PSI - Issue 37

Ekaterina Smotrova et al. / Procedia Structural Integrity 37 (2022) 257–262 E. Smotrova et al. / Structural Integrity Procedia 00 (2021) 000 – 000

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software (Dassault Systems Simulia Crop, Providence, RI, USA) using a customized UMAT subroutine for damage modelling.

Fig. 2 Stress-strain curve of material model for trabeculae

3. Results and discussion The brittle elastic model demonstrated different stress-strain behaviors when subjected to tensile and compressive loading regimes. Although in both loading cases the trabecular ligaments, directed primarily along the axis of load, acted as stress concentrators, the locations of these stress concentrators and, therefore, the damage regions were different in tension and compression. Interestingly, the differences in the stress fields in both loading cases had no effect on the load magnitude corresponding to the onset of failure in the brittle elastic material model. However, more damaged regions were observed in the model subjected to tension than when subjected to the same value of compressive load (Fig. 3). At the moment when the maximum principal stress exceeded the ultimate tensile strength of the material, the applied external load was equal to 182 N (for both tension and compression) and the area of the unit cell’s surface perpendicular to the applied load was 15.84 mm 2 , resulting in the value of effective strength of the trabecular bone tissue equal to 11.49 MPa. The obtained value of effective strength this accounts for 8.33% of the ultimate tensile strength of trabeculae. The ductile elastoplastic model demonstrated no differences in the fields of von Mises stress when subjected to tension and compression. At the moment when this stress exceeded the yield point of the trabecular material, the loading level corresponded to 78.3 N (for both tension and compression), giving the value of the corresponding effective strength of trabecular bone tissue of 4.94 MPa. The obtained value of effective strength accounts for 8.37% of the yield stress of the material. The obtained values of tensile and compressive strengths for two used material formulations are given in Table 1. For both loading cases, the numerical results obtained with the brittle elastic and ductile elastoplastic models provided values of effective strengths well within the ranges of ultimate stress of HTB obtained by direct mechanical testing in the previously published studies: 1 – 19 MPa for tension (Carter et al. 1980, Røhl et al. 1991) and 1 – 50 MPa for compression (Carter et al. 1980, Linde et al. 1989, Rincón-Kohli and Zysset 2009). These findings suggest that the trabecular bone material can be modelled using either brittle or ductile material model. The use of the specific material formulation should depend on results of additional studies aimed at assessment of the extent of ductility (i.e., post-yield deformation) in specific trabecular bone tissues depending on the age and health state of the patients.