3D plastic collapse and brittle fracture surface models of trabecular bone from asymptotic homogenization method

The objective of this work is to develop an adequate three-dimensional model for describing the multiaxial yield and failure behavior of trabecular bone, and to set up criteria for the brittle and ductile collapse based on micromechanical approaches. The yield and failure properties of trabecular bone are of key interest in understanding and predicting the fracture of bones and bone implant systems. The discrete homogenization technique is presently developed as a convenient micromechanical approach to construct the plastic yield surfaces of 2D and 3D bending-dominated periodic lattices of articulated beams considered as prototype topologies for cancellous bones. The initial lattice is replaced by an effective Cauchy continuous medium at an intermediate scale, endowed with effective properties representative of an identified representative unit cell within the structure. The cell walls of the bone microstructure are modeled as Timoshenko thick beams taking into account stretching, transverse shearing, and bending deformations. In the case of plastic yielding, the cell struts of trabecular are assumed to behave in an elastic-perfectly plastic manner. The proposed methodology is quite general, as the representative bone unit cell includes internal nodes and no assumption of uniform deformation is needed. The effective strength of trabecular bone is evaluated in the two situations of fully brittle (fracture with no tissue ductility) and fully ductile failure (yield with no tissue fracture) of the trabecular tissue. Finite element analyses are performed to validate the plastic collapse stresses of trabecular bone in a 2D situation under uniaxial, shear, and biaxial tensile loadings. A size-dependent non-classical plastic yield criterion is finally developed relying on the reduced Cosserat theory to capture the size-dependency of the trabecular bone structures. When the characteristic size of the bone sample is comparable to the bending length, a significant difference is shown between the results based on the non-classical theory and those obtained by the classical theory.