Constitutive equations for fibrous connective tissues

Lanir, Y.

Affiliations

¹Department of Biomechanics, College of Osteopathic Medicine, Michigan State University, East Lansing, MI 48824, U.S.A.

Abstract

A general multiaxial theory for the constitutive relations in fibrous connective tissues is developed on the basis of microstructural and thermodynamic considerations. It is compatible with existing general material theories. In elastic tissues, the theory considers the strain-energy function to be the sum of strain-energies of the tissue's components. The stresses are derived from this strain-energy function. Viscoelastic constitutive relations are obtained in an analogous manner. Few examples are developed in detail.

The results of the present strain-energy based theory are identical with those of the author's previous structural models (Lanir, 1979a, b) which are based on detailed equilibrium analysis. It turns out, however, that the analytical work involved in solving boundary value problems is considerably shorter if the present theory is used.

The advantages of structural theories in avoiding ambiguity in material characterization and in offering an insight into the function, structure and mechanics of tissue components are discussed.

Links

DOI: 10.1016/0021-9290(83)90041-6

PubMed: 6833305

WoS: A1983QE15900001

History

Received: 1982-04-02

Cited Works (24)

Year	Entry
1968	Jamison CE, MarangonI RD, Glaser AA. Viscoelastic properties of soft tissue by discrete model characterization. J Biomech. January 1968;1(1):33-46.
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.
1979	Bingham DN, DeHoff PH. A constitutive equation for the canine anterior cruciate ligament. J Biomech Eng. February 1979;101(1):15-22.
1964	Hashin Z, Rosen BW. The elastic moduli of fiber-reinforced materials. J Appl Mech. June 1964;31(2):223-232.
1980	Woo SL-Y, Simon BR, Kuei SC, Akeson WH. Quasi-linear viscoelastic properties of normal articular cartilage. J Biomech Eng. May 1980;102(2):85-90.
1980	Kastelic J, Palley I, Baer E. A structural mechanical model for tendon crimping. J Biomech. 1980;13(10):887-893.
1969	Haut RC, Little RW. Rheological properties of canine anterior cruciate ligaments. J Biomech. 1969;2(3):289-298.
1979	Lanir Y. A structural theory for the homogeneous biaxial stress-strain relationships in flat collagenous tissues. J Biomech. 1979;12(6):423-436.
1967	Abrahams M. Mechanical behaviour of tendon in vitro: a preliminary report. Med Biol Eng. September 1967;5(5):433-443.
1847	Wertheim MG. Mémoire sur l'élasticité et la cohésion des principaux tissus du corps humain. Ann Chim Phys. December 1847;21:385-414.
1980	Lanir Y. A microstructure model for the rheology of mammalian tendon. J Biomech Eng. November 1980;102(4):332-339.
1980	Decraemer WF, Maes MA, Vanhuyse VJ. An elastic stress-strain relation for soft biological tissues based on a structural model. J Biomech. 1980;13(6):463-468.
1972	Haut RC, Little RW. A constitutive equation for collagen fibers. J Biomech. September 1972;5(5):423-430.
1976	Wu H-C, Yao R-F. Mechanical behavior of the human annulus fibrosus. J Biomech. 1976;9(1):1-7.
1974	Jenkins RB, Little RW. A constitutive equation for parallel-fibered elastic tissue. J Biomech. September 1974;7(5):397-402.
1972	Demiray H. A note on the elasticity of soft biological tissues. J Biomech. May 1972;5(3):309-311.
1972	Diamant J, Keller A, Baer E, Litt M, Arridge RG. Collagen: ultrastructure and its relation to mechanical properties as a function of ageing. Proc R Soc B. March 14, 1972;180(1060):293-315.
1980	Decraemer WF, Maes MA, Vanhuyse VJ, Vanpeperstrafte P. A non-linear viscoelastic constitutive equation for soft biological tissues, based upon a structural model. J Biomech. 1980;13(7):559-564.
1976	Comninou M, Yannas IV. Dependence of stress-strain nonlinearity of connective tissues on the geometry of collagen fibres. J Biomech. 1976;9(7):427-433.
1973	Barbenel JC, Evans JH, Finlay JB. Stress-strain-time relations for soft connective tissues. In: Kenedi RM, ed. Perspectives in Biomedical Engineering: Proceedings of a Symposium Organised in Association With the Biological Engineering Society and Held in the University of Strathclyde, Glasgow, June 1972. Baltimore, MD: University Park Press; 1973:165-172.
1965	Ridge MD, Wright V. A rheological study of skin. In: Kenedi RM, ed. Biomechanics and Related Bio-Engineering Topics: Proceedings of a Symposium Held in Glasgow, September 1964. Oxford, UK: Pergamon Press; 1965:165-175.
1976	Cohen RE, Hooley CJ, McCrum NG. Viscoelastic creep of collagenous tissue. J Biomech. 1976;9(4):175-184.
1965	Kenedi RM, Gibson T, Daly CH. Bio-engineering studies of the human skin II. In: Kenedi RM, ed. Biomechanics and Related Bio-Engineering Topics: Proceedings of a Symposium Held in Glasgow, September 1964. Oxford, UK: Pergamon Press; 1965:147-158.
1967	Fung YCB. Elasticity of soft tissues in simple elongation. Am J Physiol. December 1967;213(6):1532-1544.

Cited By (64)

Year	Entry
2007	Cowin SC, Doty SB. Kinematics and mechanics of large deformations. In: Tissue Mechanics. Boca Raton, FL: Springer; 2007:507-557.
1993	Fung YC. Biomechanics: Mechanical Properties of Living Tissues. 2nd ed. New York, NY: Springer-Verlag; 1993.
2008	Quinn KP, Winkelstein BA. Regional changes in collagen fiber alignment may identify the onset of damage in the facet capsular ligament. In: Proceedings of the 36th International Workshop on Human Subjects for Biomechanical Research. November 2, 2008; San Antonio, TX.
2003	Elliott DM, Robinson PS, Gimbel JA, Sarver JJ, Abboud JA, Iozzo RV, Soslowsky LJ. Effect of altered matrix proteins on quasilinear viscoelastic properties in transgenic mouse tail tendons. Ann Biomed Eng. May 2003;31(5):599-605.
2000	Woo SL-Y, Debski RE, Zeminski J, Abramowitch SD, Chan Saw SS, Fenwick JA. Injury and repair of ligaments and tendons. Annu Rev Biomed Eng. 2000;2:83-118.
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.
2012	Wright RM, Ramesh KT. An axonal strain injury criterion for traumatic brain injury. Biomech Model Mechanobiol. January 2012;11(2):245-260.
1996	Weiss JA, Maker BN, Govindjee S. Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput Meth Appl Mech Eng. August 15, 1996;135(1-2):107-128.
2001	Weiss JA, Gardiner JC. Computational modeling of ligament mechanics. Crit Rev Biomed Eng. 2001;29(3):303-371.
1984	Mow VC, Holmes MH, Lai WM. Fluid transport and mechanical properties of articular cartilage: a review. J Biomech. 1984;17(5):377-394.
1984	Butler DL, Grood ES, Noyes FR, Zernicke RF, Brackett K. Effects of structure and strain measurement technique on the material properties of young human tendons and fascia. J Biomech. 1984;17(8):579-596.
1987	Oomens CWJ, van Campen DH, Grootenboer HJ. A mixture approach to the mechanics of skin. J Biomech. 1987;20(9):877-885.
1999	Liao H, Belkof SM. A failure model for ligaments. J Biomech. 1999;32(2):183-188.
2006	Thomopoulos S, Marquez JP, Weinberger B, Birman V, Genin GM. Collagen fiber orientation at the tendon to bone insertion and its influence on stress concentrations. J Biomech. 2006;39(10):1842-1851.
2023	Turčanová M, Fischer J, Hermanová M, Bednařík Z, Skácel P, Burša J. Biaxial stretch can overcome discrepancy between global and local orientations of wavy collagen fibres. J Biomech. December 2023;161:111868.
1992	Ault HK, Hoffman AH. A composite micromechanical model for connective tissues, I: theory. J Biomech Eng. February 1992;114(1):137-141.
1993	Woo SL-Y, Johnson GA, Smith BA. Mathematical modeling of ligaments and tendons. J Biomech Eng. November 1993;115(4B):468-473.
1996	Johnson GA, Livesay GA, Woo SL-Y, Rajagopal KR. A single integral finite strain viscoelastic model of ligaments and tendons. J Biomech Eng. May 1996;118(2):221-226.
1997	Hurschler C, Loitz-Ramage B, Vanderby R Jr. A structurally based stress-stretch relationship for tendon and ligament. J Biomech Eng. November 1997;119(4):392-399.
1998	Wren TAL, Carter DR. A microstructural model for the tensile constitutive and failure behavior of soft skeletal connective tissues. J Biomech Eng. February 1998;120(1):55-61.
1998	Puso MA, Weiss JA. Finite element implementation of anisotropic quasi-linear viscoelasticity using a discrete spectrum approximation. J Biomech Eng. February 1998;120(1):62-70.
1998	Quapp KM, Weiss JA. Material characterization of human medial collateral ligament. J Biomech Eng. December 1998;120(6):757-763.
2000	Soltz MA, Ateshian GA. A conewise linear elasticity mixture model for the analysis of tension-compression nonlinearity in articular cartilage. J Biomech Eng. December 2000;122(6):576-586.
2007	Ateshian GA, Ellis BJ, Weiss JA. Equivalence between short-time biphasic and incompressible elastic material responses. J Biomech Eng. June 2007;129(3):405-412.
2009	Ateshian GA, Rajan V, Chahine NO, Canal CE, Hung CT. Modeling the matrix of articular cartilage using a continuous fiber angular distribution predicts many observed phenomena. J Biomech Eng. June 2009;131(6):061003.
2013	Groves RB, Coulman SA, Birchall JC, Evans SL. An anisotropic, hyperelastic model for skin: experimental measurements, finite element modelling and identification of parameters for human and murine skin. J Mech Behav Biomed Mater. February 2013;18:167-180.
2006	Gasser TC, Ogden RW, Holzapfel GA. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface. February 22, 2006;3(6):15-35.
2005	Weiss JA, Gardiner JC, Ellis BJ, Lujan TJ, Phatak NS. Three-dimensional finite element modeling of ligaments: technical aspects. Med Eng Phys. December 2005;27(10):845-861.
2015	Stender ME. Osteochondral Tissue Modeling of Damage, Poroelasticity, and Remodeling in Osteoarthritis [PhD thesis]. University of Colorado; 2015.
2000	Soltz MA. Investigation of a Boundary Friction Model for Articular Cartilage: Effects of Interstitial Fluid Pressurization and Surface Topography [PhD thesis]. Columbia University; 2000.
2016	Yoshida K. Characterizing the Structure-Function Relationships of the Mouse Cervix in Pregnancy: Towards the Development of a Hormone-Mediated Material Model for Cervical Remodeling [PhD thesis]. Columbia University; 2016.
2017	Fernandez MJ. The Biomechanics of Pregnancy: Simulating Pregnancy Mechanics, Evaluating Preterm Delivery Interventions, and Measuring in-Vivo Mechanical Properties [PhD thesis]. Columbia University; 2017.
2018	Hou C. Implementation and Validation of Finite Element Framework for Passive and Active Membrane Transport in Deformable Multiphasic Models of Biological Tissues and Cells [PhD thesis]. Columbia University; 2018.
2021	Golman M. Toughening Mechanisms for the Attachment of Architectured Materials: The Mechanics of the Tendon Enthesis [PhD thesis]. Columbia University; 2021.
2021	Shi L. On the Mechanical Experiments and Modeling of Human Cervix [PhD thesis]. Columbia University; 2021.
2018	Li Q. Role of Meniscus Micromechanics in Joint Function and Osteoarthritis [PhD thesis]. Drexel University; December 2018.
1999	Ankersen J. Quantifying the Forces in Stabbing Incidents [PhD thesis]. Glasgow, UK: University of Glasgow; May 1999.
2019	Groetsch A. Micro- and Nanomechanics of Mineralised Collagen Fibre Elasto-Plasticity [PhD thesis]. Ebinburgh, Scotland: Heriot-Watt University; November 2019.
2021	Notermans T. Computational Modeling of Mechanobiology in Intact and Healing Rat Achilles Tendon [PhD thesis]. Lund, Sweden: Lund University; June 18, 2021.
1986	Pintar FA. The Biomechanics of Spinal Elements [PhD thesis]. Marquette University; December 1986.
2007	Lavagnino M. The Mechanotransduction Response of Tendon Cells to Tensile Loading [PhD thesis]. East Lansing, MI: Michigan State University; 2007.
2005	De Vita R. Structural Constitutive Models for Knee Ligaments [PhD thesis]. Pittsburgh, PA: University of Pittsburgh; 2005.
2010	Fisher MB. Functional Tissue Engineering of the Healing Anterior Cruciate Ligament: A Combined Experimental and Computational Approach [PhD thesis]. University of Pittsburgh; 2010.
2011	Wagnac É. Expérimentation et modélisation détaillée de la colonne vertébrale pour étudier le rôle de facteurs anatomiques et biomécaniques sur les traumatismes rachidiens [PhD thesis]. École polytechnique de Montréal; November 2011.
2019	Jannesar S. The Mechanics of Contusion Spinal Cord Injury: Towards Patient-Specific Assessments of Mechanical Loading and Injury [PhD thesis]. Simon Fraser University; Summer 2019.
1997	Wren TAL. Structure-Function Relationships for Soft Skeletal Connective Tissues [PhD thesis]. Stanford University; June 1997.
1988	Lam T. The Mechanical Properties of the Maturing Medial Collateral Ligament [PhD thesis]. Calgary, AB: University of Calgary; March 1988.
2003	Miller-Young JE. Factors Affecting Human Heel Pad Mechanics: A Finite Element Study [PhD thesis]. Calgary, AB: University of Calgary; April 2003.
2012	Ní Annaidh A. The Mechanics of Stabbing: A Combined Experimental and Numerical Study of Sharp Force Injury [PhD thesis]. Dublin, Ireland: University College Dublin; July 2012.
1999	Kukreti U. Structure-Function Relationship in Connective Tissues: A Study of Rabbit Medial Collateral Ligaments [PhD thesis]. Baltimore County, University of Maryland; 1999.
2010	Shahmirzadi D. Experimental Characterization of Vascular Tissue Viscoelasticity With Emphasis on Elastin's Role [PhD thesis]. University of Maryland; 2010.
1997	Richards M. Strain Environment Effects on New Bone Formation and Precursor Tissue Differentiation During Distraction [PhD thesis]. University of Michigan; 1997.
2000	Fedewa MM. Determinants of Microstructural Load Transfer in Cartilage Tissue from Chondrocyte Culture [PhD thesis]. University of Minnesota; August 2000.
2019	Bosnic M. Sensitivity of Bulk Leg Tissue Stiffness to Muscle Activation [Master's thesis]. University of Guelph; September 2019.
1998	Bain AC. In Vivo Mechanical Thresholds for Morphological and Functional Axonal Injury [PhD thesis]. Philadelphia, PA: University of Pennsylvania; 1998.
2001	Mazuchowski EL II. Investigation of the Biomechanical Properties of the Human Spinal Cord and Its Connective Tissue [PhD thesis]. Philadelphia, PA: University of Pennsylvania; 2001.
2008	Lee KE. An Engineering Approach to Understanding Painful Facet -Mediated Injury: Insights on Biomechanics and Neuropeptide Responses in a Rat Model [PhD thesis]. Philadelphia, PA: University of Pennsylvania; 2008.
2009	Lake SP. Anisotropic, Inhomogeneous and Nonlinear Structure-Function of Human Supraspinatus Tendon [PhD thesis]. Philadelphia, PA: University of Pennsylvania; 2009.
2016	Peloquin JM. Tensile Mechanics of the Knee Meniscus in the Context of Cracks: Failure and Fracture Mechanisms, Strain Concentrations, and the Effect of Specimen Shape [PhD thesis]. Philadelphia, PA: University of Pennsylvania; 2016.
2002	Gardiner JC. Computational Modeling of Ligament Mechanics [PhD thesis]. University of Utah; May 2002.
2008	Lucas SR. A Microstructural Viscoelastic Ligament Failure Model [PhD thesis]. Charlottesville, VA: University of Virginia; January 2008.
1997	Hurschler C. Collagen Matrix in Normal and Healing Ligaments: Microstructural Behavior, Biological Adaptation, and a Structural Mechanical Model [PhD thesis]. University of Wisconsin – Madison; 1997.
1997	Rapoff AJ. Tensile Fatigue of Intervertebral Disc Annulus and a Probabilistic Rule of Mixtures for Elastic Moduli [PhD thesis]. University of Wisconsin – Madison; August 1997.
2003	Provenzano PP. Damage and Disuse in Collagenous Extracellular Matrices: Biological Responses and Mechanical Consequences [PhD thesis]. University of Wisconsin – Madison; 2003.