Page 215 - Advances in Biomechanics and Tissue Regeneration

210 10. DETERMINATION OF THE ANISOTROPIC MECHANICAL PROPERTIES OF BONE TISSUE

10000 5082 5144
926 926
1000
Time [s] log 10 100 94 100

6
4
10
2 4
1
FEM NNRPIM FEM NNRPIM FEM NNRPIM FEM NNRPIM FEM NNRPIM
1×1 2×2 3×3 4×4
Homogeneous Heterogenous model
model
FIG. 10.14 Computational cost (in seconds) of each analysis.

Each analysis has its own computational cost. In Fig. 10.14, the time lapse of each structural analysis is shown.
Observing the computational cost of each analysis, it is possible to understand that the analysis of the homogenized
RVE is much faster than heterogeneous RVEs. Generally, the multiscale techniques use highly discretized RVEs, with a
high computational cost associated. As this example shows, the proposed homogenization methodology is capable of
reducing the cost of the multiscale analysis, enabling more demanding simulations.

10.4 CONCLUSIONS

The presented methodology allows defining the mechanical properties of a micro-CT RVE patch without any a prior
knowledge. Using a morphologic methodology to acquire the ODF data from the micro-CT RVE, and combining this
data with a phenomenological material law, a methodology was defined that allows defining a homogeneous material
that is equivalent to a heterogeneous material. The defined material mechanical properties are directly related with the
trabecular anisotropy encoded in the fabric tensor and with the material law developed by Belinha et al. [40]. It was
demonstrated that the methodology is stable and provides good results, even when considering different RVE scales
and different material principal directions. It was also shown that the NNRPIM is capable of producing accurate and
smooth microscale variable fields (at the RVE scale), which allows obtaining accurate final homogenized variable
fields. Multiscale techniques usually use highly discretized RVEs. The homogenization technique here proposed
showed that, when combined with meshless methods, it is capable of reducing the cost of analyzing highly heteroge-
neous domains. Thus, using this methodology in multiscale analyses will allow simulating more complex problems
with lower costs. Also, as bone is a hierarchal material, this methodology might be a powerful tool to understand the
remodeling process, using a multiscale approach, where the mechanical properties of trabecular bone can be defined at
the microscale, considering the trabeculae architecture.

Acknowledgments
¸
The authors acknowledge the funding provided by Minist erio da Ci^ encia, Tecnologia e Ensino Superior—Fundacão para a Ci^ encia e a Tecnologia
(Portugal), under Grants SFRH/BD/110047/2015, and by project funding MIT-EXPL/ISF/0084/2017. Additionally, the authors gratefully acknowl-
edge the funding of Project NORTE-01-0145-FEDER-000022—SciTech—Science and Technology for Competitive and Sustainable Industries, cofi-
nanced by Programa Operacional Regional do Norte (NORTE2020), through Fundo Europeu de Desenvolvimento Regional (FEDER).

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I. BIOMECHANICS