# Statistical mechanics of bone damage: a constitutive model

## TL;DR

This paper introduces a statistical mechanics-based model to predict how bone behaves and cracks under stress, validated with extensive experiments.

## Contribution

A thermodynamically consistent, nonlinear constitutive model for cortical bone microcracking based on statistical mechanics.

## Key findings

- The model accurately predicts stress–strain relationships and microcracking progression in cortical bone.
- Entropy calculations reveal a critical strain threshold beyond which microcracking increases significantly.
- Constitutive parameters correlate with anthropometric variables, showing model robustness across diverse samples.

## Abstract

After the elastic regime is surpassed, cortical bone exhibits significant microcracking in its post-elastic mechanical behavior. This work develops a thermodynamically consistent, nonlinear constitutive model based on statistical mechanics, designed to predict the stress–strain relationship and the progression of inter-osteon microcracking. To assess the model’s sufficiency, precise tensile and bending tests were performed in comparison to empirical curves that illustrated theoretical predictions of constitutive relationships. Moreover, entropy increases were quantitatively assessed using model parameters refined through experimental data. A large-size sample was utilized, comprising 51 dog-bone-shaped cortical bone specimens from the 4th ribs of various subjects for uniaxial tensile tests, and 15 complete fourth ribs for bending tests. Displacement and strain fields were meticulously recorded using digital image correlation and video analysis. The model demonstrated robustness, accurately fitting the data from all experimental specimens and revealing correlations between constitutive parameters and anthropometric variables. Entropy calculations provide insights into the behavior of the bone under varying strains: microcracking is minimal at low strains with stress nearly proportional to strain, escalating significantly beyond a critical threshold, thus challenging the linear relationship between stress and strain.