# CT-Based Micro-Mechanical Approach to Predict Response of Closed-Cell Porous Biomaterials to Low-Velocity Impact

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Tests

#### 2.1. Material

^{3}. The diameter of the pores was measured from the CT-scan images. The average pore size in the specimens was 1.5 mm. The foam plate was cut into several specimens with dimensions around 3 × 3 × 4 cm

^{3}. (Figure 2). In each specimen, coordinate z represents the longer direction of the specimen. The mean density of the prepared specimens was in the range of 600–800 kg/m

^{3}. The relative large range of densities was due to the presence of very large cavities in some specimens with sizes dozens of times larger than the pore size (For example see Figure 2 right).

#### 2.2. Compressive Quasi-Static Tests

#### 2.3. Drop Hammer Impact Tests

## 3. FE Modeling

#### 3.1. Micro-Scale Model

_{s}which represents the failure strain and was used to eliminate the failed elements after being highly distorted. The graphical view of the micro-structural FE model of a specimen is shown in Figure 6.

#### 3.2. Macro-Scale Model

_{0}is the center of yield ellipse on the p-axis, p

_{c}is the yield stress in hydrostatic compression, and p

_{t}is the strength of the material in hydrostatic tension [25].

## 4. Results and Discussions

#### 4.1. Validation of the Micro-Scale FE Models

#### 4.2. Comparison of the Results of Micro- and Macro-Scale Models

#### 4.3. Effect of Drop Weight Initial Height and Mass

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Gibson, L.J.; Ashby, M.F. Cellular Solids: Structure and Properties; Cambridge University Press: Hongkong, China, 1997. [Google Scholar]
- Hedayati, R.; Sadighi, M.; Mohammadi-Aghdam, M.; Zadpoor, A.A. Mechanical properties of regular porous biomaterials made from truncated cube repeating unit cells: Analytical solutions and computational models. Mater. Sci. Eng. C
**2016**, 60, 163–183. [Google Scholar] [CrossRef] [PubMed] - Warren, W.; Kraynik, A. Linear elastic behavior of a low-density kelvin foam with open cells. J. Appl. Mech.
**1997**, 64, 787–794. [Google Scholar] [CrossRef] - Hedayati, R.; Sadighi, M.; Mohammadi Aghdam, M.; Zadpoor, A.A. Mechanical properties of additively manufactured thick honeycombs. Materials
**2016**, 9, 613. [Google Scholar] [CrossRef] [PubMed] - Weaire, D.; Phelan, R. Cellular structures in three dimensions. Philos. Trans. R. Soc. Lond. A: Math. Phys. Eng. Sci.
**1996**, 354, 1989–1997. [Google Scholar] [CrossRef] - Veyhl, C.; Belova, I.; Murch, G.; Fiedler, T. Finite element analysis of the mechanical properties of cellular aluminium based on micro-computed tomography. Mater. Sci. Eng. A
**2011**, 528, 4550–4555. [Google Scholar] [CrossRef] - Miedzińska, D.; Niezgoda, T.; Gieleta, R. Numerical and experimental aluminum foam microstructure testing with the use of computed tomography. Comput. Mater. Sci.
**2012**, 64, 90–95. [Google Scholar] [CrossRef] - Bock, J.; Jacobi, A.M. Geometric classification of open-cell metal foams using X-ray micro-computed tomography. Mater. Charact.
**2013**, 75, 35–43. [Google Scholar] [CrossRef] - Hedayati, R.; Sadighi, M.; Mohammadi-Aghdam, M.; Zadpoor, A. Mechanical behavior of additively manufactured porous biomaterials made from truncated cuboctahedron unit cells. Int. J. Mech. Sci.
**2016**, 106, 19–38. [Google Scholar] [CrossRef] - Hedayati, R.; Sadighi, M.; Mohammadi-Aghdam, M.; Zadpoor, A.A. Mechanics of additively manufactured porous biomaterials based on the rhombicuboctahedron unit cell. J. Mech. Behav. Biomed. Mater.
**2016**, 53, 272–294. [Google Scholar] [CrossRef] [PubMed] - Stress Shielding. Available online: https://en.wikipedia.org/wiki/Stress_shielding (accessed on 10 January 2018).
- Banhart, J. Manufacture, characterisation and application of cellular metals and metal foams. Prog. Mater. Sci.
**2001**, 46, 559–632. [Google Scholar] [CrossRef] - Ashby, M.F.; Evans, T.; Fleck, N.A.; Hutchinson, J.; Wadley, H.; Gibson, L. Metal Foams: A Design Guide; Elsevier: Amsterdam, The Netherlands, 2000. [Google Scholar]
- Rajendran, R.; Moorthi, A.; Basu, S. Numerical simulation of drop weight impact behaviour of closed cell aluminium foam. Mater. Des.
**2009**, 30, 2823–2830. [Google Scholar] [CrossRef] - Li, B.; Zhao, G.; Lu, T. Low strain rate compressive behavior of high porosity closed-cell aluminum foams. Sci. China Tech. Sci.
**2012**, 55, 451–463. [Google Scholar] [CrossRef] - Zhang, C.; Tang, L.; Yang, B.; Zhang, L.; Huang, X.; Fang, D. Meso-mechanical study of collapse and fracture behaviors of closed-cell metallic foams. Comput. Mater. Sci.
**2013**, 79, 45–51. [Google Scholar] [CrossRef] - Liu, Y.; Gong, W.; Zhang, X. Numerical investigation of influences of porous density and strain-rate effect on dynamical responses of aluminum foam. Comput. Mater. Sci.
**2014**, 91, 223–230. [Google Scholar] [CrossRef] - Fang, Q.; Zhang, J.; Zhang, Y.; Liu, J.; Gong, Z. Mesoscopic investigation of closed-cell aluminum foams on energy absorption capability under impact. Compos. Struct.
**2015**, 124, 409–420. [Google Scholar] [CrossRef] - Wang, P.; Xu, S.; Li, Z.; Yang, J.; Zhang, C.; Zheng, H.; Hu, S. Experimental investigation on the strain-rate effect and inertia effect of closed-cell aluminum foam subjected to dynamic loading. Mater. Sci. Eng. A
**2015**, 620, 253–261. [Google Scholar] [CrossRef] - Andrews, E.; Gioux, G.; Onck, P.; Gibson, L. Size effects in ductile cellular solids. Part II: Experimental results. Int. J. Mech. Sci.
**2001**, 43, 701–713. [Google Scholar] [CrossRef] - Mohammadi Nasrabadi, A.; Hedayati, R.; Sadighi, M. Numerical and experimental study of the mechanical response of aluminum foams under compressive loading using ct data. J. Theor. Appl. Mech.
**2016**, 54, 1357–1368. [Google Scholar] [CrossRef] - Li, Z.; Xi, C.; Jing, L.; Wang, Z.; Zhao, L. Effect of loading rate on the compressive properties of open-cell metal foams. Mater. Sci. Eng. A
**2014**, 592, 221–229. [Google Scholar] [CrossRef] - ANSYS Mechanical APDL Basic Analysis Guide. Available online: http://allaboutmetallurgy.com/wp/wp-content/uploads/2016/12/ANSYS-Mechanical-APDL-Basic-Analysis-Guide.pdf (accessed on 10 January 2018).
- Tanwongwan, W.; Carmai, J. Finite element modelling of titanium foam behaviour for dental application. In Proceedings of the World Congress on Engineering 2011, London, UK, 6–8 July 2011. [Google Scholar]
- ABAQUS Theory Manual Version 6.11-1 (R20). Available online: http://130.149.89.49:2080/v6.11/pdf_books/THEORY.pdf (accessed on 10 January 2018).
- Irausquín, I.; Teixeira-Dias, F.; Miranda, V.; Pérez-Castellanos, J.L. Numerical modeling of the compression a closed-cell aluminum foam. In Iberian Conference on Fracture and Structural Integrity; Emerald Group Publishing Limited: Porto, Portugal, 2010. [Google Scholar]
- Khalkhali, A.; Mousavi, S. Multi-objective crashworthiness optimization of the aluminum foam-filled tubes. Int. J. Automot. Eng.
**2012**, 2, 193–206. [Google Scholar] - Rizov, V.I. Elastic–plastic response of structural foams subjected to localized static loads. Mater. Des.
**2006**, 27, 947–954. [Google Scholar] [CrossRef] - McKown, S.; Mines, R. Impact behaviour of metal foam cored sandwich beams. In Fracture of Nano and Engineering Materials and Structures; Springer: Berlin, Germany, 2006. [Google Scholar]
- Cho, J.U.; Hong, S.J.; Lee, S.K.; Cho, C. Impact fracture behavior at the material of aluminum foam. Mater. Sci. Eng. A
**2012**, 539, 250–258. [Google Scholar] [CrossRef] - Wang, J.; Waas, A.M.; Wang, H. Experimental and numerical study on the low-velocity impact behavior of foam-core sandwich panels. Compos. Struct.
**2013**, 96, 298–311. [Google Scholar] [CrossRef] - Rajaneesh, A.; Sridhar, I.; Rajendran, S. Numerical modeling of low velocity impact response on metal foam cored sandwich panels. In Proceeding of the 18 th International Conference on Composite Materials, Jeju Island, Korea, 21–26 August 2011. [Google Scholar]
- Flores-Johnson, E.; Li, Q.; Mines, R. Degradation of elastic modulus of progressively crushable foams in uniaxial compression. J. Cell. Plast.
**2008**, 44, 415–434. [Google Scholar] [CrossRef] - Motz, C.; Pippan, R. Deformation behaviour of closed-cell aluminium foams in tension. Acta Mater.
**2001**, 49, 2463–2470. [Google Scholar] [CrossRef] - Li, Z.; Zhang, J.; Fan, J.; Wang, Z.; Zhao, L. On crushing response of the three-dimensional closed-cell foam based on voronoi model. Mech. Mater.
**2014**, 68, 85–94. [Google Scholar] [CrossRef] - Zou, Z.; Reid, S.; Tan, P.; Li, S.; Harrigan, J. Dynamic crushing of honeycombs and features of shock fronts. Int. J. Impact Eng.
**2009**, 36, 165–176. [Google Scholar] [CrossRef]

**Figure 8.**Comparison of stress-strain curves obtained for impacts from axial (Z) and lateral (Y) directions.

**Figure 9.**Acceleration-time and force-displacement curves obtained from experimental tests and microstructural FE models in specimens (

**a**) N15, (

**b**) N18, and (

**c**) N20.

**Figure 11.**(

**a**) Acceleration-time and (

**b**) force-displacement diagrams obtained from experimental tests, microstructural FE model, and macrostructural FE model of specimen N15.

**Figure 12.**(

**a**) Acceleration-time and (

**b**) force-displacement diagrams obtained from two experimental tests with equal initial energy but with different drop weight masses.

**Figure 13.**(

**a**) Acceleration-time and (

**b**) force-displacement diagrams obtained from two micro-structural FE models (N15 and N20) with equal initial energy but with different drop weight masses.

**Figure 14.**(

**a**) Acceleration-time and (

**b**) force-displacement diagrams of macro-structural FE analysis of two impacts with equal drop weight initial energy but with different masses.

Property | Value |
---|---|

Density (kg/m^{3}) | 2685 |

Elastic modulus (GPa) | 71 |

Yield stress (MPa) | 165 |

Tangent modulus (GPa) | 0.487 |

Poisson’s ratio | 0.33 |

Elongation | 2–3.5% |

Specimen # | Mass (g) | Dimension in $\mathit{x}$ Direction (mm) | Dimension in $\mathit{y}$ Direction (mm) | Dimension in $\mathit{z}$ Direction (mm) | Density $\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^{3}\right)$ | Static Test | Impact Test | ||
---|---|---|---|---|---|---|---|---|---|

Elastic | Elastic-Plastic | Drop Weight Mass (kg) | Drop Weight Initial Height (cm) | ||||||

N1 | 25.09 | 31.57 | 32.33 | 40.18 | 612 | * | |||

N2 | 25.92 | 32.5 | 31.9 | 39.65 | 630 | * | |||

N3 | 28.7 | 32.44 | 32.14 | 40.37 | 682 | * | |||

N4 | 25.74 | 31.38 | 31.78 | 39.7 | 650 | * | |||

N5 | 25.49 | 31.78 | 32.22 | 39.9 | 624 | * | |||

N6 | 23.79 | 30.47 | 32.35 | 39.5 | 610 | * | |||

N7 | 24.9 | 32.75 | 31.8 | 39.84 | 600 | * | |||

N8 | 32.96 | 32.76 | 31.58 | 39.97 | 797 | * | |||

N9 | 27.5 | 31.57 | 32.36 | 40.18 | 670 | * | |||

N10 | 27.7 | 31.98 | 32.55 | 39.78 | 669 | * | |||

N11 | 27.7 | 31.9 | 32.65 | 40 | 665 | * | |||

N12 | 27.53 | 32 | 32.69 | 39.64 | 664 | * | |||

N13 | 27.5 | 31.64 | 32.13 | 40 | 676 | 13.5 | 60 | ||

N14 | 25.7 | 32.3 | 30.75 | 40.2 | 644 | 13.5 | 60 | ||

N15 | 27.6 | 32.53 | 32.55 | 39.8 | 655 | 9 | 60 | ||

N16 | 27.45 | 32.66 | 32.34 | 39.8 | 653 | 9 | 60 | ||

N17 | 32.3 | 30.95 | 32.41 | 40.3 | 800 | 13.5 | 40 | ||

N18 | 26.9 | 32.61 | 32.46 | 39.66 | 641 | 13.5 | 40 | ||

N19 | 30.72 | 32.96 | 32.45 | 40.32 | 714 | 13.5 | 40 | ||

N20 | 27.7 | 32.1 | 32.53 | 39.83 | 666 | 13.5 | 40 |

Specimen # | Density (kg/m^{3}) | Relative Density (%) | Measured Elastic Modulus (GPa) |
---|---|---|---|

N9 | 670 | 25 | 0.118 |

N10 | 669 | 24.8 | 0.153 |

N11 | 665 | 24.88 | 0.154 |

N12 | 664 | 24.7 | 0.152 |

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**MDPI and ACS Style**

Koloushani, M.; Hedayati, R.; Sadighi, M.; Mohammadi-Aghdam, M.
CT-Based Micro-Mechanical Approach to Predict Response of Closed-Cell Porous Biomaterials to Low-Velocity Impact. *J. Imaging* **2018**, *4*, 49.
https://doi.org/10.3390/jimaging4030049

**AMA Style**

Koloushani M, Hedayati R, Sadighi M, Mohammadi-Aghdam M.
CT-Based Micro-Mechanical Approach to Predict Response of Closed-Cell Porous Biomaterials to Low-Velocity Impact. *Journal of Imaging*. 2018; 4(3):49.
https://doi.org/10.3390/jimaging4030049

**Chicago/Turabian Style**

Koloushani, Mehrdad, Reza Hedayati, Mojtaba Sadighi, and Mohammad Mohammadi-Aghdam.
2018. "CT-Based Micro-Mechanical Approach to Predict Response of Closed-Cell Porous Biomaterials to Low-Velocity Impact" *Journal of Imaging* 4, no. 3: 49.
https://doi.org/10.3390/jimaging4030049