Fracture initiation in heterogeneous human cortical bone tissues predicted by an AT1 phase field model

Predicting fracture initiation in human femurs by computed tomography-based finite element analysis (FEA) is of high interest. Newly introduced phase-field models (PFMs), when combined with FEA, may offer a promising tool for predicting fracture initiation and crack paths. PFMs require a material length parameter, ℓ₀, which remains undetermined for heterogeneous cortical bone tissue. This parameter depends on the critical strain and may vary with bone density. Here, ℓ₀ is investigated within the AT1 PFM framework by comparing PF-FEA predictions with experimental observations from macroscopic three-point bending (3PB) tests on cortical bone specimens. A literature-based critical strain value of ε_c=7300μstrain provided good agreement with the experimentally measured yield forces when used within the AT1 PFM framework. This value is lower than the yield strain measured from the 3PB experiments (as documented in past experiments on cortical bone tissues). Combining this critical strain with experimentally determined fracture toughness and Young's modulus correlations yields ℓ₀ values in the range 0.20-0.22 mm. The resulting nearly constant ℓ₀ values over the investigated ash density range, ρ_ash=0.85-1.40g/cm³, led to PF-FEA predictions of the experimental yield forces within approximately ±10%. The small magnitude of ℓ₀ relative to the specimen dimensions, and even more so at the organ level, suggests that it may be approximated as a constant (homogeneous) parameter for fracture initiation and crack propagation analyses. These results provide, to our knowledge, the first tissue-level experimental framework for evaluating ℓ₀ in cortical bone and establish a basis for future organ-level validation studies of fracture initiation in human long bones using PFMs.