5 Conclusion
We have proposed an efficient, invariant-based alternative to structural constitutive equations that accounts for statistical dispersion of fibers. In contrast to existing models, our new invariant theory easily handles a 3D fiber population with a single mean preferred direction. The invariant theory is based on a novel closed-form ‘splay invariant’ that requires a single parameter in the 2D case, and two parameters in the 3D case. The proposed model is polyconvex, and fits biaxial data for aortic valve tissue better than existing aortic-valve models Billiar and Sacks (2000). A modification in the fiber stress-strain law requires no re-formulation of the constitutive tangent matrix, making the model flexible for different types of soft tissues. Most importantly, the model is computationally expedient in a finite element analysis.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Billiar, K. L., and Sacks, M. S. (2000). Angioscopic findings during coronary angioplasty of coronary occlusions.biaxial mechanical properties of the natural and glutaraldehyde-treated aortic valve cusp — Part II: A structural constitutive model. J. Biomech. Eng. 122:327–335.
Einstein, D. R., Freed, A. D., Stander, N., Fata, B., and Vesely, I. (2005). Inverse parameter fitting of biological tissues: A response surface approach. submitted.
Einstein, D. R. (2002). Nonlinear Acoustic Analysis of the Mitral Valve. Ph.D. Dissertation, University of Washington, Seattle.
Flory, P. J. (1961). Thermodynamic relations for high elastic materials. Trans. Faraday Soc. 57:829–838.
Freed, A. D., and Doehring, T. C. (2005). Elastic model for crimped collagen fibrils. J. Biomech. Eng. in press.
Holzapfel, G. A., Gasser, T. C., and Stadler, M. (2002). A structural model for the viscoelastic behavior of arterial walls: Continuum formulation and finite element analysis. Eur. J. Mech. A/Solids 21:441–463.
Lanir, Y. (1983). Constitutive equations for fibrous connective tissues. J. Biomech. 16:1–12.
Malkus, D. S., and Hughes, T. J. R. (1978). Mixed finite element methods — reduced and selective integration techniques: A unification of concept. Comput. Meth. Appl. Mech. Eng. 15:63–81.
Sacks, M. S. (2000). Biaxial mechanical evaluation of planar biological materials. J. Elasticity 61:199–246.
Sacks, M. S. (2003). Incorporation of experimentally-derived fiber orientation into a structural consitutive model for planar collagenous tissues. J. Biomech. Eng. 125:280–287.
Simo, J. C., and Hughes, T. J. R. (1998). Computational Inelasticity. New York: Springer-Verlag.
Spencer, A. J. M. (1972). Deformations of Fibre-reinforced Materials. Oxford: Clarendon Press.
Sussman, T., and Bathe, K.-J. (1987). A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comput. Struct. 26:357–409.
Zioupos, P., and Barbenel, J. C. (1994). Mechanics of native bovine pericardium. i. the multiangular behaviour of strength and stiffness of the tissue. Biomaterials 15:366–373.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Einstein, D.R., Freed, A.D., Vesely, I. (2006). Invariant Formulation for Dispersed Transverse Isotropy in Tissues of the Aortic Outflow Tract. In: Holzapfel, G.A., Ogden, R.W. (eds) Mechanics of Biological Tissue. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31184-X_25
Download citation
DOI: https://doi.org/10.1007/3-540-31184-X_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25194-1
Online ISBN: 978-3-540-31184-3
eBook Packages: EngineeringEngineering (R0)