Viscoelastic properties of wet cortical bone, III:​ a non-linear constitutive equation

Lakes, Roderic S.; Katz, J. Lawrence

Affiliations

¹Biomedical Engineering Program, College of Engineering, University of Iowa, Iowa City, IA 52242, U.S.A.

²Center for Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12181, U.S.A.

Abstract

A constitutive equation is proposed to characterize the viscoelastic properties of wet human compact bone measured in torsion at body temperature. The equation is developed, using a spectrum of relaxation times, H(τ)=logτ τ₁≤τ≤τ₂, 0 τ<τ₁,τ>τ₂ called the ‘triangle’ spectrum. The transient and dynamic viscoelastic moduli are derived from this. Applications of this distribution to other viscoelastic data as well as to dielectric systems, are discussed. Nonlinear effects observed in bone are described using the first two terms of a multiple-integral expansion. The dynamic properties of two types of nonlinear solid are obtained from constitutive equations based on transients, and applications to bone mechanics are discussed.

Links

DOI: 10.1016/0021-9290(79)90018-6

PubMed: 489636

WoS: A1979HM85200005

History

Received: 1977-08-22

Cited Works (8)

Year	Entry
1976	Yoon HS, Katz JL. Ultrasonic wave propagation in human cortical bone, II: measurements of elastic properties and microhardness. J Biomech. 1976;9(7):459-464.
1979	Lakes RS, Katz JL, Sternstein SS. Viscoelastic properties of wet cortical bone, I: torsional and biaxial studies. J Biomech. 1979;12(9):657-678.
1969	Lang SB. Elastic coefficients of animal bone. Science. July 18, 1969;165(3890):287-288.
1973	Radin EL, Parker HG, Pugh JW, Steinberg RS, Paul IL, Rose RM. Response of joints to impact loading, III: relationship between trabecular microfractures and cartilage degeneration. J Biomech. January 1973;6(1):51-57.
1972	Fung Y-CB. Stress-strain-history relations of soft tissues in simple elongation. In: Fung YC, Perrone N, Anliker M, eds. Biomechanics: Its Foundations and Objectives. Symposium on Biomechanics, its Foundations and Objectives; July 29-31, 1970. Englewood Cliff, NJ: Inc., Prentice-Hall International; 1972:181-208.
1975	Lakes RS. Viscoelastic and Dielectric Relaxation in Cortical Bone [PhD thesis]. Troy, NY: Rensselaer Polytechnic Institute; July 1975.
1971	Thompson GA. Experimental Studies of Lateral and Torsional Vibrations of Intact Dog Radii [PhD thesis]. Stanford University; 1971.
1968	Pipkin AC, Rogers TG. A non-linear integral representation for viscoelastic behaviour. J Mech Phys Solids. January 1968;16(1):59-72.

Cited By (36)

Year	Entry
1981	Katz JL. Composite material models for cortical bone. In: Cowin SC, ed. Mechanical Properties of Bone. The Joint ASME-ASCE Applied Mechnics, Fluids Engineering and Bioengineering Conference; June 22-24, 1981; Boulder, CO. New York, NY: American Society of Mechical Engineers; 1981:171-184.
2001	Lakes R. Viscoelastic properties of cortical bone. In: Cowin SC, ed. Bone Mechanics Handbook. 2nd ed. Boca Raton, FL: CRC Press; 2001:11-1–11-15.
2001	Lucchinetti E. Composite models of bone properties. In: Cowin SC, ed. Bone Mechanics Handbook. 2nd ed. Boca Raton, FL: CRC Press; 2001:12-1–12-19.
1990	Cowin SC. Properties of cortical bone and theory of bone remodeling. In: Mow VC, Ratcliffe A, Woo SL-Y, eds. Biomechanics of Diarthrodial Joints. Vol 2. New York, NY: Springer-Verlag; 1990:119-153.
1993	Fung YC. Biomechanics: Mechanical Properties of Living Tissues. 2nd ed. New York, NY: Springer-Verlag; 1993.
1998	Martin RB, Burr DB, Sharkey NA. Skeletal Tissue Mechanics. New York, NY: Springer-Verlag; 1998.
2001	Provenzano P, Lakes R, Keenan T, Vanderby R Jr. Nonlinear ligament viscoelasticity. Ann Biomed Eng. October 2001;29(10):908-914.
2002	Provenzano PP, Lakes RS, Corr DT, Vanderby R. Application of nonlinear viscoelastic models to describe ligament behavior. Biomech Model Mechanobiol. June 2002;1(1):45-57.
1979	Lakes RS, Katz JL, Sternstein SS. Viscoelastic properties of wet cortical bone, I: torsional and biaxial studies. J Biomech. 1979;12(9):657-678.
1987	Katz JL, Meunier A. The elastic anisotropy of bone. J Biomech. 1987;20(11-12):1063-1070.
1993	Sasaki N, Yoshikawa M. Stress relaxation in native and EDTA-treated bone as a function of mineral content. J Biomech. 1993;26(1):77-83.
1993	Sasaki N, Nakayama Y, Yoshikawa M, Enyo A. Stress relaxation function of bone and bone collagen. J Biomech. December 1993;26(12):1369-1376.
1994	Deligianni DD, Maris A, Missirlis YF. Stress relaxation behaviour of trabecular bone specimens. J Biomech. December 1994;27(12):1469-1476.
1996	Cristofolini L, Viceconti M, Cappello A, Toni A. Mechanical validation of whole bone composite femur models. J Biomech. April 1996;29(4):525-535.
2006	Guedes RM, Simões JA, Morais JL. Viscoelastic behaviour and failure of bovine cancellous bone under constant strain rate. J Biomech. 2006;39(1):49-60.
2010	Cristofolini L, Conti G, Juszczyk M, Cremonini S, Sint Jan SV, Viceconti M. Structural behaviour and strain distribution of the long bones of the human lower limbs. J Biomech. March 22, 2010;43(5):826-835.
2011	Juszczyk MM, Cristofolini L, Viceconti M. The human proximal femur behaves linearly elastic up to failure under physiological loading conditions. J Biomech. August 11, 2011;44(12):2259-2266.
1999	Lakes RS, Vanderby R. Interrelation of creep and relaxation: a modeling approach for ligaments. J Biomech Eng. December 1999;121(6):612-615.
2000	Garner E, Lakes R, Lee T, Swan C, Brand R. Viscoelastic dissipation in compact bone: implications for stress-induced fluid flow in bone. J Biomech Eng. April 2000;122(2):166-172.
2002	Haut Donahue TL, Hull ML, Rashid MM, Jacobs CR. A finite element model of the human knee joint for the study of tibio-femoral contact. J Biomech Eng. June 2002;124(3):273-280.
2011	Shepherd TN, Zhang J, Ovaert TC, Roeder RK, Niebur GL. Direct comparison of nanoindentation and macroscopic measurements of bone viscoelasticity. J Mech Behav Biomed Mater. November 2011;4(8):2055-2062.
2010	Cristofolini L, Schileo E, Juszczyk M, Taddei F, Martelli S, Viceconti M. Mechanical testing of bones: the positive synergy of finite–element models and in vitro experiments. Philos Trans R Soc A-Math Phys Eng Sci. June 13, 2010;368(1920):2725-2763.
2024	Szabo E. Insights Into the Mechanical Behavior of Maturing Cortical Bone: A Porcine Model [PhD thesis]. Cleveland, OH: Case Western Reserve University; May 2024.
2004	İnceoğlu S. Failure of Pedicle Screw-Bone Interface: Biomechanics of Pedicle Screw Insertion and Pullout [PhD thesis]. Cleveland State University; December 2004.
2011	Abdel-Wahab AA-GM. Experimental and Numerical Analysis of Deformation and Fracture of Cortical Bone Tissue [PhD thesis]. Loughborough University; August 2011.
2013	Li S. Cutting of Cortical Bone Tissue: Analysis of Deformation and Fracture Process [PhD thesis]. Loughborough University; June 2013.
1994	Yetkinler DN. Viscoelastic Behavior of Acrylic Bone Cements [PhD thesis]. The Ohio State University; 1994.
1983	Rouhana SW. Fluid Flow and Mechanical Damping in Bovine Cortical Bone [PhD thesis]. Troy, NY: Rensselaer Polytechnic Institute; February 1983.
1991	Pattin CAG. Cyclic Mechanical Property Degradation in Bone During Fatigue Loading [PhD thesis]. Stanford University; March 1991.
1987	Turner CH. On the Relationship Between the Elastic Properties of Cancellous Bone and Its Structure [PhD thesis]. New Orleans, LA: Tulane University; 1987.
2016	Gustafson HM. Quantifying the Response of Vertebral Bodies to Compressive Loading Using Digital Image Correlation [PhD thesis]. Vancouver, BC: University of British Columbia; October 2016.
2000	Haut Donahue TL. A Finite Element Model of the Human Knee Joint for the Study of Meniscal Replacements [PhD thesis]. Davis, University of California; 2000.
2001	Lee T. Viscoelastic Properties of Biological and High-Damping Composite Materials [PhD thesis]. University of Wisconsin – Madison; 2001.
2009	García-Rodríguez S. Mechanical Behavior of Trabecular Bone [PhD thesis]. University of Wisconsin – Madison; 2009.
2001	Brown CU. Time-Dependent Circumferential Deformation of Cortical Bone Subjected to Internal Radial Loading [PhD thesis]. West Virginia University; 2001.
2001	Parsamian GP. Damage Mechanics of Human Cortical Bone [PhD thesis]. West Virginia University; 2001.