An alternative model for anisotropic elasticity based on fabric tensors

Zysset, P. K.<sup>1</sup>; Curnier, A.<sup>1</sup>

Affiliations

¹Laboratory of Applied Mechanics, Swiss Federal Institute of Technology, CH-1015 Lausanne, Switzerland

Abstract and keywords

Motivated by the mechanical analysis of multiphase or damaged materials, a general approach relating fabric tensors characterizing microstructure to the fourth rank elasticity tensor is proposed. Using a Fourier expansion in spherical harmonics, the orientation distribution function of a positive, radially symmetric microstructural property is approximated by a scalar and a symmetric, traceless second rank tensor. Following this approximation, a general expression of the elastic free energy potential is derived from representation theorems for anisotropic scalar functions. Based on a homogeneity assumption for the elastic constitutive law with respect to the microstructural property, a particular elasticity model is developed that involves three independent constants beside the fabric tensors. Strict positive definiteness of the corresponding elasticity tensor is ensured under explicit conditions on the independent constants for arbitrary fabric tensors.

Keywords: Anisotropy; Continuum mechanics; Elasticity; Fabric; Microstructure

Links

DOI: 10.1016/0167-6636(95)00018-6

WoS: A1995TE05700001

History

Received: 1995-03-06

Revised: 1995-07-22

Cited Works (3)

Year	Entry
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Cited By (53)

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2009	Shi X, Wang X, Niebur GL. Effects of loading orientation on the morphology of the predicted yielded regions in trabecular bone. Ann Biomed Eng. 2009;37(2):354-362.
2001	Keaveny TM, Morgan EF, Niebur GL, Yeh OC. Biomechanics of trabecular bone. Annu Rev Biomed Eng. 2001;3:307-333.
2009	Garcia D, Zysset PK, Charlebois M, Curnier A. A three-dimensional elastic plastic damage constitutive law for bone tissue. Biomech Model Mechanobiol. April 2009;8(2):149-165.
2010	Charlebois M, Jirásek M, Zysset PK. A nonlocal constitutive model for trabecular bone softening in compression. Biomech Model Mechanobiol. 2010;9(5):597-611.
2011	Varga P, Dall’Ara E, Pahr DH, Pretterklieber M, Zysset PK. Validation of an HR-pQCT-based homogenized finite element approach using mechanical testing of ultra-distal radius sections. Biomech Model Mechanobiol. July 2011;10(4):431-444.
2022	Simon M, Indermaur M, Schenk D, Hosseinitabatabaei S, Willie BM, Zysset P. Fabric-elasticity relationships of tibial trabecular bone are similar in osteogenesis imperfecta and healthy individuals. Bone. February 2022;155:116282.
2008	Chevalier Y, Charlebois M, Pahr D, Varga P, Heini P, Schneider E, Zysset P. A patient-specific finite element methodology to predict damage accumulation in vertebral bodies under axial compression, sagittal flexion and combined loads. Comput Methods Biomech Biomed Eng. October 2008;11(5):477-487.
2009	Lukes J, Otahal S. Nanoindentation of porcine bone lamellae. Comput Methods Biomech Biomed Eng. 2009;12(suppl 1):177-178.
2009	Pahr DH, Zysset PK. From high-resolution CT data to finite element models: development of an integrated modular framework. Comput Methods Biomech Biomed Eng. 2009;12(1):45-57.
2019	Kalajahi SMH, Nazemi SM, Johnston JD. Separate modeling of cortical and trabecular bone offers little improvement in FE predictions of local structural stiffness at the proximal tibia. Comput Methods Biomech Biomed Eng. 2019;22(16):1258-1268.
1996	Zysset PK, Curnier A. A 3D damage model for trabecular bone based on fabric tensors. J Biomech. December 1996;29(12):1549-1558.
2003	Zysset PK. A review of morphology–elasticity relationships in human trabecular bone: theories and experiments. J Biomech. October 2003;36(10):1469-1485.
2003	Hengsberger S, Enstroem J, Peyrin F, Zysset P. How is the indentation modulus of bone tissue related to its macroscopic elastic response? a validation study. J Biomech. October 2003;36(10):1503-1509.
2009	Varga P, Baumbach S, Pahr D, Zysset PK. Validation of an anatomy specific finite element model of Colles’ fracture. J Biomech. August 7, 2009;42(11):1726-1731.
2011	Garrison JG, Gargac JA, Niebur GL. Shear strength and toughness of trabecular bone are more sensitive to density than damage. J Biomech. November 10, 2011;44(16):2747-2754.
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2013	Hazrati Marangalou J, Ito K, Cataldi M, Taddei F, van Rietbergen B. A novel approach to estimate trabecular bone anisotropy using a database approach. J Biomech. September 27, 2013;46(14):2356-2362.
2013	Kersh ME, Zysset PK, Pahr DH, Wolfram U, Larsson D, Pandy MG. Measurement of structural anisotropy in femoral trabecular bone using clinical-resolution CT images. J Biomech. October 18, 2013;46(15):2659-2666.
1998	Zysset PK, Goulet RW, Hollister SJ. A global relationship between trabecular bone morphology and homogenized elastic properties. J Biomech Eng. October 1998;120(5):640-646.
2009	Franzoso G, Zysset PK. Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation. J Biomech Eng. February 2009;131(2):021001.
2009	Chevalier Y, Pahr D, Zysset PK. The role of cortical shell and trabecular fabric in finite element analysis of the human vertebral body. J Biomech Eng. November 2009;131(11):111003.
2010	Carnelli D, Gastaldi D, Sassi V, Contro R, Ortiz C, Vena P. A finite element model for direction-dependent mechanical response to nanoindentation of cortical bone allowing for anisotropic post-yield behavior of the tissue. J Biomech Eng. August 2010;132(8):081008.
2015	Oftadeh R, Perez-Viloria M, Villa-Camacho JC, Vaziri A, Nazarian A. Biomechanics and mechanobiology of trabecular bone: a review. J Biomech Eng. January 2015;137(1):010802.
2020	Remache D, Semaan M, Rossi JM, Pithioux M, Milan J. Application of the Johnson-Cook plasticity model in the finite element simulations of the nanoindentation of the cortical bone. J Mech Behav Biomed Mater. January 2020;101:103426.
1998	Smit TH, Schneider E, Odgaard A. Star length distribution: a volume-based concept for the characterization of structural anisotropy. J Microsc (Oxford). September 1998;191(3):249-257.
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2004	Wang X. Measurement and Analysis of Microdamage in Bone [PhD thesis]. University of Notre Dame; December 2004.
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2003	Fox JC. Biomechanics of the Proximal Femur: Role of Bone Distribution and Architecture [PhD thesis]. Berkeley, CA: Berkeley, University of California; 2003.
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