Isolating The Role Of Bone Lacunar Morphology On Static And Fatigue Fracture Progression Through Numerical Simulations Part 2
Sep 01, 2023
3.3. Fatigue Analysis of Lacuna-Embedded Geometries
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With the analogous purpose of locating the most critical lacunar network for damage initiation but under fatigue loading conditions, FeSafe high cycle fatigue analyses were carried out. For each geometry, we considered the number of cycles to crack initiation (log-life) as a suitable parameter to assess the critical sites for crack onset. Figure 6a reports the most critical lacunae in each geometry, with a specific reference to the log-life. In 80% of the cases, the β region appears as the most prominent zone for crack initiation, with damage appearing at a lower number of cycles in the OP configuration concerning other geometries (Figure 6b).
4. Discussion
To address the intimate cross-talks existing between human bone lacunae and micro-cracks, our approach started by isolating lacunar morphology in osteopenic and osteoporotic subjects. This choice specifically resides in different features exhibited by OP and PET bone micro-scale architecture, resulting in opposite effects on bone mineral density and strength. Computational XFEM static and fatigue analyses were conducted on six 3D porous geometries, succeeding in evaluating and localizing critical damage initiation and progression sites. In detail, we deepened the separate effects of lacunar density, size, and orientation on the mechanical strength of bone-inspired AISI 316L samples. Furthermore, we considered the realistic 3D shape of lacunae, and we analyzed damage initiation sites in the absence of pre-cracking, overcoming the simplifications highlighted in the current state-of-the-art when schematizing lacunae as perfect ellipses or adopting fictitious crack onset sites to speed up the convergence.
Regarding the number of failed elements in XFEM simulations, the OP specimen shows a percentage of failed elements of 5.71%, which is mainly located 4 mm from the traction surface and is lower than the PET case (Figure 5). All the elements that are not depicted in red or light blue (dark blue and black) experience less than a 20% reduction in cohesive properties. For PET, twice the number of PET failed elements were identified 4 mm away from the traction surface.
After properly tuning the computational parameters and quantifying their effect in static XFEM analyses, we focused on the detailed investigation of lacunar features on the specimen’s mechanical strength by referring to the force–displacement curves (Figure 7). Interesting parallels could be performed with behavior detected in human bones that are subjected to both static and fatigue loads.
The predominant parameter affecting the loss of mechanical strength is an increase in lacunar density, with an exception represented by PETna [13]. This model, however, is the only one characterized by a single damaged plane with a loss in cohesive properties of around 40% (Figure 7). Therefore, this loss in cohesive strength is not enough to cause an overall critical reduction in the mechanical strength of the model; hence, partial damage extended by 20% was not found to be critical for the specimen strength. However, we believe that the formation of secondary partially damaged regions, as shown in all the other lacunar-embedded categories, is a more realistic condition since lacunae alone should act mainly as stress raisers (as highlighted in human bone damage [13]), resulting in damaged elements around them (Figure 6a). Therefore, OP2 with four lacunae appears as the most resistant specimen; by increasing the lacunar number to 13 (therefore increasing the porosity), PET2 shows a reduction of 1.8% in the displacement at failure. An additional drop of 9% is visible in the 20 lacunae specimen, i.e., OP. When comparing PET2 and OP, this value becomes 8.1% with a 35% rise in the lacunar number.
The lacunar size is responsible for the limited reduction of about 2% in mechanical strength (Figure 7). Indeed, the overall lacunar surface area in the case of OP2 is 22 mm2, the one related to PET2 is 49.4 mm2, and the one linked to OP is 110 mm2. Even if the ratio between OP2 and PET2 surface area and the one between PET2 and OP is quite the same, the actual magnitude of these values plays the main role; that is, passing from OP2 to PET2 means increasing the overall lacunar surface area by 27.4 mm2, whereas passing from OP2 to OP this value rises to 88 mm2 and from PET2 to OP it becomes 60.6 mm2. We, therefore, believe that variations in lacunar size and density are strongly interconnected since changing one or both of them still has the same effect of altering the total porosity of the models. This consideration is also supported by the fact that PET and OP have the same lacunar density but a different lacunar size, and PET fails at higher traction values concerning OP. As mentioned, the lacunar surface area of OP is 110 mm2, whereas the one related to PET is 76 mm2; therefore, the overall OP porosity is higher concerning the one of PET.
The influence of a random lacunar alignment on mechanical strength is, instead, less evident, starting from the aforementioned considerations regarding the predicted traction at failure for PET2na. Moreover, neither PET2na nor PETna experiences heavily damaged elements (Table S2, Supplementary Materials); this observation can be justified by considering that, in the case of PETna, the misalignment of the lacunae can split the crack path, hence demanding more energy to produce multiple fracture surfaces, which realistically happens in human bone micro-damage. By relating PET2 and PET2na, we are led to think that misalignment of the lacunae leads to a slower damage progression.
Regarding the influence of lacunar morphological and densitometric parameters on fatigue resistance, we specifically discussed the number of cycles required to initiate the primary and secondary cracks. By analyzing Figure 6b, we observed that the failure order is analogous to the one related to static XFEM analysis, always except PET2na, in which damage initiation is predicted to happen after OP and PET. Moreover, all critical lacunae predicted in the fatigue analysis are related to damage initiation and progression even in static XFEM analysis (Table S2 of Supplementary Materials and Figure 6b).
By referring to damage progression patterns, we hypothesize that the most extended and interconnected damaged zones for each category correspond to the most probable fracture surfaces. No significant deviations from planar surfaces, whose normal is parallel to the loading axis, have been detected; it can be assumed that fracture of these geometries would occur under tensile opening mode I. We underline that this output is not fictitiously forced by the employment of specific computational parameters; on the contrary, the damage initiation criterion, MAXPS, was selected because it is a solution-dependent criterion. These lacunar arrangements could potentially lead to crack attraction sites (Figure 8a) and could also deviate from the crack path (Figure 8b, left).
By considering Figure 6b, OP and PET, all with twenty lacunae, face likely fracture at the same location: they tend to potentially break in the middle—4 mm from the traction surface—and they are characterized by the same lacunar disposition in that region (see Figure 8). Since the three models in question have different lacunar sizes and alignments, we believe that this disposition, with the centers of the lacunae belonging to the same ZY plane, is the most critical one, regardless of the morphological parameters and the distance from the traction surface. We can indeed discuss that in the remaining models, which are not characterized by this pattern, the predicted fracture plane lies elsewhere. We can highlight from Figure 8c,d on the left that a similar arrangement but with different inter lacunar distances is present in the γ-region near the traction surface. However, this seems not to be crucial to model failure, mainly due to the higher inter-lacunar distances. Indeed, it is interesting that our models could be qualitatively compared with real bone micro-scale synchrotron images [13] (Figure 8), obtaining very similar crack patterns. This could be a prominent result, demonstrating that, independently from the material, lacunar voids play a role in fracture initiation and progression, and specific toughening lacunar patterns could be later exploited for practical biomedical applications.
5. Conclusions
In summary, our study provides a quantitative computational framework to investigate lacunae-micro-cracks existent interlinks by combining static XFEM and fatigue analyses. Furthermore, the work succeeds in demonstrating cross-talks between the lacunar network and damage initiation while highlighting the specific effect of both lacunar morphological and densitometric parameters on mechanical strength. An increase in lacunar density (as evidenced in OP2, PET2, and Pet2na), indeed, leads to a loss in mechanical strength al lower traction values, resulting as the most influencing parameter among the studied ones. Lacunar size (PET and Op categories), on the contrary, has a lower effect on mechanical strength, reducing it by 2%. Lacunar alignment (PET and PETna) has the main role of splitting the crack path.
Limitations could be linked to the reduced number of pores considered in the analysis, which is, however, linked to the significant computational power required to conduct XFEM analyses.
As future insights, we plan to realize the described morphologies via laser powder bed fusion using AISI 316L and later by exploiting other biomedical materials such as titanium. Since we have evidenced interesting toughening phenomena in our numerical analysis that are due to lacunar-like arrangements, we plan to translate these findings to the realization of biomedical products that could benefit from the lighter void-embedded geometry. The obtained results also indicate the potential of the developed approaches to shed some light on still obscure micro-damage phenomena when isolating micro-scale features as potential candidates for damage occurrence.
Author Contributions: Conceptualization, F.B., S.B, and L.M.V.; Methodology, F.B., S.B., and L.M.V.; Validation, F.B., S.B., and L.M.V.; Formal Analysis, F.B., F.C., and M.G.; Investigation, F.B., F.C., R.M., and M.G.; Resources, L.M.V.; Data Curation, F.B., F.C., and M.G.; Writing—Original Draft Preparation, F.B.; Writing—Review and Editing, F.B., S.B., and L.M.V.; Visualization, F.B., and F.C.; Supervision, L.M.V. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data are contained within the article.
Conflicts of Interest: The authors declare no conflict of interest.
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