# Analysis of Damage Models for Cortical Bone
## 1 Introduction  
Damage models arise in order to describe how mechanical properties of materials degrade over time. This degradation can be caused both because of the loading it is subjected to, and due to external causes (such as crack formation due to thermal shock or chemical attack). These models have been deeply studied for structures, usually concrete structures, where the progressive wear of the materials can be critical to its integrity. In this field, works about damage exist since the decade of 1980 [1]. A later work from [PERSON] and [PERSON] in 1996 [2] became a reference for new concrete damage models based on the continuum elastoplastic damage approach [3]. In this approach, the principle of virtual power is modified, including the damage in the term of the power of the internal forces. Also, damage is represented by a scalar field.  
The study of engineering concepts in the field of biology and medicine is more recent, but it is growing fast. In the particular case of bone tissue, biological aspects such as bone remodelling were studied also since the decade of 1980 [4]. The model proposed by [PERSON], [PERSON] and [PERSON] in 1992 [5] started the possibility of simulating this effect numerically, and led to a large number of contributions in this field. The effect of damage described previously can be seen in bones too. In bone tissue, both loading and external causes (now related to illness such as osteoporosis) produce again a growing deterioration of its elastic properties [6]. The first studies of damage were focused on cumulative damage caused by cyclic loading [7; 8]. This approach allows for fatigue estimations of the number of cycles that a probe can withstand, but it is not effective for its implementation in numerical simulations. In 1999, [PERSON], [PERSON] and [PERSON] proposed a model that reproduced thetensile behaviour of cortical bone [9]; however, it is limited to one-dimensional simulations, obtaining bending results by applying beam theory assumptions. A recent work from and [PERSON] et al. in 2009 [10] introduced rheological bone specific models that reproduced uniaxial and cyclic tests and could be extended to three dimensional simulations. Other works about bone damage can be seen in [11; 12; 13; 14].  
In this work, the Fremond-Nedjar model is compared against the [PERSON] et al. model in order to assess its capabilities to reproduce bone tissue behaviour. Indeed, one of the main novelties of this work is its application in the simulation of the bone damage process. The selection and the interest of analysing the Fremond-Nedjar model relies on the fact that its formulated as a set of partial differential equations (in particular a subdifferential inclusion) that couples the damage evolution with the well known linear elasticity model. This allows for its numerical analysis and its implementation in a finite element simulation. Also, it is not restricted to one dimensional simulations, like the previously mentioned [PERSON] k, Bahniuk and Davy model.  
Furthermore, since the formulation is similar to the Weinans-Huiskes-Grootenboer model of bone remodelling, it would allow for a direct coupling of these models, a future objective of the authors. Although an existing work presents a coupling between damage and a remodelling model [15], it is based on a simple remodelling rule and the numerical analysis is not performed.  
The paper is structured as follows. In Section 2, the formulations of the studied models are presented, then, in Section 3, the implementation and results obtained are shown and discussed, followed by some conclusions in Section 4.