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TECHNICAL PAPERS: Soft Tissue

Experimentally Tractable, Pseudo-elastic Constitutive Law for Biomembranes: II. Application

[+] 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), 100-105 (Feb 14, 2003) (6 pages) doi:10.1115/1.1535192 History: Received November 01, 2001; Revised September 01, 2002; Online February 14, 2003
Copyright © 2003 by ASME
Topics: Stress , Biomembranes
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References

Figures

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Green strain components for the biaxial tests with M and S as the stretch directions. The solid lines represent tests with EMM/ESS=tanθ and θ=0,7.5,15,[[ellipsis]], 90. The dashed lines represent trajectories with γ12=tanθ. The ‘o’ indicate data points from loading tests that attempted to hold EMM/ESS=tanθ with θ=0,18.4,26.6,45,63.4,71.6,90 (equivalently, EMM:ESS=1:0,3:1,2:1,1:1,1:2,1:3, 0:1). Note that the solid and dash lines are similar and differ to a lesser extent than the amount by which the data depart from their ideal trajectories.
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Second Piola-Kirchoff component SMM for each loading state. The ‘×’ represent the actual data and the surface is our piecewise polynomial fit to the data. The surface is C1 continuous, monotonic and representative of the pseudo-elastic loading behavior of this membrane (RMS deviation of data from surface is 1.42 kPa or 5.3% of the mean SMM value.)
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Second Piola-Kirchoff component SSS for each loading state. The ‘×’ represent the actual data and the surface is our piecewise polynomial fit to the data. The surface is C1 continuous, monotonic and representative of the pseudo-elastic loading behavior of this membrane (RMS deviation of data from surface is 0.68 kPa or 4.2% of the mean SSS value.)
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The work done Ω (per unit reference volume) when loaded with γ12 held at a constant ratio. The work to get to each point ‘o’ was calculated by assuming that the SMM and SSS values at each state on the loading path are respectively given by the surfaces in Figs. 2 and 3. The lines represent the Ω function given by (4). Note the accuracy of the fit.
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Intercepts of the Ω surface in Fig. 4 with the planes γ2=0 (left) and γ1=0 (right). Note that the scale is different because the predominate direction is almost twice as stiff as the cross-predominate direction.
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The resulting cross-dependence of Ω on γ1 and γ2 when the contributions of the functions in Fig. 5 (i.e., Ω(γ1,0,0) and Ω(0,γ2,0)) are subtracted from Ω(γ12,0). The lines are from the polynomial fit indicated.
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The dependence of the stress remainder function ω on γ1 or γ2 when the other is held at zero. Note that they are nearly equal but opposite.
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The stress response data can be accurately represented by Ω in (4) and tR in (5) with (7). Equations (8a8b) were used to calculate the predicted values of SMM and SSS at the particular strain state of the measured values of SMM and SSS. For the top panel tR was as in (5) with (7). For comparison, the bottom panel shows how the prediction worsens when tR is null. All lines are lines of identity which represent perfect prediction. The lines are offset for each separate testing trajectory with the ratio EMM:ESS given between the top and bottom panels.
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Magnification of the small stress region of Fig. 8. Note that the minor stresses cannot be accurately predicted if tR is null. The minor stresses are those stress values that are associated with the lesser stretch direction (e.g., SMM is the minor stress when EMM is held at zero). Note that equibiaxial stretch (1:1) is accurately represented with and without tR.

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