A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization

Journal of the Mechanical Behavior of Biomedical Materials

Volumn 16, Issue 1, 2012, Pages 87-108

  Goda, I ^a   Assidi, M ^b   Belouettar, S ^b   Ganghoffer, J F ^a  

^a UNIVERSITÉ DE LORRAINE   (France)

^b LUXEMBOURG INSTITUTE OF SCIENCE AND TECHNOLOGY   (Luxembourg)

Author keywords

Anisotropic cosserat continuum; Bone fracture; Cancellous bone; Discrete homogenization; Linear cosseratelement

Indexed keywords

ANISOTROPY; BIOMECHANICS; CONTINUUM MECHANICS; MECHANICAL PROPERTIES; MICROSTRUCTURE; SHEAR DEFORMATION; SHEAR FLOW;

BONE FRACTURE; CANCELLOUS BONE; COSSERAT CONTINUUM; DISCRETE HOMOGENIZATIONS; LINEAR COSSERATELEMENT;

BONE;

ANISOTROPIC COSSERAT CONTINUUM; ARTICLE; BIOMECHANICS; BONE STRUCTURE; CELL WALL; CONTROLLED STUDY; DENSITY; PRIORITY JOURNAL; SHEAR STRESS; TRABECULAR BONE; YOUNG MODULUS;

EID: 84869098790     PISSN: 17516161     EISSN: 18780180     Source Type: Journal    
DOI: 10.1016/j.jmbbm.2012.07.012     Document Type: Article

References (49)

1. Bakhvalov, N., Panasenko, G., 1984. Homogenisation: Averaging Processes in Periodic Media. Kluwer Academic Publisher.
2. Bowman, S.M., et al., 1998. Creep contributes to the fatigue behavior of bovine trabecular bone. Journal of Biomechanical Engineering 120, 647-654.
3. Broek, D., 1974. Elementary Fracture Mechanics. Noordhoff.
4. Bucalem, M.L., Bathe, K.J., 2011 The Mechanics of Solids and Structures- Hierarchical Modeling and Finite Element Solution (Computational Fluid and Solid Mechanics). Springer, New-York, NY.
5. Chang, C.S., Kuhn, M.R., 2005. Mechanics of solids and structures: hierarchical modeling and the finite element solution. On virtual work and stress in granular media. International Journal of Solids and Structures 42 (13), 3773-3793.
6. Cosserat, E., Cosserat, F, 1909. Theorie des Corps Deformables. A. Hermann et Fils, Paris.
7. Cowin, S.C., Doty, S.B., 2007. Tissue Mechanics. Springer, New York.
8. Dos Reis, F, 2010. Homogenization automatique de milieux discrets periodiques. Applications aux mousses polymeres et aux milieux auxetique. Ph.D, Thesis, Institut National Poly-technique de Lorraine.
9. Dos Reis, F, Ganghoffer, J.F, 2012. Equivalent mechanical properties of auxetic lattices from discrete homogenization. Computational Materials Science 51, 314-321.
10. Eringen, A.C., 1966. Linear theory of micropolar elasticity. Journal of Mathematics and Mechanics 15, 909-923.
11. Eringen, A.C., 1968. Theory of micropolar elasticity. In: Liebowitz, H. (Ed.), Fracture, vol. II. Academic Press, New York, pp. 621-729.
12. Eringen, A.C., 1976. Continuum Physics-Non-local Field Theories. Academic Press, New York.
13. Eringen, A.C., 1999. Microcontinuum Field Theories: I Foundations and Solids. Springer Verlag, New York.
14. Fang, Z., 2005. Image-Guided Modeling, Fabrication and Micro-mechanical Analysis of Bone and Heterogeneous Structure. Ph.D. Thesis, Drexel University.
15. Fatemi, J., van Keulen, F, Onck, PR., 2002. Generalized continuum theories: application to stress analysis in bone. Meccanica 37, 385-396.
16. Fatemi, J, Onck, PR, Poort, G., Van Keulen, F, 2003. Cosserat moduli of anisotropic cancellous bone: a micromechanical analysis. Journal de Physique IV France 105, 273-280.
17. Ford, C. M., Gibson, L.J., 1998. Uniaxial strength asymmetry in cellular materials: an analytical model International Journal of Mechanical Sciences 40, 521 531.
18. Gibson, L.J., Ashby, M.F, Schajer, G.S., Robertson, C.I., 1982. The mechanics of two-dimensional cellular materials. Proceedings of the Royal Society of London, A 382, 25-42.
19. Gibson, L.J., 1985. The mechanical behavior of cancellous bone. Journal of Biomechanics 18, 317-328.
20. Gibson, L.J., Ashby, M.F, 1997. Cellular Solids: Structures and Properties. Cambridge University Press, Cambridge, UK.
21. Gitman, I.M., Askes, H., Kuhl, K., Aifantis, E.C., 2010. Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity. International Journal of Solids and Structures 47, 1099-1107.
22. Gonella, S., Ruzzene, M., 2008. Homogenization and equivalent in-plane properties of two-dimensional periodic lattices. International Journal of Solids and Structures 45, 2897-2915.
23. Guo, X-D.E., McMahon, T.A., Keaveny, T.M., Hayes, W.C., Gibson, L.J., 1994. Finite element modelling of damage accumulation in trabecular bone under cyclic loading. Journal of Biomechanics 27, 145-155.
24. Kaplan, S., Hayes, W., Stone, J., Beaupre, G., 1985. Tensile strength of bovine trabecular bone. Journal of Biomechanics 18, 723.
25. Keaveny, T.M., Wachtel, E.F., Ford, C.M., Hayes, W.C., 1994. Differences between the tensile and compressive strengths of bovine tibial trabecular bone depend on modulus. Journal of Biomechanics 27, 1137.
26. Koiter, W.T., 1964. Couple stress in the theory of elasticity. In: Proceeding of the Koninklijke Nederland Akademie van Wettenschappen B-67, pp. 17-44.
27. Lakes, R., Nakamura, S., Behiri, J., Bonfield, W., 1990. Fracture mechanics of bone with short cracks. Journal of Biomechanics 23, 967-975.
28. Lakes, R., 1993. Materials with structural hierarchy. Nature 361, 511-515.
29. Lakes, R., 1995. Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Muhlhaus, H-B. (Ed.), Continuum Models for Materials with Microstructure. Wiley, New York, pp. 1-22.
30. Masters, I.G., Evans, K.E., 1996. Models for the elastic deformation of honeycombs. Composite Structures 35, 403-422.
31. Mindlin, R.D., 1964. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis 16, 51-78.
32. Mindlin, R.D., Tiersten, H.F., 1962. Effects of couple stresses in linear elasticity. Archive for Rational Mechanics and Analysis 11, 415-448.
33. Mourad, A., Caillerie, D.A., Raoult, A., 2003. A nonlinearly elastic homogenized constitutive law for the myocardium. Computational Fluid and Solid Mechanics, 1779-1781.
34. Muhlhaus, H.B., Oka, F., 1996. Dispersion and wave propagation in discrete and continuous models for granular materials. International Journal of Solids and Structures 33, 2841-2858.
35. Panasenko, G.P., 1983. Averaging of processes in frame constructions with random properties". Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 23, 1098-1109.
36. Park, H.C., Lakes, R.S., 1987. Torsion of a micropolar elastic prism of square cross section. International Journal of Solids Structures 23, 485-503.
37. Sab, K., Pradel, F., 2009. Homogenisation of periodic Cosserat media. International Journal of Computer Applications in Technology 34, 60-71.
38. Sanchez-Palencia, E., 1980. Non-Homogeneous Media and Vibration Theory. Lecture notes in Physics, vol. 127. Springer-Verlag, Berlin.
39. Shmoylova, E., Potapenko, S., Rothenburg, L., 2007. Stress distribution around a crack in plane micropolar elasticity. Journal of Elasticity 86, 19-39.
40. Silva, M.J., Hayes, W.C., Gibson, L.J., 1995. The effects of non-periodic microstructure on the elastic properties of two-dimensional cellular solids. International Journal of Mechanical Sciences 37, 1161-1177.
41. Silva, M.J., Gibson, L.J., 1997. Modeling the mechanical behavior of vertebral trabecular bone: effects of age-related changes in microstructure. Bone 21, 191-199.
42. Suiker, A.S.J., de Borst, R., Chang, C.S., 2001. Micro-mechanical modelling of granular material. Part 1: derivation of a second gradient micro-polar constitutive theory. Acta Mechanica 149, 161-180.
43. Tanaka, M., Adachi, T., 1999. Lattice continuum model for bone remodeling considering microstructural optimality of trabecular architecture. In: Pedersen, P., Bendsoe, M.P. (Eds.), Proceedings of the IUTAM Symposium on Synthesis in Bio Solid Mechanics, Kluwer Academic Publishers, The Netherlands, pp. 43-54.
44. Taylor, M., Cotton, J., Zioupos, P, 2002. Finite element simulation of the fatigue behaviour of cancellous bone. Meccanica 37, 419-429.
45. Warren, W.E., Byskov, E., 2002. Three-fold symmetry restrictions on two-dimensional micropolar materials. European Journal of Mechanics A/Solids 21, 779-792.
46. Warren, W.E., Kraynik, A.M., 1987. Foam mechanics: the linear elastic response of two dimensional spatially periodic cellular materials. Mechanics of Materials 6, 27-37.
47. Yang, J.F.C., Lakes, R.S., 1981. Transient study of couple stress effects in compact bone: torsion. Journal of Biomechanical Engineering 103, 275-279
48. Zhu, H.X., 2010. Size-dependent elastic properties of micro- and nanohoneycombs. Journal of the Mechanics and Physics of Solids 58, 679-696.
49. Zhu, H.X., Hobdell, J.R., Windle, A.H., 2001. Effects of cell irregularity on the elastic properties of 2D Voronoi honeycombs. Journal of the Mechanics and Physics of Solids 49, 857-870.