lattice spacing of a strucutre is measured to determine lattice strain utilizing Bragg’s law (Ref 8).

Ahmed et al. (Ref 16) conducted residual stress measurements for welded structures using the contour method, which validated results obtained using the inherent strain method. The study showed a higher tensile residual stress at the edges of samples and compressive residual stress toward the middle surface areas. The University of Virginia has imposed digital image correlation successfully to evaluate the residual stress field on air plasma spray coating (Ref 17, 18). Digital image correlation is another non-destructive experimental technique that is utilized to determine changes of specimen deformation, surface contour, curvature, and positions (Ref 17).

The objective of a study performed by Jun et al. (Ref 19) was to relate the deformation and the stress state of a component without significant limitations during measurement. A semi-empirical approach based on ERM was proposed to evaluate the residual stresses and strains of a component after AM processing. The approach was applied to two cases of interest: shot peening and stir welding. The study concluded that ERM is useful to find complete strain/stress states in entire components.

ERM was used to analyze the residual stress for the cross-sectional locations of a superalloy compressor blade in the study conducted by Salvati et al. (Ref 13). The compressor blade was subjected to shot peening after manufacturing. Two experimental techniques, focused ion beam ring core milling (FIB-DIC) and synchrotron x-ray powder diffraction (SXRPD), were used to validate the model and the residual stress predictions produced by ERM (Ref 13). The values found utilizing ERM were in good agreement with the experimental results.

In the work by Fransen (Ref 20), the strain source of a selective laser melting (SLM) product was mapped at a layer scale, utilizing ERM. A numerical reference model was developed to find a multi-axial residual stress field of a printed build. The ERM model was validated with the reconstruction of residual stress fields for a line, layer, and ten-layer geometries (Ref 20). It was concluded that the ERM approach is suitable to reconstruct residual stresses induced by SLM, as the results obtained via ERM were comparable to the results obtained using a numerical reference model.

To analyze the residual stresses in defective and sound welds, a defect weld of predetermined size and shape was created by Hill on the subsurface of a welded plate (Ref 12). The comparison of the defective and sound welds was completed by the simulation of FEA and the eigenstrain method. For verification of the results obtained, the residual stresses of the welds were also estimated with a swage process (Ref 12).

Levkulich measured the residual stress in specimens built with the LPBF process utilizing XRD, hole drilling, and contour methods (Ref 21). The XRD method was conducted with a proto system and nickel plating to analyze the scatter pattern and angle of diffraction of x-ray beams after contact with the specimens. During the hole drilling method, holes were drilled into the surface of the specimen being analyzed. Stresses were determined at the location of the drilled holes with rosette gauges. To complete residual stress analysis via the contour method, the displacements of the specimen were found using a scanning profilometer and applied to a finite element analysis to estimate the residual stress.

It becomes important that the features leading to material defects be recognized. Once again, the University of Virginia has produced results including infrared thermography (Ref 22,23,24). They also showed how digital image correlation could be used to evaluate microstructural defects.

2.2 Simufact Additive^© Simulation Methods

Simufact Additive^©, a software package created by Hexagon AB (Stockholm, Sweden), uses nonlinear finite element analysis to calculate the temperature, deformation, and residual stress in a simulated 3-dimensional model (Ref 25). The software program considers defined build and process parameters that are applied to the simulation file by the user. The build and process parameters include the printer selection, laser speed, laser power, initial temperatures, material density, minimum wall thickness, maximum particle size, layer thickness, material composition, and in some cases, build supports. Fabrication of lattice structure follows from the mathematical concepts of unit cells.

Trigonometric equations characterize the fabrication of Schwarz-Diamond, Schwarz-Primitive, Schon I-WP, gyroid, and lidinoid structures (Ref 26). Fabrication of the lattice structures can be implemented within various software packages/languages, such as Matrix Laboratory (MATLAB). The structures are then analyzed for any impurities or open geometries within the created files, utilizing other software packages such as Simufact Additive^©.

To simulate a layer-by-layer approach, the user input parameters to create a voxel mesh, which is composed of multiple voxel layers and up to 1.2 million “voxel nodes.” A voxel is a small, defined pixel that contains the characteristics of a three-dimensional model within the voxel dimensions. It should be noted that a voxel pixel is characterized at the intersection of voxel grid lines; it is not a finite volume but describes a space within a volume as it undergoes movement. A “voxel cube,” referred to as “voxel” from this point forward, remains constant throughout the simulation with its initial “volume.” Work is done in the voxel grid volume, and the result of that work is transferred to the surface mesh. The surface mesh is characterized by finite element in which the solution is carried out. The response of the material described in the voxel is dictated by functional expressions. Each voxel layer consists of several physical layers of the build material which is also determined by user input values. In turn, as a layer of voxels are analyzed, the nonlinear finite element analysis of multiple physical layers takes place and are averaged together to calculate the displayed results. This process is reiterated until all voxel layers are analyzed for a full model simulation.

The thermal configuration utilizes the governing equations of heat transfer to calculate the temperature field of the build, support structures, and base plate during simulation. The thermal evaluation is predominately heat conduction with temperature-dependent properties as no stress and fluid flow are considered. The thermomechanical configuration is a small strain elastoplastic analysis with von Mises criteria; it produces coupled mechanical and thermal results sequentially. The thermal results for each layer are analyzed and used to perform the mechanical analysis to find the temperature, distortion, and stresses of the model, support structures, and base plate during LPBF (Ref 27).

The thermal results for a given layer are produced considering the ambient temperature of the printer during print time as well as the thermal gradients produced by the printing of previous layers. The material of a previous layer will be of slightly different temperatures given the cooling rate of the material, speed of the print nozzle, and the length of the part being created. As the nozzle adds material from one side of the build layer to the other, the material will have an amount of time to cool down to the ambient temperature before the next layer is added. Given a hatch pattern that will add material from one end of the build to another in a sweeping motion, the ends of a layer will have more time for cooling before another layer is added than the middle of a layer. Thus, this creates a differing temperature gradient on the edges of each layer in comparison to the center of a layer, which leads to stress created within the part during the additive manufacturing process.

A voxel mesh is created which sections the build into smaller elements for analysis. The thickness of a voxel is the thickness of multiple physical layers of a build. The result of each voxel element is averaged with the surrounding voxel elements. The simulation process for LBPF is also cut into a user defined value of time steps. This feature is a user defined increment value to represent when a thermal calculation will take place during the simulation. The time steps of a simulation process include exposure time to build and cooling, and removal stages. The minimum number of time steps required by Simufact Additive^© for a simulation is ten (Ref 25).

The analysis carried out is entirely dependent upon Simufact Additive^© software, every part of the analysis is developed within their so-called widgets. The thermomechanical analysis is based upon a transient approach incorporating the usual heat transfer equations which includes the density, specific heat capacity, and conductivity, all a function of temperature and time. The boundary conditions of heat transfer are a by-product of the printer’s method of manufacturing. Once the material is chosen and the initiation of the manufacturing parameters are followed, the time-dependent heat is placed into the formulation. The time process for each thermal and mechanical parameters is carried out throughout the analysis. A table is included within the software depicting the appropriate values of the given material. The temperature per time increment is fed into software allowing the constitutive relationships to be evaluated based upon the time in which the amount of plasticity is determined. The next information considered is the model surface mesh, including the 3D hexahedral element chosen. The voxel mesh is then chosen within the analysis widget. The analysis is initially in the voxel and transferred to the so-called surface mesh. The finite element surface mesh is an over-load mesh on top of the voxel mesh. The voxel is a point at the intersection in voxel grid. Three simplifications are made:

• One element layer includes multiple real powder layers.

• All elements in one layer gets melted at once.

• After melting, a secondary energy input takes place.

Because of these simplifications, the thermal and thermomechanical processes have to be recalculated with the addition of each voxel layer. During material application, points that get melted when the laser moves over them, cools down when the laser moves away, and reheats when the laser moves over them again. A simplified thermal cycle approach is used as the energy output gets divided into two fractions. The primary fraction melts the powder, and the secondary fraction compensates for the heating and cooling after the material has melted. For a thermomechanical simulation, there is a volumetric expansion factor which scales the thermal strains.

As the voxel point is displaced, a transformation is carried out to the surface mesh. The voxel element stays the same; the surface mesh is changing with deformation. Finally, the software material site dictates all the material parameters including functions of temperature and phase change characteristics.

3 Validation of Simufact Additive^© for Residual Stress Analysis

3.1 Basic Plate Structures

To validate the use of Simufact Additive^© for the analysis of residual stresses generated in lattice structures fabricated by LPBF, numerical results for residual stress are compared to experimental results found in a previous study (Ref 21). A plate-shaped model is recreated in solidworks and is characterized by a physical size of 25.4 × 25.4 × 1.6 mm. The material of the plate is selected as Ti-6Al-4V metal powder. The machine type selected to further characterize the simulation process is the printer EOS M290. The plate model is uploaded into the Simufact process file and centered on the build plate.

Next, the process parameters for printing are incorporated into the simulation program. The laser power is set to 280 W, laser speed to 400 mm/s, laser beam width to 3 mm, and layer height to 50 μm, which are the process parameters used in the experimental study by Levkulich (Ref 21). The efficiency factor is set at 60%, Poisson’s ratio at 0.26, yield strength at 1140 MPa, and the tensile strength at 1290 MPa. The efficiency factor accounts for all energy lost prior to laser beam and metal powder contact as well as during adsorption (Ref 25). Figure 5 is the plate described above where the largest voxel mesh size is in view. The length and width of the structure are divided into 50 voxel elements and the height into 4 voxel elements. The figure also shows how the volume of each voxel is represented as fully occupied by a section of the structure; there are no void areas within the voxels. This is useful to determine a reasonable voxel size for models for a more accurate simulation. If there were void areas in a voxel, the affected voxel area would appear as a differing color on the volume fraction legend. The voxel fraction of the build plate is uniformly green which indicates that each voxel shares the same volumetric characteristics and representation.

Fig. 5

Volume fraction of a plate structure; 50 × 50 × 4 voxel size

The plate structure was analyzed with three voxel sizes: 50 × 50 × 4, 100 × 100 × 8, and 200 × 200 × 16. The three voxel sizes are represented in Fig. 6. The 50 × 50 × 4 voxel size is shown at the top of the image and the voxel size is refined to the 200 × 200 × 16 voxel size located at the bottom of the image. Note as the size of the voxels in a voxel mesh increases, the number of numerical data points available decreases. As such, results corresponding to a smaller voxel size will produce results with higher accuracy.

Fig. 6

(a) 50 × 50 × 4 voxel size; (b) 100 × 100 × 8 voxel size; (c) 200 × 200 × 16 voxel size

To reach the voxels in the center of the plates for stress analysis, all models were cut in the x-direction and y-direction to create a quarter model as shown in Fig. 7. The red and white boxes mark the cuts at the midplane in both the x and y directions to reach the center edge of the plate, which is circled in blue. After results were collected along the center of the plates and tabulated, plots were created in MATLAB for further comparison of the resulting stress values.

Fig. 7

A quarter plate model. Circled is the middle of the plate

In all the calculations, no consideration is made relative to the effect of the tooling process (toolpath). It is the software package that runs the process of manufacturing. It is obvious that the features of the melting process, the layer thickness, the scan speed, the scan pattern, and laser power all fit into the overall residual stress effect.

3.2 Basic Cubic Structures

To further understand the use of the Simufact Additive^© Software Modeling Approach and to verify the software for residual stress analysis, cubic structures having dimensions of 25.4 × 25.4 × 25.4 mm were also simulated. The build parameters, material, machine, as well as simulation numerical parameters regarding the matrix solver and step size remained the same as used with the plate geometry. However, the voxel sizes utilized for the cube geometries differ to those of the plate geometries. For the cubic geometries, voxel sizes of 10 × 10 × 10, 20 × 20 × 20, and 40 × 40 × 40 are considered, as shown in Fig. 8. The gray “mesh” in the background of the figure is a visual aid used by Simufact Additive^© to represent the boundaries and size of the 3D printer chamber that surrounds the build-plate selected for analysis. The reduction of voxel size is also evident as shown in the figure. The volume fraction of the cubic structure and build-plate, as well as their positioning, is shown in Fig. 9. The model and build plate are shown as one color, indicating the volume fraction of the voxel mesh created is the same throughout. This volume fraction result is the same for all voxel size cases considered.

Fig. 8

Voxel size meshes for the cubic structures

Fig. 9

Volume fraction of cubic structure

To investigate convergence of results, the average normal stress and z-displacement values for all voxel size cases are computed within Simufact Additive^© for the voxel nodes through the middle of the model. An array of values consisting of the z location for each node was created for each of the voxel sizes; the array starting at zero which represents the bottom surface of the cubes that would be in contact with the build plate during manufacturing, to the top surface of the cubic structure—the array being captured without the build-plate being of consideration. The array length varied by voxel height; the value is represented between the two columns of the array for each case. The voxel height for the arrays of voxel sizes 10 × 10 × 10, 20 × 20 × 20, and 40 × 40 × 40 are 2.54 mm, 1.27 mm, and 0.635 mm, respectively. Thus, even the smallest voxel size considered for this study consists of over 12 physical layers of the physical build, which are only 50 μm in all directions.

4 Triply Periodic Minimal Surface (TPMS) Lattice Structure File Creation and Importation

For the residual stress analysis of complex lattice structures during the LPBF process, the thermomechanical configuration is utilized. It should be pointed out that to further define the thermomechanical configuration, flow stress equations presented for a material are used to evaluate the plastic strains, which are then subtracted from total strains to yield the elastic strains; these elastic strains are then used to determine the residual stresses as described previously through the use of Hooke’s Law. Flow stress curves of two materials, Ti-6Al-4V and Inconel 718, are used for the simulation of the models within this study. The flow stress curves of the two materials are displayed in the following figures. The top red line represents the flow stress in relation to the effective plastic strain of the materials at room temperature (20 °C). As the temperature increases, the flow stress decreases, as seen by the flow stress curves of other colors.

Figure 10 displays the flow stress curve in units of MPa for material Ti-6Al-4V as a function of effective plastic strain and temperature measured in Celsius. The flow stress at room temperature and zero effective plastic strain is roughly 1190 MPa (Ref 25). Peak flow stress is estimated to be 1300 MPa with an effective plastic strain of 0.25 for the material at room temperature.

Fig. 10

Flow stress curve of Ti-6Al-4V

Figure 11 displays the flow stress in units of MPa for material Inconel 718 as a function of effective plastic strain and temperature measured in Celsius. The flow stress at room temperature and zero effective plastic strain is roughly 780 MPa and the yield strength is 758 MPa (Ref 25). The peak flow stress at room temperature is 1050 MPa and would occur with an effective plastic strain greater than or equal to 0.075.

Fig. 11

Flow stress curve of inconel 718

All five lattice structures that can be created with the MATLAB code provided by Spear were simulated using Simufact Additive^©. The length, width, and height of the lattices are 32 mm. The TPMS surface thickness value and cell fineness parameter values were not changed; they are 0.25 mm and 20, respectively. Cell densities of both build length and build height did not change either; they are 8 mm. The TPMS cell size was modified to become 4 mm. The MATLAB code was run for all five lattice structures and the Standard Tessellation Language files (.stl files) were saved to a drive for file storage.

Once an .stl file is saved for a lattice structure, the file can be uploaded directly to Simufact Additive^© for simulation or can be refined by another software package before upload. A model file can also be imported as another file type, such as a .step file. Although many types of files can be uploaded, it is recommended to upload the model as a .step file to guarantee a “cleaner” geometry surface for simulation. For the lattice geometries, the surface mesh created with MATLAB was not “cleaned;” the files were not checked for multiple surfaces, inverted face normals, or other types of geometrical errors (Ref 6). As the files were not checked for errors, a few geometrical inconsistencies are noticeable in the visual representation of the lattices in Simufact Additive^©. However, these inconsistencies are small and do not affect the calculations for residual stresses created during the LPBF process. An intermediate step to clean the files may be conducted with third-party software. 

All five lattice structures were simulated with the machine type, material system, and process parameters mentioned in the motivating study (Ref 6). The machine selected is the Concept Laser M2 Cusing, the material system is Inconel 718, and the process parameters include a laser power of 130 W, a laser scan speed of 1,300 mm/s, and a layer height of 40 μm. Poisson’s ratio is 0.3, the strain rate is 758 MPa, and the tensile strength of the material is 1040 MPa. A 20 × 20 × 20 voxel size was chosen for the comparison of the lattice structures. Once the voxel meshes were created, the mesh volume fraction was viewed as shown in Figures.

Figure 12 displays the volume fraction of the primitive lattice structure with the 20 × 20 × 20 voxel size. The blue voxels represent the voxels with a low volume being occupied by the material of the physical geometry being simulated. There are also regions where no voxels are present; they were not created as there was no material within that area to be analyzed.

Fig. 12

Volume fraction of primitive lattice structure: 20 × 20 × 20 voxel size

The volume fraction for the I-WP lattice structure is presented in Fig. 13. The side faces of the voxel mesh are completely blue, while a faint green is seen on the top face of the mesh. As mentioned for the primitive lattice structure, little material occupies the volume of the voxels along the edges of the structure. To further show the volume fraction of this structure, Fig. 14 displays the volume fraction for a quarter model I-WP lattice. Note that the dark blue is only displayed at the far edges of the build. The voxels toward the center of the build consist mostly of a volume fraction of 50-70%. There are also areas across the build where voxels consist of material for 20 to 30 percent of their volume.

Fig. 13

Volume fraction of I-WP lattice structure: 20 × 20 × 20 voxel size

Fig. 14

Volume fraction of a quarter model I-WP lattice structure

The volume fractions of the diamond, gyroid, and lidinoid lattice structures are shown in Fig. 15. Like the I-WP lattice structure, the volume fraction is more distributed toward the inside of the lattice structures, which can be viewed if the models were cut across the midplanes. If the voxel meshes created for the simulation processes were further refined, the surfaces displayed would be different, especially with smaller voxels being representative of a more distributed volume fraction. However, the voxel mesh can only be refined up to the software capability of 1.2 million voxels.

Fig. 15

(a) Volume fraction of diamond lattice structure; (b) volume fraction of gyroid lattice structure; (c) volume fraction of lidinoid lattice structure

After the simulation set-up process was complete, all models were saved into their respective process files and were simulated. The time taken for the simulation process for each of these lattice structures was no more than 2 hours. After all simulations were complete, the structures were ready for residual stress comparison.

As mentioned, the five lattice models generated are I-WP, diamond, primitive, gyroid, and lidinoid. All lattice structures before the simulation process are shown in Fig. 16. A close observation shows the geometrical complexity of the structures. The geometry along the corner edges of the lattices is very thin given the values of the surface thicknesses of the models. The residual equivalent stress of all five models were compared to each other to determine which structures might be better for given applications, such as compression and fatigue testing.

Fig. 16

(a) I-WP lattice structure; (b) primitive lattice structure; (c) diamond lattice structure; (d) gyroid lattice structure; (e) lidinoid lattice structure

5 Results

5.1 Plate Simulation Results for Convergence Study

To check for convergence of stress created in the plate models, normal stress data are obtained from the center nodes of the plate models using course, medium, and fine voxel meshes. The normal stress values obtained for the plate structure simulations were compared to the work of Levkulich (Ref 21). In the previous work, the residual stress along the top surface of the physical size 25.4 × 25.4 × 1.6 mm plate geometry was analyzed in two different directions. The values found for the residual stress along the top of the geometry were plotted by Levkulich, which is shown in Fig. 17. The graph also displays the residual stress values found for the other two geometry cases that were considered. For the top surface of the 25.4 × 25.4 × 1.6 mm plates, the highest residual stress value recorded is close to 400 MPa.

Fig. 17

Reproduced from the effect of process parameters on residual stress evolution and distortion in the laser powder bed fusion of Ti-6Al-4V, N. Levkulich, S. Semiatin, J. Gockel, J. Middendorf, A. DeWald, and N. Klingbeil, Additive Manufacturing, Vol 28, Copyright 2019, with permission from Elsevier

XRD measured principal stresses of LPBF deposits with different plan areas, presented by Levkulich (Ref 21).

A contour plot of the normal stress in the x-direction after removal from the base plate for the voxel size case 50 × 50 × 4 is shown in Fig. 18. The area with the highest variance of normal stress in the x-direction is along the edges of the plate. The plate appears warped along the edges as a higher stress concentration is along the left and right sides of the plate, while the center region experiences a negative stress concentration. For all results shown in this section, the maximum and minimum values are displayed under the colored stress value legend. The values displayed are not necessarily the maximum and minimum values found for the structure at the time of removal from the build plate. These values could have been calculated during any time increment of the simulation process—during the printing of any given layer of material. The values mostly serve as an “apples-to-apples” comparison for multiple simulations.

Fig. 18

Results for normal stress in X direction for plate geometry of 50 × 50 × 4 voxel size

A contour plot of the normal stress in the x-direction after removal for the voxel size case 100 × 100 × 8 is shown in Fig. 19. After refining the voxel mesh and simulating the plate model a second time, the results show an increase of normal stress in the x-direction along the left and right sides of the plate. A larger area of the midsection for the top surface of the plate is now experiencing an x-normal stress closer in value to zero MPa. The center of the top surface is still experiencing negative normal stress.

Fig. 19

Results for normal stress in X direction for plate geometry of 100 × 100 × 8 voxel size

A contour plot of the normal stress in the x-direction after removal for the voxel size case 200 × 200 × 16 is shown in Fig. 20. As the voxel mesh is successively refined, the simulation is better able to capture the peak residual stress values at the top and bottom surfaces and near the edges of the plate. The edges of the top surface experience x-normal stress within the range of 240 to 300 MPa.

Fig. 20

Results for normal stress in X direction for plate geometry of 200 × 200 × 16 voxel size

Table 1 lists the values of x-normal stress at a distance along the z-direction for plate structures before and after removal from the build plate. The results are obtained from the middle of the plates for all voxel sizes, starting where the plates have contact with the build plate to the top surface of the build (1.6 mm).

Table 1 Normal stresses of X direction analyzed through the Z direction for plate geometry using 3 voxel size meshes [MPa]

Figure 21 shows the normal stress in the x-direction as a function of z for plate geometries before removal from the build plate. The blue line represents the largest voxel size of 50 × 50 × 4, the orange line represents the voxel size of 100 × 100 × 8, and the black line represents the 200 × 200 × 16 voxel size. As the voxel size becomes smaller, the normal stress in the x-direction approaches 600 MPa over a shorter distance from the build plate, remaining in the 550-610 MPa range elsewhere.

Fig. 21

Normal Stress in X as a function of Z before build plate removal for plate geometries, values observed through the center of the plate

Figure 22 depicts the normal stress in the x-direction as a function of z for the plate geometries after removal from the build plate. For the results shown at the top of the plate (z = 1.6 mm), the first two voxel sizes show a decreasing trend, while the finest voxel size is of the same magnitude as the largest voxel size. The figure also presents the peak stress value occurring in the voxel layers leading to the center of the build in the z-direction, as the voxel mesh becomes finer. However, the 100 × 100 × 8 voxel size is found to experience the highest peak stress compared to the other voxel sizes, indicating that the results were not fully converged.

Fig. 22

Normal stress in X as a function of Z after removal of build plate for plate geometries, values observed through the center of the plate

Given the difficulty in obtaining convergence for plate-shaped geometries, it was decided to run simulations for geometries having less residual stress-induced distortion. To do this, cubic geometries were modeled to match those considered in the experimental study by Levkulich (Ref 21), for which the experimental results could be directly compared to results obtained from simulation with Simufact Additive^©.

5.2 Cube Simulation Results for Convergence Study

An array of values consisting of the z-location for each voxel node was created for the voxel sizes; the array starting at zero which represents the bottom surface of the cubes in contact with the build plate, to the top surface of the cubic structure. The array length varied by voxel height; the value represented between two nodes of the array for each case. The voxel height for the arrays of voxel sizes 10 × 10 × 10, 20 × 20 × 20, and 40 × 40 × 40 are 2.54 mm, 1.27 mm, and .635 mm, respectively. Thus, even the smallest voxel size considered for this study consists of over 12 physical layers of the physical build, which are only 50 μm in all directions.

The cubic structure analyzed with the 10 × 10 × 10 voxel size is shown in Fig. 23. The highest normal stress of the x-direction is shown to appear on the top surface of the cube toward the center of the face. The normal stress decreases to the left and right edges of the top surface. The front and rear faces display various results. There are two areas in the faces with a normal stress within the 100 to 300 MPa range, one area at the top, the other at the bottom. There is also an area in the middle of the faces experiencing normal stress within the range of − 200 to − 100 MPa. The left and right faces are stress-free, as they are normal to the x-direction.

Fig. 23

Results for normal stress in X direction for cubic geometry of 10 × 10 × 10 voxel size

The machine and build parameters used for the simulation process of the 10 × 10 × 10 voxel mesh size is copied to create simulation processes for the other voxel mesh sizes. Figure 24 shows the contour plot of the normal stress results for the 20 × 20 × 20 voxel mesh size. The increase in stress can really be seen in the front and top surfaces of the cube. The blue area for the center section of the front face is now green and surrounded by stresses represented in the 100 to 300 MPa range. A slight orange line is also appearing at the bottom of the front face.

Fig. 24

Results for normal stress in X direction for cubic geometry of 20 × 20 × 20 voxel size

The simulation process was copied once more, and a new voxel mesh size of 40 × 40 × 40 was created for the third simulation of the cubic structure. The results of the third simulation displayed in Fig. 25 show the same trends as the first two cubic structure simulations. In the given isometric view, the highest x-normal stress is seen along the top and front faces of the cube. Stress values increased across the face of the structure except at the four corners, represented to be within the green stress range. The top surface appears to be completely within the red stress range except for the left and right edges, which are represented by yellow.

Fig. 25

Results for normal stress in X direction for cubic geometry of 40 × 40 × 40 voxel size

Figure 26 displays the contour of the x normal stress for a quarter model of the cubic structure of voxel size 20 × 20 × 20. Midplane cuts are along the x- and y-directions, with the midplane cut of the x-direction shown in gray. The quarter model provides a detailed representation of the stresses in the interior of the cubic geometry. The peak normal stresses are along the upper and lower regions of the cubic structure, which are the top and bottom faces. The lowest normal stress region is in the middle of the structure, represented as the dark blue region.

Fig. 26

A quarter model of a cubic structure. Nearest edge is center of the cube

Figure 27 presents residual stress measurements found in the previous study (Ref 21) via XRD and hole drilling methods along the top surface of builds with various heights. The data to the far right of the chart represents the residual stress analyzed along the top surface of a 25.4-mm cubic structure in two directions. The values presented as black diamonds in the legend were found by XRD along the top surface of the build, considering substrate overhang. The data presented as vertical lines represent values found by XRD without considering substrate overhang. The data presented in red on the far-right hand side represent the residual stress for the top surface of the build by utilizing the hole drilling method and considering substrate overhang. The highest residual stress value recorded for the top surface of the cubic structure is close to 300 MPa.

Fig. 27

Reproduced from the effect of process parameters on residual stress evolution and distortion in the laser powder bed fusion of Ti-6Al-4 V, N. Levkulich, S. Semiatin, J. Gockel, J. Middendorf, A. DeWald, and N. Klingbeil, Additive Manufacturing, Vol 28, Copyright 2019, with permission from Elsevier

Residual stress of plate and cubic structures, found by x-ray diffraction and hole drilling. Data presented by Levkulich (Ref 21).

For further comparison with the previous work by Levkulich, the x-normal stress on the midplane of the cubic structure is compared to a contour plot from the previous literature, in which the residual stress was measured with the substrate still attached (Ref 21). The contour plot of the cubic structure in midplane view presented in the previous literature is also present here as Fig. 28. The highest stress value occurs along the bottom of the substrate and around the edges of the specimen (the square build on top of the substrate). The lowest residual stress values appear at the farthest edges of the substrate and the location where the substrate and specimen meet in the center of the figure.

Fig. 28

Reproduced from the effect of process parameters on residual stress evolution and distortion in the laser powder bed fusion of Ti-6Al-4V, N. Levkulich, S. Semiatin, J. Gockel, J. Middendorf, A. DeWald, and N. Klingbeil, Additive Manufacturing, Vol 28, Copyright 2019, with permission from Elsevier

Contour method residual stress measurements from Levkulich (Ref 21).

A cubic structure simulated with a 20 × 20 × 20 voxel size in Simufact Additive^© is shown in Fig. 29. The model is cut at the midplane across the y-axis and is shown with the build plate still attached. A moderate value of x-normal stress appears along the top and bottom surface of the build as well as the top surface of the build plate. The lowest x-normal stress appears in the center of the model. The area represented by green is estimated to have little to no stress.

Fig. 29

Midplane of cubic structure before removal from build plate. The model is cut in y-axis plane

Figure 30 is the x-normal stress of the build after removal from the build plate. The stress contour is similar to the contour seen before removal. The stress seen at the upper-center and lower-center regions of the model are shown to have slightly increased after removal. There is still an area of little to no stress surrounding the center of the cube as well as the left and right edges of the midplane. The center of the model still appears to have the lowest stress value.

Fig. 30

Midplane of cubic structure after removal from build plate. The model is cut in y-axis plane.

X-normal stress for the top surface of a specimen with a height of 25.4 mm is within the range of 250-300 MPa. Results obtained from Simufact Additive^© are in the range of 290-670 MPa. Noting that some of the results of the previous study may slightly differ, the overall results obtained with Simufact Additive^© for the cubic structure are reasonably comparable to the experimental results of the previous study (Ref 21).

The results obtained with Simufact Additive^© determined that the normal stresses of the structures top surface are increasing toward the yield strength of Ti-6Al-4V. It was also determined that finding the stresses in an actual layer was not computationally feasible, as it would require over 2 million elements. However, the average stress values of the basic geometries converged and were generally similar in magnitude to the results presented by Levkulich (Ref 21). Thus, Simufact Additive^© was successfully validated as a reasonable tool for predicting residual stress generation in LPBF AM processes.

5.3 Residual Stress Distributions of TPMS Lattice Structures

The residual stress distributions for all five lattice structures are shown in the figures below. The residual stress values analyzed throughout the lattice structures are within a considerably significant range, given the room temperature yield point of Inconel 718 is 758 MPa. The highest residual stresses observed for the lattice structures appear in the center of the four corner edges on the outside of the model in the z-direction. The next highest residual stresses are along the surfaces of the lattice structures and increasing toward the center of the faces. The lowest residual stress appears in the center of the build.

Figure 31 shows the full model representation of the primitive lattice structure. The residual stress for this lattice structure appears to be averaging close to 350 MPa. The highest residual stress seen along the outside edges of the structure is close to 800 MPa. Figure 32 is a quarter model representation of the primitive lattice structure. The outside surfaces are found to have an equivalent stress of 450-500 MPa. The center of the structure is found to have an equivalent stress in the range of 300-400 MPa. The equivalent stress distribution shown next to the legend of the contour plot becomes increasingly verifiable when the center of the structure is viewed; most of the model volume is represented to be within the equivalent stress range displayed as blue.

Fig. 31

Residual stress distribution of primitive lattice structure, 20 × 20 × 20 voxel size

Fig. 32

Residual stress distribution of primitive lattice structure, 20 × 20 × 20 voxel size—quarter model

Figure 33 is the equivalent stress results for the I-WP lattice structure. The equivalent stresses appear to be more uniform across the faces of the geometry, estimated to be 600-700 MPa. The equivalent stresses along the outside edges appear to differ. The lowest residual stresses appear along the bottom surface that would touch the build plate, roughly 200-300 MPa. Higher residual stresses are also observed in the corners of the top surface of the model, estimated to be within the 400-500 MPa range.

Fig. 33

Residual stress distribution of I-WP lattice structure, 20 × 20 × 20 voxel size

The quarter model of the I-WP lattice structure is shown in Fig. 34. As with the primitive lattice structure, the center of the I-WP lattice structure is found to experience a lower value of equivalent stress than the outer surfaces. The equivalent stress in the middle of the structure is within a range of 200 to 400 MPa, except for one small area that is estimated to experience equivalent stress of 100 to 150 MPa.

Fig. 34

Residual stress distribution of I-WP lattice structure, 20 × 20 × 20 voxel size—quarter model

The diamond lattice structure is displayed in Fig. 35. The equivalent stress distribution is very similar to the I-WP lattice. The equivalent stress average for this model is close to 500 MPa. The distribution also appears to be of a tighter, normal distribution around that average value.

Fig. 35

Residual stress distribution of diamond lattice structure, 20 × 20 × 20 voxel size

As shown in Fig. 36, the values within the center of the diamond lattice are within the greater portion of the equivalent stress distribution and are in more of a uniform pattern compared to the I-WP lattice structure. The equivalent stress throughout the build appears to be no more than 800 MPa. The lowest equivalent stress is still found along the bottom of the structure where it is evaluated on the build plate. The heat from the first layer is removed from the lattice by contact with the build plate to reach an equilibrium temperature between the two during build time. As the lattices may take hours to build, the thermal gradients acting on this layer will not be of a large range as other locations of the lattice geometry, ultimately leading to lower stress within the region.

Fig. 36

Residual stress distribution of diamond lattice structure, 20 × 20 × 20 voxel size—quarter model

Figure 37 and 38 displays equivalent stress results for the gyroid lattice structure. The results appear to be similar to the diamond lattice structure; the contour plots look alike. However, the distribution is slightly tighter than the distribution of the diamond lattice. The range of equivalent stress for the gyroid structure is 250-750 MPa. Also, some of the geometrical impurities mentioned earlier are visible in the contour plot for the gyroid lattice. They are the little black lines visible throughout the structure.

Fig. 37

Residual stress distribution of gyroid lattice structure, 20 × 20 × 20 voxel size

Fig. 38

Residual stress distribution of gyroid lattice structure, 20 × 20 × 20 voxel size—quarter model

As shown in Fig. 38, there are a few areas of a lower equivalent stress within the center of the build. Otherwise, the inside of the build is within 400-750 MPa. The equivalent stress does not appear to have a uniform pattern. There are blotches of differing equivalent stress values throughout.

Figure 39 is the full model for the lidinoid lattice structure, while Fig. 40 is the quarter model. This structure is found to have a wider equivalent stress distribution, but the average equivalent stress is still within the range of 500-600 MPa. The quarter model shows a similar equivalent stress distribution through the center of the model like the I-WP lattice structure. The lowest equivalent stress is shown at the very center of the lattice structure and the very bottom surface layer that would in contact with the build plate.

Fig. 39

Full model of the lidinoid lattice structure

Fig. 40

Quarter model of the lidinoid lattice structure

For convenience, comparisons of equivalent stress for full and quarter lattice models of all 5 geometries are displayed in Fig. 41 and 42, respectively. The displayed range of stress values is the same as previously used in earlier figures: 0-1000 MPa.

Fig. 41

Equivalent Stress distributions for whole models of all 5 lattice structures. Top row, left to right: diamond lattice, I-WP lattice, primitive lattice. Bottom row, left to right: gyroid lattice, lidinoid lattice

Fig. 42

Equivalent stress distributions for quarter models of all 5 lattice structures. Top row, left to right: diamond lattice, I-WP lattice, primitive lattice. Bottom row, left to right: gyroid lattice, lidinoid lattice

5.4 Comparison of Lattice Geometries

Of the five lattice structures analyzed, the primitive lattice structure was shown to experience the lowest average for equivalent stress, 350 MPa. The equivalent stress for all other lattice structures averaged within 500-550 MPa. The largest residual stress value for all lattice structures is estimated to fall in the range of 700-750 MPa. Although the equivalent stress distribution for the Primitive lattice structure is in a lower range, the shape of the distribution is comparable to the distributions of the diamond and gyroid lattice structures. The distributions for the I-WP and lidinoid lattice geometries are large compared to the other distributions; they span a wider range of equivalent stress values than the other lattice geometries.

Comparisons between the five lattice structures were made for the 20 × 20 × 20 voxel size after removal from the build plate. As mentioned for the basic geometry stress results, the maximum and minimum values displayed in the legend are not necessarily the maximum and minimum values found for the structure at the current moment in time for the simulation process. The values could have been calculated at any increment of the simulation process and they mostly serve as an “apples-to-apples” comparison for multiple simulations.

6 Discussion

As shown in the previous figures, the lattice geometry seems to have a significant impact on the distribution of stresses throughout the geometry volume. The primitive lattice structure showed a lower value of equivalent stress created throughout the model during manufacturing, whereas the gyroid structure showed a higher equivalent stress throughout its volume. The equivalent stress values for the lattice structures are similar in magnitude to the equivalent stress for the solid cube geometry, although they are for a different material system. Based on the stress averages, the amount of stress created during the LPBF process would be near 2/3 of the elastic limit of Inconel 718; before the material reaches the yield point of 758 MPa and the start of plastic deformation begins. However, the flow curves for the material properties within Simufact Additive^© predict a higher value of plasticity to occur and ultimately a higher value of residual stress than that of the motivating study. This suggests the residual stress of the manufactured lattices of the motivating study are lower than the values obtained from the simulations. Therefore, residual stress generated by the LPBF additive manufacturing process is suggested to be considered in assessing the mechanical performance of lattice structures in some applications, although residual stress created during LPBF may not be of significance. Post-processing heat treatment is recommended for all geometries created with AM processes to reduce any effects of residual stresses that may be developed during manufacturing (Ref 28). The decrease in stress will allow the structures to endure higher service loads before the onset of plastic deformation.

7 Conclusions

To determine the significance of residual stresses created in complex lattice geometries during the LPBF additive manufacturing process, five lattice structures were modeled with a MATLAB code utilized for the motivating study presented by Spear (Ref 6). The five lattice structures, primitive, I-WP, diamond, gyroid, and lidinoid, were simulated with the Simufact Additive^© Software Program for residual stress analysis. To validate the use of Simufact Additive^©, in-plane normal stress was predicted in basic plate and cubic geometries for comparison with a previous experimental study by Levkulich (Ref 21). Experimental methods such as x-ray diffraction, hole drilling, and contour method were utilized by Levkulich to determine residual stresses in these geometries.

After the use of Simufact Additive^© was validated, five lattice structures were imported into the software program for simulation. The process parameters mentioned in the motivating study by Spear were used to further define the build conditions for the simulation (Ref 6). All five lattice structures with a voxel size of 20 × 20 × 20 were successfully simulated. The numerical results of the lattices were compared to each other to find new insights into the role of lattice geometry on residual stress generation in the LPBF additive manufacturing process. The average equivalent stress in the lattice structures was found to be around 500 MPa, which accounts for roughly 66% of the elastic regime for Inconel 718, the material considered in the previous study conducted by Spear (Ref 6). However, it is found that the Inconel 718 flow stress curves used within Simufact Additive^© are meant for specimens that are prismatic and do not relate to complex lattice geometries. It is further observed that the stress-strain curves produced within the motivating study for a given equivalent stress are much lower than those predicted within the numerical simulations. More plasticity occurs and is accounted for during the simulations; the increase in plasticity leads to a smaller amount of elastic strain. Since the elastic strain tensor leads directly to residual stress, the residual stress observed in the motivational study is less than the amount of residual stress predicted by Simufact Additive^©. The results produced by Spear when not considering residual stress are within reason and display the limited amount of residual stress created during manufacturing. Overall, it can be concluded that the residual stresses created during the LPBF process may not be of significance to the mechanical performance of complex lattice geometries, although the consideration of residual stress is still suggested when using LPBF to create structures for various applications.