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TECHNICAL PAPERS: Soft Tissues

A Sub-Domain Inverse Finite Element Characterization of Hyperelastic Membranes Including Soft Tissues

[+] Author and Article Information
Padmanabhan Seshaiyer

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042

Jay D. Humphrey

Department of Biomedical Engineering, Texas A&M University, College Station, TX 77843-3120

J Biomech Eng 125(3), 363-371 (Jun 10, 2003) (9 pages) doi:10.1115/1.1574333 History: Received April 01, 2002; Revised February 01, 2003; Online June 10, 2003
Copyright © 2003 by ASME
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References

Kavanaugh,  K. T., and Clough,  R. W., 1971, “Finite Element Applications in the Characterization of Elastic Solids,” Int. J. Solids Struct., 7, pp. 11–23.
Iding,  R. H., Pister,  K. S., and Taylor,  R. L., 1974, “Identification of Nonlinear Elastic Solids by a Finite Element Method,” Comput. Methods Appl. Mech. Eng., 4, pp. 121–142.
Kyriacou,  S. K., Shah,  A. D., and Humphrey,  J. D., 1997, “Inverse Finite Element Characterization of Nonlinear Hyperelastic Membranes,” J. Appl. Mech., 64, pp. 257–262.
Humphrey,  J. D., 1998, “Computer Methods in Membrane Biomechanics,” Computer Methods in Biomechanics and Biomedical Engineering, 1, pp. 171–210.
Twizell,  E. H. and Ogden,  R. W., 1983, “Nonlinear Optimization of the Material Constants in Ogden’s Stress-deformation Function for Incompressible Isotropic Elastic Materials,” J. Austral. Math. Soc., 24, pp. 424–434.
Humphrey,  J. D., Strumpf,  R. K., and Yin,  F. C. P., 1990, “Determination of a Constitutive Relation for Passive Myocardium: II. Parameter Estimation,” ASME J. Biomech. Eng., 112, pp. 340–346.
Humphrey, J. D., 2002, Cardiovascular Solid Mechanics: Cells, Tissues, and Organs, Springer-Verlag, NY.
Humphrey,  J. D., Strumpf,  R. K., and Yin,  F. C. P., 1992, “A Constitutive Theory for Biomembranes: Application to Epicardium,” ASME J. Biomech. Eng., 114, pp. 461–466.
Hsu,  F. P. K., Schwab,  C., Rigamonti,  D., and Humphrey,  J. D., 1994, “Identification of Response Functions for Nonlinear Membranes via Axisymmetric Inflation Tests: Implications for Biomechanics,” Int. J. Solids Struct., 31, pp. 3375–3386.
Oden,  J. T., and Sato,  T., 1967, “Finite Strains and Displacements of Elastic Membranes by the Finite Element Method,” Int. J. Solids Struct., 3, pp. 471–488.
Oden, J. T., 1972, Finite Elements of Nonlinear Continua., McGraw-Hill, NY.
Wriggers,  P., and Taylor,  R. L., 1990, “A Fully Nonlinear Axisymmetrical Membrane Element for Rubber-like Materials,” Eng. Comput., 7, pp. 303–310.
Gruttmann,  F., and Taylor,  R. L., 1992, “Theory and Finite Element Formulation of Rubberlike Membrane Shells Using Principal Stretches,” Int. J. Numer. Methods Eng., 35, pp. 1111–1126.
Kyriacou,  S. K., Schwab,  C., and Humphrey,  J. D., 1996, “Finite Element Analysis of Nonlinear Orthotropic Hyperelastic Membranes,” Computational Mechanics, 18, pp. 269–278.
Hsu,  F. P. K., Downs,  J., Liu,  A. M. C., Rigamonti,  D., and Humphrey,  J. D., 1995, “A Triplane Video-based Experimental System for Studying Axisymmetrically Inflated Biomembranes,” IEEE Trans. Biomed. Eng., 42, pp. 442–449.
Shah,  A. D., Harris,  J. L., Kyriacou,  S. K., and Humphrey,  J. D., 1997, “Further Roles of Geometry and Properties in the Mechanics of Saccular Aneurysms,” Computer Methods in Biomechanics and Biomedical Engineering, 1, pp. 109–121.
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Figures

Grahic Jump Location
Schema of a general membrane inflation. Panel A shows five markers that can be tracked experimentally using a video system whereas Panel B shows the associated four-element computational “sub-domain” Ωs⊂Ω. The sub-domain can be made as small as allowed experimentally, and can be repeated at multiple locations on the specimen to explore possible material heterogeneities.
Grahic Jump Location
Schema showing a possible four element sub-domain on an inflated sphere (panel A) and an axisymmetrically inflated membrane (panel B). The latter also shows a three noded element used in the forward problem in an axisymmetric finite element solution.
Grahic Jump Location
Results for the inflation of a neo-Hookean spherical membrane as a function of the in-plane stretch λ. The top and middle panels show the material response for moderate stretches, less than that associated with a limit point instability (λ=71/6). The bottom panel shows the estimated value of the neo-Hookean parameter for increasing values of λ, which includes increasing numbers of equilibrium configurations.
Grahic Jump Location
Estimated parameter values versus number of equilibrium configurations for a neo-Hookean sphere inflated to a stretch of 1.33; noise is 0.01 mm.
Grahic Jump Location
Similar to Figure 4 except for a Mooney-Rivlin sphere inflated to a stretch of 1.33 (Γ=0.89); noise is 0.01 mm.
Grahic Jump Location
Similar to Figure 4 except for a Fung-exponential sphere inflated to a stretch of 1.1 (α=0.2,β=1.0); noise is 0.01 mm.
Grahic Jump Location
Estimated parameter values as a function of increasing experimental noise for a neo-Hookean sphere inflated to a stretch of 1.33 via 10 equilibrium configurations.
Grahic Jump Location
Similar to Figure 7 except for a Mooney-Rivlin sphere inflated to a stretch of 1.33 via 10 equilibrium configurations.
Grahic Jump Location
Similar to Figure 7 except for a Fung-exponential sphere inflated to a stretch of 1.05 via 10 equilibrium configurations.
Grahic Jump Location
Estimated parameter values for an orthotropic Fung-exponential material over element 10 (left panels) and over element 17 (right panels). See text for details.

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