Abstract
Cosserat models of trabecular bone are constructed in 2D and 3D situations, based on micromechanical approaches to investigate microstructure-related scale effects on the macroscopic properties of bone. The effective mechanical properties of cancellous bones considered as cellular solids are obtained thanks to the discrete homogenization technique. The cell walls of the bone microstructure are modeled as Timoshenko thick beams. An anisotropic micropolar equivalent continuum model is constructed, the effective mechanical properties of which are identified. Closed form expressions of the equivalent properties are obtained versus the geometrical and mechanical microparameters, accounting for the effects of bending, axial, and transverse shear deformations; torsion is additionally considered for a 3D geometry. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the micropolar material constants. The stress distribution in a cracked bone sample is computed based on the effective micropolar model, highlighting the regularizing effect of the Cosserat continuum in comparison to a classical elasticity continuum model.
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References
Bakhvalov, N., Panasenko, G.: Homogenisation: averaging processes in periodic media. Kluwer Academic Pub, Dordrecht (1984)
Bouyge, F., Jasiuk, I., Boccara, S., Ostoja-Starzewski, M.: A micromechanically based couple-stress model of an elastic orthotropic two-phase composite. Eur. J. Mech. A. Solids. 21, 465–481 (2002)
Bowman, S.M., et al.: Creep contributes to the fatigue behavior of bovine trabecular bone’. J. Biomech. Eng. 120, 647–654 (1998)
Broek, D.: Elementary fracture mechanics. Noordhoff, Leyden (1974).
Chang, C.S., Kuhn, M.R.: Mechanics of Solids and Structures: Hierarchical Modeling and the Finite Element Solution. On virtual work and stress in granular media. Int. J. Solids. Struct. 42(13), 3773–3793 (2005)
Cosserat, E., Cosserat, F.: Théorie Des Corps Déformables. A. Hermann et Fils, Paris (1909)
Cowin, S.C., Doty, S.B.: Tissue Mechanics. Springer, New York (2007)
Dos Reis, F.: Homogenization automatique de milieux discrets périodiques. Applications aux mousses polymères et aux milieux auxétique. Ph.D. Thesis, Institut National Polytechnique de Lorraine (2010)
Dos Reis, F., Ganghoffer, J.F.: Equivalent mechanical properties of auxetic lattices from discrete homogenization. Comput. Mater. Sci. 51, 314–321 (2012)
Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966)
Eringen, A.C.: Theory of Micropolar Elasticity. In: Liebowitz, H. (ed.) Fracture, vol. II, pp. 621–729. Academic Press, New York (1968)
Eringen, A.C.: Continuum Physics - Non-local Field Theories. Academic Press, New York (1976)
Eringen, A.C.: Microcontinuum field theories: i foundations and solids. Springer Verlag, New York (1999)
Fang, Z.: Image-guided modeling, fabrication and micromechanical analysis of bone and heterogeneous structure. PhD thesis Drexel University, Philadelphia (2005)
Fatemi, J., van Keulen, F., Onck, P.R.: Generalized continuum theories: application to stress analysis in bone. Meccanica. 37, 385–396 (2002)
Fatemi, J., Onck, P.R., Poort, G., Van Keulen, F.: Cosserat moduli of anisotropic cancellous bone: a micromechanical analysis. J. Phys. IV France. 105, 273–280 (2003)
Ford, C.M., Keaveny, T.M.: The dependence of shear failure properties of trabecular bone on apparent density and trabecular orientation. J Biomech 29, 1309 (1996)
Ford, C.M., Gibson, L.J.: Uniaxial strength asymmetry in cellular materials: an analytical model Int. J. Mech. Sci. 40(521), 531 (1998)
Gibson, L.J., Ashby, M.F., Schajer, G.S., Robertson, C.I.: The mechanics of two-dimensional cellular materials. Proc. Roy. Soc. Lond. A 382, 25–42 (1982)
Gibson, L.J., Ashby, M.F.: Cellular Solids: Structures and Properties. Cambridge University Press, Cambridge (1997)
Gonella, S., Ruzzene, M.: Homogenization and equivalent in-plane properties of two dimensional periodic lattices. Int. J. Solids. Struct. 45, 2897–2915 (2008)
Kim, H.S., Al-Hassani, S.T.S.: A morphological model of vertebral trabecular bone. J. Biomech. 35, 1101–1114 (2002)
Koiter, W.T.: Couple stress in the theory of elasticity. Proc. Koninklijke Nederland Akademie van Wettenschappen B 67, 17–44 (1964)
Lakes, R., Nakamura, S., Behiri, J., Bonfield, W.: Fracture mechanics of bone with short cracks. J. Biomech. 23, 967–975 (1990)
Lakes, R.: Materials with structural hierarchy. Nature 361, 511–515 (1993)
Lakes, R.: Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Muhlhaus, H.-B. (ed.) Continuum Models for Materials with Microstructure, pp. 1–22. Wiley, New York (1995)
Liu, S.X., Zhang, H.X., Guo, E.X.: Contributions of trabecular rods of various orientations in determining the elastic properties of human vertebral trabecular bone. Bone 45, 158–163 (2009)
Masters, I.G., Evans, K.E.: Models for the elastic deformation of honeycombs. Compos. Struct. 35, 403–422 (1996)
Miller, Z., Fuchs, M.B.: Effect of trabecular curvature on the stiffness of trabecular bone. J. Biomech. 38, 1855–1864 (2005)
Mindlin, R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3, 1–7 (1963)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78 (1964)
Mindlin, R.D., Tiersten, H.F.: Effects of couple stresses in linear elasticity. Arch. Rational Mech. Anal. 11, 415–448 (1962)
Mourad, A., Caillerie, D.A., Raoult, A.: A nonlinearly elastic homogenized constitutive law for the myocardium, pp. 1779–1781. Computational Fluid and Solid Mechanics, Cambridge (2003)
Muhlhaus, H.B., Oka, F.: Dispersion and wave propagation in discrete and continuous models for granular materials. Int. J. Solids Struct. 33, 2841–2858 (1996)
Panasenko, G.P.: Averaging of processes in frame constructions with random properties. Zh. Vychisl. Mat. Mat. Fiz. 23, 1098–1109 (1983)
Park, H.C., Lakes, R.S.: Torsion of a micropolar elastic prism of square cross section. Int. J. Solids. Struct. 23, 485–503 (1987)
Rovati, M., Veber, D.: Optimal topologies for micropolar solids. Struct. Multidisc. Optim. 33, 47–59 (2007)
Sab, K., Pradel, F.: Homogenisation of periodic Cosserat media. Int. J. Comput. Appl. Technol. 34, 60–71 (2009)
Sanchez-Palencia, E.: Non-homogeneous media and vibration theory, Lecture notes in Physics, 127. Springer-Verlag, Berlin (1980)
Shmoylova, E., Potapenko, S., Rothenburg, L.: Stress distribution around a crack in plane micropolar elasticity. J. Elast. 86, 19–39 (2007)
Silva, M.J., Hayes, W.C., Gibson, L.J.: The effects of non-periodic microstructure on the elastic properties of two-dimensional cellular solids. Int. J. Mech. Sci. 37, 1161–1177 (1995)
Suiker, A.S.J., de Borst, R., Chang, C.S.: Micro-mechanical modelling of granular material. Part 1: derivation of a secondgradient micro-polar constitutive theory. Acta. Mech. 149, 161–180 (2001)
Tanaka, M., Adachi, T.: Lattice continuum model for bone remodeling considering microstructural optimality of trabecular architecture. In: Pedersen, P., Bendsoe, M.P. (eds.) IUTAM Symposium on Synthesis in Bio Solid Mechanics, pp. 43–54. Kluwer Academic Publishers, The Netherlands (1999)
Taylor, M., Cotton, J., Zioupos, P.: Finite element simulation of the fatigue behaviour of cancellous bone. Meccanica. 37, 419–429 (2002)
Warren, W.E., Byskov, E.: Three-fold symmetry restrictions on two-dimensional micropolar materials. Eur. J. Mech. A. Solids. 21, 779–792 (2002)
Warren, W.E., Kraynik, A.M.: Foam mechanics: the linear elastic response of two dimensional spatially periodic cellular materials. Mech. Mater. 6, 27–37 (1987)
Yang, J.F.C., Lakes, R.S.: Transient study of couple stress effects in compact bone: Torsion. J. Biomech. Engng. 103, 275–279 (1981)
Yoo, A., Jasiuk, I.: Couple-stress moduli of a trabecular bone idealized as a 3D periodic cellular network. J. Biomech. 39, 2241–2252 (2006)
Zhu, H.X.: Size-dependent elastic properties of micro- and nanohoneycombs. J. Mech. Phys. Solids. 58, 679–696 (2010)
Zhu, H.X., Hobdell, J.R., Windle, A.H.: Effects of cell irregularity on the elastic properties of 2D Voronoi honeycombs. J. Mech. Phys. Solids. 49, 857–870 (2001)
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Goda, I., Assidi, M., Ganghoffer, JF. (2013). Cosserat Anisotropic Models of Trabecular Bone from the Homogenization of the Trabecular Structure: 2D and 3D Frameworks. In: Altenbach, H., Forest, S., Krivtsov, A. (eds) Generalized Continua as Models for Materials. Advanced Structured Materials, vol 22. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36394-8_7
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