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Research Papers

Evaluation of Global Load Sharing and Shear-Lag Models to Describe Mechanical Behavior in Partially Lacerated Tendons

[+] Author and Article Information
Marco Pensalfini

Department of Industrial Engineering,
University of Bologna,
Bologna 33, 40126Italy;
Department of Biomedical Engineering,
University of Wisconsin-Madison,
Madison, WI 53705

Sarah Duenwald-Kuehl

Department of Biomedical Engineering,
University of Wisconsin-Madison,
Madison, WI 53705;
Department of Orthopedic and Rehabilitation,
University of Wisconsin-Madison,
Madison, WI 53705

Jaclyn Kondratko-Mittnacht

Department of Biomedical Engineering,
University of Wisconsin-Madison,
Madison, WI 53705;
Department of Orthopedic and Rehabilitation,
University of Wisconsin-Madison,
Madison, WI 53705

Roderic Lakes

Department of Engineering Physics,
University of Wisconsin-Madison,
Madison, WI 53705;
Department of Materials Science,
University of Wisconsin-Madison,
Madison, WI 53705

Ray Vanderby

Department of Biomedical Engineering,
University of Wisconsin-Madison,
Madison, WI 53705;
Department of Orthopedic and Rehabilitation,
University of Wisconsin-Madison,
Madison, WI 53705;
Materials Science Program,
University of Wisconsin-Madison,
Madison, WI 53705
e-mail: vanderby@ortho.wisc.edu

1Corresponding author.

2Present address: 5059 Wi Institute Medical Research, 1111 Highland Avenue, Madison, WI 53705.

Manuscript received February 6, 2014; final manuscript received May 14, 2014; accepted manuscript posted May 21, 2014; published online July 15, 2014. Assoc. Editor: Guy M. Genin.

J Biomech Eng 136(9), 091006 (Jul 15, 2014) (12 pages) Paper No: BIO-14-1068; doi: 10.1115/1.4027714 History: Received February 06, 2014; Revised May 14, 2014; Accepted May 21, 2014

The mechanical effect of a partial thickness tear or laceration of a tendon is analytically modeled under various assumptions and results are compared with previous experimental data from porcine flexor tendons. Among several fibril-level models considered, a shear-lag model that incorporates fibril–matrix interaction and a fibril–fibril interaction defined by the contact area of the interposed matrix best matched published data for tendons with shallow cuts (less than 50% of the cross-sectional area). Application of this model to the case of many disrupted fibrils is based on linear superposition and is most successful when more fibrils are incorporated into the model. An equally distributed load sharing model for the fraction of remaining intact fibrils was inadequate in that it overestimates the strength for a cut less than half of the tendon's cross-sectional area. In a broader sense, results imply that shear-lag contributes significantly to the general mechanical behavior of tendons when axial loads are nonuniformly distributed over a cross section, although the predominant hierarchical level and microstructural mediators for this behavior require further inquiry.

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Figures

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Fig. 1

K3 (postlaceration stress as a percentage of prelaceration value) predictions versus q% (percentage of cut fibers) using the GLS model. A simple linear relation between reduction in stress and percentage of broken fibers is obtained.

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Fig. 2

K3 predictions versus q% (percent cut fibers for the 100 fiber case) according to the Hedgepeth model. An asymptotic shape is obtained. Note that values of K3 depend directly on the absolute number of cut fibers in the material rather than on the percent of cut fibers.

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Fig. 3

Model composite demonstrating intact and broken fibers. ϑ ranges from 0, along the line from the center of the intact fiber to the center of the broken fiber, to ϑMAXi = arccos(ρi-1), at the intersection between the fiber radius and the tangent drawn from the center of the broken fiber, where ρi = 2Ri/r0. Adapted from Ref. [20].

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Fig. 4

Model two-dimensional composite; this example contains multiple intact and one broken fibril. Inset illustrates the linear cross-sectional profile (compared to traditional, elliptical, three-dimensional profile of true tendon).

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Fig. 5

Sensitivity of z99.9% to λc (both normalized to d0) according to the Cox and Nairn models. The value assumed for λc is meaningful for λc≤100·d0.

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Fig. 6

K3 versus fibril position (number of fibrils away from the notch) and distance z along the fibril normalized to fibril diameter d0 using the (a) Cox model or (b) Nairn model shear-lag parameter with 20 cut fibrils in a simplified 100-fibril tendon model. K3 increases with increased fibril position and distance along the fibril from the cut. The dependence on z/d0 is stronger in Nairn's model.

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Fig. 7

K3 at the first intact fibril next to the notch (fibril position = 21) versus distance z (along the fiber) normalized to fibril diameter d0 calculated using Cox and Nairn models with 20 cut fibrils in a 100-fibril tendon model. The stronger dependence of Nairn's curve on z/d0 can be observed.

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Fig. 8

Minimum K3 versus fibril position at z/d0 = 0 calculated using Cox and Nairn models with 20 cut fibrils in a 100-fibril tendon model. The curves obtained with Cox's and Nairn's models are the same at the position z/d0 = 0.

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Fig. 9

Comparison between theoretical models with 100 aligned fibrils and published experimental results [16]. Wagner and Eitan's model (Wagner) is preferred over the others in the shallow (<50% of tendon CSA) cut region. GLS is a better approximation available for deeper cuts.

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Fig. 10

Effect of the number of fibrils included in Wagner and Eitan's model (Wagner) on the computed K3 for cuts of depths (a) 8.2%, (b) 19.8%, (c) 47.2%, and (d) 63.8% of tendon CSA. The corresponding experimental value is also shown (mean experimental value (solid line) and one standard deviation above/below the mean (dotted lines)). The analytical result better approaches the experimental values when more fibrils are incorporated into the model (>300 fibrils), particularly in the midsize (19.8%, 47.2% tendon CSA) cut depths.

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Fig. 11

Comparison between theoretical models with 1000 aligned fibrils and published experimental results [16]. Wagner and Eitan's model (Wagner) performed best of the investigated models in the pre-50% cut region. The model has an asymptotic shape, precluding it from accurately predicting values at very high cut depths (e.g., predicted values do not go to zero after 100% of the fibrils are cut).

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Fig. 12

Effect of fibril volume fraction on K3 computed with Wagner and Eitan's model (Wagner) for cut depths of (a) 8.2%, (b) 19.8%, (c) 47.2%, and (d) 63.8% of the tendon's CSA. No convergence between the model-predicted values and the experimental data is observed when fibril volume fraction is increased between 10% and 80%.

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