We propose a second gradient, two-solids, continuum mixture model with variable masses to describe the effect of micro-structure on mechanically-driven remodelling of bones grafted with bio-resorbable materials. A one-dimensional numerical simulation is addressed showing the potentialities of the proposed generalized continuum model. In particular, we show that the used second gradient model allows for the description of some micro-structure-related size effects which are known to be important in hierarchically heterogeneous materials like reconstructed bones. Moreover, the influence of the introduced second gradient parameters on the final percentages of replacement of artificial bio-material with natural bone tissue is presented and discussed.
Accepted:
Published online:
Angela Madeo 1, 2; D. George 3; T. Lekszycki 4, 5, 2; Mathieu Nierenberger 3; Yves Rémond 3, 2
@article{CRMECA_2012__340_8_575_0, author = {Angela Madeo and D. George and T. Lekszycki and Mathieu Nierenberger and Yves R\'emond}, title = {A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling}, journal = {Comptes Rendus. M\'ecanique}, pages = {575--589}, publisher = {Elsevier}, volume = {340}, number = {8}, year = {2012}, doi = {10.1016/j.crme.2012.05.003}, language = {en}, }
TY - JOUR AU - Angela Madeo AU - D. George AU - T. Lekszycki AU - Mathieu Nierenberger AU - Yves Rémond TI - A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling JO - Comptes Rendus. Mécanique PY - 2012 SP - 575 EP - 589 VL - 340 IS - 8 PB - Elsevier DO - 10.1016/j.crme.2012.05.003 LA - en ID - CRMECA_2012__340_8_575_0 ER -
%0 Journal Article %A Angela Madeo %A D. George %A T. Lekszycki %A Mathieu Nierenberger %A Yves Rémond %T A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling %J Comptes Rendus. Mécanique %D 2012 %P 575-589 %V 340 %N 8 %I Elsevier %R 10.1016/j.crme.2012.05.003 %G en %F CRMECA_2012__340_8_575_0
Angela Madeo; D. George; T. Lekszycki; Mathieu Nierenberger; Yves Rémond. A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling. Comptes Rendus. Mécanique, Volume 340 (2012) no. 8, pp. 575-589. doi : 10.1016/j.crme.2012.05.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2012.05.003/
[1] Mechanical factors in bone growth and development, Bone, Volume 18 (1996), p. S5-S10
[2] Temporal evolution of skeletal regenerated tissue: What can mechanical investigation add to biological?, Med. Biol. Eng. Comput., Volume 48 (2010), pp. 811-819
[3] Effects of mechanical forces on maintenance and adaptation of form in trabecular bone, Nature, Volume 405 (2000), pp. 704-706
[4] A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery, C. R., Méc., Volume 339 (2011), pp. 625-640
[5] Optimality conditions in modeling of bone adaptation phenomenon, J. Theoret. Appl. Mech., Volume 37 (1999) no. 3, pp. 607-624
[6] Modeling of bone adaptation based on an optimal response hypothesis, Meccanica, Volume 37 (2002), pp. 343-354
[7] Functional adaptation of bone as an optimal control problem, J. Theoret. Appl. Mech., Volume 43 (2005) no. 3, pp. 120-140
[8] Das Gesetz der Transformation der Knochen, Hirschwald Verlag, Berlin, 1892
[9] The Law of Bone Remodelling, Springer-Verlag, Berlin, 1986
[10] Studies on constitutive equation that models bone tissue, Acta Bioeng. Biomech., Volume 10 (2008) no. 4, pp. 39-47
[11] A theoretical framework for strain-related trabecular bone maintenance and adaptation, J. Biomech., Volume 38 (2005), pp. 931-941
[12] Bone remodeling II: Small strain adaptive elasticity, J. Elast., Volume 6 (1976), pp. 337-352
[13] Optimal-tuning PID control of adaptive materials for structural efficiency, Struct. Multidiscipl. Optim., Volume 43 (2011), pp. 43-59
[14] Prediction of micromotion initiation of an implanted femur under physiological loads and constraints using the finite element method, Proc. Inst. Mech. Eng., H J. Eng. Med., Volume 223 (2009), pp. 589-605
[15] Prediction of bone adaptation using damage accumulation, J. Biomech., Volume 27 (1994), pp. 1067-1076
[16] Anisotropic bone remodelling model based on a continuum damage-repair theory, J. Biomech., Volume 35 (2002), pp. 1-17
[17] Resorption of synthetic porous hydroxyapatite and replacement newly formed bone, J. Orthop. Sci., Volume 6 (2001), pp. 444-447
[18] Bioinert, biodegradable and injectable polymeric matrix composites for hard tissue replacement: State of the art and recent developments, Compos. Sci. Technol., Volume 64 (2004), pp. 789-817
[19] Resorbability of bone substitute biomaterials by human osteoclasts, Biomaterials, Volume 25 (2004), pp. 3963-3972
[20] Review: Biomaterials and bone mechanotransduction, Biomaterials, Volume 22 (2001), pp. 2581-2593
[21] A.S. Greenwald, S.D. Boden, V.M. Goldberg, Y.K. Cato, T. Laurencin, R.N. Rosier, Bone graft substitutes: Facts, fictions & applications, in: 68th Annual Meeting American Academy of Orthopaedic Surgeons, San Francisco, California, February 28–March 4, 2001.
[22] Bioceramic bone graft substitutes: Influence of porosity and chemistry, Int. J. Appl. Ceram. Technol., Volume 2 (2005) no. 3, pp. 184-199
[23] Ceramic bioactivity and related biomimetic strategy, Curr. Opin. Solid State Mater. Sci., Volume 7 (2003), pp. 289-299
[24] Treatment strategies in thoracolumbar vertebral fractures: Are there indications for biomaterials?, European J. Trauma (2007), pp. 253-257
[25] A bone replaceable artificial bone substitute: Morphological and physiochemical characterizations, Yonsei Med. J., Volume 41 (2000) no. 4, pp. 468-476
[26] Process of hip joint prosthesis design including bone remodeling phenomenon, Comput. Struct., Volume 81 (2003) no. 8–11, pp. 887-893
[27] Modelling of bone–implant interaction, Comput. Methods Biomech. Biomed. Eng., Volume 2 (1998), pp. 289-298
[28] Load transfer between elastic hip implant and viscoelastic bone, Comput. Methods Biomech. Biomed. Eng., Volume 2 (1998), pp. 123-130
[29] Bioceramics composition modulate resorption of human osteoclasts, J. Mater. Sci., Mater. Med., Volume 16 (2005), pp. 1199-1205
[30] Modelling bioactivity and degradation of bioactive glass based tissue engineering scaffolds, Int. J. Solids Struct., Volume 48 (2010), pp. 257-268
[31] Micro–macro numerical modelling of bone regeneration in tissue engineering, Comput. Methods Appl. Mech. Eng., Volume 197 (2008), pp. 3092-3107
[32] A continuum model for remodeling in living structures, J. Mater. Sci., Volume 42 (2007) no. 21, pp. 8811-8823
[33] Bone poroelasticity, J. Biomech., Volume 32 (1999), pp. 217-238
[34] Bone remodeling I: Theory of adaptive elasticity, J. Elast., Volume 6 (1976), pp. 313-326
[35] A continuum treatment of growth in biological tissue: The coupling of mass transport and mechanics, J. Mech. Phys. Solids, Volume 52 (2004), pp. 1595-1625
[36] Framework for optimal design of porous scaffold microstructure by computational simulation of bone regeneration, Biomaterials, Volume 27 (2006) no. 21, pp. 3964-3972
[37] Microstructure design of biodegradable scaffold and its effect on tissue regeneration, Biomaterials, Volume 32 (2011) no. 22, pp. 5003-5014
[38] Size effects in the elasticity and viscoelasticity of bone, Biomech. Model. Mechanobiol., Volume 1 (2003), pp. 295-301
[39] Limitations of the continuum assumption in cancellous bone, J. Biomech., Volume 21 (1988), pp. 269-275
[40] Dynamical study of couple stress effects in human compact bone, J. Biomech. Eng., Volume 104 (1981), pp. 6-11
[41] Cosserat micromechanics of human bone: Strain redistribution by a hydration sensitive constituent, J. Biomech., Volume 19 (1986) no. 53, pp. 85-97
[42] Experimental study of micropolar and couple stress elasticity in compact bone in bending, J. Biomech., Volume 15 (1982) no. 2, pp. 91-98
[43] The material bone: Structure-mechanical function relations, Annu. Rev. Mater. Sci., Volume 28 (1998), pp. 271-298
[44] Modeling of trabecular bone as a hierarchical material, Comput. Fluid Solid Mech., Volume 1 (2003) no. 2, pp. 1727-1728
[45] The TEM characterization of the lamellar structure of osteoporotic human trabecular bone, Micron, Volume 36 (2005), pp. 653-664
[46] Transient study of couple stress in compact bone torsion, J. Biomech. Eng., Volume 103 (1981), pp. 275-279
[47] Modeling of trabecular bone as a couple stress continuum, Adv. Bioeng. ASME (2003), pp. 41-42
[48] Couple-stress moduli of a trabecular bone idealized as a 3D periodic cellular network, J. Biomech., Volume 39 (2006), pp. 2241-2252
[49] Generalized continuum theories: Application to stress analysis in bone, Meccanica, Volume 37 (2002) no. 4–5, pp. 385-396
[50] Continua with microstructure: Second gradient theory. Theory, examples and computational issues, Eur. J. Environ. Civ. Eng., Volume 14 (2010) no. 8–9, pp. 1031-1050
[51] Strain gradient interpretation of size effects, Eur. J. Mech. A, Solids, Volume 95 (1999) no. 3, pp. 299-314
[52] Microstructure in linear elasticity and scale effects: A reconsideration of basic rock mechanics and rock fracture mechanics, Tectonophysics, Volume 335 (2001), pp. 81-109
[53] La méthode des puissances virtuelles en mécanique des milieux continus, premiere partie, théorie du second gradient, J. Méc., Volume 12 (1973) no. 2, pp. 234-274
[54] The relationship between edge contact forces, double force and interstitial working allowed by the principle of virtual power, C. R. Acad. Sci. Paris, Ser. IIb, Volume 321 (1995), pp. 303-308
[55] Radius and surface tension of microscopic bubbles by second gradient theory, C. R. Acad. Sci. Paris, Ser. IIB, Volume 320 (1995), pp. 211-216
[56] Nucleation of spherical shell-like interfaces by second gradient theory: Numerical simulations, Eur. J. Mech. B, Fluids, Volume 15 (1996) no. 4, pp. 545-568
[57] Edge contact forces and quasi-balanced power, Meccanica, Volume 32 (1997), pp. 33-52
[58] Generalized Hookeʼs law for isotropic second gradient materials, Proc. R. Soc. A, Volume 465 (2009) no. 2107, pp. 2177-2196
[59] Elastic materials with couple-stresses, Arch. Ration. Mech. Anal., Volume 11 (1962), pp. 385-414
[60] An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures, Acta Mech., Volume 160 (2003), pp. 45-60
[61] A variational deduction of second gradient poroelasticity I: General theory, J. Mech. Mater. Struct., Volume 3 (2008) no. 3, pp. 507-526
[62] Perturbation methods for bifurcation analysis from multiple nonresonant complex eigenvalues, Nonlinear Dyn., Volume 14 (1997), pp. 193-210
[63] Multiple scale analysis for divergence-Hopf bifurcation of imperfect symmetric systems, J. Sound Vib., Volume 218 (1998) no. 3, pp. 527-539
[64] Multiple time scales analysis for 1:2 and 1:3 resonant Hopf bifurcations, Nonlinear Dyn., Volume 34 (2003), pp. 269-291
[65] Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams, Int. J. Non-Linear Mech., Volume 42 (2007), pp. 88-98
[66] A revival of electric analogs for vibrating mechanical systems aimed to their efficient control by PZT actuators, Int. J. Solids Struct., Volume 39 (2002), pp. 5295-5324
[67] Piezoelectric passive distributed controllers for beam flexural vibrations, J. Vib. Control, Volume 10 (2004), p. 625
[68] Damping of bending waves in truss beams by electrical transmission lines with PZT actuators, Arch. Appl. Mech., Volume 68 (1998), pp. 626-636
[69] Multiscale modeling of elastic properties of cortical bone, Acta Mech., Volume 213 (2010) no. 1–2, pp. 131-154
[70] Characterization of anisotropy in porous media by means of linear intercept measurements, Int. J. Solids Struct., Volume 40 (2003) no. 5, pp. 1243-1264
[71] Youngʼs modulus of trabecular and cortical bone material: Ultrasonic and microtensile measurements, J. Biomech., Volume 26 (1993) no. 2, pp. 111-119
[72] On the derivation of thermomechanical balance equations for continuous systems with a nonmaterial interface, Int. J. Eng. Sci., Volume 25 (1987), pp. 1459-1468
[73] A micro-structured continuum modelling compacting fluid-saturated grounds: The effects of pore-size scale parameter, Acta Mech., Volume 127 (1998), pp. 165-182
[74] A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghiʼs effective stress principle, Arch. Appl. Mech., Volume 70 (2000), pp. 323-337
[75] An extension of Kelvin and Bredt formulas, Math. Mech. Solids, Volume 1 (1996), pp. 243-250
[76] Generalized continua and non-homogeneous boundary conditions in homogenization methods, ZAMM, Volume 91 (2011), pp. 90-109
[77] Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua, Int. J. Solids Struct., Volume 47 (2010), pp. 3367-3376
[78] Milieux continus généralisés et matériaux hétérogènes, Les presses de lʼEcole des Mines, Paris, Avril 2006 (ISBN: 2-911762-67-3)
[79] A phenomenological approach to phase transition in classical field theory, Int. J. Eng. Sci., Volume 25 (1987), pp. 1469-1475
[80] Scale and boundary conditions effects on the apparent elastic moduli of trabecular bone modeled as a periodic cellular solid, J. Biomech. Eng., Trans. ASME, Volume 131 (2009) no. 12, p. 121008
[81] Second gradient of strain and surface tension in linear elasticity, Int. J. Solids Struct., Volume 1 (1965), pp. 417-438
Cited by Sources:
Comments - Policy