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Research Papers

Biaxial Tension of Fibrous Tissue: Using Finite Element Methods to Address Experimental Challenges Arising From Boundary Conditions and Anisotropy

[+] Author and Article Information
Nathan T. Jacobs

Department of Mechanical Engineering
and Applied Mechanics,
University of Pennsylvania,
229 Towne Building, 220 South 33rd Street,
Philadelphia, PA 19104

Daniel H. Cortes

Biomedical Engineering Program,
University of Delaware,
125 E. Delaware Ave.,
Newark, DE 19716

Edward J. Vresilovic

Penn State Hershey Bone and Joint Institute,
Pennsylvania State University,
Suite 2400, Building B, 30 Hope Drive,
Hershey, PA 17033

Dawn M. Elliott

Biomedical Engineering Program,
University of Delaware,
125 E. Delaware Ave.,
Newark, DE 19716
e-mail: delliott@udel.edu

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received November 12, 2012; final manuscript received January 15, 2013; accepted manuscript posted January 29, 2013; published online February 11, 2013. Editor: Victor H. Barocas.

J Biomech Eng 135(2), 021004 (Feb 11, 2013) (10 pages) Paper No: BIO-12-1551; doi: 10.1115/1.4023503 History: Received November 12, 2012; Revised January 15, 2013; Accepted November 29, 2013

Planar biaxial tension remains a critical loading modality for fibrous soft tissue and is widely used to characterize tissue mechanical response, evaluate treatments, develop constitutive formulas, and obtain material properties for use in finite element studies. Although the application of tension on all edges of the test specimen represents the in situ environment, there remains a need to address the interpretation of experimental results. Unlike uniaxial tension, in biaxial tension the applied forces at the loading clamps do not transmit fully to the region of interest (ROI), which may lead to improper material characterization if not accounted for. In this study, we reviewed the tensile biaxial literature over the last ten years, noting experimental and analysis challenges. In response to these challenges, we used finite element simulations to quantify load transmission from the clamps to the ROI in biaxial tension and to formulate a correction factor that can be used to determine ROI stresses. Additionally, the impact of sample geometry, material anisotropy, and tissue orientation on the correction factor were determined. Large stress concentrations were evident in both square and cruciform geometries and for all levels of anisotropy. In general, stress concentrations were greater for the square geometry than the cruciform geometry. For both square and cruciform geometries, materials with fibers aligned parallel to the loading axes reduced stress concentrations compared to the isotropic tissue, resulting in more of the applied load being transferred to the ROI. In contrast, fiber-reinforced specimens oriented such that the fibers aligned at an angle to the loading axes produced very large stress concentrations across the clamps and shielding in the ROI. A correction factor technique was introduced that can be used to calculate the stresses in the ROI from the measured experimental loads at the clamps. Application of a correction factor to experimental biaxial results may lead to more accurate representation of the mechanical response of fibrous soft tissue.

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References

Figures

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Fig. 4

Effect of fiber angle. XISO with fibers oriented 25 deg to the x loading axis (top) and ORTH with fibers aligned ±25 deg to the x and y axes (bottom). (a) Diagram of fiber orientation with solid lines representing fibers. (b) Color map of von Mises stress concentration; color map depicts ratio of stress to ROI stress. (c) Plot of ROI stress versus clamp stress for the complete FE simulation, with all stress values normalized to peak clamp stress.

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Fig. 3

XISO (top) and ORTH (bottom) response for square geometry. (a) Diagram of fiber orientations with solid lines representing fibers. (b) Von Mises stress concentration; color map depicts ratio of stress to ROI stress. (c) Plot of ROI stress versus clamp stress for the complete FE simulation, with all stress values normalized to peak clamp stress. The solid line is obtained through linear regression, and the correction factor is the corresponding slope of the regression. Correction factors closer to 1 indicate ROI stresses that are more similar to the applied clamp stresses.

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Fig. 2

(a) Uniaxial tension stress-strain data from O'Connell 2009. Filled circles are individual sample responses, while the open circles and solid line represent the average of the experimental dataset (n = 5). (b) Average experimental stress-strain curve from (a) with Matlab fit and FEBio prediction, demonstrating excellent agreement between experimental data and FE prediction for the uniaxial response.

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Fig. 1

Biaxial tension response for isotropic material. (a) Square and (b) cruciform color map of von Mises stress concentration with color scale representing the ratio of stress to the average ROI stress. Internal arrows point to regions of peak stress concentration. The ROI is denoted by a dotted square, bold arrows indicate loading boundary conditions at the clamps, and the vertical hash line represents the symmetry implemented in the FE simulations. (c) Stress-strain curve for the loading clamp (square and cruciform) and ROI with all responses plotted against the ROI Lagrange strain. Lower stresses are experienced in the ROI than applied at the clamp for both square and cruciform geometries. (d) Stress transfer to the ROI as a percentage of the total stress at the loading clamp for both square and cruciform geometries.

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Fig. 5

Sensitivity of correction factor to the initial choice of material properties. X-axis is the factor change in initial material property, and y-axis is the resulting factor change in correction factor (new correction factor ÷ initial correction factor). (a) Correction factor for ISO material is independent of modulus (square) and dependent on Poisson ratio (diamond). (b) Correction factor for XISO-X is independent of fiber modulus c4 (square) and fiber nonlinearity c5 (triangle). (c) Correction factor for XISO-Y is linearly dependent on both fiber modulus c4 (square) and nonlinearity c5 (triangle). (d) Correction factor for ORTH is independent of fiber modulus c4 (square) and nonlinearity c5 (triangle).

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