# Analysis of Damage Models for Cortical Bone

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Damage Models

#### 2.1. Frémond–Nedjar Model

**g**represent volume and traction forces, respectively, and we include the subdifferential of the indicator function ${I}_{[{\beta}_{\ast},1]}$ into (4) to ensure that the damage function belongs to the interval $[{\beta}_{\ast},1]$. Here, ${\beta}_{\ast}>0$ is a positive constant, assumed small, and it is introduced for mathematical reasons. In any case, when damage becomes zero, the material is dense with microcracks and modelling it as elastic ceases to make sense (see [16] for details). Finally, we note that damage function $\varphi $ is given in (1), where constants ${\lambda}_{d}$, ${\lambda}_{u}$ and ${\lambda}_{w}$ are constitutive parameters.

**Theorem**

**1.**

**Proof.**

#### 2.2. García et al. Model

## 3. Numerical Results and Discussion

#### 3.1. Comparison in a One-Dimensional Problem

#### 3.2. A Two-Dimensional Problem Solved Using Frémond–Nedjar Model

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Stress–strain curve obtained with the Frémond–Nedjar model in a tensile test (

**up**) and evolution of the damage variable in the tensile test (

**down**).

**Figure 4.**Comparative of both models with experimental data. Source of the experimental data: [18].

$\mathit{h}\downarrow \mathit{k}\to $ | 0.1 | 0.01 | 0.001 | 0.0001 |
---|---|---|---|---|

${2}^{-2}$ | 13.5490 | 13.5613 | 13.561268 | 13.561268 |

${2}^{-3}$ | 6.6596 | 6.6679 | 6.667863 | 6.667863 |

${2}^{-4}$ | 3.3080 | 3.3054 | 3.305386 | 3.305386 |

${2}^{-5}$ | 1.6641 | 1.6457 | 1.645661 | 1.645661 |

${2}^{-6}$ | 0.8460 | 0.8210 | 0.821036 | 0.821036 |

${2}^{-7}$ | 0.4370 | 0.4099 | 0.409942 | 0.409942 |

${2}^{-8}$ | 0.2328 | 0.2046 | 0.204565 | 0.204565 |

${2}^{-9}$ | 0.1315 | 0.1017 | 0.101652 | 0.101653 |

${2}^{-10}$ | 0.0822 | 0.0496 | 0.049597 | 0.049596 |

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**MDPI and ACS Style**

Baldonedo, J.; Fernández, J.R.; López-Campos, J.A.; Segade, A.
Analysis of Damage Models for Cortical Bone. *Appl. Sci.* **2019**, *9*, 2710.
https://doi.org/10.3390/app9132710

**AMA Style**

Baldonedo J, Fernández JR, López-Campos JA, Segade A.
Analysis of Damage Models for Cortical Bone. *Applied Sciences*. 2019; 9(13):2710.
https://doi.org/10.3390/app9132710

**Chicago/Turabian Style**

Baldonedo, Jacobo, José R. Fernández, José A. López-Campos, and Abraham Segade.
2019. "Analysis of Damage Models for Cortical Bone" *Applied Sciences* 9, no. 13: 2710.
https://doi.org/10.3390/app9132710