Role of matrix behavior in compressive fracture of bovine cortical bone

In compressive fracture of dry plexiform bone, we examine the individual roles of overall mean porosity, the connectivity of the porosity network, and the elastic as well as the failure properties of the nonporous matrix, using a random spring network model (RSNM). Porosity network structure is shown to reduce the compressive strength by up to 30%. However, the load-bearing capacity increases with an increase in either of the matrix properties—the elastic modulus or the failure strain threshold. To validate the porosity-based RSNM model with available experimental data, bone-specific failure strain thresholds for the ideal matrix of similar elastic properties were estimated to be within 60% of each other. Further, we observe the avalanche size exponents to be independent of the bone-dependent parameters as well as the structure of the porosity network.

Figure 1

Flow chart detailing the steps involved in developing a porosity-based RSNM.

Figure 2

(a) Cubic sample of size $5\mathrm{mm}\times 5\mathrm{mm}\times 5\mathrm{mm}$ with faces perpendicular to the longitudinal, radial, and transverse directions. (b) Optical micrograph of the longitudinal face showing plexiform bone. CT scan images (tangential view) of the specimen (c) before fracture and (d) after fracture, where blue points indicate higher porosity. Data are for sample III.

Figure 3

(a) Stack of representative domains, each spanning over $500\mu \text{m}$ in the tangential direction, obtained from porosity analysis. (b) Spring stiffness is based on the % porosity at the node (pixel). (c) A typical representative domain and (d) corresponding homogenized configuration for sample III.

Figure 4

(a) Schematic diagram of the spring network model. (b) Unit cell of the network showing the linear springs attached to a site. Also shown is an example of an angle ${\theta }_{ijk}$ whose distortion is resisted by a bending spring.

Figure 5

Variation of (a) the elastic modulus, (b) Poisson's ratio, and (c) failure strain with porosity for a different modulus and failure strain of the matrix.

Figure 6

(a) Macroscopic response of sample III and (b) for a typical representative domain. The corresponding failure paths at the first load drop with (c) base values, (d) 10% $E$, and (e) 10% failure strain as input parameters.

Figure 7

Variation of load-bearing capacity with mean porosity levels and $\pm 10\%$ variation in the elastic modulus, $E$, accounting for (a) the porosity network and (b) the homogenized distribution of porosity.

Figure 8

Variation of load-bearing capacity with mean porosity levels and $\pm 10\%$ variation in failure strain, ${\epsilon }_f$, accounting for (a) the porosity network and (b) the homogenized distribution of porosity.

Figure 9

Probability density of element counts from EDS line scans of bone samples.

Figure 10

Comparison of macroscopic response from simulations and experiments of bovine samples, where (a) is for sample I, (b) is for sample II, and so on.

Figure 11

Load-bearing capacity from experiments and simulations: (a) variation with porosity and (b) correlation between experiment and simulations.

Figure 12

Avalanche distribution $P\left(s\right)$ for sample III from simulations.