The multiscale meso-mechanics model of viscoelastic cortical bone

Abstract

Cortical bone is a complex hierarchical structure consisting of biological fiber composites with transversely isotropic constituents, whose microstructures deserve extensive study to understand the mechanism of living organisms and explore development of biomimetic materials. Based on this, we establish a three-level hierarchical structure from microscale to macroscale and propose a multiscale micromechanics model of cortical bone, which considers Haversian canal, osteonal lamellae, cement line and interstitial lamellae. In order to study the microstructural effect on the elastic behavior of hierarchical structures, the Mori–Tanaka model and locally exact homogenization theory are introduced for the homogenization of heterogeneous materials of microstructure at each level. Within sub-microscale, Haversian canal and Osteonal lamella are treated as fiber and matrix, whose homogenization is surrounded with cement line matrix in microstructure (or what we called “osteon”) for the second homogenization; finally, osteon and interstitial lamella establish the meso-structure for the third homogenization, predicting the effective moduli of cortical bone. The correctness of the model in this paper is verified against the data in literature with good agreement. Finally, the dynamic viscoelastic response of cortical bones is investigated from a multiscale perspective, where the measured data are substituted into the present models to study the hydration and aging effect on bones’ stiffness and viscoelasticity. It is demonstrated that the hydration is much more influential in affecting the storage and loss moduli of cortical bone than the aging effect. We also present a few numerical investigations on microstructural material and geometric parameters on the overall mechanical properties of cortical bone.

Appendix

The following is the calculation formula of CCA model.

$$E_{ao} = E_{m} \times vf_{m} + E_{f} \times vf_{f} + \frac{{4(v_{f} - v_{m} )^{2} \times vf_{f} \times vf_{m} }}{{\frac{{vf_{m} }}{{k_{f} }} + \frac{{vf_{f} }}{{k_{m} }} + \frac{1}{{G_{m} }}}}$$

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$$k_{to} = \frac{{k_{m} \times vf_{m} \times (k_{f} + G_{m} ) + k_{f} \times vf_{f} \times (k_{m} + G_{m} )}}{{(k_{f} + G_{m} ) \times vf_{m} + (k_{m} + G_{m} ) \times vf_{f} }}$$

(21)

$$v_{ao} = v_{m} \times vf_{m} + v_{f} \times vf_{f} + \frac{{(v_{f} - v_{m} ) \times (\frac{1}{{k_{m} }} - \frac{1}{{k_{f} }}) \times vf_{m} \times vf_{f} }}{{\frac{{v_{m} }}{{k_{f} }} + \frac{{v_{f} }}{{k_{m} }} + \frac{1}{{G_{m} }}}}$$

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$$G_{ao} = G_{m} \times \frac{{G_{m} \times vf_{m} + G_{f} \times (1 + vf_{f} )}}{{G_{m} \times (1 + vf_{f} ) + G_{f} \times f_{m} }}$$

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$$G_{to} ( + ) = G_{m} + \frac{{vf_{m} }}{{\frac{1}{{G_{m} }} + \frac{{(k_{m} + 2G_{m} ) \times vf_{f} }}{{2G_{m} \times (k_{m} + G_{m} )}}}}$$

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$$G_{to} ( - ) = G_{m} + \frac{{vf_{f} }}{{\frac{1}{{G_{f} - G_{m} }} + \frac{{(k_{m} + 2G_{m} ) \times vf_{m} }}{{2G_{m} \times (k_{m} + G_{m} )}}}}$$

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where o, t, a, f and m represent osteons, transverse, axial, fiber and matrix, respectively. E, k, v and G donate Young’s modulus、bulk modulus、Poisson’s ratio and shear modulus, respectively. The subscript “vf” means volume fraction.