Isotropic Failure Criteria Are Not Appropriate for Anisotropic Fibrous Biological Tissues

Abstract

The von Mises (VM) stress is a common stress measure for finite element models of tissue mechanics. The VM failure criterion, however, is inherently isotropic, and therefore may yield incorrect results for anisotropic tissues, and the relevance of the VM stress to anisotropic materials is not clear. We explored the application of a well-studied anisotropic failure criterion, the Tsai–Hill (TH) theory, to the mechanically anisotropic porcine aorta. Uniaxial dogbones were cut at different angles and stretched to failure. The tissue was anisotropic, with the circumferential failure stress nearly twice the axial (2.67 ± 0.67 MPa compared to 1.46 ± 0.59 MPa). The VM failure criterion did not capture the anisotropic tissue response, but the TH criterion fit the data well (R2 = 0.986). Shear lap samples were also tested to study the efficacy of each criterion in predicting tissue failure. Two-dimensional failure propagation simulations showed that the VM failure criterion did not capture the failure type, location, or propagation direction nearly as well as the TH criterion. Over the range of loading conditions and tissue geometries studied, we found that problematic results that arise when applying the VM failure criterion to an anisotropic tissue. In contrast, the TH failure criterion, though simplistic and clearly unable to capture all aspects of tissue failure, performed much better. Ultimately, isotropic failure criteria are not appropriate for anisotropic tissues, and the use of the VM stress as a metric of mechanical state should be reconsidered when dealing with anisotropic tissues.

Fig. 1

(a) Outlines of dogbone sample geometries are shown along the axial length of the vessel (not drawn to scale). Angles were taken to be relative to the circumferential orientation (0 deg). Scale bar shown in white. (b) A representative stress–stretch curve for one uniaxial sample, with corresponding tissue images during testing. (c) Outline of the shear lap sample geometry (not drawn to scale). (d) A representative force–displacement curve for one shear lap sample. Failure initiated near the overlap region of the sample and propagated across the overlap region (lap across failure).

Fig. 2

Finite element mesh for one shear lap sample with applied boundary conditions. The nodes on the right face were fixed in all directions, while the nodes on the left face were fixed in the vertical and out of plane directions, and given prescribed displacements based on the experiment.

Fig. 3

Failure stresses at each sample angle (n > 9 for each angle). ANOVA showed that change in sample angle had a statistically significant effect on failure stress (p = 0.0003). Error bars show 95%CI's.

Fig. 4

Experiment (points) and failure criteria fits. (a) The von Mises failure criterion (solid green line, 95%CI shaded) fit to the mean peak stresses does not capture the anisotropic response of the tissue. (b) Tsai–Hill maximum-work theory model (solid line, 95%CI shaded). Black error bars indicate 95%CI's on experimental points.

Fig. 5

Strain tracking results from one shear lap sample. Large shear strains (∼40%) were exhibited in the overlap region of the sample.

Fig. 6

(a) Representative force–displacement curve for one shear lap sample (black dots), with a simulation force–displacement curve (red line) using optimized parameters. (b) Failure propagation for one shear lap sample, shown at three different displacements. The onset of failure began near the overlap region of the sample (indicated by the arrow) and propagated across the center (lap across failure). (c) Failure simulation using the Tsai–Hill criterion. Propagation occurred through the overlap region of the sample and eventually tore in the overlap region (lap across failure). (d) Failure simulation using the von Mises criterion, where ${\sigma }_{\text{yield}}={\sigma }_{\text{avg}}$. Failure propagated across the sample arm and tore the arm off (arm failure). Failure simulations are shown at similar failure points to the experiment but not at the same displacement as the experiment. (See supplementary videos, which are available under the “Supplemental Materials” tab for this paper on the ASME Digital Collection.)

Fig. 7

Area fraction for the experimental shear lap samples, along with the Tsai–Hill and von Mises (avg) failure cases. Averages shown with 95%CI bars.

Fig. 8

(a) One experimental sample immediately prior to total failure. (b) Sample in the undeformed domain. White dotted line indicates calculated crack propagation location and direction in undeformed domain. Lap arm failure occurred in the experimental sample. (c) and (d) Typical failure comparison between the Tsai–Hill and von Mises failure criteria in the undeformed domain. The Tsai–Hill failure criterion predicted lap arm failure, while the von Mises failure criterion predicted arm failure.

Fig. 9

Average failure location (dots) and crack propagation angle (solid line) with 95%CI (dotted lines and shaded region) for experimental samples, Tsai–Hill, and von Mises (avg) failure simulations. Shown in black is the average shear lap sample geometry calculated using radius-based averaging from sample outlines (linear approximation was used for noisy regions of the average sample outline). Samples were rotated (if needed) so that failure occurred in the left arm for comparison purposes.