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TECHNICAL PAPERS: Soft Tissue

Experimentally Tractable, Pseudo-elastic Constitutive Law for Biomembranes: I. Theory

[+] Author and Article Information
John C. Criscione

Dept. of Bioengineering, University of California San Diego, La Jolla, CA 92093

Michael S. Sacks

Dept. of Bioengineering, University of Pittsburgh, Pittsburgh, PA 15260

William C. Hunter

Dept. of Biomedical Engrng., The Johns Hopkins University, Baltimore, MD 21205

J Biomech Eng 125(1), 94-99 (Feb 14, 2003) (6 pages) doi:10.1115/1.1530770 History: Received November 01, 2001; Revised September 01, 2002; Online February 14, 2003
Copyright © 2003 by ASME
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References

Sacks,  M. S., 2000, “Biaxial Mechanical Evaluation of Planar Biological Materials,” J. Elast., 61, pp. 199–246.
Fung, Y. C., 1993, Biomechanics: Mechanical Properties of Living Tissues (2nd Ed.), New York, Springer-Verlag.
Humphrey,  J. D., Strumpf,  R. K., Yin,  F. C. P., 1992, “A Constitutive Theory for Biomembranes: Application to Epicardial Mechanics,” J. Biomech. Eng., 114, pp. 461–466.
Kang,  T., Humphrey,  J. D., Yin,  F. C. P., 1996, “Comparison of Biaxial Mechanical Properties of Excised Endocardium and Epicardium,” Am. J. Physiol., 270(6 PART 2), pp. H2169–H2176.
Criscione,  J. C., Humphrey,  J. D., Douglas,  A. S., Hunter,  W. C., 2000, “An Invariant Basis for Natural Strain which Yields Orthogonal Stress Response Terms in Isotropic Hyperelasticity,” J. Mech. Phys. Solids, 48, pp. 2445–2465.
Criscione,  J. C., Douglas,  A. S., Hunter,  W. C., 2001, “Physically Based Strain Invariant Set for Materials Exhibiting Transversely Isotropic Behavior,” J. Mech. Phys. Solids, 49, pp. 871–897.
Criscione, J. C., McCulloch, A. D., Hunter, W. C., 2001, “Constitutive Framework Optimized for Myocardium and Other High-strain, Laminar Materials with One Fiber Family,” J. Mech. Phys. Solids, in press, accepted October, 2001.
Sacks,  M. S., Smith,  D. B. , 1997, “A Small Angle Light Scattering Device for Planar Connective Tissue Microstructural Analysis,” Ann. Biomed. Eng., 25(4), pp. 678–89.
Sacks,  M. S., Chuong,  C. J., 1998, “Orthotropic Mechanical Properties of Chemically Treated Bovine Pericardium,” Ann. Biomed. Eng., 26, pp. 892–902.
Truesdell,  C., 1955, “Hypoelasticity,” J. Rational Mech. Anal., 4 , 83–133 (1019–1020).
Truesdell,  C., 1955, “The Simplest Rate Theory of Pure Elasticity,” Commun. Pure Appl. Math., 8, pp. 123–132.
Criscione, J. C., Sacks, M. S., Hunter, W. C., 2002, “Experimentally Tractable, Pseudo-elastic Constitutive Law for Biomembranes: II. Application,” Submitted to J. Biomech. Eng.

Figures

Grahic Jump Location
Reference configurations (left panels) and current configurations (right panels) of an anisotropic membrane. The dashed lines represent the predominate axis of anisotropy which is in the M and m directions for the reference and current configurations, respectively. N and n are normal to the reference and current membrane planes, respectively. S and s are the in-plane directions orthogonal to M and m , respectively. The three orientations are orthonormal in the reference and current configurations. In the lower panels, a small patch of membrane of unit dimensions in the reference configuration is the view perpendicular to the plane before and after deformation. The size of the patch is assumed to be small (i.e., the unit of measure is small) such that the curvature of the membrane is negligible. The kinematic parameters λMS, and ϕMS are as depicted whereas λH, the ratio of current thickness to reference thickness, is not.
Grahic Jump Location
The dependence of work done on the path taken. Using the load surfaces in 12, the work done to get to point γ1=b1 and γ2=b2 is calculated for two separate paths: (1) stretch in M then stretch in S ; (2) stretch in S then stretch in M . To calculate the “% dependence of work on path” at each point, subtract work (2) from work (1), multiply by 100, and divide by their average. The points with either γ1=0 or γ2=0 must vanish because paths (1) and (2) are identical.
Grahic Jump Location
Significant hypoelastic behavior is evident in the data itself because the mixed partials of W are not equal when EMM is about 0.15 and ESS is small. The left panel plots SSS vs EMM when ESS is held at zero. The slope indicated is 126 kPa. The right panel plots SMM vs EMM when ESS is either held at zero or at one-third of the EMM value. Since the SMM response is the same or even falls when ESS is increased, the mixed partial is zero or negative—it is not 126 kPa when EMM is 0.15.

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