PSI - Issue 77

Zihao Liu et al. / Procedia Structural Integrity 77 (2026) 190–197 Z. Liu et al./ Structural Integrity Procedia 00 (2026) 000–000 where is the distance from the neighbour to site . Let ( ) denote the set of surface bone elements within 0 of site . The distance-weighted average is ( )= ∫ ( ) ( ) ∫ ( ) ( ) (4) where ( ) is the trabecular surface with elements within 0 from site x . The mechanical non-uniformity stimulus is calculated with equation: = ln + ( ) + (5) where is a small positive number used to avoid division by zero. The logarithmic ratio normalizes different magnitudes and units for various stimuli, is symmetric for increasing and decreasing changes and converts multiplicative contributions into additive parameters that are easy to use. If the local mechanical stimulus of the bone element is lower than the resorption threshold, this element turns into a marrow one (for a marrow element, an opposite transformation happens). To compare and quantify the effects for different mechanical stimuli for the BA models in this research, the same threshold is applied in all the models. 2.2.2 Decomposition of strain energy density For small strains in linear, isotropic elasticity, the elastic energy density splits into volumetric and deviatoric parts: = 1 2 : = + (6) where is the volumetric part: = 1 2 2 = 2 2 (7) where is the bulk modulus, is the volumetric strain, and is the pressure. , the deviatoric part, has the following form: = 1 2 : = : (8) where is the shear modulus. Here, = (9) = 1 3 (10) 2.3 Implantation in USDFLD subroutine The developed mechanical model calculates the stress non-uniformity based on the Abaqus stress output for each element, at each iteration. Then it determines the phase change in the element and inputs the resultant new element properties to Abaqus for the next iteration of mechanical-state calculations. The mechanical model workflow is presented in Fig. 2. 193 4

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