Finite element implementation of incompressible, transversely isotropic hyperelasticity. (English) Zbl 0893.73071

Summary: This paper describes a three-dimensional constitutive model for biological soft tissues and its finite element implementation for fully incompressible material behavior. The necessary continuum mechanics background is presented, along with derivations of the stress and elasticity tensors for a transversely isotropic, hyperelastic material. A particular form of the strain energy for biological soft tissues is motivated, and a finite element implementation of this model based on a three-field variational principle (deformation, pressure and dilation) is discussed. Numerical examples are presented that demonstrate the effectiveness of this approach.


74S05 Finite element methods applied to problems in solid mechanics
74B20 Nonlinear elasticity
74L15 Biomechanical solid mechanics
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[1] Babuska, I., The finite element method with Lagrangian multipliers, Numer. math., 20, 179-192, (1973) · Zbl 0258.65108
[2] Bathe, K-J., Finite element procedures in engineering analysis, (1982), Prentice-Hall Englewood Cliffs, NJ
[3] Choung, C.J.; Fung, Y.C., Residual stress in arteries, (), 117-179
[4] Flory, P.J., Thermodynamic relations for high elastic materials, Trans. Faraday soc., 57, 829-838, (1961)
[5] Fung, Y.C., Foundations of solid mechanics, (1965), Prentice-Hall Englewood Cliffs, NJ
[6] Fung, Y.C., Elasticity of soft tissues in simple elongation, Am. J. physiol., 213, 1532-1544, (1967)
[7] Fung, Y.C., Biomechanics: mechanical properties of living tissues, (1981), Springer-Verlag New York
[8] Govindjee, S.; Simo, J.C., Mullins’ effect and the strain amplitude dependence of the storage modulus, Int. J. solids struct., 29, 1737-1751, (1992) · Zbl 0764.73073
[9] Green, A.E., Large elastic deformations, (1970), Clarendon Press Oxford, UK · Zbl 0227.73067
[10] Guccione, J.M.; McCulloch, A.D.; Waldman, L.K., Passive material properties of intact ventricular myocarcium determined from a cylindrical model, ASME J. biomech. engrg., 113, 42-55, (1991)
[11] Horowitz, A.; Sheinman, I.; Lanir, Y., Nonlinear incompressible finite element for simulating loading of cardiac tissue—part II: three-dimensional formulation for thick ventricular wall segments, ASME J. biomech. engrg., 110, 62-68, (1988)
[12] Hughes, T.J.R., Generalization of selective integration procedures to anisotropic and nonlinear media, Int. J. numer. methods engrg., 15, 1413-1418, (1980) · Zbl 0437.73053
[13] Humphrey, J.D.; Strumph, R.K.; Yin, F.C.P., Determination of a constitutive relation for passive myocardium, I. A new functional form, ASME J. biomech. engrg., 112, 333-339, (1990)
[14] Humphrey, J.D.; Yin, F.C.P., On constitutive relations and finite deformations of passive cardiac tissue: I. A pseudostrain-energy approach, ASME J. biomech. engrg., 109, 298-304, (1987)
[15] Huyghe, J.M.; van Campen, D.H.; Arts, T.; Heethaar, R.M., The constitutive behavior of passive heart muscle tissue: A quasi-linear viscoelastic formulation, J. biomech., 24, 841-849, (1991)
[16] Hvidberg, E., Investigations into the effect of mechanical pressure on the water content of isolated skin, Acta pharmac. (kobenhavn), 16, 245-259, (1960)
[17] Key, S.W., A variational principle for incompressible and nearly-incompressible anisotropic elasticity, Int. J. solids struct., 5, 951-964, (1969) · Zbl 0175.22101
[18] Lanir, Y., Constitutive equations for fibrous connective tissues, J. biomech., 16, 1-12, (1983)
[19] Maker, B.N.; Ferencz, R.M.; Hallquist, J.O., Nike3d: A nonlinear, implicit, three-dimensional finite element code for solid and structural mechanics, Lawrence livermore national laboratory technical report, UCRL-MA-105268, (1990)
[20] Marsden, J.E.; Hughes, T.J.R., The mathematical foundations of elasticity, (1983), Prentice-Hall Englewood Cliffs, NJ
[21] Matthies, H.; Strang, G., The solution of nonlinear finite element equations, Int. J. numer. methods engrg., 14, 1613-1626, (1979) · Zbl 0419.65070
[22] Mooney, M., A theory of large elastic deformation, J. appl. phys., 11, 582-592, (1940) · JFM 66.1021.04
[23] Oden, J.T.; Kikuchi, N., Finite element methods for constrained problems in elasticity, Int. J. numer. methods engrg., 18, 701-725, (1982) · Zbl 0486.73068
[24] Simo, J.C., On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects, Comput. methods appl. mech. engrg., 60, 153-173, (1987) · Zbl 0588.73082
[25] Simo, J.C.; Taylor, R.L., Quasi-incompressible finite elasticity in principal stretches: continuum basis and numerical algorithms, Comput. methods appl. mech. engrg., 85, 273-310, (1991) · Zbl 0764.73104
[26] Simo, J.C.; Taylor, R.L.; Pister, K.S., Variational and projection methods for the volume constraint in finite deformation elastoplasticity, Comput. methods appl. mech. engrg., 51, 177-208, (1985) · Zbl 0554.73036
[27] Smith, G.F.; Rivlin, R.S., Integrity bases for vectors. the crystal classes, Arch. rat. mech. anal., 15, 169-221, (1994) · Zbl 0133.26305
[28] Spencer, A.J.M., Continuum theory of the mechanics of fibre-reinforced composites, (1984), Springer-Verlag New York · Zbl 0559.00015
[29] Sussman, T.; Bathe, K-J., A finite element formulation for nonlinear incompressible elastic and inelastic analysis, Comput. struct., 26, 357-409, (1987) · Zbl 0609.73073
[30] Washizu, K., Variational methods in elasticity and plasticity, (1974), Pergamon Oxford · Zbl 0164.26001
[31] Weiss, J.A., A constitutive model and finite element representation for transversely isotropic soft tissues, ()
[32] Weiss, J.A.; Maker, B.N.; Schauer, D.A., Treatment of initial stress in hyperelastic finite element models of soft tissues, (), 105-106
[33] Woo, S.L-Y.; Buckwalter, J.A., Injury and repair of the musculoskeletal soft tissues, (1988), American Academy of Orthopaedic Surgeons Park Ridge, Illinois
[34] Yin, F.C.P.; Strumph, R.K.; Chew, P.H.; Zeger, S.L., Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading, J. biomech., 20, 577-589, (1987)
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