Engineering Transactions, 49, 2-3, pp. 213–240, 2001

Modelling Elastic Behaviour of Soft Tissues. Part I. Isotropy

S. Jemioło
Warsaw University Of Technology

J.J. Telega
Institute of Fundamental Technological Research Polish Academy of Sciences

Soft tissues mostly exhibit strongly nonlinear behaviour while undergoing large strains. New macroscopic models for both isotropic and transversely isotropic soft tissues have been proposed. The models developed are suitable for finite element formulation. It the first part of the paper isotropic models have been proposed. Implementation in the FEM programme ABAQUS has been discussed. Muscle contraction models have also been concisely reviewed.
Full Text: PDF


ABAQUS, Theory manual, Version 5.7; ABAQUS/Standard Example problems manual, Version 5.7; ABAQUS/Standard User's manual, Version 5.7; ABAQUS/Standard Verification manual, Version 5.7., Hibbitt, Karlsson and Sorensen, Inc., Pawtucket, 1997.

T. ADACHI, M. TANAKA, Y. TOMITA, Uniform stress state in bone structure with residual stress, J. Biomech. Engng., 120, 342–347, 1998.

M.-G. ASCENZI, A first estimation of prestress in so-called circularly fibered osteonic lamellae, J. Biomech., 32, 935–942, 1999.

C. ATKINSON, On a possible theory for the design of tissue expanders, Quart. J. Mech. Appl. Math., 41, 301–317, 1988.

T.S. ATKINSON, R.C. HAUT, N.J. ALTIERO, A poroelastic model that predicts some phenomenological responses of ligaments and tendons, J. Biomech. Engng., 119, 400–405, 1997.

M F. BEATTY, Topics in finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues – with examples, Appl. Mech. Reviews, 40, Part 1, 1699–1734, 1987.

M.F. BEATTY, Introduction to nonlinear elasticity, [In:] Nonlinear effects in fluids and solids, M.M. CARROLL and M. HAYES [Eds.], Plenum Press, 13–112, New York 1996.

J. BONET, R.D. WOOD, Nonlinear continuum mechanics for finite element analysis, Cambridge University Press, Cambridge 1997.

G.H. BROWN, J.J. WOLKEN, Liquid crystal and biological structures, Academic Press, New York 1979.

P.G. CIARLET, Mathematical elasticity, North-Holland, Amsterdam 1988.

S.C. COWIN, How is a tissue built?, J. Biomech. Engng., 122, 553–569, 2000.

A. DELFINO, N. STERGIOPULOS, J.E. MOORE, J.-J. MEISTER, Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation, J. Biomech., 30, 777–786, 1997.

W. EHLERS, B. MARKERT, A linear viscoelastic biphasic model for soft tissues based on the theory of porous media, Bericht aus dem Institut für Mechanik (Bauwesen), No. 99–11–3, 1999.

L.E. FORD, A.F. HUXLEY, R.M. SIMMONS, Tension responses to sudden length change in simulated frog muscle fibers near slack length, J. Physiol. (Cambridge), 269, 441–515, 1977.

Y.C. FUNG, Biomechanics: mechanical properties of living tissues, Springer-Verlag, New York 1993.

Y.C. FUNG, Biomechanics: circulation, Springer-Verlag, New York 1997.

L. GLASS, P. HUNTER, A. MCCULLOCH [Eds.], Theory of heart, Springer-Verlag, New York 1991

J.E. HEUSER, R. COOKE, Actin-myosin interactions visualized by the quick-freeze, deep-etch replica technique, J. Mol. Biology, 169, 97–122, 1983.

A.V. HILL, The heat shortening and the dynamic constants of muscle, Proc. R. Soc. London, Ser. B. Biol. Sci., 126, 136–195, 1938.

A.V. HILL, The energetics of relaxation in a muscle twitch, Proc. R. Soc. London, Ser. B. Biol. Sci., 136, 211–219, 1949.

A.V. HILL, The instantaneous elasticity of active muscle, Proc. R. Soc. London, Ser. B. Biol. Sci., 141, 161–178, 1953.

D.K. HILL, Tension due to interaction between the sliding filaments in resting striated muscle, the effect of simulation, J. Physiology, 155, 657–684, 1968.

A. HOFFMAN, Determining the material properties of soft tissues, Preprint, Mechanical Engineering Department, Worcester Polytechnic Institute, 1998.

A. HOGER, On the residual stress possible in an elastic body with material symmetry, Arch. Rat. Mech. Anal., 88, 271–289, 1985.

G.A. HOLZAPFEL, Biomechanics of soft tissues, [In:] Handbook of material behaviour: nonlinear models and properties, J. LEMAITRE [Ed.], Academic Press, (in press).

J.D. HUMPHREY, Computer methods in membrane biomechanics, Comp. Meth. Biomech. Biomed. Engng., 1, 171–210, 1998.

J.D. HUMPHREY, F.C.P. YIN, A new constitutive formulation for characterizing the mechanical behavior of soft tissue, Biophys. J., 52, 563–570, 1987.

J.D. HUMPHREY, R.K. STRUMPF, F. C. P. YIN, Determination of a constitutive relation for passive myocardium: I. A new functional form, J. Biomech. Engng., 112, 333–339, 1990; II. Parameter estimation, ibid., pp. 340–346.

J.D. HUMPHREY, R.K. STRUMPF, F.C.P. YIN, A constitutive theory for biomembranes: application to epicardial mechanics, J. Biomech., 114, 461–466, 1992.

C. HURSCHLER, B. LOITZ-RAMAGE, R. VANDERBY, A structurally based stress-stretch relationship for tendon and ligament, J. Biomech. Engng., 119, 392–399, 1997.

A.F. HUXLEY, Muscle structure and theories of contraction, Prog. Biophys. Biophys. Chem., 7, 255–318, 1957.

H.E. HUXLEY, The cross-bridge mechanism of muscle contraction and its implications, J. Exp. Biology, 115, 17–30, 1985.

H.E. HUXLEY, J. HANSON, Changes in cross-striations of muscle during contraction and stretch and their structural interpretation, Nature, 173, 973–976, 1954.

A.F. HUXLEY, R. NIEDERGERKE, Interference microscopy of living muscle fibers, Nature, 173, 971–973, 1954.

A.F. HUXLEY, R.M. SIMONS, Proposed mechanism of force generation in striated muscle, Nature, 233, 533–538, 1971.

S. JEMIOŁO, A. SZWED, Implementation of subroutine UHYPER of ABAQUS finite element program for hyperelastic Blatz–Ko material, [In:] Theoretical Foundations of Civil Engineering, W. SZCZĘŚNIAK [Ed.], ZG OW PW, 251–262, Warsaw 1999.

S. JEMIOŁO, J.J. TELEGA, Representations of tensor functions and applications in continuum mechanics, IFTR Reports, 3/1997, Warsaw.38. S. JEMIOŁO, J. J. TELEGA, Implementation of isotropic hyperelastic models of soft biological tissues in finite element program ABAQUS, [In:] Theoretical Foundations of Civil Engineering, W. SZCZĘŚNIAK [Ed.], ZG OW PW, 269–280, Warsaw 1999.

S. JEMIOŁO, J.J. TELEGA, Isotropic models of hyperelastic soft tissues: implementation in FEM ABAQUS, Acta Bioeng. Biomech., 1, Supplement 1, 207–210, 1999.40. F. JOUVE, Modélisation de l'œil en élasticité non linéaire, Masson, Paris 1993.41. P. VAN KEMENADE, Water and ion transport through intact and damaged skin, Ph. D. thesis, Technische Universiteit Eindhoven 1998.

P. KOWALCZYK, Numerical analysis of stress distribution in lung parenchyma when interaction between tissue matrix and flowing air is taken into account [in Polish], IFTR Reports 1/1993, Warsaw.

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

Y. LANIR, Constitutive equations for fibrous connective tissue, J. Biomech., 16, 1–12, 1983.45. Y. LANIR, Constitutive equations for the lung tissue, J. Biomech. Engng., 105, 374–380, 1983.

G. LAMM, A. SZABO, Langevin nodes of macromolecules, J. Chem. Phys., 85, 7334–7348, 1986.

P. LE TALLEC, Numerical methods for nonlinear three–dimensional elasticity, [In:] Handbook of numerical analysis, P. G. CIARLET and J. L. LIONS [Eds.], Elsevier Science, Vol. III, 465–622, Amsterdam 1994.

G.C. LEE, Solid mechanics of lungs, J. Engng. Mech. Div., 104, 177–199, 1978.

G. LEWIS, K.M. SHAW, Modelling the tensile behavior of human Achilles tendon, BioMedical Mat. Engng., 7, 231–244, 1997.

D.H.S. LIN, F.C.P. YIN, A multiaxial constitutive law for mammalian left ventricular myocardium in steady-state barium contracture in tetanus, J. Biomech. Engng., 120, 504517, 1998.

J.-T. Liu, G.C. LEE, Static finite deformation analysis of the lung, J. Engng. Mech. Div., 104, 225–238, 1978.

W. MAUREL, W. YIN, N.M. THALMAN, D. THALMAN, Biomechanical models for soft tissues simulation, Springer-Verlag, Berlin 1998.

K. MAY-NEWMAN, F.C.P. YIN, A constitutive law for mitral valve tissue, J. Biomech. Engng., 120, 38–47, 1998.

K. MEIJER, Muscle mechanics: the effect of stretch and shortening on skeletal muscle force, P.D. thesis, Universiteit Twente, 1998.

K. MEIJER, H.J. GROOTENBOER, H.F.J. M. KOOPMAN, B.J.J.J. van der LINDEN, P.A. HUIJING, A Hill type model of rat medial gastrocnemius muscle that accounts for shortening history effects, J. Biomech., 31, 555–563, 1998.

T. MITSUI, H. OSHIMA, A self-induced translation model of myosin head motion in contracting muscle. I. Force-velocity relation and energy liberation, J. Muscle Res. Cell Motil., 9, 52–62, 1988.

E. NAKAMACHI, J. TSUKAMOTO, Y. TAMURA, Molecular dynamics simulation of skeletal muscle contraction, [In:] Computational Biomechanics, K. HAYASHI and H. ISHIKAWA [Eds.], Springer-Verlag, 89–114, Tokyo 1976.

R.W. OGDEN, Nonlinear elastic deformations, Ellis Horwood, Chichester 1984.

J.M. PRICE, Biomechanics of smooth muscle, [In:] Frontiers in biomechanics, G. W. SCHMID-SCHONBEIN, S. L.-Y. Woo and B. W. ZWEIFACH [Eds.], Springer-Verlag, 51–61, New York 1986.

M.A. Puso, J.A. WEISS, Finite element implementation of anisotropic quasi-linear viscoelasticity using a discrete spectrum approximation, J. Biomech. Engng., 120, 62–70, 1998.

M.K. REEDY, R.C. HOLMES, R.T. TREGEAR, Induced changes in orientation of the cross-bridges of glycerinated insect flight muscle, Nature, 207, 1276–1281, 1965.

M. RENARDY, D.L. RUSSELL, Formability of linear elastic structures with volume-type actuation, Arch. Rat. Mech. Anal., 149, 97–122, 1999.

B.R. SIMON, Multiphase poroelastic finite element models for soft tissue structures, Appl. Mech. Rev., 45, 191–218, 1992.

B.R. SIMON, M.V. KAUFMAN, M.A. MCAFEE, A.L. BALDWIN, Porohyperelastic finite element analysis of large arteries using ABAQUS, J. Biomech. Engng., 120, 296–298, 1998.

B.R. SIMON, M.V. KAUFMAN, M.A. MCAFEE, A.L. BALDWIN, L.M. WILSON, Identification and determination of material properties for porohyperelastic analysis of large arteries, J. Biom. Engng., 120, 188–194, 1998.

L. SKUBISZAK, Mechanism of muscle contraction, Technol. Health Care, 1, 133–142, 1993.

L. SKUBISZAK, L. KOWALCZYK, Computer system modelling muscle work, Technol. Health Care, 6, 139–149, 1998.

L. SKUBISZAK, L. KOWALCZYK, Relation between the mechanical properties of muscles and their structure on the molecular level, Engng. Trans., 49, 2–3, 191–212, 2001.

P. TONG, Y.C. FUND, The stress-strain relationship for the skin, J. Biomech., 9, 649–657, 1976.

A. TORELLI, Study of a mathematical model for muscle contraction with deformable elements, Rend. Sem. Mat. Univ. Pol. Torino, 55, 241–271, 1997.

T.Q.P. UYEDA, H.M. WARRICK, S.J. KRON, J.A. SPUDICH, Quantized velocities at low myosin densities in an in vitro motility assay, Nature, 352, 307–311, 1991.

C.C. VAN DONKELAAR, Skeletal muscle mechanics: a numerical and experimental approach to spatial phenomena, Universiteit Maastricht, 1999.

P. VENA, R. CONTRO, R. PIETRABISSA, L. AMBROSIO, Design of materials subject to biomechanical compatibility constraints, [In:] Synthesis in bio-solid mechanics, P. PEDERSEN and M.P. BENDSOE [Eds.}, Kluwer Academic Publ., 67–78, Dordrecht 1999.

D.V. VORP, D.H.-J. WANG, Use of finite elasticity in abdominal aneurysm research, [In:] Mechanics in biology, J. CASEY and G. BAO [Eds.], 157–171, AMD – vo. 242, The American Society of Mechanical Engineers, New York 2000.

J. VOSSOUGHI, A. TOZEREN, Determination of an effective shear modulus of aorta, Russian J. Biomech., No 1–2, 20–35, 1998.

V. VUSKOVI, M. KAMER, J. DUAL, M. BAJKA, Method and device for in-vivo measurement of elasto-mechanical properties of soft biological tissues, Machine Graphics Vision, 8, 637654, 1999.

H.W. WEIZSACKER, J.G. PINTO, Isotropy and anisotropy of the arterial wall, J. Biomech., 21, 477–487, 1988.

S.L.-Y. Woo, J.S. WAYNE, Mechanics of the anterior cruciate ligament and its contribution to knee kinematics, Appl. Mech. Reviews, 43, Part 2, 142–149, 1990.

S.L.-Y. Woo, G.A. JOHNSON, R.E. LEVINE, K.R. RAJAGOPAL, Viscoelastic models for ligaments end tendons, Appl. Mech. Reviews, 47, Part 2, 282–286.

H. YAMADA, Strength of biological materials, The Williams & Wilkins Company, Baltimore 1970.

T. YANAGIDA, T. ARATA, F. OOSAWA, Sliding distance of actin filament induced by a myosin cross-bridge during one ATP hydrolysis cycle, Nature, 316, 366–369, 1985.

H.-L. YEH, T. HUANG, R.A. SCHACHAR, A closed shell structured eyeball model with application to radial keratotomy, J. Biomech. Engng., 122, 505–510, 2000.

G.I. ZAHALAK, S.P. MA, Muscle activation and contraction: constitutive relations based directly on cross-bridge kinetics, J. Biomech. Engng., 112, 52–62, 1990.

G.I. ZAHALAK, I. MOTABARZADEH, A re-examination of calcium activation in the Huxley cross-bridge model, J. Biomech. Engng., 119, 20–29, 1997.

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