A Modelling Approach for Micro Crack Healing in Bones by Flexoelectricity-induced Surface Growth
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Micro cracks naturally occur in bones under mechanical loading and are healed by a mechanism called bone remodelling. The initiation of bone remodelling is attributed to different phenomena in the literature which are closely related to electromechanical coupling. One of them is flexoelectricity which was experimentally observed in cortical bone [1,2] and describes a coupling between strain gradients and electric polarisation. Due to its size-dependency, flexoelectricity is particularly relevant in the region of micro and nano cracks. It leads to the induction of electric fields in response to mechanical loading and, thereby, causes osteocyte apoptosis which is a pivotal event for bone remodelling. In this contribution, a modelling framework for the simulation of flexoelectricity-induced bone remodelling is proposed. More specifically speaking, a globally C1-continuous Isogeometric Analysis framework is employed to resolve the flexoelectric initiation of the remodelling process whereas the subsequent bone cell diffusion as well as the resulting crack healing is accounted for in a classic Finite Element framework with a moving mesh approach to incorporate surface growth, i.e. the production of new bone material. Bone cell concentrations are considered as independent field variables and anisotropic diffusion tensors describe their migration through the bone towards the remodelling site. Additionally, signalling mechanisms are incorporated and coupled to the cell migration by source terms and boundary conditions. The proposed framework is applied to a bone excerpt with a narrow micro crack and the healing of the crack is studied based on flexoelectric initiation. The presented work is based on the model proposed in [3] and extended with regard to surface growth and the modelling of the diffusion processes. [1] F. Vasquez-Sancho, A. Abdollahi, D. Damjanovic, and G. Catalan, Flexoelectricity in bones Adv. Mater., 30, 1801413, 2018. DOI: 10.1002/adma.201801413 [2] R. Nunez-Toldra, F. Vasquez-Sancho, N. Barroca, G. Catalan, Investigation of the cellular response to bone fractures: Evidence for flexoelectricity Sci. Rep., 10, 254, 2020. DOI: 10.1038/s41598-019-57121-3 [3] C. Witt, T. Kaiser, and A. Menzel, Modelling and numerical simulation of remodelling processes in cortical bone: An IGA approach to flexoelectricity-induced osteocyte apoptosis and subsequent bone cell diffusion J. Mech. Phys. Solids, 173, 105194, 2023. DOI: 10.1016/j.jmps.2022.105194