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TECHNICAL BRIEFS

Stress Distribution in a Circular Membrane With a Central Fixation

[+] Author and Article Information
Daisuke Mori, Guido David, Jay D. Humphrey

Department of Biomedical Engineering,  Texas A&M University 337 Zachry Engineering Center, 3120 TAMU College Station, TX, 77843-3120

James E. Moore1

Department of Biomedical Engineering,  Texas A&M University 337 Zachry Engineering Center, 3120 TAMU College Station, TX, 77843-3120

1

Corresponding author; email: jmoorejr@tamu.edu (James E. Moore, Jr.)

J Biomech Eng 127(3), 549-553 (Jan 31, 2005) (5 pages) doi:10.1115/1.1894389 History: Received August 25, 2004; Revised December 28, 2004; Accepted January 31, 2005

Clinical interventions can change the mechanical environment of the tissues targeted for therapy. In order to design better procedures, it is important to understand cellular responses to altered mechanical stress. Rigid fixation is one example of a constraint imposed on living tissues as a result of implanted devices. This results in disturbed stress and strain fields, with potentially strong gradients. Herein, we numerically solve the governing nonlinear ordinary differential equation for the stress distribution in a finitely deformed anisotropic circular membrane with a concentric fixation by applying a zero-displacement condition at the inner circumference. Results show that rigid fixations yield distributions of stress and strain that are markedly different from tissue defects with traction-free boundaries. Moreover, the material anisotropy plays a significant role in the manner the stress redistributes regardless of the size of fixation. The present study will contribute to the design of experiments to determine cellular reactions involved in the failure of interventional treatments.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic illustration of an elastic response of a circular membrane subjected to uniform radial traction along its outer circumference. The membrane has a central circular fixation. A material particle at (R,Θ) in the reference configuration is mapped to (r,θ) in the deformed configuration. The deformed inner radius ri equals to the referential inner radius Ri because of the central fixation. The initial outer radius Ro is increased to the deformed outer radius ro with the application of a load. The nonzero components of stress, which act on an infinitesimal element on the membrane, are radial stress trr and circumferential stress tθθ.

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Figure 4

Distributions of stresses and stretches for circumferentially stiffer membranes with a radius of fixation of 0.001, 0.5, 1.0, or 1.5, and an initial outer radius of 2.0 stretched by 10%. Material parameters are c1=0.5,c2=1.0, and c3=0.005. In the small fixation (Ri=0.001), the stresses and stretches near the fixation were much smaller than in the isotropic case (Fig. 3). In the large fixation (Ri=1.5), the maximum radial stresses at the inner boundaries were 0.37 lower than for the isotropic case. The changes were large in comparison with those for middle-sized fixations (Ri=0.5 and 1.0).

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Figure 5

Distributions of stresses and stretches for radially stiffer membranes with a radius of fixation of 0.001, 0.5, 1.0, or 1.5, and an initial outer radius of 2.0 stretched by 10%. Material parameters are c1=1.0,c2=0.5, and c3=0.005. In the small fixation (Ri=0.001), the radial stiffness of the membrane resulted in very high values in the stresses and stretches near the fixation. Furthermore, the large fixation (Ri=1.5) remarkably increased the radial stress, while the stresses in the middle sized fixations (Ri=0.5 and 1.0) were not changed much in value, in comparison with the isotropic case.

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Figure 3

Distributions of stresses and stretches for isotropic membranes with a radius of fixation of 0.001, 0.5, 1.0, or 1.5, and an initial outer radius of 2.0 stretched by 10%. Material parameters are c1=0.75,c2=0.75, and c3=0.005. The radius of fixation of 0.001 resulted in a concentration of the radial stress trr at the inner radius. The radial stress increased with increasing fixation size.

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Figure 2

Distributions of stresses and stretches for isotropic membranes stretched 5%, 10%, 15%, 20%, or 25%. Material parameters are c1=0.75,c2=0.75, and c3=0.005. The radius of fixation Ri is 1.0, and the outer undeformed radius Ro is 2.0. The radial stress trr (upper right) and the circumferential stress tθθ (lower right) monotonically decreased and increased, respectively, with radial position. The radial stress exhibited a maximum value and larger gradient at the inner radius.

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