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Research Papers

Crack Propagation Versus Fiber Alignment in Collagen Gels: Experiments and Multiscale Simulation

[+] Author and Article Information
Sarah M. Vanderheiden, Mohammad F. Hadi

Department of Biomedical Engineering,
University of Minnesota,
Minneapolis, MN 55455

V. H. Barocas

Department of Biomedical Engineering,
University of Minnesota,
Minneapolis, MN 55455
e-mail: baroc001@umn.edu

1Corresponding author.

Manuscript received November 20, 2014; final manuscript received September 1, 2015; published online October 27, 2015. Assoc. Editor: Michael Detamore.

J Biomech Eng 137(12), 121002 (Oct 27, 2015) (7 pages) Paper No: BIO-14-1571; doi: 10.1115/1.4031570 History: Received November 20, 2014; Revised September 01, 2015

It is well known that the organization of the fibers constituting a collagenous tissue can affect its failure behavior. Less clear is how that effect can be described computationally so as to predict the failure of a native or engineered tissue under the complex loading conditions that can occur in vivo. Toward the goal of a general predictive strategy, we applied our multiscale model of collagen gel mechanics to the failure of a double-notched gel under tension, comparing the results for aligned and isotropic samples. In both computational and laboratory experiments, we found that the aligned gels were more likely to fail by connecting the two notches than the isotropic gels. For example, when the initial notches were 30% of the sample width (normalized tip-to-edge distance = 0.7), the normalized tip-to-tip distance at which the transition occurred from between-notch failure to across-sample failure shifted from 0.6 to 1.0. When the model predictions for the type of failure event (between the two notches versus across the sample width) were compared to the experimental results, the two were found to be strongly covariant by Fisher’s exact test (p < 0.05) for both the aligned and isotropic gels with no fitting parameters. Although the double-notch system is idealized, and the collagen gel system is simpler than a true tissue, it presents a simple model system for studying failure of anisotropic tissues in a controlled setting. The success of the computational model suggests that the multiscale approach, in which the structural complexity is incorporated via changes in the model networks rather than via changes to a constitutive equation, has the potential to predict tissue failure under a wide range of conditions.

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Figures

Grahic Jump Location
Fig. 1

Typical samples: (a) alignment plot of a nominally isotropic sample cut from a cruciform mold. The direction of the vectors indicates the primary direction of alignment as measured by quantitative polarized-light microscopy [33], and the length of the vector indicates the degree of alignment. Although there is some alignment, particularly near the edges (near the arms of the original cruciform), the sample is close to isotropic. (b) Alignment plot for an anisotropic sample from a uniaxial (dogbone) mold. Very strong alignment is seen throughout the sample. (c) Photograph of a sample under initial tension showing the notch tip-to-edge and notch tip-to-tip distances, which were used to characterize samples.

Grahic Jump Location
Fig. 2

Failure mechanisms: Samples failed either by a crack connecting the two notches (“between,” left side image) or by a crack propagating across the sample from one notch (“across,” right side image)

Grahic Jump Location
Fig. 3

Model networks: Model networks were generated with different degrees of fiber alignment ranging from isotropic to strongly anisotropic. The values of Ω11 for the networks shown are 0.33, 0.4, 0.5, and 0.62 from left to right.

Grahic Jump Location
Fig. 4

Failure behavior of simulated gels: Depending on the size and separation of the notches, simulated gels failed either by forming a single crack running BETWEEN the two notches (left-hand column) or by forming two separate cracks that ran ACROSS the sample (right-hand column)

Grahic Jump Location
Fig. 5

Model results: Model predictions for BETWEEN (triangles) and ACROSS (squares) failure are given for isotropic (Ω = 0.33, solid symbols) and anisotropic (Ω = 0.62, solid symbols) samples. The dotted line separates BETWEEN from ACROSS conditions for the isotropic simulations, and the dashed line separates BETWEEN from ACROSS conditions for the anisotropic simulations. In both cases, samples with the notches deeper into the tissue (lower down on the plot) and samples with the notches farther apart (to the right on the plot) tended to fail ACROSS the sample, whereas those with shorter and closer notches tended to fail BETWEEN the notches. The dashed line has a slightly higher slope and is shifted considerably to the right when compared to the dotted line, showing that the aligned model is more likely to fail between the notches.

Grahic Jump Location
Fig. 6

Force trace: Typical force trace for isotropic (dashed line) and anisotropic (solid line) samples. As expected based on Ref. [18], there is a discernable toe region, a linear region, and a relatively flat peak region, and finally, a drop to zero force when the sample failed. The quantitative features of the force trace depended on the alignment of the gel and on the size and positions of the notches.

Grahic Jump Location
Fig. 7

Between versus across failure in experiments: In both plots, the triangles indicate failure between the notches, and the squares indicate failure across the sample (same symbols as in Fig. 5). Each symbol corresponds to a single, independent experiment. (a) For isotropic samples, the between/across results are largely consistent with the model prediction (dotted line repeated from Fig. 5(a)). (b) For anisotropic samples, as expected, the division between BETWEEN and ACROSS is shifted to the right. The dashed line is the Ω = 0.62 simulation result from Fig. 5(b), and the solid gray line is the simulation result for a less strongly aligned case (Ω = 0.5).

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