Fig 1.
Schematics for the experiments.
A: The experimental apparatus for uniaxial unconfied compression tests. B: Locations of the excised skin on mouse.
Fig 2.
Example run of the compressive test procedure for one skin specimen when varying strain rates.
A: Position of the compression platen over time, as measured by its distance from the fixed platform. B: Reaction force at the compression platen. C: Magnified view of reaction force and platen position for Compression 6, demonstrating that “skin thickness” was defined by “contact point” as determined from the force trace. The platen was moved into the skin with an acceleration of 0.06 s−1 for each of the first 10 repetitions. Then, 10 additional compressions were performed at 22.88 s−1. The 6th compression was analyzed in each sequence of 10 compressions.
Fig 3.
There is no correlation between the measured steady-state stretch (λ∞) and thickness.
A: Skin thickness naturally varies between about 200 and 800 μm when a single, consistent stretch level of about 0.6 is delivered to each specimen in the first experiment. B: Skin thickness naturally varies when multiple stretch levels (λ∞ from about 0.2 to 0.8) are delivered to each of the skin specimens.
Fig 4.
Example fit of stress over time by one-term (left column) and two-term (right column) QLV models at three strain rates (rows A: 0.06 s−1; B: 4.29 s−1; C: 35.34 s−1).
Black line shows the modeled prediction, and gray data points show the experimental data. The average weighted R2 value for the one-term case for the three strain-rates is 0.86 while the R2 value using the two-term case is 0.93. Therefore, the tradeoff is that the number of free parameters increased from 1 to 2, versus attaining a slight improvement in the fitting, and for this reason we chose the one-term case.
Table 1.
Median parameters from model fits to data for all experimental conditions.
Fig 5.
Data from 341 experimental runs (n = 41 specimens) each stimulated an average of eight stretch levels.
Distributions are shown of A: skin thickness measurements; B: steady-state skin stretches (λ∞) applied; C: time constants from fitting stress versus time to the one-term QLV model; D: the steady-state residual stress ratio G∞. Note that each of the four variables exhibits high variability.
Fig 6.
Correlations between skin thickness/stretch level and residual stress ratio (G∞).
A: In the first experiment where only thickness varied, the steady-state residual stress ratio (G∞) correlates with increasing skin thickness, n = 44. Linear regression (solid line) with residual stress ratio G∞ as the dependent variable was performed, which returns p < 0.001 for independent variable thickness l0, and G∞ = 9.997 × 10−4 μm−1 ⋅ l0 + 0.077. In the second experiment where both thickness and strain level varied, the residual stress ratio (G∞) correlates with both B: stretch and C: skin thickness. Note that the two correlations are independent from each other because there is no correlation between stretch and skin thickness. Multilinear regression with residual stress ratio G∞ was also performed, which returns p < 0.001 for independent variable stretch λ∞, p < 0.001 for independent variable thickness l0, and G∞ = 0.810 ⋅ λ∞ + 4.25 × 10−4 μm−1 ⋅ l0–0.074. Note that in B and C, solid lines are single-linear regressions for residual stress ratio with respect to stretch and thickness respectively.
Fig 7.
Values of time constant (τ1) and residual stress ratio (G∞) at three strain rates ().
A: Overall, the time constants are significantly smaller under faster strain rates, while there is no systematic trend in the change of residual stress ratio. The boxes range from the lower quartile to upper quartile, the centerlines denote the medians, the whiskers denote extreme values and crosses denote outliers. B-D: Detailed views of the distributions of time constants and residual stress ratios from all data points at B: strain rate 0.06 s−1, n = 54; C: strain rate 3.54 s−1, n = 44; D: strain rate 22.88 s−1, n = 54.
Fig 8.
Comparison of the reduced relaxation function (Equation 5) from measurement of different skin samples.
The solid line shows median data from work presented here on mouse hindlimb skin fitted to the one-term model, in the first experiment with median strain rate . The dotted-dash line gives a measurement from pig dorsal skin [11]. The dashed and dotted lines are both from rat skin, but the dashed line function is attributable only to collagen elements in the skin while the dotted function is only elastin elements [4]. Note that the skin in compression relaxes more than skin in tension.
Fig 9.
Schematic and illustration of the constitutive model.
A: Rheological representation of the viscoelastic model, where the material consists of parallel chains, and each chain consists of an elastic component (denoted by a spring) and a viscous component (denoted by a dashpot). Usually, the steady-state response of a viscoelastic solid is represented by a chain with no viscous component (i.e., τ = ∞). Here, in addition to one solid chain, models including one and two chains with viscous components are evaluated and denoted as one-term and two-term models. B: Illustration of how stimuli with a low strain-rate may lose information from low time-constant QLV chains. The solid line is the response of a typical two-term viscoelastic solid with time constants τ1 and τ2 under a step load. Two dashed lines represent the response of the material under slower strain-rates. For the slowest strain-rate, , stress relaxation properties of the faster chain (τ1) may not show up because its relaxation for the faster chain takes place within its ramp phase. Therefore, this will not be captured by curve fitting, which simply characterizes the material as a one-term QLV solid and only calculates the slower chain (with time constant τ2). Thus, we can eliminate the two extra parameters (G1, τ1) if we only care about low strain-rate situations. In other words, for low strain-rate cases, the single term model is sufficient and therefore more appropriate than the two-term models because of the reduced number of free parameters.
Fig 10.
Typical distribution of vertical compressive stress (S22 in ABAQUS) from the axisymmetric FE simulation (therefore only the right half of the skin middle-section is shown), with friction coefficient of 0.3 and skin thickness of 400 μm.
Note that there is only minor edge effect around the periphery.
Fig 11.
FE simulations for a skin thickness of 400 μm, at stretch levels of 0.5, 0.6 and 0.7, under frictionless (solid lines) and rough (dashed lines) friction conditions.
A: displacement stimuli to achieve desired stretch level; B: responsive force traces for three stretch levels under different frictional conditions; C: calculated relaxation function for force traces shown in B. Note that while frictional conditions have an impact on responsive force traces, they do not impact the calculated viscoelastic reduced relaxation function.
Fig 12.
FE analysis shows minimal frictional edge effects on the calculated residual stress ratio, A: when skin thickness changes, plotted on top of Fig. 6A; and B: when strain level changes, plotted on top of Fig. 6B.
Table 2.
Value of dermis/epidermis thickness ratio r previously measured [12].
Fig 13.
Validation experiment in a secondary context, using a 1.5 mm diameter tip and 8 mm diameter skin specimen, to demonstrate the applicability of the measured QLV parameters.
A: Schematic drawing of the experimental set-up; B: Finite element model with the contact region magnified; C: FE analysis shows good agreement between numerical prediction and actual experimental measurement.