# An Overview of Selected Material Properties in Finite Element Modeling of the Human Femur

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Types of Material Properties

#### 1.1.1. Isotropic Materials

#### 1.1.2. Anisotropic Materials

#### 1.1.3. Orthotropic Materials

_{1}, E

_{2}, E

_{3}, G

_{12}, G

_{23}, G

_{31}>0

_{11}, C

_{22}, C

_{33}, C

_{44}, C

_{55}, C

_{66}> 0

_{23}U

_{32}). (1 − U

_{13}U

_{31}). (1 − U

_{12}U

_{21}) > 0

_{12}U

_{21}) − (U

_{23}U

_{32}) − (U

_{31}U

_{13}) − 2(U

_{21}U

_{32}U

_{13}) > 0

#### 1.2. Physical and Mechanical Properties of the Human Femur

^{3}for cortical tissue and 0.4 g/cm

^{3}for trabecular tissue. This is in addition to the mechanical properties indicated.

#### 1.2.1. Material Properties for Trabeculae

- For cancellous bone tissue, the ratios of the directional Young’s moduli exhibit a relationship similar to that reported in [36], where E
_{1}/E_{2}and E_{1}/E_{3}are equal to approximately 2 (here, equal to 1.4 and 2.0, respectively), and E_{3}/E_{2}is equal to approximately 0.6 (here, 0.07). - The typical Young’s modulus is 1.0 GPa, which is in accordance with what has been documented in the literature.

#### 1.2.2. Material Properties for Cortical Bone

_{1}= 8.69 GPa (longitudinal), E

_{2}= 4.19 GPa (transverse), and E

_{3}= 3.76 GPa (radial).

_{1}is substantially lower than that of the majority of other studies: E

_{2}/E

_{1}= 48%, E

_{3}/E

_{2}= 90%, and E

_{3}/E

_{1}= 43%. The longitudinal tensile and bending moduli for wet human cortical bone specimens, principally from the femur, are listed in detail in Table 5 [18]. As can be observed, the average longitudinal Young’s modulus throughout these experiments is 16.0 GPa. Using the aforementioned percentages, E

_{2}and E3 are equivalent to roughly 6.8 GPa and 6.3 GPa, respectively, with E1 equal to 16.0 GPa. Shear moduli are found in Schuster [11] and are comparable to an average shear modulus of 3.36 GPa as reported by Reilly and Burnstein [17]. In another study, Mirzaali et al. [37] evaluated the physical and mechanical properties of cortical bone in different cases. It was written that the axial hardness modulus for osteonal, interstitial, and pooled bone are 408 ± 69, 503 ± 56, and 455 ± 78, respectively. Therefore, transverse hardness moduli for female, male, and pooled bone are 367 ± 91, 428 ± 75, and 395 ± 89 respectively. In uniaxial tests, the moduli are 18.16 ± 1.88 and 18.97 ± 1.84 under uniaxial tension and compression, respectively. The Poisson’s ratios for cortical tissue recorded by Reilly and Burnstein [17] were 0.62 for “radial specimens” and 0.40 for “longitudinal specimens”. Since Poisson ratios greater than 0.5 are not permitted in the infinitesimal theory, these values led to problems in the constitutive equations used for the orthotropic material model. As a result, the ratios were reduced while maintaining their relative magnitudes. The radial and longitudinal Poisson’s ratios are scaled to 0.45 and 0.30, respectively. These are comparable to the average Poisson’s ratio for femoral cortical bone tissue reported by Katsamanis and Raftopoulos [38] of 0.36. The transverse direction of Poisson’s ratio in this study is also calculated using the Poisson’s ratio for the radially harvested specimen (set to 0.3). Table 6 lists the final nine elastic material constants used for the orthotropic cortical bone. The orthotropic material model description with E

_{2}= E

_{3}= 6.30 GPa and G

_{12}= G

_{13}= 3.30 is all that is required to describe the transversely isotropic material model for cortical tissue, as illustrated in Table 7. Table 8 displays the isotropic material model description for cortical tissue. The material constants for the isotropic piecewise linear- plasticity material model of cortical bone are displayed in Table 9. The yield stress is the average of the tension values for specimens tested in by Reilly and Burnstein [17], and the elastic modulus and Poisson’s ratio are the same used for the elastic isotropic model. The tangent modulus is 5% of the elastic modulus [20].

#### 1.2.3. Material Properties for Marrow

## 2. Results

^{2}(based on the mid-diaphysis cross-sectional characteristics of the femur, the proportional limit deflection, and load) and the current investigation reveals a value of 18.0 kN/mm

^{2}that is similar.

^{2}, or less than half of the ultimate torsion strength of 45.3 N/mm

^{2}. The maximum twist angle is roughly 1.5°. According to Cristofolini et al. [16], fresh-frozen femur samples have an elastic stiffness in the torsion range of 6.5–10.5 Nm/deg. The isotropic FE femur model has a higher stiffness of 19.4 Nm/deg, and the orthotropic and transversely isotropic femur models show stiffness in torsion that is closer to the literature at 11.64 Nm/deg. Each FE femur model’s elastic whole bone torsion stiffness is listed in Table 12. The transversely isotropic model and the orthotropic model have identical whole bone elastic torsion stiffnesses, with the isotropic model having a 50% greater stiffness.

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Young’s Moduli (MPa) | Shear Moduli (MPa) | Poisson’s Ratios |
---|---|---|

E_{1} = 1352 | G_{12} = 292 | V_{12} = 0.30 |

E_{2} = 968 | G_{23} = 370 | V_{23} = 0.30 |

E_{3} = 676 | G_{13} = 505 | V_{13} = 0.30 |

Young’s Moduli (MPa) | Shear Moduli (MPa) | Poisson’s Ratios |
---|---|---|

E_{1} = 1352 | G_{12} = 399 | V_{12} = 0.30 |

E_{2} = 822 | G_{23} = 370 | V_{23} = 0.30 |

E_{3} = 822 | G_{13} = 399 | V_{13} = 0.30 |

Young’s Moduli (GPA) | Poisson’s Ratios |
---|---|

E_{1} = 1 | V_{12} = 0.30 |

Elastic Modulus (MPa) | Tangent Modulus (MPa) | Poisson’s Ratios | Yield Stress (MPa) |
---|---|---|---|

E = 1000 | E_{tan} = 1000 | 0.3 | 7.5 |

Young’s Moduli (GPa) | Shear Moduli (GPa) | Poisson’s Ratios |
---|---|---|

E_{1} = 16 | G_{12} = 3.2 | V_{12} = 0.30 |

E_{2} = 6.88 | G_{23} = 3.6 | V_{23} = 0.45 |

E_{3} = 6.30 | G_{13} = 3.3 | V_{13} = 0.30 |

Young’s Moduli (GPa) | Shear Moduli (GPa) | Poisson’s Ratios |
---|---|---|

E_{1} = 16 | G_{12} = 3.3 | V_{12} = 0.30 |

E_{2} = 6.30 | G_{23} = 3.6 | V_{23} = 0.45 |

E_{3} = 6.30 | G_{13} = 3.3 | V_{13} = 0.30 |

Young’s Moduli (GPa) | Poisson’s Ratios |
---|---|

E_{1} = 16 | V_{12} = 0.36 |

Elastic Modulus (GPa) | Tangent Modulus (MPa) | Poisson’s Ratios | Yield Stress (MPa) |
---|---|---|---|

E = 16 | E_{tan} = 800 | 0.36 | 108 |

**Table 9.**Effective Young’s Modulus comparisons for samples from the same bone using in vitro methods.

Marrow Sample Temperature | Rheology (kPa) | Indentation (kPa) | Cavitation (kPa) |
---|---|---|---|

25 °C | 20 °C | 20 °C | |

1 | 52.1 ±10.2 | 30.3 ± 4.0 | 64.3 ± 0.2 |

2 | 4.0 ± 0.9 | 5.7 ± 0.3 | 9.0 ± 0.01 |

3 | 0.7 ± 0.3 | 0.9 ± 0.2 | 0.9 ± 0.2 |

4 | 3.2 ± 1.9 | 2.1 ± 0.3 | 14.4 ± 10.0 |

5 | 84.4 ± 6.5 | 35.3 ± 4.9 | no data |

6 | 135.6 ± 25.6 | 37.1 ± 6.3 | no data |

7 | 69.0 ± 21.4 | — | — |

8 | — | 12.2 ± 2.8 | — |

9 | — | — | 16.0 ± 1.6 |

Average | 38.77 | 13.73 | 11.52 |

**Table 10.**Hip contact forces’ resulting components (F

_{r}, F

_{x}, F

_{y}, and F

_{z}) in (%BW) were predicted to have the following values for isotropic and orthotropic models.

Forces | Material | F_{y} | F_{z} | F_{r} |
---|---|---|---|---|

Predicted | Isotropic | 59 | − 319 | 73 |

Orthotropic | 53 | − 306 | 71 |

**Table 11.**For the first third (0–33%), the last third (66–100%), and the entire width (0–100%) of the slice, the root mean squared error (RMSE,%) and Pearson’s product-moment coefficient (r, p 0.0001) between the two distinct predictions (isotropic and orthotropic) and the CT scan profiles were calculated.

Slice | Region | Model | 0–100% | 0–33% | 66–100% | |||
---|---|---|---|---|---|---|---|---|

RMSE (%) | r | RMSE (%) | r | RMSE (%) | r | |||

1 | 5% femoral head | Iso | 32.48 | 0.49 | 17.31 | 0.77 | 22.90 | − 0.12 |

Ortho | 29.23 | 0.49 | 17.08 | 0.77 | 20.74 | − 0.02 | ||

2 | 20% shaft | Iso | 75.83 | 0.74 | 43.09 | 0.77 | 57.81 | 0.59 |

Ortho | 51.27 | 0.88 | 25.92 | 0.88 | 38.92 | 0.72 | ||

3 | 40% shaft | Iso | 107.50 | 0.29 | 65.73 | 0.37 | 78.23 | − 0.66 |

Ortho | 82.32 | 0.54 | 35.04 | 0.72 | 64.87 | − 0.09 | ||

4 | 60% shaft | Iso | 63.95 | 0.67 | 28.38 | 0.86 | 55.40 | 0.37 |

Ortho | 64.03 | 0.65 | 36.38 | 0.89 | 48.41 | 0.74 | ||

5 | 80% shaft | Iso | 72.34 | 0.53 | 34.80 | 0.73 | 53.69 | 0.60 |

Ortho | 68.29 | 0.46 | 27.07 | 0.69 | 51.83 | 0.81 | ||

6 | 95% shaft | Iso | 66.15 | 0.43 | 30.57 | 0.85 | 42.81 | 0.64 |

Ortho | 66.10 | 0.25 | 21.43 | 0.89 | 45.24 | 0.80 | ||

7 | Neck | Iso | 25.65 | 0.72 | 18.53 | 0.89 | 17.21 | 0.68 |

Ortho | 12.29 | 0.88 | 9.56 | 0.93 | 5.38 | 0.89 | ||

8 | Greater trochanter | Iso | 26.67 | 0.58 | 22.48 | 0.82 | 12.47 | − 0.14 |

Ortho | 30.72 | 0.55 | 26.08 | 0.81 | 14.90 | − 0.13 | ||

9 | Femoral head | Iso | 30.06 | 0.40 | 23.60 | 0.46 | 16.81 | 0.26 |

Ortho | 25.87 | 0.50 | 19.31 | 0.40 | 15.98 | 0.24 | ||

10 | Femoral head | Iso | 28.72 | 0.55 | 20.48 | 0.73 | 17.35 | 0.17 |

Ortho | 24.53 | 0.60 | 18.01 | 0.73 | 14.25 | 0.24 | ||

11 | Femoral shaft | Iso | 82.43 | 0.57 | 45.73 | 0.67 | 63.81 | 0.10 |

Ortho | 65.87 | 0.69 | 32.45 | 0.83 | 50.73 | 0.46 | ||

12 | Femoral condyles | Iso | 69.25 | 0.48 | 32.69 | 0.79 | 48.25 | 0.62 |

Ortho | 67.20 | 0.35 | 24.25 | 0.79 | 48.54 | 0.80 | ||

13 | Whole femur | Iso | 55.63 | 0.54 | 31.61 | 0.72 | 39.70 | 0.25 |

Ortho | 47.79 | 0.58 | 24.21 | 0.77 | 34.03 | 0.44 |

Material Properties | Elastic Bending Stiffness of the Bone | Elastic Torsion Stiffness of the Bone |
---|---|---|

Isotropic | 267 | 19.4 |

Orthotropic | 278 | 11.6 |

Anisotropic | 278 | 11.6 |

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**MDPI and ACS Style**

Bazyar, P.; Baumgart, A.; Altenbach, H.; Usbeck, A.
An Overview of Selected Material Properties in Finite Element Modeling of the Human Femur. *Biomechanics* **2023**, *3*, 124-135.
https://doi.org/10.3390/biomechanics3010012

**AMA Style**

Bazyar P, Baumgart A, Altenbach H, Usbeck A.
An Overview of Selected Material Properties in Finite Element Modeling of the Human Femur. *Biomechanics*. 2023; 3(1):124-135.
https://doi.org/10.3390/biomechanics3010012

**Chicago/Turabian Style**

Bazyar, Pourya, Andreas Baumgart, Holm Altenbach, and Anna Usbeck.
2023. "An Overview of Selected Material Properties in Finite Element Modeling of the Human Femur" *Biomechanics* 3, no. 1: 124-135.
https://doi.org/10.3390/biomechanics3010012