Abstract
Materials with inherent microstructures like granular media, foams or spongy bones often show a complex constitutive behaviour on the macroscale while the microscopic constitutive equations may be formulated in a simple fashion. Applying homogenization procedures allows to transfer the information from the microlevel to the macrolevel.
In the present contribution the porous structure of hard biological tissues, i.e. of spongy bones, is investigated. On the macroscale the approach is embedded into an extended continuum mechanical setting in order to capture size effects. The constitutive equations are formulated on the microscopic level taking into account growth and reorientation of the microstructural elements. By application of a strain-driven numerical homogenization procedure the macroscopic stress response is obtained.
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References
Aşkar A (1986) Lattice dynamical foundations of continuum theories. World Scientific Publication, Singapore
Aşkar A, Çakmak A (1968) A structural model of a micropolar continuum. Int J Eng Sci 6:583–589
Adachi T, Tomita Y, Tanaka M (1999) Three-dimensional lattice continuum model of cancellous bone for structural and remodeling simulation. JSME Int J 42(3):470–480
Adachi T, Tsubota K, Tomita Y, Hollister SJ (2001) Trabecular surface remodeling simulation for cancellous bone using microstructural voxel finite element models. ASME J Biomech Eng 123: 403–409
Bardet J, Vardoulakis I (2001) The asymetry of stress in granular media. Int J Solids Struct 38:353–367
Carter D, Beaupre G (2001) Skeletal Function and Form: mechanobiology of skeletal development, aging and regeneration. Cambridge University Press, Cambridge
Cosserat E, Cosserat F (1909) Théorie des corps déformables. A. Hermann et Fils, Paris
Cowin S, Hegedus D (1976) Bone remodeling I: Theory of adaptive elasticity. J Elasticity 6:313–326
Diebels S, Ehlers W (2001). Homogenization method for granular assemblies. In: Wall W, Bletzinger K-U, Schweizerhof K (eds). Proceedings of trends in computational structural Mechanics. CIMNE, Barcelona, Spain, pp. 79–88
Diebels S, Steeb H (2002) The size effect in foams and its theoretical and numerical investigation. In: Proceedings of the Royal Society London A, vol 458, pp 2869–2883
Diebels S, Steeb H (2003) Stress and couple stress in foams. Comp Math Sci 28:714–722
Ebinger T, Steeb H, Diebels S (2004) Modeling macroscopic extended continua with the aid of numerical homogenization schemes. Comp Math Sci 32:337–347
Ebinger T, Steeb H, Diebels S (2005) Modeling and homogenization of foams. Comp Assisted Mech Eng Sci 12:49–63
Ehlers W, Ramm E, Diebels S, D’Addetta GA (2003) From particle ensembles to Cosserat continua: Homogenization of contact forces towards stresses and couple stresses. Int J Solids Struct 40: 6681–6702
Eringen C (1999) Microcontinuum Field Theories, vol I, Foundations and solids. Springer, Berlin Heidelberg New York
Feyel F, Chaboche J (2000) FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fiber SiC/Ti composite materials. Comp Meth Appl Mech Eng 183:309–330
Forest S (1998) Mechanics of generalized continua: construction by homogenization. J Phys IV:39–48
Forest S, Sab K (1998) Cosserat overall modeling of heterogeneous materials. Mech Res Commun 25:449–454
Geers M, Kouznetsova V, Brekelmans W (2003) Multi-scale first-order and second-order computational homogenization of microstructures towards continua. Int J Multiscale Comput Eng (in press)
Gibson L, Ashby M (1997) Cellular solids. Structure and properties. Cambridge solid state science series. Cambridge University Press, Cambridge
Goldstein SA (1987) The mechanical properties of trabecular bone: dependence on anatomical location and function. J Biomech 20:1055–1061
Goulet RW, Goldstein SA, Ciarelli MJ, Kuhn JL, Brown MB, Feldkamp LA (1994) The relationship between the structural and mechanical properties of trabecular bone. J Biomech 27:375–389
Günther W (1958) Zur Statik und Kinematik des Cosseratschen Kontinuums. Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft 10:195–213
Hashin Z (1983) Analysis of composite materials – a survey. J Appl Mech 50:481–505
Hohe J, Becker W (2001) An energetic homogenisation procedure for the elastic properties of general cellular sandwich cores. Composites: Part B 32:185–197
Hollister SJ, Fyrhie DP, Jepsen KJ, Goldstein SA (1991) Application of homogenization theory to the study of trabecular bone mechanics. J Biomech 24:825–839
Huet C (1997) An integrated micromechanics and statistical continuum thermodynamics approach for studying the fracture behaviour of microcracked heterogeneous materials with delayed response. Eng Fracture Mech 58:459–556
Huiskes R, Ruimerman R, van Lenthe G, Janssen J (2000) Effects of mechanical forces on maintenance and adaption of form in trabecular bone. Nature 405:704–706
Huiskes R, Weinans H, Dalstra M (1989) Adaptive bone remodeling and biomechanical design considerations for noncemented total hip arthroplasty. Orthopedics 12:1255–1267
Kouznetsova V, Brekelmans W, Baaijens F (2001) An approach to micro-macro modeling of heterogeneous materials. Comp Mech 37–48
Kuhl E (2004) Theory and numerics of open system continuum thermodynamics – spatial and material settings. Habilitation-thesis, Chair of Applied Mechanics, Technical University of Kaiserslautern
Kuhl E, Steinmann P (2003) Theory and numerics of geometrically non-linear open system mechanics. Int J Numer Meth Eng 58:1593–1615
Lakes R (1995). Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Mühlhaus H (eds). Continuum methods for materials with microstructures. Wiley, Chichester, pp. 1–25
van Lenthe GH, Willems MMM, Verdonschot N, de Waal Malefijt MC, Huiskes R (2002) Stemmed femoral knee prostheses. Acta Orthop Scand 73:630–637
Mullender M, Huiskes R (1995) Proposal for the regulatory mechanism of Wolff’s law. J Orthop Res 13:503–512
Nackenhorst U (1997) Numerical simulation of stress stimulated bone remodeling. Technische Mechanik 17:31–40
Nauenberg T, Bouxsein M, Mikić B, Carter D (1993) Using clinical data to improve computational bone remodeling theory. Trans Orthop Res Soc 18:123
Nemat-Nasser S, Hori M (1993) Micromechanics. North-Holland, Amsterdam
Nowacki W (1986) Thermoelasticity. Pergamon Press, Oxford
Onck P, Andrews E, Gibson L (2001) Size effects in ductile cellular solids. Part I: modeling. Int J Mech Sci 43:681–699
Papka S, Kyriakides S (1998) Experiments and full-scale numerical simulations of in-plane crushing of a honeycomb. Acta Mater 46:2765–2776
Pettermann HE, Reiter TJ, Rammerstorfer FG (1997) Computational simulation of internal bone remodeling. Arch Comput Methods Eng 4:295–323
Pistoia W, van Rietbergen B, Laib A, Rüegsegger P (2001) High-resolution three-dimensional-pqct images can be an adequate basis for in vivo μfe analysis of bone. ASME J Biomech Eng 123:176–183
Roux W (1881) Der Kampf der Teile im Organismus. Engelmann, Leipzig
Ruimerman R, Hilbers P, van Rietbergen B, Huiskes R (2005) A theoretical framework for strain-related trabecular bone maintenance and adaptation. J Biomech 38:931–941
Sanchez-Palencia E (1980) Non-homogeneous meida and vibration theory. Springer, Berlin Heidelberg New York
Schaefer H (1967) Das Cosserat-Kontinuum. Z Angew Math Mech 47:485–498
Steeb H, Ebinger T, Diebels S (2005) Microscopically motivated model describing growth and remodeling of spongy bones. In: Ehlers W (ed) Proceedings of 1st GAMM symposium on continuum biomechanics, 24 - 26 November 2004, Freudenstadt- Lauterbad, Glückauf, Essen
Warren W, Byskov E (2002) Three-field symmetry restrictions on two-dimenional micropolar materials. Eur J Mech A/Solids 21:779–792
Weinans H, Huiskes R, Grootenboer HJ (1992) Effects of material properties of femoral hip components on bone remodeling. J Orthop Res 10:845–853
Wolff J, (1892) Das Gesetz der Transformation der Knochen. Hirschwald Verlag, Berlin
Zohdi T, Wriggers P (2005) Introduction to computational micromechanics. Lecture Notes in Applied and Computational Mechanics. Springer, Berlin Heidelberg New York
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Ebinger, T., Diebels, S. & Steeb, H. Numerical Homogenization Techniques Applied to Growth and Remodelling Phenomena. Comput Mech 39, 815–830 (2007). https://doi.org/10.1007/s00466-006-0071-8
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DOI: https://doi.org/10.1007/s00466-006-0071-8