# Numerical and Experimental Investigation of Cumulative Fatigue Damage under Random Dynamic Cyclic Loads of Lattice Structures Manufactured by Laser Powder Bed Fusion

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Experimental Methods

#### 2.1. Material, Cell Topology and Specimen Shape and Dimension

#### 2.2. Experimental Tests and Setup

#### 2.2.1. Sweep Frequency Test

#### 2.2.2. Endurance Test

#### 2.2.3. Random Test

## 3. Numerical Models

#### Numerical Analyses for Failure Identification

## 4. Experimental Results

#### 4.1. Endurance Test

#### 4.2. Random Tests

## 5. Fatigue Life Analysis

#### 5.1. Fatigue Life Evaluation under Resonant Conditions

#### 5.2. Damage Evaluation under Random Fatigue Loads

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Tancogne-Dejean, T.; Diamantopoulou, M.; Gorji, M.B.; Bonatti, C.; Mohr, D. 3D Plate-Lattices: An Emerging Class of Low-Density Metamaterial Exhibiting Optimal Isotropic Stiffness. Adv. Mater.
**2018**, 30, 1–6. [Google Scholar] [CrossRef] - Kolken, H.M.A.; Zadpoor, A.A. Auxetic mechanical metamaterials. RSC Adv.
**2017**, 7, 5111–5129. [Google Scholar] [CrossRef] [Green Version] - Scalzo, F.; Totis, G.; Vaglio, E.; Sortino, M. Experimental study on the high-damping properties of metallic lattice structures obtained from SLM. Precis. Eng.
**2021**, 71, 63–77. [Google Scholar] [CrossRef] - Tancogne-Dejean, T.; Spierings, A.B.; Mohr, D. Additively-manufactured metallic micro-lattice materials for high specific energy absorption under static and dynamic loading. Acta Mater.
**2016**, 116, 14–28. [Google Scholar] [CrossRef] - Maconachie, T.; Leary, M.; Lozanovski, B.; Zhang, X.; Qian, M.; Faruque, O.; Brandt, M. SLM lattice structures: Properties, performance, applications and challenges. Mater. Des.
**2019**, 183, 108137. [Google Scholar] [CrossRef] - Lira, C.; Scarpa, F.; Rajasekaran, R. A gradient cellular core for aeroengine fan blades based on auxetic configurations. J. Intell. Mater. Syst. Struct.
**2011**, 22, 907–917. [Google Scholar] [CrossRef] - Spadoni, A.; Ruzzene, M.; Scarpa, F. Dynamic response of chiral truss-core assemblies. J. Intell. Mater. Syst. Struct.
**2006**, 17, 941–952. [Google Scholar] [CrossRef] [Green Version] - Colombo, C.; Biffi, C.; Fiocchi, J.; Scaccabarozzi, D.; Saggin, B.; Tuissi, A.; Vergani, L. Modulating the damping capacity of SLMed AlSi10Mg trough stress-relieving thermal treatments. Theor. Appl. Fract. Mech.
**2020**, 107, 1–6. [Google Scholar] [CrossRef] - Fiocchi, J.; Biffi, C.A.; Scaccabarozzi, D.; Saggin, B.; Tuissi, A. Enhancement of the Damping Behavior of Ti6Al4V Alloy through the Use of Trabecular Structure Produced by Selective Laser Melting. Adv. Eng. Mater.
**2020**, 22, 1–6. [Google Scholar] [CrossRef] - Liu, J.; Guo, K.; Sun, J.; Sun, Q.; Wang, L.; Li, H. Compressive behavior and vibration-damping properties of porous Ti-6Al-4V alloy manufactured by laser powder bed fusion. J. Manuf. Process.
**2021**, 66, 1–10. [Google Scholar] [CrossRef] - Zadeh, M.N.; Alijani, F.; Chen, X.; Dayyani, I.; Yasaee, M.; Mirzaali, M.J.; Zadpoor, A.A. Dynamic characterization of 3D printed mechanical metamaterials with tunable elastic properties. Appl. Phys. Lett.
**2021**, 118, 211901. [Google Scholar] [CrossRef] - Benedetti, M.; du Plessis, A.; Ritchie, R.; Dallago, M.; Razavi, S.; Berto, F. Architected cellular materials: A review on their mechanical properties towards fatigue-tolerant design and fabrication. Mater. Sci. Eng. R Rep.
**2021**, 144, 100606. [Google Scholar] [CrossRef] - Yavari, S.A.; Ahmadi, S.; Wauthle, R.; Pouran, B.; Schrooten, J.; Weinans, H.; Zadpoor, A.A. Relationship between unit cell type and porosity and the fatigue behavior of selective laser melted meta-biomaterials. J. Mech. Behav. Biomed. Mater.
**2015**, 43, 91–100. [Google Scholar] [CrossRef] - Van Hooreweder, B.; Apers, Y.; Lietaert, K.; Kruth, J.-P. Improving the fatigue performance of porous metallic biomaterials produced by Selective Laser Melting. Acta Biomater.
**2017**, 47, 193–202. [Google Scholar] [CrossRef] [PubMed] - Ahmadi, S.; Hedayati, R.; Li, Y.; Lietaert, K.; Tümer, N.; Fatemi, A.; Rans, C.; Pouran, B.; Weinans, H.; Zadpoor, A.; et al. Fatigue performance of additively manufactured meta-biomaterials: The effects of topology and material type. Acta Biomater.
**2018**, 65, 292–304. [Google Scholar] [CrossRef] [Green Version] - Zhao, S.; Li, S.; Wang, S.; Hou, W.; Li, Y.; Zhang, L.; Hao, Y.; Yang, R.; Misra, R.; Murr, L. Compressive and fatigue behavior of functionally graded Ti-6Al-4V meshes fabricated by electron beam melting. Acta Mater.
**2018**, 150, 1–15. [Google Scholar] [CrossRef] - Yavari, S.A.; Wauthle, R.; van der Stok, J.; Riemslag, A.; Janssen, M.; Mulier, M.; Kruth, J.; Schrooten, J.; Weinans, H.; Zadpoor, A.A. Fatigue behavior of porous biomaterials manufactured using selective laser melting. Mater. Sci. Eng. C
**2013**, 33, 4849–4858. [Google Scholar] [CrossRef] [PubMed] - Abad, E.M.K.; Khanoki, S.A.; Pasini, D. Fatigue design of lattice materials via computational mechanics: Application to lattices with smooth transitions in cell geometry. Int. J. Fatigue
**2013**, 47, 126–136. [Google Scholar] [CrossRef] [Green Version] - Dobson, S.D.; Starr, T.L. Powder characterization and part density for powder bed fusion of 17-4 PH stainless steel. Rapid Prototyp. J.
**2020**, 27, 53–58. [Google Scholar] [CrossRef] - Donik, Č.; Kraner, J.; Paulin, I.; Godec, M. Influence of the Energy Density for Selective Laser Melting on the Microstructure and Mechanical Properties of Stainless Steel. Metals
**2020**, 10, 919. [Google Scholar] [CrossRef] - Ullah, A.; Wu, H.; Rehman, A.U.; Zhu, Y.; Liu, T.; Zhang, K. Influence of laser parameters and Ti content on the surface morphology of L-PBF fabricated Titania. Rapid Prototyp. J.
**2020**, 27, 71–80. [Google Scholar] [CrossRef] - Pal, S.; Lojen, G.; Gubeljak, N.; Kokol, V.; Drstvensek, I. Melting, fusion and solidification behaviors of Ti-6Al-4V alloy in selective laser melting at different scanning speeds. Rapid Prototyp. J.
**2020**, 26, 1209–1215. [Google Scholar] [CrossRef] - Pawlak, A.; Szymczyk-Ziółkowska, P.; Kurzynowski, T.; Chlebus, E. Selective laser melting of magnesium AZ31B alloy powder. Rapid Prototyp. J.
**2019**, 26, 249–258. [Google Scholar] [CrossRef] - Liang, C.; Hu, Y.; Liu, N.; Zou, X.; Wang, H.; Zhang, X.; Fu, Y.; Hu, J. Laser Polishing of Ti6Al4V Fabricated by Selective Laser Melting. Metals
**2020**, 10, 191. [Google Scholar] [CrossRef] [Green Version] - Dallago, M.; Raghavendra, S.; Luchin, V.; Zappini, G.; Pasini, D.; Benedetti, M. The role of node fillet, unit-cell size and strut orientation on the fatigue strength of Ti-6Al-4V lattice materials additively manufactured via laser powder bed fusion. Int. J. Fatigue
**2020**, 142, 105946. [Google Scholar] [CrossRef] - Boniotti, L.; Beretta, S.; Patriarca, L.; Rigoni, L.; Foletti, S. Experimental and numerical investigation on compressive fatigue strength of lattice structures of AlSi7Mg manufactured by SLM. Int. J. Fatigue
**2019**, 128, 105181. [Google Scholar] [CrossRef] - Yang, L.; Yan, C.; Cao, W.; Liu, Z.; Song, B.; Wen, S.; Zhang, C.; Shi, Y.; Yang, S. Compression–compression fatigue behaviour of gyroid-type triply periodic minimal surface porous structures fabricated by selective laser melting. Acta Mater.
**2019**, 181, 49–66. [Google Scholar] [CrossRef] - Kelly, C.N.; Francovich, J.; Julmi, S.; Safranski, D.; Guldberg, R.E.; Maier, H.J.; Gall, K. Fatigue behavior of As-built selective laser melted titanium scaffolds with sheet-based gyroid microarchitecture for bone tissue engineering. Acta Biomater.
**2019**, 94, 610–626. [Google Scholar] [CrossRef] - Bobbert, F.; Lietaert, K.; Eftekhari, A.A.; Pouran, B.; Ahmadi, S.; Weinans, H.; Zadpoor, A.A. Additively manufactured metallic porous biomaterials based on minimal surfaces: A unique combination of topological, mechanical, and mass transport properties. Acta Biomater.
**2017**, 53, 572–584. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Richard, B.; Pellicone, D.; Anderson, W.G. Progress on the Development of a 3D Printed Loop Heat Pipe. In Proceedings of the 2019 35th Semiconductor Thermal Measurement, Modeling and Management Symposium (SEMI-THERM), San Jose, CA, USA, 18–22 March 2019; pp. 1–11. [Google Scholar]
- Schmitz, A.; Horst, P. A finite element unit-cell method for homogenised mechanical properties of heterogeneous plates. Compos. Part A Appl. Sci. Manuf.
**2014**, 61, 23–32. [Google Scholar] [CrossRef] - Tancogne-Dejean, T.; Mohr, D. Elastically-isotropic truss lattice materials of reduced plastic anisotropy. Int. J. Solids Struct.
**2018**, 138, 24–39. [Google Scholar] [CrossRef] - Sausto, F.; Carrion, P.; Shamsaei, N.; Beretta, S. Fatigue failure mechanisms for AlSi10Mg manufactured by L-PBF under axial and torsional loads: The role of defects and residual stresses. Int. J. Fatigue
**2021**, submit. [Google Scholar] - Lee, Y.L.; Tjhung, T. Rainflow Cycle Counting Techniques; Elsevier Inc.: Amsterdam, The Netherlands, 2012. [Google Scholar] [CrossRef]
- Nourian-Avval, A.; Fatemi, A. Variable amplitude fatigue behavior and modeling of cast aluminum. Fatigue Fract. Eng. Mater. Struct.
**2021**, 44, 1611–1621. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**,

**b**) FCC and Diamond specimen modelled with nTopology. (

**c**,

**d**) FCC and Diamond lattice specimens made by L-PBF.

**Figure 2.**(

**a**) Lateral view of the electrodynamic shaker with mounted FCC specimen. (

**b**) Axonometric view of the structure. A set of accelerometers is bonded on the tip, base and shaker piston, while a pair of masses have been rigidly connected on the tip to tailor the desired dynamic behavior.

**Figure 5.**Detailed view of the damage families considered in this work for the FCC lattice configuration. (

**a**) Broken strut and (

**b**) missing node. (

**c**,

**d**) FRF of the as-designed (black line) and induced-damage geometry (red and blue lines) for FCC and diamond cell respectively.

**Figure 6.**Accelerations measured in correspondence of the base and tip during the endurance tests. (

**a**) FCC specimen number 7 and (

**b**) Diamond specimen 4.

**Figure 7.**Spectrogram relative to the FCC specimen number 4 tested under random vibration. (

**a**) Axonometric and (

**b**) top view. The frequency trend is tracked and displayed in (

**c**) for a better visualization of the failure time.

**Figure 8.**Spectrogram relative to the Diamond specimen number 7 tested under random vibration. (

**a**) Axonometric view and (

**b**) top view. (

**c**) Peak amplitude tracked in time to show the frequency drop.

**Figure 10.**SN curve of the AlSi10Mg bulk material with superimposed SN data points associated to the FCC and Diamond configuration for resonant condition tests.

Material | Elastic Modulus [MPa] | Shear Modulus [MPa] | Poisson’s Ratio | Yield Stress [MPa] |
---|---|---|---|---|

AlSi10Mg | 68,000 | 26,154 | 0.3 | 240 |

Frequency [Hz] | Amplitude [g^{2}/Hz] |
---|---|

15 | 0.04 |

100 | 0.04 |

300 | 0.17 |

1000 | 0.17 |

2000 | 0.05 |

**Table 3.**Homogenized mechanical elastic properties and yield strength of the FCC and Diamond unit cells.

Cell Type | ρ [%] | Elastic Modulus [MPa] | Shear Modulus [MPa] | Poisson’s Ratio | Yield Stress [MPa] |
---|---|---|---|---|---|

FCC | 7.3 | 1371.4 | 314.4 | 0.19 | 5.74 |

Diamond | 7.0 | 174.1 | 178.8 | 0.46 | 1.65 |

**Table 4.**First natural frequency and FRF of the numerical models and the results of the sweep tests.

Damping Ratio [%] | ${\mathit{f}}_{\mathbf{0}}\mathbf{\left[}\mathbf{Hz}\mathbf{\right]}$ | $\mathbf{\left|}\mathit{A}\mathbf{\left(}{\mathit{\omega}}_{\mathbf{0}}\mathbf{\right)}\mathbf{\right|}$ | ||
---|---|---|---|---|

FCC | Numerical (Homogenized) | 0.21 | 410.7 | 244.5 |

Numerical (Discrete) | 0.21 | 420.8 | 243.9 | |

Experimental * | 0.21 (0.03) | 399.2 (2.7) | 256.2 (31.7) | |

Diamond | Numerical (Homogenized) | 0.31 | 368.7 | 166.2 |

Numerical (Discrete) | 0.31 | 383.8 | 166.0 | |

Experimental * | 0.31 (0.08) | 353.8 (3.8) | 176.0 (25.5) |

**Table 5.**Reduction of frequency and tip acceleration under resonant conditions through numerical models.

As-Designed | Missing Strut | Missing Node | ||
---|---|---|---|---|

FCC | ${f}_{0}$ [Hz] | 420.8 | 420.2 | 418.7 |

Tip acceleration [g] | 243.7 | 201.4 | 92.9 | |

Diamond | ${f}_{0}$ [Hz] | 420.8 | 420.2 | 418.7 |

Tip acceleration [g] | 243.7 | 201.4 | 92.9 |

Specimen No. | Tip Displacement [mm] | Excitation Frequency [Hz] | Failure Time [s] | Cycle to Failure |
---|---|---|---|---|

7 | 0.31 | 398 | 88.2 | 35104 |

8 | 0.3 | 400 | 99.2 | 39680 |

9 | 0.32 | 392.5 | 71.4 | 28024 |

10 | 0.2 | 400.4 | 598.1 | 239480 |

11 | 0.18 | 396.3 | 610.9 | 242100 |

Specimen No. | Tip Displacement [mm] | Excitation Frequency [Hz] | Failure Time [s] | Cycle to Failure |
---|---|---|---|---|

1 | 0.23 | 352.6 | 236.1 | 83249 |

2 | 0.25 | 357.3 | 421.1 | 150460 |

3 | 0.33 | 353.9 | 55.4 | 19606 |

4 | 0.32 | 349.6 | 102.2 | 35729 |

Specimen No. | PSD Type | Test Duration [h] | Initial Sweep Freq. [Hz] | Mid-Sweep Freq. [Hz] | Final Sweep Freq. [Hz] | Initial FRF | Time to Failure [s] |
---|---|---|---|---|---|---|---|

2 | /15 | 2 | 398.9 | 398.1 | 397.5 | 273.5 | 2400 |

3 | /10 | 1 | 399.9 | / | 398.5 | 198.2 | 2060 |

4 | /10 | 1 | 400.8 | / | 397 | 259.2 | 2035 |

5 | /15 | 2 | 401.4 | 401.4 | 400.5 | 273.8 | 4355 |

Specimen No. | PSD Type | Test Duration [h] | Initial Sweep Freq. [Hz] | Mid-Sweep Freq. [Hz] | Final Sweep Freq. [Hz] | Initial FRF | Time to Failure [s] |
---|---|---|---|---|---|---|---|

6 | /10 | 1 | 358.9 | / | 357.8 | 122.6 | 1620 |

7 | /10 | 1 | 351.9 | / | 350.2 | 169.3 | 855 |

8 | /15 | 2 | 353.7 | 352.9 | 352.6 | 172.8 | Undamaged |

9 | /15 | 2 | 357.1 | 356.3 | 355.9 | 151.8 | 2430 |

Specimen No. | 2 | 3 | 4 | 5 |
---|---|---|---|---|

Time to failure [s] | 2400 | 2060 | 2035 | 4355 |

Damage to failure | 0.9 | 0.94 | 1.55 | 1.31 |

Specimen No. | 6 | 7 | 8 | 9 |
---|---|---|---|---|

Time to failure [s] | 1620 | 855 | / | 2430 |

Damage to failure | 1.37 | 0.93 | / | 1.18 |

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**MDPI and ACS Style**

Pisati, M.; Corneo, M.G.; Beretta, S.; Riva, E.; Braghin, F.; Foletti, S.
Numerical and Experimental Investigation of Cumulative Fatigue Damage under Random Dynamic Cyclic Loads of Lattice Structures Manufactured by Laser Powder Bed Fusion. *Metals* **2021**, *11*, 1395.
https://doi.org/10.3390/met11091395

**AMA Style**

Pisati M, Corneo MG, Beretta S, Riva E, Braghin F, Foletti S.
Numerical and Experimental Investigation of Cumulative Fatigue Damage under Random Dynamic Cyclic Loads of Lattice Structures Manufactured by Laser Powder Bed Fusion. *Metals*. 2021; 11(9):1395.
https://doi.org/10.3390/met11091395

**Chicago/Turabian Style**

Pisati, Marco, Marco Giuseppe Corneo, Stefano Beretta, Emanuele Riva, Francesco Braghin, and Stefano Foletti.
2021. "Numerical and Experimental Investigation of Cumulative Fatigue Damage under Random Dynamic Cyclic Loads of Lattice Structures Manufactured by Laser Powder Bed Fusion" *Metals* 11, no. 9: 1395.
https://doi.org/10.3390/met11091395