Micromechanics of elastoplastic porous polycrystals: Theory, algorithm, and application to osteonal bone

Since its advent in the 1960s, elastoplastic micromechanics has been confronted by continuous challenges, as the classical incremental elastoplastic tangents are known to deliver unrealistically stiff material responses. As a complement to the various “secant” approximations targeting this challenge, we here develop a theoretical framework based on an extension of Dvorak's transformation field analysis, comprising the derivation of concentration and influence tensors. We thereby overcome the problem of actually non-homogeneous stress distributions across finite (often spherical) material phases, through consideration of infinitely many (non-spherical) solid phases oriented in all space directions, arriving at a micro-elastoplasticity theory of porous polycrystals. The resulting governing equations are discretized in time and space, and then solved in the framework of a new return mapping algorithm, the realization of which we exemplify by means of Mohr-Coulomb plasticity at the solid phase level. The corresponding homogenized material law is finally shown to satisfactorily represent the behavior of the porous hydroxyapatite polycrystals making up the so-called cement lines in osteonal bone. This is experimentally validated through strength and ultrasonic tests on hydroxyapatite, as well as through mass density, light microscopy, chemical composition, and osteon pushout tests on bone.