Engineering Transactions, 49, 2-3, pp. 241–281, 2001

Modelling Elastic Behaviour of Soft Tissues. Part II. Transverse Isotropy

S. Jemioło
Warsaw University of Technology

J.J. Telega
Polish Academy of Sciences

New constitutive relationships for hyperelastic transversely isotropic materials have been proposed. The well-known isotropic hyperelastic model due to OGDEN 11.581 has been extended to transverse isotropy. It has been shown that some models intended to describe the nonlinear elastic behaviour of soft tissues are oversimplified and lead to incorrect results. An overview of soft tissue modelling, being a continuation of the one started in [481, has also been given.
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E.S. ALMEIDA, R.L. SPILKER, Finite element formulation for hyperelastic transversely isotropic biphasic soft tissues, Comput. Methods Appl. Mech. Engng., 151, 513–538, 1998.

M. BATHE, R.D. KAMM, A fluid-structure interaction finite element analysis of pulsatile blood flow through a compliant stenotic artery, J. Biomech. Engng., 121, 361–369, 1999.

D. BEATTIE, C. XU, R. VITO, S. GLAGOV, M.C. WHANG, Mechanical analysis of heterogeneous, atherosclerotic human aorta, J. Biomech. Engng., 121, 602–607, 1998.

M.K.-E. BENAOUDA, J.J. TELEGA, On existence of minimizers for Saint-Venant Kirchhoff bodies: placement boundary condition, Bull. Pol. Acad. Sci., Tech. Sci., 45, 211–223, 1997.

K.L. BILLIAR, M.S. SACKS, Biaxial mechanical properties of the natural and glutaraldehyde treated aortic valve cups. Part I: Experimental results, J. Biomech. Engng., 122, 23–30, 2000; Part II: A structural constitutive model, ibid., 327–335, 2000.

P. BURSAE, C.V. MCGRATH, S.R. EISENBERG, D. STAMENOVIO, A microstructural model of elastostatic properties of articular cartilage in confined compression, J. Biomech. Engng., 122, 347–353, 2000.

M.B. CANNEL, J.R. BERLIN, W.J. LEDERER, Effect of membrane potential changes on the calcium transient in single rat cardiac muscle cells, Science, 238, 1419–1423, 1987.

E.D. CAREW, E.A. TALMAN, D. R. BOUGHNER, I. VESELY, Quasi-linear viscoelastic theory applied to internal shearing of porcine aortic valve leaflets, J. Biomech. Engng., 121, 386–392, 1999.

S.S. CHEN, J.D. HUMPHREY, Heat-induced changes in the mechanics of a collagenous tissue: pseudoelastic behavior at C, J. Biomech., 31, 211–216, 1998.

A. CHERKAEV, Variational methods for structural optimization, Springer Verlag, New York 2000.

C.J. CHUONG, Y.C. FUNG, Three-dimensional stress distribution in arteries, J. Biomech. Engng., 105, 268–274, 1983.

C.J. CHUONG, Y.C. FUNG, Residual stress in arteries, [In:] Frontier in biomechanics, G.W. SCHMID-SCHÖNBEIN, S.L.-Y. WOO and B. W. ZWEIFACH [Eds.], pp. 117–129, Springer Verlag, New York 1986.

C.J. CHUONG, Y.C. FLING, Residual stress in arteries, J. Biomech. Engng., 108, 189–192, 1986.

K.D. COSTA, F.C.P. YIN, Analysis of indentation: implications for measuring mechanical properties with atomic force microscopy, J. Biomech. Engng., 121, 462–471, 1999.

H. DEMIRAY, R.P. VITO, A layered cylindrical shell model for an aorta, Int. J. Engng. Sci., 29, 47–54, 1991.

E. DENNY, R.C. SCHROTER, Viscoelastic behavior of a lung alveolar duct model, J. Biomech. Engng., 122, 143–151, 2000.

K.A. DERWIN, L.J. SOSLOWSKY, A quantitative investigation of structure-function relationships in a tendon fascile model, J. Biomech. Engng., 121, 598–604, 1999.

G.V. DIMITROV, N.A. DIMITROVA, Fundamentals of power spectra of extracellular potentials produced by a skeletal muscle fibre of finite length. Part I: Effect of fibre anatomy, Med. Engng. Physics, 20, 580–587, 1998, Part II: Effect of parameters altering with functional state, ibid., 702–707.

G.V. DIMITROV, N.A. DIMITROVA, Effect of parameters altering with muscle fibre functional state on the power spectra of spatially filtered extracellular potentials, J. Med. Engng. Technol., 2001, in press.

N.A. DIMITROVA, G.V. DIMITROV, O.A. NIKITIN, Longitudinal variations of characteristic frequencies of skeletal muscle fibre potentials detected by a bipolar electrode or multi–electrode, J. Med. Engng. Tech., 2001, in press.

S. DOKOS, I.J. LEGRICE, B.H. SMAILL, J. KAR, A.A. YOUNG, A triaxial-measurement shear device for soft biological tissues, J. Biomech. Engng., 122, 471–478, 2000.

R. EBERLEIN, G.A. HOLZAPFEL, C.A.J. SCHULZE-BAUER, An anisotropic constitutive model for annulus tissue and enhanced finite element analyses of intact lumbar disc bodies, Comp. Meth. Biomech. Biomed. Engng., in press.

D.M. ELLIOT, L.A. SETTON, A linear model for fiber-induced anisotropy of the anulus fibrosus, J. Biomech. Engng., 122, 173–179, 2000.

D.M. FLYNN, G.D. PEURA, P. GRIGG, A.H. HOFFMAN, A finite element based method to determine the properties of planar soft tissue, J. Biomech. Engng., 120, 202–210, 1998.

M. FORTIN, J. SOULHAT, A. SHIRAZI-ADL, E.B. HUNZIKER, M.D. BUSCHMANN, Unconfined compression of articular cartilage: nonlinear behavior and comparison with a fibre-reinforced biphasic model, J. Biomech. Engng., 122, 189–195, 2000.

Y.C. FUNG, S.Q. Liu, Change of residual strains in arteries due to hypertrophy caused by aortic constriction, Circulation Res., 65, 1340–1349, 1989.

Y.C. FUNG, S.Q. Liu, Changes of zero-stress state of rat pulmonary arteries in hypoxic hypertension, J. Appl. Physiol., 70, 2455–2470, 1991.

J.R. FUNK, G.W. HALL, J.R. CRANDALL, W.D. PILKEY, Linear and quasi-linear viscoelastic characterization of ankle ligaments, J. Biomech. Engng., 122, 15–22, 2000.

A. GAŁKA, J.J. TELEGA, R. WOJNAR, Electromechanical modelling of cartilage, Engng. Trans., this issue.

S.E. GREENWALD, J.E. MOORE, A. RACHEV, T.P.C. KANE, J.-J. MEISTER, Experimental investigation of the distribution of residual strains in the artery wall, J. Biomech. Engng., 119, 438–444, 1997.

H. GREGERSEN, T.C. LEE, S. CHIEN, R. SKALAK, Y.C. FUNG, Strain distribution in the layered wall of the esophagus, J. Biomech. Engng., 121, 442–448, 1999.

J.M. GUCCIONE, A.D. MCCULLOCH, Finite element modeling of ventricular mechanics [In:] Theory of heart, L. GLASS, P. HUNTER and A. MCCULLOCH [Eds.], 121–144, Springer Verlag, New York 1991.

M. HAJJAR, W. MÊME, C. LÉOTY, Measurement of sarcomere length during fast contraction of muscle fibers by digital image analysis, J. Biomechanics, 32, 737–742, 1999.

H.C. HAN, Y.C. FUNG, Direct measurement of transverse residual strain in aorta, Amer. J. Physiol., 270, (Heart Circ. Physiol., 39), H750–H759, 1996.

G.A. HOLZAPFEL, R. EBERLEIN, P. WRIGGERS, H.W. WEIZSÄCKER, Large strain analysis of soft biological membranes: formulation and finite element analysis, Compt. Methods Appl. Mech. Engng., 132, 45–61, 1996.

G.A. HOLZAPFEL, R. EBERLEIN, P. WRIGGERS, H.W. WEIZSÄCKER, A new axisymmetrical membrane element for anisotropic finite strain analysis of arteries, Comm. Numer. Meth. Engng., 12, 507–517, 1996.

G.A. HOLZAPFEL, T.C. GASSER, A viscoelastic model for fiber–reinforced composites at finite strain: Continuum basis, computational aspects and applications, Comput. Methods Appl. Mech. Engng., in press.

G.A. HOLZAPFEL, T.C. GASSER, R. W. OGDEN, A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. Elasticity, in press.

G.A. HOLZAPFEL, C.A.J. SCHULZE-BAUER, M. STADLER, Mechanics of angioplasty: wall balloon and scent [In:] Mechanics in biology, J. CASEY and G. BAO [Eds.], 141–156, AMD-vol. 242, The American Society of Mechanical Engineers, New York 2000.

G.A. HOLZAPFEL, H. W. WEIZSÄCKER, Biomechanical behavior of the arterial wall and its numerical characterization, Computers in Biol. Med., 28, 377–392, 1998.

A. HOROWITZ, Structural considerations in formulating material laws for the myocardium [In:] Theory of heart, L. GLASS, P. HUNTER and A. MCCULLOCH [Eds.], 31–58, Springer Verlag, New York 1991.

J.D. HUMPHREY, Remodeling of a collagenous tissue at fixed lengths, J. Biomech. Engng., 121, 591–597, 1999.

J.D. HUMPHREY, An evaluation of pseudoelastic descriptors used in arterial mechanics, J. Biomech. Engng., 121, 259–262, 1999.

J.M. HUYGHE, T. ARTS, D.H. VAN CAMPEN, R.S. RENEMAN, Porous medium finite element model of the beating left ventricle, Amer. J. Physiol., 262, (Heart Circ. Physiol., 31), H1256–H1267, 1992.

J.C. IATRIDIS, L.A. SETTON, R.J. FOSTER, B.A. RAWLINS, M. WEIDENBAUM, V.C. MOW, Degeneration affects the anisotropic and nonlinear behaviours of human anulus fibrosus in compression, J. Biomechanics, 31, 535–544, 1998.

S. JEMIOŁO, J.J. TELEGA, C. MICHALAK, Hyperelastic anisotropic models of soft tissues, Acta Bioeng. Biomech., 2, Supplement 1, 235–240, 2000.

S. JEMIOŁO, J.J. TELEGA, On finite anisotropic hyperelasticity and applications to soft tissue modelling [In:] Polish–Ukrainian Transactions " Theoretical Foundations in Civil Engineering, W. SZCZEŚNIAK [Ed.], 319–330, Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa 2000.

S. JEMIOŁO, J.J. TELEGA, Modelling elastic behaviour of soft tissues. Part I. Isotropy, Engng. Trans., 49, 2–3, 213–240, 2001.

A.D. KARAKAPLAN, M.P. BIENIEK, R. SKALAK, A mathematical model of lung parenchyma, J. Biomech. Engng., 102, 124–136, 1980.

V.A. KAS'YANOV, A.I. RACHEV, Deformation of blood vessels upon stretching, intramural pressure and torsion, Mech. Comp. Mater., 16, 76–80, 1980.

S.M. KLISCH, J.C. LUTZ, A special theory of biphasic mixtures and experimental results for human annulus fibrosus tested in confined compression, J. Biomech. Engng., 122, 181–188, 2000.

E. KIMMEL, R.D. KAMM, A.H. SHAPIRO, A cellular model of lung elasticity, J. Biomech. Engng., 109, 126–131, 1987.

A. KIRPALANI, H. PARK, J. BUTANY, K.W. JOHUSTON, M. OJHA, Velocity and wall shear stress patterns in the human right coronary artery, J. Biomech. Engng., 121, 370–375, 1999.

W.M. LAI, V.C. MOW, D.D. SUN, G.A. ATESHIAN, On the electric potentials inside a charged hydrated biological tissue: streaming potential versus diffusion potential, J. Biomech. Engng., 122, 336–346, 2000.

R.S. LAKES, R. VANDERBY, Interrelation of creep and relaxation: a modeling approach for ligaments, J. Biomech. Engng., 121, 612–615, 1999.

C.S. LEE, J.M. TARBELL, Wall shear rate distribution in an abdominal aortic bifurcation model: effects of vessel compliance and phase angle between pressure and flow waveforms, J. Biomech. Engng., 119, 333–342, 1997.

J.D. LEMMON, A.P. YOGANATHAN, Three-dimensional computational model of left heart diastolic function with fluid-structure interaction, J. Biomech. Engng., 122, 109–117, 2000.

J.D. LEMMON, A.P. YOGANATHAN, Computational modelling of left heart diastolic function: examination of ventricular dysfunction, J. Biomech. Engng., 122, 297–303, 2000.

W.Y.W. LEW, Functional consequences of regional heterogeneity in the left ventricle [In:] Theory of heart, L. GLASS, P. HUNTER and A. MCCULLOCH [Eds.], 209–237, Springer Verlag, New York 1991.

S.Q. LIU, Y.C. FUNG, Zero–stress state of arteries, J. Biomech. Engng., 110, 82–84, 1988.

S.Q. LIU, Y.C. FuNG, Relationship between hypertension, hypertrophy, and opening angle of zero-stress state of arteries following aortic constriction, J. Biomech. Engng., 111, 325– 335, 1989.

A.D. MCCULLOCH, J.H. OMENS, Factors affecting the regional mechanics of the diastolic heart [In:] Theory of heart, L. GLASS, P. HUNTER and A. MCCULLOCH [Eds.], 87–119, Springer Verlag, New York 1991.

G.W. MILTON, The theory of composites, Laminated Publishers, New York, in press.

B.S. MYERS, C.T. WOOLLEY, T.L. SLOTTER, W.E. GARRETT, T.M. BEST, The influence of strain rate on the passive and stimulated engineering stress-large strain behavior of the rabbit tibialis anterior muscle, J. Biomech. Engng., 120, 126–132, 1998.

P. NIELSEN, I. HUNTER, Identification of the time–varying properties of the heart [In:] Theory of heart, L. GLASS, P. HUNTER and A. MCCULLOCH [Eds.], 77–86, Springer Verlag, New York 1991.

R.W. OGDEN, C.A.J. SCHULZE-BAUER, Phenomenological and structural aspects of the mechanical response of arteries [In:] Mechanics in biology, J. CASEY and G. BAO [Eds.], 125–140, AMD–vol. 242, The American Society of Mechanical Engineers, New York 2000.

R.W. OGDEN, P.G. ROXBURGH, A pseudo-elastic model for the Mullins effect in filled rubber, Proc. R. Soc., A455, 2861–2877, London 1999.

R.J. OKAMOTO, M.J. MOULTON, S.J. PETERSON, D. LI, M.K. PASQUE, J.M. GUCCIONE, Epicardial suction: a new approach to mechanical testing of the passive ventricular wall, J. Biomech. Engng., 122, 479–487, 2000.

J.H. OMENS, Y.-C. FUNG, Residual strain in rat left ventricle, Circulation Res., 66, 37–45, 1990.

E.M. ORTT, D.J. DOSS, E. LEGALL, N.T. WRIGHT, J.D. HUMPHREY, A device for evaluating the multiaxial finite strain thermomechanical behavior of elastomers and soft tissues, J. Appl. Mech., 67, 465–471, 2000.

S.J. PETERSON, R J. OKAMOTO, Effect of residual stress and heterogeneity on circumferential stress in the arterial wall, J. Biomech. Engng., 122, 454–456, 2000.

K.M. QUAPP, J.A. WEISS, Material characterization of human medial collateral ligament, J. Biomech. Engng., 120, 757–763, 1998.

A. RACHEV, N. STERGIOPULOS, J.-J. MEISTER, A model for geometric and mechanical adaptation of arteries to sustained hypertension, J. Biomech. Engng., 120, 9–17, 1998.

T.F. ROBINSON, L. COHEN-GOULD, S.M. FACTOR, Skeletal framework of mammalian heart muscle: arrangement of inter-and peri-cellular connective tissue structures, Lab. Invest., 49, 482–498, 1983.

T.F. ROBINSON, S.M. FACTOR, E.H. SONNENBLICK, The heart as a suction pump, Scientific Amer., 254, 84–91, 1986.

M.B. RUBIN, S.R. BODNER, N.S. BINUR, An elastic–viscoplastic model for excised facial tissues, J. Biomech. Engng., 120, 686–689, 1998.

M.S. SACKS, A method for planar biaxial mechanical testing that includes in–plane shear, J. Biomech. Engng., 121, 551–555, 1999.

M. S. SACKS, Tissue-level structural constitutive models [In:] Mechanics in biology, J. CASEY and G. BAO [Eds.], 113–124, AMD–vol. 242, The American Society of Mechanical Engineers, New York 2000.

L.A. SETTON, H. TOHYAMA, V.C. MOW, Swelling and curling behaviors of articular cartilage, J. Biomech. Engng., 120, 355–361, 1998.

B.H. SMAILL, P.J. HUNTER, Structure and function of the diastolic heart [In:] Theory of heart, L. GLASS, P. HUNTER and A. MCCULLOCH [Eds.], 1–29, Springer Verlag, New York 1991.

S.R. SUMMEROUR, J.L. EMERY, B. FAZELI, J.H. OMENS, A.D. MCCULLOCH, Residual strain in ischemic ventricular myocardium, J. Biomech. Engng., 120, 710–714, 1998.

L.A. TABER, Biomechanics of growth remodeling, and morphogenesis, Appl. Mech. Reviews, 48, 487–545, 1995.

L.A. TABER, A model for aortic growth based on fluid shear and fiber stresses, J. Biomech. Engng., 120, 348–354, 1998.

E. TANAKA, H. YAMADA, S. MURAKAMI, Inelastic constitutive modeling of arterial and ventricular walls [In:] Computational biomechanics, K. HAYASHI and H. ISHIKAWA [Eds.], 137–163, Springer Verlag, Tokyo 1996.

D. TANG, J. YANG, C. YANG, D.N. KU, A nonlinear axisymmetric model with fluid-wall interactions for steady viscous flow in stenotic tubes, J. Biomech. Engng., 121, 494–501, 1999.

L. K. WALDMAN, Multidimensional measurement of regional strains in the intact heart [In:] Theory of heart, L. GLASS, P. HUNTER and A. MCCULLOCH [Eds.], 145–174, Springer Verlag, New York 1991.

T.A.L. WREN, D.R. CARTER, A microstructural model for the tensile constitutive and failure behavior of soft skeletal connective tissues, J. Biomech. Engng., 120, 55–61, 1998.

K. TAKAMIZAWA, K. HAYASHI, Constitutive law of the arterial wall and stress distribution [In:] Computational mechanics 86: theory and applications, G. YAGAWA and S.N. ATLURI [Eds.], (IV–149)—(IV–154), Springer–Verlag, Tokyo 1986.

K. TAKAMIZAWA, K. HAYASHI, Strain energy density function and uniform strain hypothesis for arterial mechanics, J. Biomechanics, 20, 7–17, 1987.

K. TAKAMIZAWA, K. HAYASHI, Uniform strain hypothesis and thin–walled theory in arterial mechanics, Biorheology, 25, 555–565, 1988.

K. TAKAMIZAWA, T. MATSUDA, Kinematics for bodies undergoing residual stress and its applications to the left ventricle, J. Appl. Mech., 57, 321–329, 1990.

R.N. VAISHNAV, J. VOSSOUGHI, Estimation of residual strains in aortic segments [In:] Biomedical Engineering II, Recent Developments, C.W. HALL [Ed.], 330–333, Pergamon Press, New York 1983.

J. VALENTA, T. HRU, M. SOCHOR, Residual stresses in the human aorta and their influence by growth and remodelling, Bio-Med. Mat. Engng., 7, 159–169, 1997.

C.A. VAN EE, A.L. CHASSE, B.S. MYERS, Quantifying skeletal muscle properties in cadaveric test specimens: effects of mechanical loading, postmortem time and freezer storage, J. Biomech. Engng., 122, 9–14, 2000.

V.J. VANKAN, J.M. HUYGHE, J.D. JANSSEN, A. HUSON, Poroelasticity of saturated solids with an application to blood perfusion, Int. J. Engng. Sci., 34, 1019–1031, 1996.

J. VOSSOUGHI, Z. HEDJAZI, F.S. BORRIS, Intimal residual stress and strain in large arteries [In:] BED–vol. 24, Proc. ASME Summer Bioengineering Conference, Breckenridge, 434–437, 1993.

J.A. WEISS, B.N. MAKER, S. GOVINDJEE, Finite element implementation of incompressible, transversely isotropic hyperelasticity, Comp. Meth. Engng. Appl. Mech. Engng., 135, 107–128, 1996.

A. YOUNG, Epicardial deformation from coronary cineagiograms, in: Theory of heart, L. GLASS, P. HUNTER and A. McCuLuocH [Eds.], 175–207, Springer Verlag, New York 1991.

G.I. ZAHALAK, V. DE LABORDERIE, The effects of cross-fiber deformation on axial fiber stress in myocardium, J. Biomech. Engng., 121, 376–385, 1999.

S.Z. ZHAO, X.Y. XU, M.W. COLLINS, The numerical analysis of fluid-solid interactions for blood flow in arterial structures. Part 1: A review of models for arterial wall behaviour, Proc. Instn. Mech. Engrs, 212, Part H, 229–240, 1998; Part 2: Development of coupled fluid-solid algorithms, ibid., 241–252.

J. MACKERLE, A finite element bibliography for biomechanics (1987–1997), Appl. Mech. Reviews, 51, 587–634, 1998.

J.H. OMENS, S.M. VAPLON, B. FAZELI, A.D. MCCULLOCH, Left ventricular geometric remodeling and residual stress in the rat heart, J. Biomech. Engng., 120, 715–719, 1998.

DOI: 10.24423/6x8q-2p37