24 L. Nguyen et al. Table 1 Solved material parameter values Direction A B λ(s) n Normal 0.305 0.356 293 −0.117 Shear 0.448 0.162 488 −0.166 Figure 4 shows all raw data plotted on a semi-log scale. From the data, we do not observe any clear effect of the shear strain amplitude on the stress relaxation response in either the normal or shear directions. The characteristic time generally appears to remain the same regardless of the input shear strain chosen. Considering the effect of the compressive strain amplitude, we observe a generally positive correlation between the nominal stress values and the given compressive strain. However, after normalization (Figure 1), we cannot state a clear significant effect that varying compressive strain has on the relaxation behavior. Discussion Implications of Experimental Data From our normalized data, we notice there is not a significant difference in stress relaxation behavior due to varying the input normal strain nor the input shear strain. With ten samples at each combination of control variables, we believe this experiment sufficiently demonstrates that the viscoelastic response of the porcine aorta can be approximated using a quasilinear viscoelastic model. We also note the difference in the shape of the trend lines between the normal and shear stress response. This demonstrates the anisotropy of the material. In our analysis, we see that both data sets in the normal and shear directions can be fitted using a model of the form of equation 4. At the time of this publishing, we have not proved uniqueness to this model. In fact, we do not expect uniqueness from this model. Further study is required for the viscoelastic constitutive modeling of the porcine aorta. The purpose of this paper is to provide an experimental data set in which future modeling can be validated with. From this data set, we see the magnitude of the time-dependence of the material in combined normal and shear loading. Within the time frame of the experiment (1000 seconds), we can further examine the viscoelastic behavior of aortic tissue. From Figure 3, we can see that neither the normal or the shear stress plateau to a finite stress value. From longer experiments, we can determine whether the material behaves as a viscoelastic solid or viscoelastic fluid. Novelty of our experiment The novelty of this experiment comes from both the loading configuration and the combination of normal and shear forces onto the test samples. We did not test samples of porcine aortas in their natural, exhumed configuration. In fact, we do not expect normal physiological loadings in the configuration in which we are testing. Additionally, with the way the samples are prepared, we expect residual stresses to exist within the body. Thus, the nominal values of our experiments will not correlate to the stress values we expect to see in vivo. However, the purpose of this experiment is to gather an understanding of the viscoelastic response the porcine aorta shows under combined loading. Our methods allow us to take multiple samples from each aorta. This allows for a wider range of loading conditions and a larger number of samples at each combination. As a biological material, we expect each sample of porcine aorta to show large variance, and testing at least 10 samples at each loading condition is necessary for characterizing the material. While statistical accuracy is expected to be low for a biological material, we can minimize this by taking many samples out of each aorta retrieved. Implication on future modeling We discovered a clear anisotropic relaxation response within the porcine aorta. From the relaxation experiment, we determine if the material behaves as a viscoelastic solid or fluid. This will impact the choice between a Gibbs or Helmholtz free energy [27]. From our findings, we determine that future modeling of the porcine aorta must include anisotropic relaxation behavior. The rate of dissipation will need to vary in the normal and shear directions.
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