DISCUSSION

Stress-Strain vs. Time Relationship & Phase Lag

Plotted against time, shear stress and strain follow similar patterns for each of the nine experimental setups. An important relationship between the stress and strain curves is the difference between them, and the ratio of the peaks relative to one another. It is also important to note that a purely viscous material is exactly 90∘ out of phase. Thus, there is a phase lag of 90∘ or 𝜋/2 radians. Conversely, a purely elastic material is completely in

radians is considered to be viscoelastic. As seen in Figure 3, the putty exhibits a phase lag that decreases with an increasing oscillation frequency. This is observed by the shear strain curves approaching the shear stress curves as oscillation frequency increases from 0.1 [Hz] to 10 [Hz]. The decrease in phase lag indicates that the putty exhibits more elastic behavior as the oscillation frequency increases. Looking at Table 2, the phase lag for putty is above 𝜋/4 radians (45∘) at 0.1 [Hz] and 1 [Hz]. The phase lag at 0.1 [Hz] is also greater than the phase lag at 1 [Hz]. Thus, the putty initially experiences a more dominant viscous behavior. As the oscillation frequency increases, the elastic behavior of the putty becomes more and more dominant. At 10 [Hz] the phase lag of the putty is below 𝜋/4 phase with no phase lag. Any phase lag between 0∘ and 90∘ or 0 and 𝜋/4 radians (45∘) thus indicating dominant elastic behavior. Additionally, the porcine skin exhibits a phase lag that decreases with an increasing oscillation frequency at a slower rate than the putty. This is shown in Figure 3 by the shear strain curves approaching the shear stress curves at a slower rate than in the putty. In Table 2, an initial decrease in tissue phase lag from 0.1 [Hz] to 1 [Hz] and then the seemingly constant behavior of the phase lag from 1 [Hz] to 10 [Hz] can be observed. It is also important to note that the phase lag of the tissue is always below 𝜋/4 radians (45∘) from 0.1[Hz] to 10 [Hz]. This means that the tissue is already more dominant in elastic behavior and its elastic behavior becomes slightly more dominant as oscillation frequency increases. Finally, for the “chewed” gum, in Figure 3, the shear stress curves and shear strain curves appear to remain equidistant from one another as the oscillation frequency increases. This seemingly constant behavior is confirmed in Table 2 as the phase lag of the gum decreases ever so slightly from 0.1 [Hz] to 1 [Hz], then

increases again from 1 [Hz] to 10 [Hz] to the previous phase lag value. This phase lag value is slightly below 𝜋/4 radians (45∘). This means that the gum is fairly distributed in elastic and viscous behaviors. The slight increase in elastic behavior at 1 [Hz] can potentially be attributed to noise.

Storage Modulus, Loss Modulus, and Tangent of the Phase Lag

The storage and loss moduli are analyzed together as a ratio. Therefore, when the storage modulus is larger than the loss modulus, elastic properties of the material dominate, and when the loss modulus is larger than the storage modulus, the viscous properties of the material dominate. This can be seen by the tangent of the phase lag plot in Figure 4 as tan(delta) is equal to the ratio between loss and storage moduli. It is important to remember that tan() is equal to the loss modulus over the storage modulus. This can be easily visualized, as tan(delta) will be less than 1.0 [radian] when elastic properties dominate and greater than 1.0 [radian] when viscous properties dominate. Additionally, the tan(delta) can be considered analogous to the complex modulus which describes the overall stiffness of a material and is found by the ratio between the loss and storage moduli. Though not identical, tan() and the complex modulus essentially describe the same thing: the stiffness of a material due to the peak stresses and strains experienced. Using the plots in Figure 4 to quantify and analyze these values, we can prove that putty is more viscous than elastic at low and medium frequencies and more elastic than viscous at high frequencies. The tan(delta) is significantly larger than 1.0 at the low and medium oscillation frequencies, indicating that the loss modulus is larger than the elastic modulus. This then flips, and the elastic modulus is larger than the loss modulus at higher oscillation frequencies and is thus supported by a tan(delta) value of <1.0 [radians]. The porcine skin displays the most constant behavior of any material tested regarding viscosity and elasticity. This biological tissue more dominantly exhibits properties of elastic materials, as tan() is less than 1.0 [radian] for all tested oscillation frequencies. The tangent of the phase lag is closer to 0.75 [radians] at low frequencies, but still remains more elastic than viscous. The chewed gum also demonstrates prominently elastic material properties, but the degree of elasticity is less than that of the porcine skin. The tan() values fluctuate ever so slightly just under 1.0 [radian]. This fluctuation in the tan(delta) could be attributed to noise. The tan(delta) can be confirmed using the loss and storage moduli plots. Where the storage modulus is larger than the loss modulus, a tan() value of <1.0 [radian] is observed thus indicating dominant elastic behavior. When the loss modulus is larger than the storage modulus, a tan(delta) value >1.0 [radians] is observed thus indicating dominant viscous behavior. This correlation can be observed for the putty, “chewed” gum’ and porcine skin.