PSI - Issue 42

Aleksandr Shalimov et al. / Procedia Structural Integrity 42 (2022) 1153–1158 A. Shalimov et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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Fig. 1. Three-dimensional geometry of trabecular bone structure (a) obtained from CT scan and its finite-element model (b)

Trabecular bone behaves differently in tension and compression; so, it is necessary to distinguish fracture parameters separately for each of these two loading cases. For tension, the range of yield strain is from 0.33% to 0.54%, and from 0.8% to 0.83% for compression (Sabet et al., 2018). In this work, the values of yield stress were chosen 0.33% for tension and 0.81% for compression, as in (Nawathe et al., 2013; Sanyal et al., 2012; Sanyal and Keaveny, 2013). Models of the unit cells were rigidly fixed at their lower face and a perpendicular load was applied to the upper face, expressed as a uniform vertical displacement of 0.1 mm (in tension) and -0.1 mm (in compression). In order to implement the procedure for degradation of elastic properties, the internal state variable , which characterizes the development of microstructural damage, was introduced into the model. Its value increased when the degradation criterion was fulfilled, with the components of the stiffness tensor being reduced. The generalized Hooke's law then has the following form: ̂ = ̂ ̂ , (1) ̂ ∗ = (1 − ) ∙ ̂ , (2) ̂ ∗ = (1 − ) ∙ ̂ ̂ , (3) where ̂ is the stress tensor, ̂ is the stiffness tensor, ̂ is the strain tensor. A field of state-variable values varies from 0 to , with the maximum value in the elements where the criterion is fulfilled. Intermediate values of damage were obtained using a power function. If this model is compared to the maximum-stress failure model, the areas meeting the degradation criterion show the failed elements. The internal state variable in this case is defined by the following function: = { 0 ≤ , ∙ ( ) < < , ≥ , (4) where is the equivalent strain determined as = √ 2 3 ̂ ̂ ; is the critical value of the degradation factor; is the yield strength strain; is the tensile strain at failure. The respective material parameters were different for tension and compression (Table 2). In the case of the ultimate strains, the size of the finite-element must be taken into account in the numerical solution: