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Technical Brief

A Generalized Method for the Analysis of Planar Biaxial Mechanical Data Using Tethered Testing Configurations

[+] Author and Article Information
Will Zhang, Yuan Feng, Chung-Hao Lee

Department of Biomedical Engineering,
Center for Cardiovascular Simulation,
Institute for Computational Engineering and Sciences,
The University of Texas at Austin,
Austin, TX 78712-1229

Kristen L. Billiar

Department of Biomedical Engineering,
Worcester Polytechnic Institute,
Worcester, MA 01609-2280

Michael S. Sacks

W. A. “Tex” Moncrief, Jr. Simulation-Based
Engineering Science Chair I,
Professor of Biomedical Engineering,
Institute for Computational Engineering and Sciences,
Department of Biomedical Engineering,
Center for Cardiovascular Simulation,
The University of Texas at Austin,
Austin, TX 78712-1229
e-mail: msacks@ices.utexas.edu

1Present address: School of Mechanical and Electronic Engineering, Soochow University, SuZhou, Jiangsu 512000, China.

2Corresponding author.

Manuscript received November 10, 2013; final manuscript received November 9, 2014; published online April 15, 2015. Assoc. Editor: Stephen M. Klisch.

J Biomech Eng 137(6), 064501 (Jun 01, 2015) (13 pages) Paper No: BIO-13-1524; doi: 10.1115/1.4029266 History: Received November 10, 2013; Revised November 09, 2014; Online April 15, 2015

Simulation of the mechanical behavior of soft tissues is critical for many physiological and medical device applications. Accurate mechanical test data is crucial for both obtaining the form and robust parameter determination of the constitutive model. For incompressible soft tissues that are either membranes or thin sections, planar biaxial mechanical testing configurations can provide much information about the anisotropic stress–strain behavior. However, the analysis of soft biological tissue planar biaxial mechanical test data can be complicated by in-plane shear, tissue heterogeneities, and inelastic changes in specimen geometry that commonly occur during testing. These inelastic effects, without appropriate corrections, alter the stress-traction mapping and violates equilibrium so that the stress tensor is incorrectly determined. To overcome these problems, we presented an analytical method to determine the Cauchy stress tensor from the experimentally derived tractions for tethered testing configurations. We accounted for the measured testing geometry and compensate for run-time inelastic effects by enforcing equilibrium using small rigid body rotations. To evaluate the effectiveness of our method, we simulated complete planar biaxial test configurations that incorporated actual device mechanisms, specimen geometry, and heterogeneous tissue fibrous structure using a finite element (FE) model. We determined that our method corrected the errors in the equilibrium of momentum and correctly estimated the Cauchy stress tensor. We also noted that since stress is applied primarily over a subregion bounded by the tethers, an adjustment to the effective specimen dimensions is required to correct the magnitude of the stresses. Simulations of various tether placements demonstrated that typical tether placements used in the current experimental setups will produce accurate stress tensor estimates. Overall, our method provides an improved and relatively straightforward method of calculating the resulting stresses for planar biaxial experiments for tethered configurations, which is especially useful for specimens that undergo large shear and exhibit substantial inelastic effects.

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Figures

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Fig. 3

The configurations Ω0, Ω1, and Ωt for a glutaraldehyde treated aortic valve leaflet (a) and the RV myocardium (b) are shown. These represent the typical change in the reference configuration for a typical biaxial experiment due to preconditioning and other inelastic run-time effects. Note that the RV data, tissue is sheared in the negative direction during preconditioning. It is then sheared back in the positive direction during loading.

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Fig. 2

Different specimen configurations used during biax testing. Ω0 is the original stress-free and undeformed free floating state. Ω1 is an intermediate configuration due to mounting, preconditioning and other inelastic run-time effects, which can be described by the deformation  01F. Ωt is the current deformed state. The overall change in configuration is given by  0tF, where the deformation due to stress is given by  1tF.

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Fig. 1

(a) Typical biaxial mechanical test configuration and (b) schematic of the forces and dimensions. Note that f(1) acts on an area of A(1)= L2 × L3 and f(2) on A(2)= L1 × L3.

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Fig. 5.

(a) The main components of the biaxial system are constructed and simulated in FE. The (b) free floating and (c) preconditioned states are shown.

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Fig. 4

Flowchart of for the geometric rigid body moment minimization and the simulation of the biaxial geometry and stresses

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Fig. 6

The four material models used in the heterogeneous specimens. They are for percardium tissue (a) and (b) and valvular tissues (c) and (d). The specimen is rotated for normal loading (a) and (c) and shear loading (b) and (d).

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Fig. 11

Close up of side 1 of the biaxial testing device under deformation of a specimen. The normal vector n and the average traction vector T oriented at 0.85 deg and 7.6 deg are shown by the dashed arrows. The pulley system rotates about the pivot oY by the angle ϕ, and transverses along the test axis by the distance δ (not shown). The tethers are represented by the vectors vi, which attach to the tissue at the points xi= F · Xi and are tangent to the pulley shafts at the points yi= R(ϕ)Yi+ oY.

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Fig. 7

The results for the preconditioned pericardium specimen with (a) tether arrangement (A). The (b) t22 stress distribution, (c) normal stresses, and (d) shear stresses are shown.

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Fig. 8

The results for the preconditioned pericardium specimen with (a) tether arrangement (B). The (b) t22 stress distribution, (c) normal stresses, and (d) shear stresses are shown.

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Fig. 9

The results for the preconditioned pericardium specimen with (a) tether arrangement (C). The (b) t22 stress distribution, (c) normal stresses, and (d) shear stresses are shown.

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Fig. 10

The results for the preconditioned pericardium specimen with (a) tether arrangement (D). The (b) t22 stress distribution, (c) normal stresses, and (d) shear stresses are shown.

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