^{1}and H. Katsuragi

^{2}

### Abstract

A viscoelastic characterization method using low-velocity impact is experimentally studied. A steel ball is dropped from a certain height and impacts on an agar gel target with 1–4 m/s velocity. The motion of the impactor ball is captured by a high-speed camera. Instantaneous penetration depth, velocity, and acceleration of the impactor are computed from the high-speed video data. The obtained kinematic data are analyzed in terms of the equation of motion of the impactor. Specifically, we compute the impact viscosity and impact elasticity, assuming a simple impact drag force model. The impact drag force model consists of a linear viscous term, a linear elastic term, and a constant term. From the estimated impact viscosity, we confirm that the Reynolds number is relatively low (less than 10). This low Reynolds number is consistent with the simple linear viscous assumption. From the estimated impact elasticity, we can calculate the speed of sound and the strength of target agar gel. In order to examine the velocity dependence of the elasticity, we also perform very slow (less than 0.1 mm/s) penetration tests using the same agar gel samples. The comparison between impact elasticity and slow penetration elasticity reveals the weak velocity strengthening of agar gel.

The authors would like to thank H. Honjo, T. Yamaguchi, M. Nishida, and M. Tokita for useful comments. This research has been supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT), Grant-in-Aid for Challenging Exploratory Research, No. 23654134.

I. INTRODUCTION

II. EXPERIMENTAL MATERIALS AND METHODS

III. RESULTS AND ANALYSIS

A. Kinematic data set

B. Impact drag force model

C. Analysis results

D. Slow penetration

IV. DISCUSSION

V. CONCLUSION

### Key Topics

- Elasticity
- 40.0
- Gels
- 37.0
- Impact phenomena
- 33.0
- Viscosity
- 25.0
- Viscoelasticity
- 17.0

## Figures

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.

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.

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.

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.

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 .

Normalized acceleration vs. normalized velocity at some fixed penetration depth *z _{i} * 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.

Normalized acceleration vs. normalized velocity at some fixed penetration depth *z _{i} * 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.

Normalized acceleration vs. normalized penetration depth at some fixed velocity *v _{i} * 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).

Normalized acceleration vs. normalized penetration depth at some fixed velocity *v _{i} * 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).

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 .

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 .

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 .

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 .

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).

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).

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).

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).

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.

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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