Engineering Transactions, 46, 1, pp. 3–26, 1998

Fabric Tensors in Bone Mechanics

S. Jemioło
Warsaw University of Technology

J.J. Telega
Polish Academy of Sciences

Mechanical properties of cancellous and cortical bone have been investigated. The fabric tensors used in the relevant literature have been discussed. Nonlinear elastic and elastic-perfectly plastic constitutive relationships have been proposed within the framework of small deformations. To this end the theory of representation of tensor functions has been used. It has been shown that the fabric tensor plays the role of a parametric tensor. Orthotropic linear elasticity has been carefully examined from the point of view of interrelations of classical material constants with the proposed material parameters and eigenvalues of the fabric tensor. Hoffman's strength criterion has been extended by incorporating the fabric tensor. Anisotropic properties of human cancellous and cortical bones have been investigated by using the relations derived.
Full Text: PDF


A. BLINOWSKI and J. OSTROWSKA-MACIEJEWSKA, On the elastic orthotropy, Arch. Mech., 48, pp. 129–141, 1996.

J.P. BOEHLER [Ed.], Applications of tensor functions in solid mechanics, CISM Courses and Lectures, No. 292, Springer–Verlag, Wien–New York 1987.

H. CEZAYIRLIOGLU, E. BAHNIUK, D.T DAVY and K.G. HEIPLE, Anisotropic yield behavior of bone under combined axial force and torque, J. Biomechanics, 18, pp. 61–69, 1985.

S.C. COWIN, On the strength anisotropy of bone and wood, ASME Appl. Mech. Div., 46, pp. 832–838, 1979.

S.C. COWIN, The relationship between the elasticity tensor and the fabric tensor, Mech. Materials, 4, pp. 137–147, 1985.

S.C. COWIN, Fabric dependence of an anisotropic strength criterion, Mech. Materials, 5, pp. 251–260, 1986.

S.C. COWIN, Wolff's law of trabecular architecture at remodelling equilibrium, J. Biomech. Engng., 108, pp. 83–88, 1986.

S.C. COWIN [Ed.], Bone mechanics, CRC Press, Inc. Boca Raton, Florida 1989.

S.C. COWIN, A note on the micro structural dependence of the anisotropic elastic con stants of textured materials, [in:] Advances in Micromechanics of Granular Materials, H.H. SHEN et al. [Eds], Elsevier Science Publishers B.V., pp. 61–70, 1992.

S.C. COWIN and M.M. MEHRABADI, On the identification of material symmetry for anisotropic elastic materials, Q. J. Mech. Appl. Math., 40, pp. 451–476, 1987.

J. CURREY, The mechanical adaptations of bone, Princeton University Press, 1984.

L.J. GIBSON and M.F. ASHBY, Cellular solids: structure and properties, Pergamon Press, Oxford–Toronto 1988.

R.W. GOULET, S.A. GOLDSTEIN, M.J. CIARELLI, J.K. KUHN, M.B. BROWN and L.A. FELDKAM, The relationship between the structural and orthogonal compressive properties of trabecular bone, J. Biomechanics, 27, pp. 375–389, 1994.

Z.-H. GUO, Representations of orthogonal tensors, SM Arch., 6, pp. 451–466, 1981.

T. HARRIGAN and R.W. MANN, Characterisation of micro structural anisotropy in orthotropic materials using a second rank tensor, J. Mat. Sci., 19, pp. 761–767, 1984.

M. HAYES, Connexions between the moduli for anisotropic elastic materials, J. Elasticity, 2, pp. 135–141, 1972.

R. HILL, A theory of the yielding and plastic flow of anisotropic metals, Proc. Roy. Soc. London, A 193, pp. 281–297, 1948.

O. HOFFMAN, The brittle strength of orthotropic materials, J. Composite Materials, 1, 200–206, 1967.

S. JEMIOŁO, Yield conditions and failure criteria for orthotropic and transversely isotropic materials. A review and invariant formulation of constitutive relationships [in Polish], Zesz. Nauk. PW, Budown., 131, pp. 5–52, 1996.

S. JEMIOŁO and K. KOWALCZYK, Invariant formulation and canonical form of the Hoffman anisotropic failure criterion [in Polish], V Polish–Ukrainian Sem., held in Dnepropetrovsk 30.06–6.07.1997, Proc. Theoretical Foundations of Civil Engineering, W. SZCZEŚNIAK [Ed.], pp. 291–300, Oficyna Wydawnicza PW, Warszawa 1997.

S. JEMIOŁO and J.J. TELEGA, Non-polynomial representations of orthotropic tensor functions in the three-dimensional case: an alternative approach, Arch. Mech., 49, pp. 233–239, 1997.

S. JEMIOŁO and J.J. TELEGA, Representations of tensor functions and applications in continuum mechanics, IFTR Reports, Warsaw, 3/1997.

S. JEMIOŁO and J.J TELEGA, Fabric tensor and constitutive equations for a class of plastic and locking orthotropic materials, Arch. Mech., 49, 1041–1067, 1997.

K. KANATANI, Distribution of directional data and fabric tensors, Int. J. Engng. Sci., 22, pp. 149–164, 1984.

T.M. KEAVENY and W.C. HAYES, A 20-year perspective on the mechanical properties of trabecular bone, ASME J. Biomechanical Engng., 115, pp. 534–542, 1993.

M. ODA, Initial fabrics and their relations to mechanical properties of granular material, Soils Found., 12, pp. 17–36, 1972.

R.L. RAKOTOMANANA, A. CORNIER and P.P. LEYVRAZ, An objective elastic tic model and algorithm applicable to bone mechanics, Eur. J. Mech., A. Solids, 10, pp. 327–342, 1991.

J. RYCHLEWSKI, "CEIIINOSSSTTUV" Mathematical structure of elastic bodies [in Russian], Report of the Institute for Problems in Mechanics of the Academy of Sciences of the USSR, No. 217, Moscow 1983.

J. RYCHLEWSKI, Elastic energy decompositions and limit criteria [in Russian], Adv. Mech., 7, pp. 51–80, 1984.

J. RYCHLEWSKI, Unconventional approach to linear elasticity, Arch. Mech., 47, pp. 149–171, 1995.

J. RYCHLEWSKI and J.M. ZHANG, Anisotropy degree of elastic materials, Arch. Mech., 41, 697–715, 1989.

A.M. SADEGH, S.C. COWIN and G.M. LUO, Inversion related to the stress–strain–fabric relationship, Mech. Mater., 11, pp. 323–336, 1991.

J.J. TELEGA, Some aspects of invariant theory in plasticity. Part I. New results relative to representation of isotropic and anisotropic tensor functions, Arch. Mech., 36, pp. 147–162, 1984.

P.S. THEOCARIS, Failure characterization of anisotropic materials by means of the elliptic paraboloid failure criterion, Adv. Mech., 10, pp.83–102, 1987.

P.S. THEOCARIS, Weighing failure tensor polynomial criteria for composites, Int. J. Damage Mechanics, 1, pp. 4–45, 1992.

S.W. TSAI and E.M. WU, A general theory of strength for anisotropic materials, J. Composite Mater., 5, pp. 58–80, 1971.

C.H. TURNER, Yield behavior of bovine cancellous bone, J. Biomechanical Engng., 111, pp. 1–5, 1989.

C.H. TURNER and S.C. COWIN, Errors induced by off-axis measurement of the elastic properties of bone, J. Biomechanical Engng., 110, pp. 213–215, 1988.

C.H. TURNER, S.C. COWIN, J.Y. RHO, R.B. ASHMAN and J.C. RICE, The fabric dependence of the orthotropic elastic constants of cancellous bone, J. Biomechanics, 23, pp. 549–561, 1990.

W.J. WHITEHOUSE, The quantitative morphology of anisotropic trabecular bone, J. Microscopy, 101, pp. 153–168, 1974.

Q.-S. ZHENG, Theory of representations for tensor functions – A unified invariant approach to constitutive equations, Appl. Mech. Rev., 47, pp. 545–587, 1994.

P.K. ZYSSET and A. CURNIER, An alternative model for anisotropic elasticity based on fabric tensors, Mech. Mat., 21, pp. 243–250, 1995.

Copyright © 2014 by Institute of Fundamental Technological Research
Polish Academy of Sciences, Warsaw, Poland