# 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

- Mazars, J. A description of micro- and macroscale damage of concrete structures. Eng. Fract. Mech.
**1986**, 25, 729–739. [Google Scholar] [CrossRef] - Frémond, M.; Nedjar, B. Damage, gradient of damage and principle of virtual power. Int. J. Solids Struct.
**1996**, 33, 1083–1103. [Google Scholar] [CrossRef] - Nedjar, B. Elastoplastic-damage modelling including the gradient of damage: Formulation and computational aspects. Int. J. Solids Struct.
**2001**, 38, 5421–5451. [Google Scholar] [CrossRef] - Wolff, J. The Law of Bone Remodelling; Springer: Berlin/Heidelberg, Germany, 1986. [Google Scholar]
- Weinans, H.; Huiskes, R.; Grootenboer, H.J. The behavior of adaptive bone-remodeling simulation models. J. Biomech.
**1992**, 25, 1425–1441. [Google Scholar] [CrossRef][Green Version] - Fondrk, M.T.; Bahniuk, E.H.; Davy, D.T. Inelastic strain accumulation in cortical bone during rapid transient tensile loading. J. Biomech. Eng.
**1999**, 121, 616–621. [Google Scholar] [CrossRef] [PubMed] - Carter, D.R.; Caler, W.E. A cumulative damage model for bone fracture. J. Orthop. Res.
**1985**, 3, 84–90. [Google Scholar] [CrossRef] [PubMed] - Pattin, C.A.; Caler, W.E.; Carter, D.R. Cyclic mechanical property degradation during fatigue loading of cortical bone. J. Biomech.
**1996**, 29, 69–79. [Google Scholar] [CrossRef] - Fondrk, M.T.; Bahniuk, E.H.; Davy, D.T. A Damage Model for Nonlinear Tensile Behavior of Cortical Bone. J. Biomech. Eng.
**1999**, 121, 533–541. [Google Scholar] [CrossRef] [PubMed] - Garcia, D.; Zysset, P.K.; Charlebois, M.; Curnier, A. A 1D elastic plastic damage constitutive law for bone tissue. Arch. Appl. Mech.
**2009**, 80, 543–555. [Google Scholar] [CrossRef] - Ramtani, S. Damaged elastic bone-column buckling theory within the context of adaptive elasticity. Mech. Res. Commun.
**2018**, 88, 1–6. [Google Scholar] [CrossRef] - Ramtani, S.; Zidi, M. Damaged-bone remodeling theory: Thermodynamical Approach. Mech. Res. Commun.
**1999**, 26, 701–708. [Google Scholar] [CrossRef] - Hosseini, H.S.; Horák, M.; Zysset, P.K.; Jirásek, M. An over-nonlocal implicit gradient-enhanced damage-plastic model for trabecular bone under large compressive strains. Int. J. Numer. Methods Biomed. Eng.
**2015**, 31, e02728. [Google Scholar] [CrossRef] [PubMed] - Martínez, G.; García-Aznar, J.M.; Doblaré, M.; Cerrolaza, M. External bone remodeling through boundary elements and damage mechanics. Math. Comput. Simul.
**2006**, 73, 183–199. [Google Scholar] [CrossRef] - Mengoni, M.; Ponthot, J.P. An enhanced version of a bone-remodelling model based on the continuum damage mechanics theory. Comput. Methods Biomech. Biomed. Eng.
**2015**, 18, 1367–1376. [Google Scholar] [CrossRef] [PubMed] - Campo, M.; Fernández, J.R.; Kuttler, K.L.; Shillor, M.; Via no, J.M. Numerical analysis and simulations of a dynamic frictionless contact problem with damage. Comput. Methods Appl. Mech. Eng.
**2006**, 196, 476–488. [Google Scholar] [CrossRef] - Ciarlet, P.G. Basic error estimates for elliptic problems. In Handbook of Numerical Analysis; Ciarlet, P.G., Lions, J.L., Eds.; North-Holland: Amsterdam, The Netherlands, 1993; Volume II, pp. 17–351. [Google Scholar]
- García, D. Elastic Plastic Damage Laws for Cortical Bone. Ph.D. Thesis, Lausanne, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 2006. [Google Scholar]

**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