Limit Analysis of Lattices Based on the Asymptotic Homogenization Method and Prediction of Size Effects in Bone Plastic Collapse

Abstract

Lattice structures possess a huge potential for energy absorbing applications, thus it is important to develop predictive tools for their mechanical response up to collapse. Yielding is generally premonitory of structural collapse for lattice structures, so a comprehensive and quantitative understanding of lattice yielding behavior is indispensable in engineering applications. In the present work, the overall plastic yield and brittle failure behaviors of three-dimensional lattices is investigated by a microstructural modeling approach based on the homogenization of the initially discrete microstructure. The multiaxial yield behavior of the lattice is analyzed to formulate a multiaxial plastic yield criterion. Furthermore, the brittle fracture of the lattice is modeled under triaxial stress states to construct the failure surfaces, defined in the tension–tension quadrant. In plastic yielding, the analyses are performed assuming an elastic perfectly plastic lattice, and a micromechanical model based on an homogenization scheme is applied to a representative unit cell to determine the macroscopic plastic yield surfaces in stress space. This general framework is applied to evaluate the yield and failure properties of trabecular bone, which are of key interest in understanding and predicting the fracture of bones and bone implant systems. 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. At high bone volume fraction, the real strut-level ductility is sufficiently high to effectively be fully ductile but at very low bone volume fraction, the real behavior of bone may fail in a brittle mode. An adaptation and extension of the discrete homogenization method towards a micropolar effective medium is introduced in order to construct the plastic yield surfaces for which the material point of the effective continuum supports couple stresses in addition to Cauchy-type stresses. The size effects in the ductile fracture mode are addressed by considering a micropolar behavior, reflecting the influence of additional degrees of freedom and internal bending length effects on the initiation of plasticity. It is observed that when the characteristic size of the microscale structure 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.