Compared to regime I in which large fluctuations in λmax had no effect on E1, the
value of the latter in regime II was more sensitive. Furthermore, ~1% increase in residual strain due to fluctuations in regime II corresponded to 0.1 GPa increase in E1, and regime
III was the most sensitive as a 2% increase in residual strain due to fluctuations resulted in ~0.5 GPa increase in E1. A comparison of the σ - λ curves in Figure 4.13(b-f) with
those in Figure 3.1, shows that mechanical cycling in any of the three regimes (without recovery) results in linearization of the σ - λ curves up to λmax. This is evidenced in Figure
4.13(a) that shows the final loading of a fibril cycled 20 times in regime I, and Figure 4.13(b) that shows the final loading curve of iteration 2 of a fibril cycled in regime I (note that the fibril whose curve is shown in Figure 4.13(b) did not reach regime III). Due to the large overlap between cycles and the small range in which regime I was cycled, the earlier cycles are not shown in the two plots. A comparison between the curves in Figure 3.1 and Figure 4.13(a) shows no discernible difference in the shape of the σ - λ curve beyond regime I, except for a slight linearization in the behavior of regime I, which does not impact the mechanical behavior of the fibril in regimes II and III. A similar conclusion can be deduced from the plot in Figure 4.13(b).
Unlike in regime I, there is a clear difference in the mechanical behavior of a fibril after cycling into regime II. The linearization of the mechanical behavior is more pronounced in Figure 4.13(c) for a fibril only cycled in regime II, and in Figure 4.13(d) for a fibril cycled in regime II, but let to recover for 1 hr, and then cycled again in regime II. In both plots, regime I is not distinguished any longer after cycling in regime II, and the final loading curves appear linear at failure. Regime III might be distinguished in Figure 4.13(d) but it was not reached in the final loading curve in Figure 4.13(c). After cycling in regime III, Figure 4.13(e,f), the overall curve was quite linear and steeper than that of cycle 11 in Figure 4.13(e) and 1-1 in Figure 4.13(f). After cycling, the fibril whose curves are shown in Figure 4.13(e) failed at a stress similar to the maximum stress in the first cycle, and although this did not occur in many fibrils, it implies that mechanical cycling does not necessarily result in mechanical drawing and strengthening. The fibril that was allowed to recover before a second iteration of 10 cycles, Figure 4.13(f),
demonstrated all 3 regimes in the final loading curve (with regime I rather suppressed) and failed at quite high stress after experiencing over 30% strain, supporting the conclusion that cycling in regime III does not result in significant damage accumulation.
Finally, the mechanical response of a fibril that was cycled four times into regime II is presented in Figure 4.14. This test was performed differently from previous cyclic loading/recovery tests: The first cycle of each iteration was brought to a designated initial
σmax and in subsequent cycles the fibril was brought to λmax associated with the initial σmax.
The hysteresis curves shifted with increased residual strain, and this process continued into the fourth iteration of cycling reaching a great consistency in iteration 4 as shown in Figure 4.14(d). The actual residual strain, Figure 4.15(c), increased slightly within each iteration but more significantly between iterations, reaching a plateau in iterations 1, 3 and 4. Achieving a plateau in residual stress depended on the ability to cycle the fibril in each iteration by accurately reaching the same λmax in each cycle. Figures Figure
4.14(a,c,d) attest to this condition. Similarly to previous experiments in regime II, the fibril did not recover all of its strain between iterations but reached very consistent values for E1, Figure 4.15(a) and notably the value of E1 did not change after a reduction in the
first cycle despite the total of 40 cycles, Figure 4.15(a). On the other hand, E2 showed
larger variability between iterations and a consistent reduction to a steady-state value in iterations 3 and 4, Figure 4.15(b).
(a) (b)
(c) (d)
(e) (f)
Figure 4.13. Regime I: (a) Final loading until failure after 20 cycles, (b) final loading until failure after recovery and an additional 10 cycles. Regime II: (c) Initial and final loading until failure after 10 cycles, (d) final loading until failure after recovery, also showing cycles 1-1 and 2-1. Regime III: (e) Initial and final loading until failure after 10 cycles, (f) final loading until failure after recovery, also showing cycles 1-1 and 2-1.
0 100 200 300 400 500 1.0 1.1 1.2 1.3 1.4 1.5 S tr es s (M P a) Stretch Ratio, λ Cycle 21 0 100 200 300 400 500 600 700 1.0 1.1 1.2 1.3 1.4 1.5 S tr es s (M P a) Stretch Ratio, λ Cycle 2-11 0 100 200 300 400 500 600 1.0 1.1 1.2 1.3 1.4 1.5 S tr es s (M P a) Stretch Ratio Cycle 1 Cycle 21 0 200 400 600 800 1000 1200 1400 1.0 1.1 1.2 1.3 1.4 1.5 S tr es s (M P a) Stretch Ratio, λ Cycle 1-1 Cycle 2-1 Cycle 2-11 0 100 200 300 400 500 600 1.0 1.1 1.2 1.3 1.4 1.5 S tr es s (M P a) Stretch Ratio, λ Cycle 1 Cycle 11 0 200 400 600 800 1000 1200 1400 1.0 1.1 1.2 1.3 1.4 1.5 S tr es s (M P a) Stretch Ratio, λ Cycle 1-1 Cycle 2-1 Cycle 2-11
(a) (b)
(c) (d)
Figure 4.14. σ - λ curves of a fibril cycled four times in regime II. (a) Cycles 1-1, and 1-2, vs. 1-9, and 1-10, (b) cycles 1-1 and 1-10 (in grey) plotted together with cycles 2-1, 2-2, and 2-9, 2-10, (c) cycles 2-1 and 2-10 (in gray) plotted together with the first and the last two cycles of iteration 3, and (d) cycles 3-1 and 3-10 (in grey) plotted together with the first and last two cycles of iteration 4.
0 100 200 300 400 500 600 1 1.1 1.2 1.3 S tr es s (M P a) Stretch Ratio, λ Cycle 1-1 Cycle 1-2 Cycle 1-9 Cycle 1-10 0 100 200 300 400 500 600 1 1.1 1.2 1.3 S tr es s (M P a) Stretch Ratio, λ Cycle 2-1 Cycle 2-2 Cycle 2-9 Cycle 2-10 0 100 200 300 400 500 600 700 1 1.1 1.2 1.3 1.4 S tr es s (M P a) Stretch Ratio, λ Cycle 3-1 Cycle 3-2 Cycle 3-9 Cycle 3-10 0 100 200 300 400 500 600 700 1 1.1 1.2 1.3 1.4 S tr es s (M P a) Stretch Ratio, λ Cycle 4-1 Cycle 4-2 Cycle 4-9 Cycle 4-10
(a) (b)
(c) (d)
(e)
Figure 4.15. (a) E1, (b) E2, (c) residual strain, (d) λmax, and (e) σmax vs. cycle for a fibril
that was cycled four times into regime II with 1 hr of recovery between iterations. 0.0 2.0 4.0 6.0 8.0 10.0 0 2 4 6 8 10 12 E1 (G P a ) Cycle Iteration 1 Iteration 2 Iteration 3 Iteration 4 0.0 1.0 2.0 3.0 4.0 5.0 0 2 4 6 8 10 12 E2 (G P a) Cycle Iteration 1 Iteration 2 Iteration 3 Iteration 4 0 5 10 15 20 25 0 2 4 6 8 10 12 Resi d u al S tr ain (% ) Cycle Iteration 1 Iteration 2 Iteration 3 Iteration 4 1.0 1.1 1.2 1.3 1.4 0 2 4 6 8 10 12 λmax Cycle Iteration 1 Iteration 2 Iteration 3 Iteration 4 0 200 400 600 800 0 2 4 6 8 10 12 σm ax (M P a) Cycle Iteration 1 Iteration 2 Iteration 3 Iteration 4
4.5. Conclusions
This Chapter examined the effect of cyclic loading and recovery in each of the three deformation regimes distinguished in Figure 2.5. Collagen fibrils cycled into regime I did not experience drastic changes in their σ - λ behavior. Interestingly, they consistently recovered ~80% of the total applied strain, in agreement with MD simulations predicting that 80% of the deformation in regime I originates in uncoiling of tropocollagen molecules [26]. This suggests that deformation from uncoiling of tropocollagen is recoverable, while deformation from molecular sliding in regime II is not. According to the same MD simulations [26], the deformation in regime II is a combination of molecular sliding and uncoiling and is restrained by cross-links, which explains the constant value for E1 after cycling (even after 4 iterations and a total of 40
cycles) in regime II and the increase in residual strain. However, fibrils cycled into regime II also experienced an increased overall stiffness. After cycling in each of the three regimes, the σ - λ curves showed a more linear and stiffer response up to λmax in
comparison to monotonic loading. This effect was removed during recovery. When most of the residual strain was removed during recovery, the recovered fibrils consistently reached maximum strain at failure exceeding 35%.
CHAPTER 5
CONCLUSIONS
Reconstituted type I collagen fibrils with diameters in the range 70-280 nm tested monotonically at 20-30% RH and ~0.004 s-1 strain rate demonstrated a three regime σ - λ curves that were consistent in shape with previous studies [26,27]. The experimental results from dry collagen fibrils did not uphold the existence of an initial compliant regime at <2% strain that has been reported before for hydrated fibrils [26,27,63,64]. Additional experiments conducted at 40-53% RH, similarly to previous studies at 30-60% RH [32-34], demonstrated the same σ - λ curves as dry collagen (20-30% RH), thus implying that the initial compliance may be a characteristic of fully hydrated fibrils. As computed from the σ - λ curves, the dry fibrils had an initial stiffness E1 = 5.7±2.3 GPa, aregime II stiffness E2 = 2.19±1.06 GPa, tensile strength of 752±186 MPa, and ultimate
stretch ratio of 1.3±0.06. In comparison, rehydrated rat tail fibrils were shown to be much weaker with tensile strength of 71±23 MPa and elastic modulus of 326±112 MPa [35], but the tensile strengths measured in this research were closer to rehydrated human patellar collagen fibrils with a tensile strength of 540±140 MPa as calculated using the dry fibril diameter [27]. The dry state of the fibrils tested in this research resulted in engineering failure strains (31±6%) that were lower than rehydrated collagen that has been reported to have 63±21% fracture strain in rat tail tendon but only 20±1% in human patellar tendon [35].
While the difference in the results from this dissertation research and prior studies may originate in the specific details of the collagen tested, sample preparation, and measurement techniques, the data reported by all studies to date has been characterized by large variations. The results of this research clearly show that, despite following strict protocols to ensure consistency between specimens, there are still variations in the recorded mechanical response. The lack of precise control on fibril diameter is a major
contributing factor to the scatter of the experimental data. Since E1 correlated with the
fibril diameter, the effect of the initial material state or fibril diameter on the σ - λ curves could be reduced by normalizing with E1. The σ/E1 - λ curves showed very consistent
trends with the applied strain rate in the range 0.001 - 100 s-1, agreement between different plots obtained in the range at ~0.001 s-1, a clear increase in E
2 and the ultimate
tensile strength that exceeded 2.0 GPa at high strain rates. E2/E1 increased with increasing
strain rate, implying a linearization of the effective σ - λ behavior with increasing rate. The study of failure sections yielded mixed results, with most of the fibrils failing in a fibrillar manner possibly influenced by the internal fibril structure of the collagen fibrils.
The mechanical behavior under cyclic loading was characterized in each of the three regimes of deformation via cyclic and cyclic/recovery experiments at target λmax ~
1.05 in regime I, λmax ~ 1.25 in regime II, and λmax ~ 1.3 in regime III. Fibrils cycled into
regime I did not experience any drastic changes in the σ - λ curves and recovered more than half of the residual strain accumulated by the last cycle before recovery time, and ~80% of λmax. According to DePalle et al [26], intermolecular adhesion allows uniform
deformation within the fibril, with deformation in regime I driven by uncoiling of the tropocollagen molecules. Notably, mechanical cycling in regime II increased E2 and the
residual strain, but left E1 largely unaffected despite a distinct change in the shape of the σ - λ curves between cycles 1 and 2. This was also true for fibrils tested before and after 1
hr of rest at zero stress and for a fibril subjected to four iterations of 10 cycles each with 1 hr of recovery between iterations. Despite experiencing 40 loading-unloading cycles, E1
was approximately the same in cycle 4-10 as in cycle 1-2. However, although E2
increased with cycle order, it decreased with iteration order, and E2 in iteration 4 cycling
was comparable to E2 in iteration 3. Deformation in regime II is attributed to molecular
sliding and uncoiling of the collagen triple helix, with the majority of molecular sliding contained within the gap regions [35]. Applied loads are also distributed throughout the fibril via water [63,64], resulting in uniform deformation. These mechanisms explain the reasonably uniform diameter after loading to λ ~1.25, the increase in E2 and residual
strain with mechanical cycling, and the rather constant value of E1 throughout many
Cycling in regime III resulted in similar results as regime II, with an overall increase in the effective total stiffness of the fibrils, as fitted across all three regimes, and an effective linearization of the σ - λ curves. Most notable was the change in the shape of the σ - λ curve before and after recovery: Iteration 1 demonstrated all three regimes, but only regimes I and II were present in the σ - λ curves in iteration 2. The amount of possible deformation imparted in regime III is controlled by cross-link strength that can enable stretching of the molecular backbone [26].
In summary, the experimental results presented in this dissertation point out to a process of damage accumulation during cycling, as manifested by the very consistent hysteresis loops and the gradually accumulated residual strain, which however, does not affect the mechanical stiffness of regimes I and II. The latter points out to a cross-link network within the collagen fibril that maintains molecular connectivity, as well as material regions that allow for viscous sliding (supported by the increase in E2 and E2/E1
with applied strain rate) in the softening regime of the σ - λ curves without disrupting the cross-link network. The rapid recovery and restoration of the three-regime shape of the σ - λ curves of collagen fibrils also supports the existence of sacrificial bonds that reform upon recovery that is driven by residual stresses in the fiber.
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