Hierarchical elastoplasticity of cortical bone: Observations, mathematical modeling, validation

Abstract

Motivated by the water layer-coated nanoscale bone mineral crystals and the elastoplastic behavior seen at the extracellular scale, we develop a six-step hierarchical micromechanics model for the elastoplasticity of cortical bone. For that purpose, the Eshelby problem-based concentration-influence tensor concept is generalized for a multi-scale situation, quantifying the mechanical interaction between elastic and plastic strains between material phases across six orders of magnitude in observation scale. This hierarchical interaction scheme is complemented by non-associated Mohr–Coulomb plasticity assigned to the mineral crystal phases, and a return-mapping algorithm which adapts classical computational mechanics approaches for the realm of semi-analytical continuum micromechanics. Founded on elastic and strength properties of molecular collagen and hydroxyapatite, the model passes experimental validation against ultrasonic and quasi-static tests at the extrafibrillar, extracellular, extravascular, and cortical observation scales, across different tissues and species. It reveals cortical bone strength to increase nonlinearly with the vascular porosity, and to depend bi-linearly on the extracellular mass density, while elucidating plastic spreading events at the nanocrystal scale, which are fundamentally different in tensile and compressive loading.