# A Study to Derive Equivalent Mechanical Properties of Porous Materials with Orthotropic Elasticity

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## Abstract

**:**

## 1. Introduction

## 2. Prior Research and Background Theory

#### 2.1. Study on Equivalent Properties Derivation

#### 2.2. Orthotropic Elasticity

## 3. Deriving Equivalent Properties Using Simulation

#### 3.1. Computer Software

#### 3.2. Equivalence Derivation Process

#### 3.3. Determination of Unit Cell Model and Its Shape

#### 3.4. Simulation Procedure

#### 3.5. Simulation Results

## 4. Verification and Results through Experiments

#### 4.1. Experimental Equipment and Methods

#### 4.2. Shape of Test Specimen

#### 4.3. Measurement Results

#### 4.4. Comparison and Verification

## 5. Discussion

## 6. Conclusions

- In the case of a finite element method using porous or composite material, it is inefficient to perform the analysis using material modeling. The equivalent properties of a material were estimated by applying the representative volume element method.
- Working from the assumption that the pores are horizontally/vertically asymmetrical, an elastic modulus matrix of an orthogonal anisotropic material was constructed. The equivalent elastic modulus and equivalent Poisson’s ratio of a representative volumetric element were calculated using the equivalent strain and equivalent stress.
- Based on the element volume and element stress values derived from the finite element method program, the representative stress value and elastic modulus matrix were calculated using Python. In addition, the equivalent material properties were derived using the calculated elastic modulus matrix.
- A thin-plate specimen made of STS304 was etched in a specific pattern to simulate pores. The elastic modulus and Poisson’s ratio were measured using UTM and verified through comparison with simulation results.
- This research can be applied to many industries such as medicine and dentistry, which treat porous materials such as bacterial biofilm, bones, teeth, and porous tantalum. As porosity of certain materials can influence the retention and formation of bacterial biofilm, this research is very powerful for analysis for materials with porosity.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Shape of unit cell before homogenization. (

**b**) Shape of unit cell after homogenization.

**Figure 5.**Dimension and distribution of circular pores (Reprinted from Ref. [13]).

**Figure 6.**Dimension and distribution of elliptical pores (Reprinted from Ref. [14]).

**Figure 11.**The boundary condition for strain. (

**a**) X-direction tensile strain. (

**b**) XY-direction shear strain.

**Figure 12.**The distribution of stress with 0.1% strain condition. (

**a**) X-direction tensile strain. (

**b**) XY-direction shear scheme 11. S22, S33, S12, S13 and S23 (S11: tensile stress of X-direction, S22: tensile stress of Y-direction, S33: tensile stress of Z-direction, S12: shear stress of XY-direction, S13: shear stress of XZ-direction, S23: shear stress of YZ-direction) of each element were calculated and printed out. The calculation of weighted average concluded the equivalent stress of each direction at each step.

**Figure 14.**Types of test specimens. (

**a**) Specimen with circular pores. (

**b**) Specimen with elliptical pores—long axis in the tensile direction. (

**c**) Specimen with elliptical pores—short axis in the tensile direction (Reprinted from Ref. [14]).

Design | Real | |||
---|---|---|---|---|

Circular pore | Length of unit cell | 220.0 | 221.25 | |

Hole diameter | 120.0 | 116.25 | ||

Elliptical pore | Length of unit cell | 440.0 | 433.13 | |

Hole diameter | Short | 120.0 | 120.00 | |

Long | 240.0 | 233.13 |

Contents | Value | Unit |
---|---|---|

Density | 8000 | kg/m^{3} |

Modulus of elasticity | 193.0 | GPa |

Poisson’s ratio | 0.29 | - |

Contents | Value | Unit |
---|---|---|

Elastic modulus (E_{x}) | 112.3 | GPa |

Elastic modulus (E_{y}) | 112.3 | GPa |

Elastic modulus (E_{z}) | 147.9 | GPa |

Poisson’s ratio (ν_{yx}) | 0.230 | m/m |

Poisson’s ratio (ν_{xy}) | 0.230 | m/m |

Poisson’s ratio (ν_{zx}) | 0.290 | m/m |

Poisson’s ratio (ν_{zy}) | 0.290 | m/m |

Contents | Value | Unit |
---|---|---|

Elastic modulus (E_{x}) | 106.2 | GPa |

Elastic modulus (E_{y}) | 127.6 | GPa |

Elastic modulus (E_{z}) | 150.5 | GPa |

Poisson’s ratio (ν_{yx}) | 0.235 | m/m |

Poisson’s ratio (ν_{xy}) | 0.196 | m/m |

Poisson’s ratio (ν_{zx}) | 0.290 | m/m |

Poisson’s ratio (ν_{zy}) | 0.290 | m/m |

**Table 5.**Equivalent properties of Specimen Type I [13].

Test No. | Modulus of Elasticity (GPa) | Poisson’s Ratio (mm/mm) |
---|---|---|

1 | 116.0 | 0.226 |

2 | 117.0 | 0.228 |

3 | 117.0 | 0.243 |

4 | 118.0 | 0.248 |

5 | 118.0 | 0.242 |

6 | 117.0 | 0.231 |

Average | 117.17 | 0.2363 |

Standard Deviation | 0.687 | 0.0083 |

**Table 6.**Equivalent properties of Specimen Type II [14].

Test No. | Modulus of Elasticity (GPa) | Poisson’s Ratio (mm/mm) |
---|---|---|

1 | 124.0 | 0.211 |

2 | 124.0 | 0.233 |

3 | 124.0 | 0.212 |

4 | 127.0 | 0.223 |

5 | 123.0 | 0.219 |

6 | 122.0 | 0.226 |

Average | 124.0 | 0.221 |

Standard Deviation | 1.53 | 0.0077 |

**Table 7.**Equivalent properties of Specimen Type III [14].

Test No. | Modulus of Elasticity (GPa) | Poisson’s Ratio (mm/mm) |
---|---|---|

1 | 110.0 | 0.196 |

2 | 109.0 | 0.196 |

3 | 111.0 | 0.188 |

4 | 110.0 | 0.206 |

5 | 109.0 | 0.194 |

6 | 109.0 | 0.199 |

Average | 109.7 | 0.197 |

Standard Deviation | 0.75 | 0.0054 |

Modulus of Elasticity (GPa) | Poisson’s Ratio (mm/mm) | |
---|---|---|

Simulation | 112.3 | 0.232 |

Measurement | 117.2 | 0.236 |

Difference (%) | 4.18 (%) | 1.69 (%) |

Modulus of Elasticity (E_{y}, GPa) | Poisson’s Ratio (ν_{yx}, m/m) | |
---|---|---|

Simulation | 127.6 | 0.236 |

Measurement | 124.0 | 0.221 |

Difference (%) | 2.82 (%) | 6.36 (%) |

Modulus of Elasticity (E_{x}_{,} GPa) | Poisson’s Ratio (ν_{xy}, m/m) | |
---|---|---|

Simulation | 106.2 | 0.196 |

Measurement | 109.7 | 0.197 |

Difference (%) | 3.19 (%) | 0.51 (%) |

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**MDPI and ACS Style**

Pyo, C.; Kim, Y.; Kim, J.; Kang, S.
A Study to Derive Equivalent Mechanical Properties of Porous Materials with Orthotropic Elasticity. *Materials* **2021**, *14*, 5132.
https://doi.org/10.3390/ma14185132

**AMA Style**

Pyo C, Kim Y, Kim J, Kang S.
A Study to Derive Equivalent Mechanical Properties of Porous Materials with Orthotropic Elasticity. *Materials*. 2021; 14(18):5132.
https://doi.org/10.3390/ma14185132

**Chicago/Turabian Style**

Pyo, Changmin, Younghyun Kim, Jaewoong Kim, and Sungwook Kang.
2021. "A Study to Derive Equivalent Mechanical Properties of Porous Materials with Orthotropic Elasticity" *Materials* 14, no. 18: 5132.
https://doi.org/10.3390/ma14185132