The multiscale meso-mechanics model of viscoelastic cortical bone

Abdel-Wahab AA, Alam K, Silberschmidt VV (2011) Analysis of anisotropic viscoelastoplastic properties of cortical bone tissues. J Mech Behav Biomed Mater 4:807–820. https://doi.org/10.1016/j.jmbbm.2010.10.001

Ascenzi A, Bonucci E (1967) The tensile properties of single osteons. Anat Rec 158:375–386. https://doi.org/10.1002/ar.1091580403

Ascenzi A, Bonucci E (1968) The compressive properties of single osteons. Anat Rec 161:377–391. https://doi.org/10.1002/ar.1091610309

Beel JA, Groswald DE, Luttges MW (1984) Alterations in the mechanical properties of peripheral nerve following crush injury. J Biomech 17:185–193. https://doi.org/10.1016/0021-9290(84)90009-5

Cuppone M, Seedhom BB, Berry E, Ostell AE (2004) The longitudinal Young’s modulus of cortical bone in the midshaft of human femur and its correlation with CT scanning data. Calcif Tissue Int 74:302–309. https://doi.org/10.1007/s00223-002-2123-1

Dong XN, Guo XE (2004) The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity. J Biomech 37:1281–1287. https://doi.org/10.1016/j.jbiomech.2003.12.011

Dong XN, Guo XE (2006) Prediction of cortical bone elastic constants by a two-level micromechanical model using a generalized self-consistent method. J Biomech Eng 128:309–316. https://doi.org/10.1115/1.2187039

Ghanbari J, Naghdabadi R (2009) Nonlinear hierarchical multiscale modeling of cortical bone considering its nanoscale microstructure. J Biomech 42:1560–1565. https://doi.org/10.1016/j.jbiomech.2009.02.014

Gottesman T, Hashin Z (1980) Analysis of viscoelastic behaviour of bones on the basis of microstructure. J Biomech 13:89–96. https://doi.org/10.1016/0021-9290(80)90182-7

Hamed E, Lee Y, Jasiuk I (2010) Multiscale modeling of elastic properties of cortical bone. Acta Mech 213:131–154. https://doi.org/10.1007/s00707-010-0326-5

He Z, Pindera MJ (2020) Locally exact asymptotic homogenization of periodic materials under anti-plane shear loading. Eur J Mech, A/Solids. https://doi.org/10.1016/j.euromechsol.2020.103972

He Z, Pindera MJ (2021) Locally exact asymptotic homogenization of viscoelastic composites under anti-plane shear loading. Mech Mater. https://doi.org/10.1016/j.mechmat.2021.103752

He Z, Wang G, Pindera MJ (2019) Multiscale homogenization and localization of materials with hierarchical porous microstructures. Compos Struct. https://doi.org/10.1016/j.compstruct.2019.110905

Ho Ba Tho MC, Stolz C, Vanleene M, Bensamoun S, Treutenaere JM, Rey C (2005) Multi-scale characterization and modelling of human cortical bone. Materials Research Society Symposium Proceedings, pp 60–65. https://doi.org/10.1557/proc-0898-l05-15

Hogan HA (1992) Micromechanics modeling of Haversian cortical bone properties. J Biomech 25:549–556. https://doi.org/10.1016/0021-9290(92)90095-I

Iyo T, Maki Y, Sasaki N, Nakata M (2004) Anisotropic viscoelastic properties of cortical bone. J Biomech 37:1433–1437. https://doi.org/10.1016/j.jbiomech.2003.12.023

Katz JL, Van Mow C (1973) Mechanical and structural criteria for orthopaedic implants. Biomater Med Devices Artif Organs 1:575–634. https://doi.org/10.3109/10731197309118565

Knets IV (1977) Mechanics of biological tissues. Rev Polym Mech 13:434–441. https://doi.org/10.1007/BF00859428

Knet-s IV, Krauya UE, Vilks YK (1975) Acoustic emission in human bone tissue subjected to longitudinal extension. Polym Mech 11:589–593. https://doi.org/10.1007/BF00856790

Manda K, Xie S, Wallace RJ, Levrero-Florencio F, Pankaj P (2016) Linear viscoelasticity - bone volume fraction relationships of bovine trabecular bone. Biomech Model Mechanobiol 15:1631–1640. https://doi.org/10.1007/s10237-016-0787-0

Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 21:571–574. https://doi.org/10.1016/0001-6160(73)90064-3

Pindera MJ, Khatam H, Drago AS, Bansal Y (2009) Micromechanics of spatially uniform heterogeneous media: a critical review and emerging approaches. Compos B Eng 40:349–378. https://doi.org/10.1016/j.compositesb.2009.03.007

Rho JY, Ashman RB, Turner CH (1993) Young’s modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements. J Biomech 26:111–119. https://doi.org/10.1016/0021-9290(93)90042-D

Riggs BL, Melton LJ (1986) Involutional osteoporosis. N Engl J Med 314:1676–1686. https://doi.org/10.1056/NEJM198606263142605

Schuit SCE et al (2004) Fracture incidence and association with bone mineral density in elderly men and women: the Rotterdam study. Bone 34:195–202. https://doi.org/10.1016/j.bone.2003.10.001

Singleton RC, Pharr GM, Nyman JS (2021) Increased tissue-level storage modulus and hardness with age in male cortical bone and its association with decreased fracture toughness. Bone. https://doi.org/10.1016/j.bone.2021.115949

Turner CH, Rho J, Takano Y, Tsui TY, Pharr GM (1999) The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques. J Biomech 32:437–441. https://doi.org/10.1016/S0021-9290(98)00177-8

Van Buskirk WC, Ashman RB (1981) The elastic moduli of bone. In: Mechanical properties of Bone (ed. S. ,C. Cowin) American Society of Mechancial Engineers, New York, pp 131–144.

Van Buskirk WC, Cowin SC, Ward RN (1981) Ultrasonic measurement of orthotropic elastic constants of bovine femoral bone. J Biomech Eng 103:67–72. https://doi.org/10.1115/1.3138262

Vaughan TJ, McCarthy CT, McNamara LM (2012) A three-scale finite element investigation into the effects of tissue mineralisation and lamellar organisation in human cortical and trabecular bone. J Mech Behav Biomed Mater 12:50–62. https://doi.org/10.1016/j.jmbbm.2012.03.003

Wang G (2019) An efficient analytical homogenization technique for mechanical-hygrothermal responses of unidirectional composites with applications to optimization and multiscale analyses. Chin J Aeronaut 32:382–395. https://doi.org/10.1016/j.cja.2018.03.025

Wang T, Feng Z (2005) Dynamic mechanical properties of cortical bone: the effect of mineral content. Mater Lett 59:2277–2280. https://doi.org/10.1016/j.matlet.2004.08.048

Wang G, Pindera MJ (2016) On boundary condition implementation via variational principles in elasticity-based homogenization. J Appl Mech Trans ASME. https://doi.org/10.1115/1.4034227

Wesly RLR, Vaishnav RN, Fuchs JCA, Patel DJ, Greenfield JC Jr (1975) Static linear and nonlinear elastic properties of normal and arterialized venous tissue in dog and man. Circ Res 37:509–520. https://doi.org/10.1161/01.RES.37.4.509

Wu Z, Ovaert TC, Niebur GL (2012) Viscoelastic properties of human cortical bone tissue depend on gender and elastic modulus. J Orthop Res 30:693–699. https://doi.org/10.1002/jor.22001

Yamada H, Evans GF (1970) Strength of biologic materials. Williams and Wilkins, Baltimore

Yoon HS, Katz JL (1976) Ultrasonic wave propagation in human cortical bone-I. Theoretical considerations for hexagonal symmetry. J Biomech 9(407–408):409–412. https://doi.org/10.1016/0021-9290(76)90118-4

Yoon HS, Lawrence Katz J (1976) Ultrasonic wave propagation in human cortical bone-II. Measurements of elastic properties and microhardness. J Biomech 9:459–462. https://doi.org/10.1016/0021-9290(76)90089-0