Collagen is known to be a viscoelastic material and therefore is an important source of time dependent behavior in soft tissue. Though there are material models for soft tissue that rely on properties and structure of constituent collagen fibers, such models generally utilize only the elastic properties of collagen, and thus the resulting tissue level response does not possess any viscoelastic features. Here, the time dependent properties of collagen are directly incorporated into a fiber based hyperelastic model using the one dimensional theory of quasi-linear viscoelasticity within the context of a locally defined, anisotropic representation of extracellular matrix structure. The resulting model possesses seven material parameters and, using numerical and computational analysis, is shown to successfully predict many key features of soft tissue response including anisotropy, strain hardening, preconditioning, and rate independent hysteresis. A formulation is also introduced for incorporating fiber level viscoelasticity into structural models derived from a continuous representation of fiber structure.
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REFERENCES
Abramowitch, S. D., S. L.-Y. Woo, T. D. Clineff, and R. E. Debski. An evaluation of the quasi-linear viscoelastic properties of the healing medial collateral ligament in a goat model. Ann. Biomed. Eng. 32(3):329–335, 2004.
Best, T. M., J. McElhaney, W. E. Garrett, and B. S. Myers. Characterization of the passive responses of live skeletal muscle using the quasi-linear theory of viscoelasticity. J. Biomech. 27:413–419, 1994.
Billiar, K. L. and M. S. Sacks. Biaxial mechanical properties of the natural and gluteraldehyde treated aortic valve cusp—Part I: Experimental results. J. Biomech. Eng. 122:23–30, 2000a.
Billiar, K. L., and M. S. Sacks. Biaxial mechanical properties of the natural and gluteraldehyde treated aortic valve cusp—Part II: A structural constitutive model. J. Biomech. Eng. 122:327–335, 2000b.
Bischoff, J. E. Static indentation of anisotropic biomaterials using axially asymmetric indenters—A computational study. J. Biomech. Eng. 126(4):498–505, 2004.
Bischoff, J. E., E. M. Arruda, and K. Grosh. Orthotropic hyperelasticity in terms of an arbitrary molecular chain model. J. Appl. Mech. 69(2):198–201, 2002a.
Bischoff, J. E., E. M. Arruda, and K. Grosh. A microstructurally based orthotropic hyperelastic constitutive law. J. Appl. Mech. 69:570–579, 2002b.
Bischoff, J. E., E. M. Arruda, and K. Grosh. A rheological network model for the continuum anisotropic and viscoelastic behavior of soft tissue. Biomech. Modeling Mechanobiol. 3(1):56–65, 2004.
Carew, E. O., E. A. Talman, D. R. Boughnew, and I. Vesely. Quasi-linear viscoelastic theory applied to internal shearing of porcine aortic valve leaflets. J. Biomech. Eng. 121:386–392, 1999.
Decraemer, W. F., M. A. Maes, V. J. Vanhuyse, and P. Vanpeperstraete. A non-linear viscoelastic constitutive equation for soft biological tissues, based upon a structural model. J. Biomech. 13:559–564, 1980.
Doehring, T. C., E. O. Carew, and I. Vesely. The effect of strain rate on the viscoelastic response of aortic valve tissue: A direct fit approach. Ann. Biomed. Eng. 32(2):223–232, 2004.
Fung, Y. C. Stress-strain-history relations of soft tissues in simple elongation. In: Biomechanics: Its Foundations and Objectives, edited by Y. C. Fung, N. Perrone, and M. Anliker. Englewood Cliffs, Prentice-Hall, 1972.
Fung, Y. C. Biomechanics: Mechanical Properties of Living Tissues, 2nd ed. New York: Springer-Verlag, Inc., 1993.
Gutsmann, T., G. E. Fantner, J. H. Kindt, M. Venturone, S. Danielsen, and P. K. Hansma. Force spectroscopy of collagen fibers to investigate their mechanical properties and structural organization. Biophys. J. 86:3186–3193, 2004.
Haut, R. C., and R. W. Little. A constitutive equation for collagen fibers. J. Biomech. 5:423–430, 1972.
Holzapfel, G. A., and T. C. Gasser. A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications. Comp. Methods Appl. Mech. Eng. 190:4379–4403, 2001.
Johnson, G. A., D. M. Tramaglini, R. E. Levine, K. Ohno, N.-Y. Choi, and S. L. Y. Woo. Tensile and viscoelastic properties of human patellar tendon. J. Orthopaed. Res. 12:796–803, 1994.
Kwan, M. K., T. H. C. Lin, and S. L. Y. Woo. On the viscoelastic properties of the anteromedial bundle of the anterior cruciate ligament. J. Biomech. 26(4/5):447–452, 1993.
Lanir, Y. A structural theory for the homogeneous biaxial stress-strain relationships in flat collagenous tissues. J. Biomech. 12:423–426, 1979.
Lanir, Y. A microstructural model for the rheology of mammalian tendon. J. Biomech. 102:332–339, 1980.
Lanir, Y. Constitutive equations for fibrous connective tissues. J. Biomech. 16(1):1–12, 1983
Limbert, G., and J. Middleton. A transversely isotropic viscohyperelastic material—Application to the modeling of biological soft connective tissues. Int. J. Solids Struct. 41:4237–4260, 2004.
Nagatomi, J., D. C. Gloeckner, M. B. Chancellor, W. C. DeGroat, and M. S. Sacks. Changes in the biaxial viscoelastic response of the urinary bladder following spinal cord injury. Ann. Biomed. Eng. 32(10):1409–1419, 2004.
Pioletti, D. P., and L. R. Rakotomanana. Non-linear viscoelastic laws for soft biological tissues. Eur. J. Mech. A/Solids 19:749–759, 2000.
Provenzano, P., R. Lakes, T. Keenan, and R. Vanderby, Jr. Nonlinear ligament viscoelasticity. Ann. Biomed. Eng. 29:908–914, 2001.
Puso, M. A., and J. A. Weiss. Finite element implementation of anisotropic quasi-linear viscoelasticity using a discrete spectrum approximation. J. Biomech. Eng. 120:62–70, 1998.
Rubin, M. B., S. R. Bodner. A three-dimensional nonlinear model for dissipative response of soft tissue. Int. J. Solids Struc. 39:5081–5099, 2002.
Sacks, M. S. Biaxial mechanical evaluation of planar biological materials. J. Elast. 61:199–246, 2000.
Sacks, M. S. Incorporation of experimentally-derived fiber orientation into a structural constitutive model for planar collagenous tissues. J. Biomech. Eng. 125:280–287, 2003.
Sverdlik, A., and Y. Lanir. Time-dependent mechanical behavior of sheep digital tendons, including the effects of preconditioning. J. Biomech. Eng. 124:78–84, 2002.
Thornton, G. M., C. B. Frank, and N. G. Shrive. Ligament creep behavior can be predicted from stress relaxation by incorporating fiber recruitment. J. Rheol. 45(2):493–507, 2001.
Woo, S. L. Y., M. A. Gomez, and W. H. Akeson. The time and history-dependent viscoelastic properties of canine medial collateral ligament. J. Biomech. Eng. 103:293–298, 1981.
ACKNOWLEDGMENT
The author gratefully acknowledges the work of Mr. Jing Lu in acquiring the experimental data presented in this paper.
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Bischoff, J.E. Reduced Parameter Formulation for Incorporating Fiber Level Viscoelasticity into Tissue Level Biomechanical Models. Ann Biomed Eng 34, 1164–1172 (2006). https://doi.org/10.1007/s10439-006-9124-6
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DOI: https://doi.org/10.1007/s10439-006-9124-6