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Viscoelastic characterization of low-velocity impact of a solid ball on an agar gel
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View: Figures


Image of FIG. 1.
FIG. 1.

Example of raw data captured by a high-speed camera. The white spot in each image is an impactor of d = 6.35 mm. The free fall height h is 600 mm and the target agar gel concentration is C = 0.7 wt. %. Although a deceleration trend can be confirmed, direct locating of the impactor by its barycenter is difficult on account of imaging imperfections. The depth z is calculated from the top or bottom edge of the white spot. The impact time t = 0 is identified by deviation of the v(t) from free fall line in plots such as Figs. 2(b) and 3(b) . The surface level of the gel sample z = 0 is defined by z(0) = 0.

Image of FIG. 2.
FIG. 2.

The raw kinematic data, z, v, a vs. t of d = 6.35 mm and C = 0.7 wt. %. All impact events show a rebound tendency due to the elasticity of the target gel. In the large cases, the impactor settles in the target gel, whereas small h impacts result in complete rebound. The maximum deceleration occurs at t = 1–2 ms.

Image of FIG. 3.
FIG. 3.

The raw kinematic data z, v, a vs. t for d = 12.7 mm and C = 1.0 wt. %. The global structure of the data looks similar to Fig. 2 . However, all impact events show complete rebound for this case. The data fluctuation are slightly larger than for the data of Fig. 2 .

Image of FIG. 4.
FIG. 4.

Normalized acceleration vs. normalized velocity at some fixed penetration depth zi for (a) the small impactor (d = 6.35 mm, c = 0.7 wt. %) and (b) the large impactor (d = 12.7 mm, c = 1.0 wt. %) cases. While the data are noisy, we assume linear trends. The solid lines indicate the parallel lines of best fit. The slope of these parallel lines corresponds to (dimensionless impact viscosity) of the target agar gel.

Image of FIG. 5.
FIG. 5.

Normalized acceleration vs. normalized penetration depth at some fixed velocity vi for (a) the small impactor (d = 6.35 mm, c = 0.7 wt%) and (b) the large impactor (d = 12.7 mm, c = 1.0 wt. %) cases. The data scattering are larger than that in Fig. 4 . In addition, the data show a sudden deflection around z/L = (a) 0.06 and (b) 0.09. This deflection comes from the yielding of the gel samples at roughly z = (a) 4 and (b) 6 mm. Only the largest z/L data points in (b) (h = 698 mm impact) show the yielding-like deflection. The solid lines show parallel line-fitting results in the shallow regime. The slope corresponds to (dimensionless impact elasticity).

Image of FIG. 6.
FIG. 6.

Fitting parameters for lines in Fig. 4 . The dimensionless impact viscosity is calculated from the slopes of vs. for (a) the small impactor and (c) the large impactor cases, respectively. First, the data are fitted with independent lines (data points in (a) and (c)). Then, the slopes are averaged to obtain (gray level lines in (a) and (c)). Using the averaged , all data are refitted and the intercepts of the fitted parallel lines are shown in (b) for the small impactor and (d) for the large impactor cases. According to Eq. (5) , the intercept corresponds to .

Image of FIG. 7.
FIG. 7.

Fitting parameters for lines in Fig. 5 . The dimensionless impact elasticity can be calculated from the slopes of vs. for (a) the small impactor and (c) the large impactor cases. Similar to the impact viscosity analysis, the data are first independently fitted with lines (data points in (a) and (c)) and the averaged slope (gray level lines in (a) and (c)) is used to refit the data. Then, the intercepts found from the refits are shown in (b) and (d). Gray lines in (b) and (d) indicate the fits using obtained in the impact viscosity analysis (Figs. 6(a) and 6(c) ). Because the points at seem to be less reliable on account of yielding, we exclude them from the fitting. According to Eq. (5) , the intercept corresponds to .

Image of FIG. 8.
FIG. 8.

Dimensionless impact-viscoelastic parameters, , and vs. agar gel concentration C. The analyses explained in Figs. 6 and 7 are applied to all other experimental conditions and the results are compiled. The left column (a)-(c) data are calculated on the basis of vs. (Fig. 6 ). The right column (d) and (e) data are calculated from Fig. 7 and the value of from (a). δ = 1 is used to obtain (b), (c), and (e).

Image of FIG. 9.
FIG. 9.

Impact-viscoelastic parameters obtained in this experiment, η, E, , and . We believe that the direct estimates of slope in Figs. 4 and 5 to be more reliable than intercept-based estimates. Thus, we use Figs. 8(a) and 8(d) to calculate impact viscosity and impact elasticity. For the constant drag stress, we use the data from Fig. 8(e) because the data values are positive. The Maxwell time is the ratio of (a) and (b).

Image of FIG. 10.
FIG. 10.

Very slow penetration drag force measurement results. Raw data examples are displayed in (a). Yielding-like behavior can be observed at . Until yielding occurs, roughly linear elastic behavior is observed. The elasticity computed for the linear part is shown with black plot markers in (b). The impact elasticity data are also plotted for comparison (red markers). They show approximately one order of magnitude difference. The scaled results for elasticity are shown in (c). The scaling indicates weak velocity strengthening (hardening) of the agar gel.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Viscoelastic characterization of low-velocity impact of a solid ball on an agar gel