# Statistical Homogenization of Elastic and Fracture Properties of a Sample Selective Laser Melting Material

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Vertical Direction

#### 3.2. Horizontal Direction

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Carter, L.N.; Martin, C.; Withers, P.J.; Attallah, M.M. The influence of the laser scan strategy on grain structure and cracking behaviour in SLM powder-bed fabricated nickel superalloy. J. Alloys Compd.
**2014**, 615, 338–347. [Google Scholar] [CrossRef] - Qiu, C.; Adkins, N.J.; Attallah, M.M. Microstructure and tensile properties of selectively laser-melted and of HIPed laser-melted Ti–6Al–4V. Mater. Sci. Eng. A
**2013**, 578, 230–239. [Google Scholar] [CrossRef] - Zou, S.; Xiao, X.; Li, Z.; Liu, M.; Zhu, C.; Zhu, Z.; Xhen, C.; Zhu, F. Comprehensive investigation of residual stress in selective laser melting based on cohesive zone model. Mater. Today Commun.
**2022**, 31, 103283. [Google Scholar] [CrossRef] - Choo, H.; Sham, K.-L.; Bohling, J.; Ngo, A.; Xiao, X.; Ren, Y.; Depond, P.J.; Matthews, M.J.; Garlea, E. Effect of laser power on defect, texture, and microstructure of a laser powder bed fusion processed 316 L stainless steel. Mater. Des.
**2019**, 164, 107534. [Google Scholar] [CrossRef] - Andani, M.T.; Karamooz-Ravari, M.R.; Mirzaeifar, R.; Ni, J. Micromechanics modeling of metallic alloys 3D printed by selective laser melting. Mater. Des.
**2018**, 137, 204–213. [Google Scholar] [CrossRef] - Andani, M.T.; Ghodrati, M.; Karamooz-Ravari, M.R.; Mirzaeifar, R.; Ni, J. Damage modeling of metallic alloys made by additive manufacturing. Mater. Sci. Eng. A
**2019**, 743, 656–664. [Google Scholar] [CrossRef] - Shifeng, W.; Shuai, L.; Qingsong, W.; Yan, C.; Sheng, Z.; Yusheng, S. Effect of molten pool boundaries on the mechanical properties of selective laser melting parts. J. Mater. Process. Technol.
**2014**, 214, 2660–2667. [Google Scholar] [CrossRef] - Riedlbauer, D.; Scharowsky, T.; Singer, R.F.; Steinmann, P.; Körner, C.; Mergheim, J. Macroscopic simulation and experimental measurement of melt pool characteristics in selective electron beam melting of Ti-6Al-4V. Int. J. Adv. Manuf. Technol.
**2017**, 88, 1309–1317. [Google Scholar] [CrossRef] - Belotti, L.P.; Hoefnagels, J.; Geers, M.; van Dommelen, J. A modular framework to obtain representative microstructural cells of additively manufactured parts. J. Mater. Res. Technol.
**2022**, 21, 1072–1094. [Google Scholar] [CrossRef] - Pilgar, C.M.; Fernandez, A.M.; Lucarini, S.; Segurado, J. Effect of printing direction and thickness on the mechanical behavior of SLM fabricated Hastelloy-X. Int. J. Plast.
**2022**, 153, 103250. [Google Scholar] [CrossRef] - Cao, M.; Liu, Y.; Dunne, P. A crystal plasticity approach to understand fatigue response with respect to pores in additive manufactured aluminum alloys. Int. J. Fatigue
**2022**, 161, 106917. [Google Scholar] [CrossRef] - Ji, L.; Wang, S.; Wang, C.; Zhang, Y. Effect of hatch space on morphology and tensile property of laser powder bed fusion of Ti
_{6}Al_{4}V. Opt. Laser Technol.**2022**, 150, 107929. [Google Scholar] [CrossRef] - Hao, L.; Wang, W.; Zeng, J.; Song, M.; Chang, S.; Zhu, C. Effect of scanning speed and laser power on formability, microstructure, and quality of 316 L stainless steel prepared by selective laser melting. J. Mater. Res. Technol.
**2023**, 25, 3189–3199. [Google Scholar] [CrossRef] - Li, E.; Zhang, Z.; Chang, C.; Li, L.Q. Numerical homogenization for incompressible materials using selective smoothed finite element method. Compos. Struct.
**2015**, 123, 216–232. [Google Scholar] [CrossRef] - Zou, S.; Xiao, H.; Ye, F.; Li, Z.; Tang, W.; Zhu, F.; Chen, C.; Zhu, C. Numerical analysis of the effect of the scan strategy on the residual stress in the multi-laser selective laser melting. Results Phys.
**2020**, 16, 103005. [Google Scholar] [CrossRef] - Hill, R. Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids
**1963**, 11, 357–372. [Google Scholar] [CrossRef] - Hill, R. On constitutive macro-variables for heterogeneous solids at finite strain. Proceedings of the Royal Society A: Mathematical. Phys. Eng. Sci.
**1972**, 326, 131–147. [Google Scholar] - Mandel, J. Contribution th’eorique á l’étude de l’écrouissage et des lois de l’écoulement plastique. In Applied Mechanics; Springer: Berlin/Heidelberg, Germany, 1966; pp. 502–509. [Google Scholar]
- Ogden, R. On the overall moduli of non-linear elastic composite materials. J. Mech. Phys. Solids
**1974**, 22, 541–555. [Google Scholar] [CrossRef] - Sab, K. On the homogenization and the simulation of random materials on the homogenization and the simulation of random materials. Eur. J. Mech. A-Solids
**1992**, 11, 585–607. [Google Scholar] - Gitman, I.M.; Askes, H.; Sluys, L.J. Representative volume: Existence and size determination. Eng. Fract. Mech.
**2007**, 74, 2518–2534. [Google Scholar] [CrossRef] - Pelissou, C.; Baccou, J.; Monerie, Y.; Perales, F. Determination of the size of the representative volume element for random quasi-brittle composites. Int. J. Solids Struct.
**2009**, 46, 2842–2855. [Google Scholar] [CrossRef] - Ostoja-Starzewski, M. Random field models of heterogeneous materials. Int. J. Solids Struct.
**1998**, 35, 2429–2455. [Google Scholar] [CrossRef] - Ostoja-Starzewski, M.; Du, X.; Khisaeva, Z.F.; Li, W. Comparisons of the size of the representative volume element in elastic, plastic, thermoelastic, and permeable random microstructures. Int. J. Multiscale Comput. Eng.
**2007**, 5, 73–82. [Google Scholar] [CrossRef] - Kanit, T.; Forest, S.; Galliet, I.; Mounoury, V.; Jeulin, D. Determination of the size of the representative volume element for random composites: Statistical and numerical approach. Int. J. Solids Struct.
**2003**, 40, 3647–3679. [Google Scholar] [CrossRef] - Abedi, R.; Garrard, J.; Acton, K.; Soghrati, S. Effect of boundary condition and statistical volume element size on inhomogeneity and anisotropy of apparent properties. Mech. Mater.
**2022**, 173, 104408. [Google Scholar] [CrossRef] - Huyse, L.; Maes, M.A. Random field modeling of elastic properties using homogenization. J. Eng. Mech.
**2001**, 127, 27–36. [Google Scholar] [CrossRef] - Segurado, J.; Llorca, J. Computational micromechanics of composites: The effect of particle spatial distribution. Mech. Mater.
**2006**, 38, 873–883. [Google Scholar] [CrossRef] - Tregger, N.; Corr, D.; Graham-Brady, L.; Shah, S. Modeling the effect of mesoscale randomness on concrete fracture. Probabilistic Eng. Mech.
**2006**, 21, 217–225. [Google Scholar] [CrossRef] - Bazant, Z.P.; Planas, J. Fracture and Size Effect in Concrete and Other Quasi-Brittle Materials; CRC Press: Rutledge, NY, USA, 1998; Volume 16. [Google Scholar]
- Genet, M.; Couegnat, G.; Tomsia, A.P.; Ritchie, R.O. Scaling strength distributions in quasi-brittle materials from micro- to macro-scales: A computational approach to modeling nature inspired structural ceramics. J. Mech. Phys. Solids
**2014**, 68, 93–106. [Google Scholar] [CrossRef] - Strack, O.E.; Leavy, R.B.; Brannon, R.M. Aleatory uncertainty and scale effects in computational damage models for failure and fragmentation. Int. J. Numer. Methods Eng.
**2015**, 102, 468–495. [Google Scholar] [CrossRef] - Dimas, L.S.; Giesa, T.; Buehler, M.J. Coupled continuum and discrete analysis of random heterogeneous materials: Elasticity and fracture. J. Mech. Phys. Solids
**2014**, 63, 481–490. [Google Scholar] [CrossRef] - Baxter, S.C.; Graham, L.L. Characterization of random composites using moving-window technique. J. Eng. Mech.
**2000**, 126, 389–397. [Google Scholar] [CrossRef] - Graham, L.L.; Baxter, S.C. Simulation of local material properties based on moving-window GMC. Probabilistic Eng. Mech.
**2001**, 16, 295–305. [Google Scholar] [CrossRef] - Al-Ostaz, A.; Jasiuk, I. Crack initiation and propagation in materials with randomly distributed holes. Eng. Fract. Mech.
**1997**, 58, 395–420. [Google Scholar] [CrossRef] - Kozicki, J.; Tejchman, J. Effect of aggregate structure on fracture process in concrete using 2D lattice model. Arch. Mech.
**2007**, 59, 365–384. [Google Scholar] - Hanzl, P.; Zetek, M.; Bakša, T.; Kroupa, T. The Influence of Processing Parameters on the Mechanical Properties of SLM Parts. Procedia Eng.
**2015**, 100, 1405–1413. [Google Scholar] [CrossRef] - Soghrati, S.; Nagarajan, A.; Liang, B. Conforming to interface structured adaptive mesh refinement: New technique for the automated modeling of materials with complex microstructures. Finite Elem. Anal. Des.
**2017**, 125, 24–40. [Google Scholar] [CrossRef] - Soghrati, S.; Xiao, F.; Nagarajan, A. A conforming to interface structured adaptive mesh refinement technique for modeling fracture problems. Comput. Mech.
**2017**, 59, 667–684. [Google Scholar] [CrossRef] - Bahmani, B.; Yang, M.; Nagarajan, A.; Clarke, P.L.; Soghrati, S.; Abedi, R. Automated homogenization-based fracture analysis: Effects of SVE size and boundary condition. Comput. Methods Appl. Mech. Eng.
**2019**, 345, 701–727. [Google Scholar] [CrossRef] - Abaqus. Abaqus Analysis User’s Manual (V6.6); Washington University: St. Louis, MO, USA, 2006. [Google Scholar]
- Huet, C. Application of variational concepts to size effects in elastic heterogeneous bodies. J. Mech. Phys. Solids
**1990**, 38, 813–841. [Google Scholar] [CrossRef] - Hazanov, S.; Huet, C. Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume. J. Mech. Phys. Solids
**1994**, 42, 1995–2011. [Google Scholar] [CrossRef] - Hazanov, S.; Amieur, M. On overall properties of elastic heterogeneous bodies smaller than the representative volume. Int. J. Eng. Sci.
**1995**, 33, 1289–1301. [Google Scholar] [CrossRef] - Ming, Y.; Abedi, R.; Garrard, J.; Soghrati, S. Effect of microstructural variations on the failure response of a nano-enhanced polymer: A homogenization-based statistical analysis. Comput. Mech.
**2021**, 67, 315–340. [Google Scholar] - Liu, W.K.; Siad, L.; Tian, R.; Lee, S.; Lee, D.; Yin, X.; Chen, W.; Chan, S.; Olson, G.B.; Lindgen, L.E.; et al. Complexity science of multiscale materials via stochastic computations. Int. J. Numer. Methods Eng.
**2009**, 80, 932–978. [Google Scholar] [CrossRef] - Jiang, Z.; Xu, P.; Liang, Y.; Liang, Y. Deformation effect of melt pool boundaries on the mechanical property anisotropy in the SLM AlSi
_{10}Mg. Mater. Today Commun.**2023**, 36, 106879. [Google Scholar] [CrossRef] - Luo, Z.; Zhao, Y. A survey of finite element analysis of temperature and thermal stress fields in powder bed fusion Additive Manufacturing. Addit. Manuf.
**2018**, 21, 318–332. [Google Scholar] [CrossRef] - Letenneur, M.; Kreitcberg, A.; Brailovski, V. Optimization of laser powder bed fusion processing using a combination of melt pool modeling and design of experiment approaches: Density control. J. Manuf. Mater. Process.
**2019**, 3, 21. [Google Scholar] [CrossRef] - Burkhardt, C.; Soldner, D.; Mergheim, J. A comparison of material models for the simulation of selective beam melting processes. Procedia CIRP
**2020**, 94, 52–57. [Google Scholar] [CrossRef] - Luo, Z.; Zhao, Y. Numerical simulation of part-level temperature fields during selective laser melting of stainless steel 316 L. Int. J. Adv. Manuf. Technol.
**2019**, 104, 1615–1635. [Google Scholar] [CrossRef] - Sun, Z.; Tan, X.; Tor, S.B.; Yeong, W.Y. Selective laser melting of stainless steel 316 L with low porosity and high build rates. Mater. Des.
**2016**, 104, 197–204. [Google Scholar] [CrossRef]

**Figure 1.**Conceptual drawing of a melt pool [3].

**Figure 3.**Transformation of a structured mesh composed of 4-node quadrilateral elements into a conforming mesh using the CISAMR algorithm: (

**a**) h-adaptive refinement in the vicinity of material interfaces; (

**b**) r- adaptivity of the nodes of elements intersecting each interface; (

**c**) sub-triangulating the elements deformed during the r-adaptivity process, as well as elements with hanging nodes created after h-adaptivity. Red, and yellow nodes correspond to hanging nodes created during the SAMR process, and conforming background mesh nodes, respectively.

**Figure 4.**Modifications in the original CISAMR algorithm to handle the presence of sharp corners along material interfaces. Blue, red, and yellow nodes correspond to sharp corners of the material interface, hanging nodes created during the SAMR process, and conforming background mesh nodes, respectively.

**Figure 5.**(

**a**) The microstructure of the SLM material. Each color representing a unique melt pool; (

**b**) a sample SVE extracted from CISAMR mesh generated on the domain in (

**a**), with an inset showing the mesh structure.

**Figure 6.**Bilinear traction separation relation (TSR) modified from [42].

**Figure 7.**Calibration of cohesive model stiffness $k$: (

**a**) vertical elastic modulus ${E}_{h}$; (

**b**) vertical yield (max) stress as a function of $k$.

**Figure 8.**(

**a**) Domain and boundary condition for vertical loading for parameter calibration; (

**b**) vertical stress versus strain for the domain in (

**a**) with calibrated parameters.

**Figure 10.**Vertical results for ${E}_{v}$: (

**a**) PDF of ${E}_{v}$ for all SVE sizes. The red dot serves as a reference to the published value from [38]; (

**b**) the size effect plot for ${E}_{v}$.

**Figure 11.**Vertical results for ${\sigma}_{v,\mathrm{m}\mathrm{a}\mathrm{x}}$: (

**a**) PDF of ${\sigma}_{v,\mathrm{m}\mathrm{a}\mathrm{x}}$ for all SVE sizes. The red dot serves as a reference to the published value from [38]; (

**b**) the size effect plot for ${\sigma}_{v,\mathrm{m}\mathrm{a}\mathrm{x}}$.

**Figure 12.**Horizontal stress vs. strain overlay for (

**a**) ${L}_{SVE}=$ 56 μm; (

**b**) ${L}_{SVE}=$ 140 μm.

**Figure 13.**Horizontal results for ${E}_{h}$: (

**a**) PDF of ${E}_{h}$ for all SVE sizes. The red dot serves as a reference to the published value from [38]; (

**b**) the size effect plot of ${E}_{h}$ for all SVE sizes.

**Figure 14.**Horizontal results for ${\sigma}_{h,max}$ (

**a**) PDF of ${\sigma}_{h,max}$ for all SVE sizes. The red dot serves as a reference to the published value from [38]; (

**b**) Size effect plot for ${\sigma}_{h,max}$.

**Table 1.**Experimental results compared to simulation results for vertical and horizontal load cases.

Experimental [38] | Calibration Domain | |||
---|---|---|---|---|

Horizontal | Vertical | Horizontal | Vertical | |

Young’s Modulus (GPa) | 172 | 160 | 171.75 | 160.80 |

Yield (max) Strength (GPa) | 1.09 | 1.08 | 1.27 | 1.07 |

Calibrated Material Properties | |
---|---|

${E}_{m}$ | 175 GPa |

$\mathsf{\nu}$ | 0.3 |

$k$ | 72 GPa |

${\sigma}_{max}$ | 1.08 GPa |

$\varphi $ | 0.025 GPa-μm |

**Table 3.**Summary of values used to examine mean- and variation-based RVE sizes for all four homogenized properties.

Horizontal | Vertical | |||
---|---|---|---|---|

${\mathit{E}}_{\mathit{h}}$ | ${\mathit{\sigma}}_{\mathit{h},\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{E}}_{\mathit{v}}$ | ${\mathit{\sigma}}_{\mathit{v},\mathit{m}\mathit{a}\mathit{x}}$ | |

Mean-based: e(μ) | 0.000604 | 0.00766 | 0.00146 | 0.0166 |

Variation-based: COV (${L}_{SVE}=140$ μm) | 0.00460 | 0.0230 | 0.00997 | 0.0162 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Connor, R.P.; Vemparala, B.; Abedi, R.; Huynh, G.; Soghrati, S.; Feldmeier, C.T.; Lamb, K.
Statistical Homogenization of Elastic and Fracture Properties of a Sample Selective Laser Melting Material. *Appl. Sci.* **2023**, *13*, 12408.
https://doi.org/10.3390/app132212408

**AMA Style**

Connor RP, Vemparala B, Abedi R, Huynh G, Soghrati S, Feldmeier CT, Lamb K.
Statistical Homogenization of Elastic and Fracture Properties of a Sample Selective Laser Melting Material. *Applied Sciences*. 2023; 13(22):12408.
https://doi.org/10.3390/app132212408

**Chicago/Turabian Style**

Connor, Ryan P., Balavignesh Vemparala, Reza Abedi, Giang Huynh, Soheil Soghrati, Chris T. Feldmeier, and Kevin Lamb.
2023. "Statistical Homogenization of Elastic and Fracture Properties of a Sample Selective Laser Melting Material" *Applied Sciences* 13, no. 22: 12408.
https://doi.org/10.3390/app132212408