2.1. Quasi-one-dimensional front development

If the front propagation process is assumed to be quasi-one-dimensional, meaning the compacted region grows only downward in a column with depth $z_f$ and not laterally, then the volume-conservation picture Waitukaitis et al. (Reference Waitukaitis, Roth, Vitelli and Jaeger2013) can be applied to obtain

(2.1)\begin{equation} k \equiv \frac{v_f}{v} = \frac{\phi_J}{\phi_J - \phi_0}, \end{equation}

where $v_f = \textrm {d} z_f/\textrm {d} t$ is the characteristic speed of the front and $v = \textrm {d} z/\textrm {d} t$ is the intruder's speed. For a schematic, see figure 1(a). Despite the fact that no detectable compaction of the suspension occurs, this dependence on $\phi _0$ and $\phi _J$ was corroborated by Peters & Jaeger (Reference Peters and Jaeger2014) and Han et al. (Reference Han, Peters and Jaeger2016) for 2-D and 3-D impacts, where $\phi _J\approx 0.51$ is the jamming packing fraction for the cornstarch particles, low due to swelling (Chen et al. Reference Chen, Zheng, Wang and Behringer2019). This implies $k \approx 10$ when $\phi _0 = 0.46$. Since we do not vary $\phi$ in our experiments, we will assume a constant value of $k$.

Figure 1.

Three cases for front growth and ensuing intruder dynamics: (a) the original added-mass model with downward and lateral growth, (b) a viscous model with only downward front growth and (c) a front that grows downward and laterally and can experience added-mass and viscous forces. In all three panels, the dashed line indicates the initial surface at $z=0$, $z$ indicates the depth below the initial surface and $z_f$ indicates the length scale of the jammed region.

2.2. Including lateral front growth

Experiments show that the jammed region below the intruder does not grow strictly downward in a quasi-1-D column but spreads out laterally as well, albeit at a smaller speed. Experiments with small impacting objects (Peters & Jaeger Reference Peters and Jaeger2014; Han et al. Reference Han, Peters and Jaeger2016) typically find that the transverse dimension is approximately half of $z_f$, meaning that the volume of the jammed region still scales as ${z_f}^d$, where $d$ is the dimensionality of the system (2 or 3). Additionally, experimental data show that the front slows down as it moves; see figure 4 of Peters & Jaeger (Reference Peters and Jaeger2014) and figure 4 of Han et al. (Reference Han, Peters and Jaeger2016). The data from these papers appear consistent with $z_f \propto z^\varGamma$ with $\varGamma <1$. We discuss models below including this slowing-down behaviour, corresponding to $z_f \propto z^\varGamma$, where $\varGamma$ is varied to demonstrate that the results are relatively insensitive to its value. If compression-induced jamming were the sole cause, the volume of the jammed region scales as ${z_f}^d$ while the volume swept out by the intruder is linear $z$, meaning that $z_f$ scales as $z^{1/d}$. So, $\varGamma = 1/d$ represents a reasonable lower bound for $\varGamma$.

2.3. General equation of motion

To understand how the front growth, including its shape, affects the resulting dynamics, we consider a generic equation of motion that describes the dynamics of the intruder. Assuming that the growing solid-like region is rigidly connected to the intruder, then the total momentum of both is $p = [m+m_a(t)]v(t)$, where $m$ is the constant intruder mass, $m_a(t)$ is the added mass and $v(t)$ is the velocity of the intruder and solid-like region. The shape of the jammed region and its growth rate will set $m_a(t)$. To complete the equation of motion, there are three external forces to consider. Two relate to gravity: the weight of the intruder, $F_g = mg$, where g is the gravitational acceleration, and a buoyant force $F_b$ from the displaced suspension (this term is typically negligible). The third, which is not included in the added-mass model, is any viscous-like forces $F_v$ that act at the boundary of the jammed region. Newton's second law can then be written as

(2.2)\begin{equation} (m+m_a)\frac{\textrm{d} v}{\textrm{d} t} + v\frac{\textrm{d} m_a}{\textrm{d} t} = F_b + F_g + F_v. \end{equation}

Before (2.2) can be solved for $z(t)$ (including the derivatives $v(t) = \textrm {d} z/\textrm {d} t$, and the acceleration $a(t) = \textrm {d} v/\textrm {d} t$), assumptions must be made about the mathematical form of $m_a$, $F_b$ and $F_v$. Based on our front dynamics discussion, we now consider a few scenarios, shown in figure 1, and solve (2.2) numerically or, if possible, exactly. The original added-mass model, shown in (a), assumed that a solid, cylindrical plug grows straight down, but that the total added mass is some proportion of an inverted cone-shaped region that grows downward and outward at the same rate. We note that schematic in figure 1(a), which follows Waitukaitis & Jaeger (Reference Waitukaitis and Jaeger2012), does not show a cone but a truncated cone; this difference does not affect the scaling laws shown. An alternative, shown in (b) is to consider drag force $F_v$ on the growing solid plug through shear in a boundary layer with thickness $\delta$. A version of this model was proposed in appendix D of Waitukaitis (Reference Waitukaitis2014). Although the existence of this thin boundary layer was documented in Han et al. (Reference Han, Peters and Jaeger2016), the microscopic physics that set $\delta$ are not known. Some possibilities include that $\delta$ is set by $\phi$, similar to the front width in Waitukaitis et al. (Reference Waitukaitis, Roth, Vitelli and Jaeger2013), or by the particle size, perhaps through Darcy pressure. Finally, several 2-D and 3-D imaging experiments suggest that the solid-like region grows laterally. In this case, the solid region experiences a drag force that grows with its surface area, as depicted in figure 1(c). We first describe the first two cases in §§ 2.4 and 2.5. We then discuss scaling laws predicted by these models in §§ 2.6 and 2.7. Finally, we discuss the third case and its scaling laws in § 2.8.

2.4. Case 1: added-mass model, no viscous drag

First, we consider the original added-mass model (Waitukaitis & Jaeger Reference Waitukaitis and Jaeger2012), which assumed that the solidified region is an inverted cone whose height and radius grow at the same rate $v_f = k v$, yielding $m_a=C_m \rho (1/3){\rm \pi} (D/2+kz)^2kz$, where $\rho$ is the suspension mass density, $C_m$ is an added-mass coefficient found experimentally to be $C_m\approx 0.37$, and D is the intruder diameter. The fact that $C_m<1$ means that the entirety of the added-mass region is not perfectly rigidly connected to the intruder. They assumed that viscous drag was negligible or absent and set $F_v = 0$ and that $F_b$ comes from displaced fluid in a conical depression near the intruder, $F_b = 1/3{\rm \pi} \rho g z(D/2+kz)^2$. Numerical solutions to this model are qualitatively similar to experimental trajectories, as shown in figure 2 (thick black dashed line) and in Waitukaitis & Jaeger (Reference Waitukaitis and Jaeger2012).

Figure 2.

Experimental trajectories (red) for velocity $v(t)$ and acceleration $a(t)$ are compared with the solutions to the intruder equation of motions for the added-mass (thick black dashed line) and viscous (thinner black dot-dashed line) models discussed in §§ 2.4 and 2.5, respectively. (a,b) Show impacts with $v_0 \approx 4$ m s$^{-1}$ and $v_0 \approx 1$ using cylindrical intruders with $m=189$ g and $D=25.4$ mm. The fit parameters in both models can be adjusted to approximately match the experiments. The added-mass model trajectories for both panels were solved using $C_m = 0.1$, which is much smaller than the value $C_m = 0.37$ used in Waitukaitis & Jaeger (Reference Waitukaitis and Jaeger2012). The viscous model trajectories are those in (2.5) and (2.6) using $C_v = 0.4$, $\eta _s = 20$ Pa and $\delta = 1$ mm.

2.5. Case 2: cylindrical jammed region with large, viscous-like drag

While the added-mass model provides qualitative features that can be matched to experiments, it is not unique in this respect: other choices for $F_v$ and $m_a$ yield similar results and can also be calibrated to match experimental trajectories, as also shown in figure 2 (thinner black dot-dashed line). In particular, models where $F_v$ is dominant yield similar results and have the advantage of matching other features of the dynamics, as discussed below. As shown in figure 2 of Han et al. (Reference Han, Peters and Jaeger2016), the growing jammed region moves at approximately the same speed $v$ as the intruder, and it is surrounded by a thin layer of thickness $\delta \approx 5$ mm where the shear rate is $v/\delta$. Thus, on dimensional grounds, we can approximate viscous force as

(2.3)\begin{equation} F_v={-}C_v \eta_s S \frac{v}{\delta}. \end{equation}

Here, $C_v$ is a dimensionless drag coefficient, $\eta _s$ is the effective (constant) viscosity of the suspension and $S$ is the surface area of the jammed region. We reemphasize that our use of a constant viscosity $\eta _s$ does not mean that we consider the suspension to be a simple, viscous fluid. Instead, we refer to the large, nearly constant viscosity observed for $\dot {\gamma }>\dot {\gamma }_c$. For example, figure 11 of Fall et al. (Reference Fall, Bertrand, Ovarlez and Bonn2012) shows a nearly constant value of $\eta _s \approx 100$ Pa s for cornstarch suspensions in the fully shear-thickened regime for several different volume fractions. Additionally, the shear rates in this layer would be of the order of $\dot {\gamma }\sim 10 - 100$ s$^{-1}$, which is larger than $\dot {\gamma }_c \sim 1 - 10$ for cornstarch suspensions at volume fractions near $\phi = 0.4$ (Fall et al. Reference Fall, Bertrand, Ovarlez and Bonn2012).

This model is a generalized form of a model appearing in Appendix D of Waitukaitis’ Ph.D. thesis (Waitukaitis Reference Waitukaitis2014), which involves a columnar, solid-like front growing beneath the intruder with height $h_f = k z$ and thus $S = {\rm \pi}D k z + {\rm \pi}D^2 / 4$. We note that their dimensional analysis used $D$ in place of $S/\delta$, which then required $\eta _s$ to be unphysically large, $\eta _s \approx 2000$. This situation is sketched in figure 1(b). If the second term is dropped on the grounds that $D$ is much smaller than $kz$ or that the viscous-like forces only act on the sides of the growing cylinder, then the resulting dynamics is exactly solvable. This approximation is certainly not valid during the very early stages of impact, which would be particularly important for large $D$ and small $m$. In this case, (2.3) becomes

(2.4)\begin{equation} F_v={-}C_v {\rm \pi}D \eta_s k z \frac{v}{\delta}. \end{equation}

Assuming other forces can be neglected and $m_a$ is negligible, then (2.2) can be exactly solved, yielding

(2.5)\begin{gather} v(t)=v_0 \textrm{sech}^2(t/\tau), \end{gather}

(2.6)\begin{gather}a(t)={-}\sqrt{\frac{2C_v {\rm \pi}D k \eta_s}{m\delta}}{v_0}^{3/2}\textrm{sech}^2(t/\tau)\textrm{tanh}(t/\tau), \end{gather}

where $\tau = \sqrt {C_v {\rm \pi}D k \eta _s v_0/2m\delta }$. These functions are plotted in figure 2 and agree well with experiments. The viscous model solution in figure 2 uses $C_v = 0.42$, $\delta = 0.5$ cm and $\eta _s = 100$ Pa s. This agrees well with the viscosity of cornstarch and water–CsCl suspensions in the shear-thickening regime with similar values of $\phi$, as shown in figure 11 of Fall et al. (Reference Fall, Bertrand, Ovarlez and Bonn2012). This comparison demonstrates that reasonable parameter values can be used in matching to experiments, although there is some flexibility and therefore uncertainty in the values of these parameters.

2.6. Scaling laws for the added-mass model

Since both the added-mass and the viscous drag models can be reasonably matched to experimentally observed intruder trajectories, some further validation can come from comparing how $F_{max}$ and $t_{max}$ scale with $v_0$, $m$ and $D$. Such scalings have been previously used in the case of impact to connect macroscale dynamics with the microscale mechanisms that give rise to them (Uehara et al. Reference Uehara, Ambroso, Ojha and Durian2003; Walsh et al. Reference Walsh, Holloway, Habdas and de Bruyn2003; Goldman & Umbanhowar Reference Goldman and Umbanhowar2008; Clark, Petersen & Behringer Reference Clark, Petersen and Behringer2014; Zhao et al. Reference Zhao, Zhang, Tjugito and Cheng2015; Krizou & Clark Reference Krizou and Clark2020). For all experiments and theoretical models, we find that these scalings can be well approximated by

(2.7)\begin{gather} F_{max} = A {v_0}^\alpha \end{gather}

(2.8)\begin{gather}t_{max} = B {v_0}^\beta . \end{gather}

The prefactors $A$ and $B$ can vary with intruder properties, and we will examine how they depend on $m$, $D$ and, for conical intruders, cone angle $\theta$.

In the added-mass model, $F_{max}$ and $t_{max}$ can depend on four parameters: $m$, $\rho$, $v_0$, $D$. A total of three dimensionless quantities can be formed by these six total parameters, so $F_{max}/\rho {v_0}^2 D^2$ and $t_{max} v_0/D$ must each be a function of a single dimensionless quantity $m/\rho D^3$. This suggests that, when $m/\rho D^3$ is fixed, $F_{max}\propto {v_0}^2$ and $t_{max}\propto {v_0}^{-1}$. The added-mass model was solved numerically by Mukhopadhyay et al. (Reference Mukhopadhyay, Allen and Brown2018), finding that $F_{max} \propto {v_0}^2 m^{2/3}$. We also numerically solve the added-mass model and find the same result, along with $t_{max} \propto {v_0}^{-1}m^{1/3}$. We find $F_{max}$ and $t_{max}$ to be nearly independent of $D$ in the range of parameters studied here, in agreement with Mukhopadhyay et al. (Reference Mukhopadhyay, Allen and Brown2018). The lack of dependence on $D$, combined with the dimensional analysis, also requires $F_{max} \propto \rho {v_0}^2 D^2 (m/\rho D^3)^{2/3}$ and $t_{max} \propto D/{v_0} (m/\rho D^3)^{1/3}$, in agreement with the numerical solutions. Figure 3 shows numerical solutions as well as these scaling laws for the added mass model.

Figure 3.

Data in (a,b) show numerical solutions for $F_{max}$ and $t_{max}$ plotted as a function of $v_0$, where each symbol represents a different mass. All data with $v_0>0.5$ m s$^{-1}$ (to avoid the low-velocity plateau seen in (a,b), where gravity begins to play a role) are plotted in (c,d) in terms of dimensionless quantities $F_{max}/\rho {v_0}^2 D^2$, $t_{max}/(D/v_0)$ and $m/\rho D^3$.

2.7. Scaling law for viscous models

In the viscous model, there are two additional dimensionless numbers: ${Re} = \rho v_0 D/\eta _s$ and $D/\delta$. This means that the dimensional analysis no longer directly predicts a relationship between $F_{max}$ and $v_0$. However, the peak force and time at which the peak acceleration occurs can be directly calculated by differentiating $a(t)$ in (2.6), setting the result to zero, and solving for $t$. This time $t_{max}$ can be substituted back into (2.6) to calculate $a_{max} = a(t_{max})$. By this method, the peak force $F_{max} = m a_{max}$ and $t_{max}$ are found to be

(2.9)\begin{gather} F_{max}=\sqrt{2C_v D k \eta_s m/\delta}{v_0}^{3/2} \textrm{sech}^2(\beta)\tanh(\beta), \end{gather}

(2.10)\begin{gather}t_{max}=\sqrt{\frac{2\;m \delta}{C_v D k \eta_s v_0}}\beta, \end{gather}

where $\beta = \frac {1}{2}\log (2+\sqrt {3})$. We also solve the viscous model numerically and find the same result. Thus, for the viscous model where the jammed region is a cylindrical column, $F_{max} \propto {v_0}^{3/2} m^{1/2}D^{1/2}$ and $t_{max} \propto {v_0}^{-1/2} m^{1/2}D^{1/2}$. These scaling relations can also be written in terms of the dimensionless quantities given above, e.g. $F_{max}/\rho {v_0}^2 D^2 \propto {Re}^{-1/2}(D/\delta )^{1/2}(m/\rho D^3)^{1/2}$.

2.8. Case 3: hybrid models and scaling laws

The simple viscous model discussed above does not include several features that may make a comparison with experimental data more complicated. First, the solidified region grows in all three dimensions, not just straight down in a cylindrical column. This is particularly relevant when the contact point for the impacting object is point like. Second, momentum is being transferred to the solidified region, so added-mass terms must be included generally. Third, the rate at which the front is growing tends to decrease with time (i.e. $\varGamma <1$). Finally, many of the physical parameters in these models have been previously studied, so there are physical bounds on, e.g. $\eta _s$ based on experimental measurements. To check how sensitive the scaling laws are for varying shapes of the growing jammed region, we study a case where the volume of the solidified region grows as ${z_f}^3$ and the surface area grows as ${z_f}^2$, where $z_f = k z^\varGamma$. For simplicity, we approximate the growing jammed region as a hemisphere where the volume and surface area are $2{\rm \pi} {z_f}^3/3$ and $2{\rm \pi} {z_f}^2$, respectively. This situation is sketched in figure 1(c). We note that this overestimates the volume (and thus the added mass) in particular, since Han et al. (Reference Han, Peters and Jaeger2016) showed that the mass is better approximated as a half-ellipsoid with semi-minor axes of $z_f$, $z_f/2$ and $z_f/2$, meaning that the volume is $1/4$ of the hemisphere. This yields

(2.11)\begin{gather} m_a = C_m \rho \frac{2{\rm \pi}}{3} (kz^\varGamma)^3, \end{gather}

(2.12)\begin{gather}F_v ={-} 2 C_v \eta_s {\rm \pi}(kz^{\varGamma})^2 \frac{v}{\delta}. \end{gather}

We first consider the case where the added-mass term is dominant. We set $C_m=0.2$, $C_v=0$, $F_b = 0$ and numerically solve (2.2). We find that $F_{max} \propto {v_0}^{2}$ and $t_{max} \propto {v_0}^{-1}$ persist for all values of $\varGamma$ we study ($1/3<\varGamma <1$). We emphasize that these scaling exponents are required by the dimensional analysis and inconsistent with experimental data shown below. Next, we consider the case where the viscous term is dominant, setting $C_v=0.5$, $C_m=0$ and $F_b=0$. For $\varGamma = 1$, corresponding to the front moving at a constant speed in all directions, we find $F_{max} \propto {v_0}^{1.63}m^{0.69}$ and $t_{max} \propto {v_0}^{-0.67}m^{0.33}$. For $\varGamma = 1/3$, corresponding to the lower-bound case where compression-induced jamming is dominant (see § 2.2), we find $F_{max} \propto {v_0}^{1.40}m^{0.40}$ and $t_{max} \propto {v_0}^{-0.40}m^{0.60}$. The behaviour smoothly varies between these limits as $\varGamma$ is varied. For example, when $\varGamma = 0.7$, we find $F_{max} \propto {v_0}^{1.58}m^{0.59}$ and $t_{max} \propto {v_0}^{-0.60}m^{0.42}$.

We also find that, if added-mass and viscous terms are both present, then the values of the exponents fall in between the predictions of each model, depending on the relative strength of each term. To illustrate this, figure 4 shows numerical solutions using (2.11) and (2.12). We choose $C_m = 0.1$, $\rho =1630$ kg m$^{-3}$, $k=4$, $\varGamma = 0.7$, $C_v = 0.5$, $\eta _s = 20$ Pa s and $\delta = 2$ mm, all values that are based on previous experiments or within reasonable physical bounds. This yields $F_{max} \propto {v_0}^{1.61}m^{0.59}$ and $t_{max} \propto {v_0}^{-0.90}m^{0.46}$.

Figure 4.

Mixed model numerical solutions using parameters described in the text. As in figure 3, each symbol represents a different mass.

2.9. Summary of theoretical considerations

Table 1 shows a summary of all the theoretical predictions from these models, assuming the forms $F_{max} \propto {v_0}^\alpha m^{\zeta } D^\lambda$ and $t_{max} \propto {v_0}^\beta m^{\kappa } D^\mu$. These scalings can be derived analytically only for case 2, as shown above and in Waitukaitis (Reference Waitukaitis2014), as well as case 1 in the limit of small $D$ (Mukhopadhyay et al. Reference Mukhopadhyay, Allen and Brown2018). We also solve these models numerically to confirm the analytical scalings, including the non-zero values of $D$ we use in experiments for case 1. All results from case 3 are obtained numerically.

Table 1.

A table summarizing predictions of the models shown in figure 1, where $F_{max} \propto {v_0}^\alpha m^{\zeta } D^\lambda$ and $t_{max} \propto {v_0}^\beta m^{\kappa } D^\mu$. See § 2.9 for a discussion.

All added-mass models (case 1 and case 3 with $C_v = 0$) predict $\alpha = 2$ and $\beta = -1$. Viscous models (case 2 and case 3 with $C_m = 0$) predict $1.4<\alpha <1.63$ and $-0.67 < \beta < -0.4$. The exponents associated with $m$, $\zeta$ and $\kappa$, are similar for all models. The exponents associated with $D$, $\lambda$ and $\mu$, are zero for all added-mass models, but are non-zero for the viscous model, case 2, with $\lambda = 1/2$ and $\mu = -1/2$. This is due to the fact that the size of the jammed region (and therefore the surface area experiencing viscous-like drag) scales with the intruder diameter in this model.

3.1. Experimental results

Figure 5 shows $F_{max} = -ma_{max}$ and $t_{max}$ plotted as a function of $v_0$ for four representative experiments of cylinders impacting cornstarch suspensions: three from our experiments (one cylinder, one sphere and one cone) as well as the experimental data from Waitukaitis & Jaeger (Reference Waitukaitis and Jaeger2012). These quantities appear to scale with $v_0$ according to power-law relations, (2.7) and (2.8). Comparison with the fit line that is shown strongly suggests that $\alpha \approx 1.5$; linear fits to the data from our experiments confirm this, returning $1.3<\alpha <1.6$ for all intruders we study. This is consistent with the range predicted by viscous models discussed in § 2. The data for $t_{max}$ are more scattered, making clear determination of $\beta$ more difficult. Additionally $v_0>3$ m s$^{-1}$, the $t_{max}$ data from all intruders appear to flatten out and even curve upward slightly, which is not predicted by any of the theoretical models. However, best fits for impact velocities $0.5< v_0<3$ m s$^{-1}$ give $\beta \approx -0.5$. These values, $\alpha = 1.5$ and $\beta = -0.5$, agree with the viscous model well, but do not agree with the predictions of the added-mass models discussed above, $\alpha = 2$ and $\beta = -1$. This strongly suggests that viscous terms at the boundary of the dynamically jammed region play an important, and likely dominant, role in the deceleration of the intruder.

Figure 5.

The magenta stars are data from Waitukaitis and Jaeger (Waitukaitis & Jaeger Reference Waitukaitis and Jaeger2012) for impacts using a cylindrical intruder with $D = 1.86$ cm and $m=368$ g. The other three data sets come from our experiments, using cylindrical ($D = 3$ cm, $m = 189$ g), conical ($\theta = 70^{\circ }$, $D = 3$ cm, $m = 195$ g) and spherical ($D=3$ cm, $m = 199$ g) intruders. All four data sets follow power laws consistent with $F_{max} = A {v_0}^{3/2}$ and $t_{max}=B{v_0}^{-1/2}$; these are representative of all experiments. We also show $t_{max}\propto {v_0}^{-1}$, which is more consistent with low-velocity ($v_0 < 0.5$) data from Waitukaitis.

To further examine the consistency of these models with the experimental data, figure 6 shows how the prefactors $A$ and $B$ scale with $m$, $D$ and cone angle $\theta$. We measure $A$ and $B$ as the mean of $F_{max}/{v_0}^{\alpha }$ and $t_{max}/{v_0}^{\beta }$, with $\alpha = 3/2$ and $\beta = -1/2$. Given the scatter in the $t_{max}$ data, we also measure $B$ using other values of $\beta$ ranging from $-$0.4 to $-$1, and the results shown in figure 6 are insensitive to our choice of $\beta$. Error bars represent the standard error on the mean; e.g. $A$ is measured as the mean of $F_{max}/{v_0}^{\alpha }$, and the size of the error bar is given by the square root of these data divided by the number of data points. Figures 6(a) and 6(b) show $A$ and $B$ vs $m$ for three cylindrical intruders with the same diameter $D=25$ mm but with varied mass, $m \approx 80$, 150 and 230 g. Power-law fit lines are shown in black for the predictions of the original added-mass model, $A \propto m^{2/3}$ and $B \propto m^{1/3}$, and the viscous model involving a quasi-1-D cylindrical dynamically jammed region, $A \propto m^{1/2}$ and $B \propto m^{1/2}$. The data appear more consistent with the viscous model predictions, especially for $B$. However, we note that the details of the shape of the added mass, as well as how dramatically the propagating front slows down as it moves, can cause these exponents to vary somewhat.

Figure 6.

(a,b) Show $A$ and $B$ vs $m$ for four cylindrical intruders with $D = 2.5$ cm (for the three square symbols) and $D = 1.86$ cm (for the magenta stars, from Waitukaitis & Jaeger Reference Waitukaitis and Jaeger2012). (c,d) Show $A$ and $B$ vs $D$ for experimental data on cylindrical (square symbols) and spherical intruders with $m=189$ g and $m=199$ g, respectively. (e,f) Show $A$ and $B$ for conical intruders with $m=195$ g and $D = 2.5$ cm as a function of cone angle $\theta$. Error bars represent the standard error of the mean (see text).

3.2. Intruder size scaling

Figures 6(c) and 6(d) show data from cylindrical and spherical intruders of similar $m$ but varying $D$. The cylinders have $m \approx 190$ g and $D=12.5$, 25 and 50 mm, and the spheres have $m \approx 200$ g and $D=20$, 30 and 50 mm. The added-mass model predicts that, for these sizes and weights, there is very little dependence of $A$ or $B$ on $D$, i.e. $A \propto D^{0}$ and $B \propto D^{0}$. The first viscous model predicts $A \propto D^{1/2}$ and $B \propto D^{-1/2}$. Overall, the experimental data show that $A$ increases with $D$ and $B$ decreases with $D$, which is inconsistent with the added-mass model. We note that increase is clearer for the cylindrical intruders (square symbols) than for the spherical intruders (circular symbols). The cylindrical intruders appear to follow the predictions of the first viscous model, $A \propto D^{1/2}$ and $B \propto D^{-1/2}$. For spherical intruders, the hybrid model from § 2.8 may be more relevant, where the impact is more point like instead of a circular surface that makes simultaneous contact with the fluid. These models still predicted $\alpha \approx 1.5$ and $\beta \approx -0.5$, but they had no $D$ dependence.

3.3. Cone-shaped intruders

Intruder shape affects $A$ and $B$ somewhat, as can be observed from the dynamics of cone-shaped intruders. Figures 6(e) and 6(f) show conical intruder data whose mass and diameter are constant, $m\approx 195$ g and diameter $D=30$ mm, but with varied angles $\theta =0^\circ$, 20$^\circ$, 30$^\circ$, 45$^\circ$, 55$^\circ$ and 70$^\circ$. Here, $\theta = 0^\circ$ corresponds to a flat cylinder and $\theta = 90^\circ$ is the maximum possible value. We observe $F_{max}\propto {v_0}^{1.5}$ and $t_{max}\propto {v_0}^{-0.5}$ for all cone angles, suggesting that viscous-like forces are again dominant. However, $A$ decreases with increasing $\theta$, while $B$ increases with increasing $\theta$. One hypothesis for this behaviour is that larger $\theta$ corresponds to a smaller contact area, equivalent to smaller $D$. Another explanation could come from the fact that increasing $\theta$ means that the dynamically jammed region transitions from being generated primarily by normal compression (for $\theta = 0$) to being generated primarily through shear jamming. As shown by Han et al. (Reference Han, Wyart, Peters and Jaeger2018), the value of $k$ is smaller for fronts created by shear jamming. Our data are inconclusive on this question, except for the fact that we consistently find $\alpha \approx 1.5$ and $\beta \approx -0.5$ for all values of $\theta$, which is consistent with the class of viscous models discussed in § 2.

3.4. Relaxation after peak deceleration

Finally, we consider the intruder dynamics after the peak deceleration, which also provides information on the microscopic physics of suspension dynamics. While the dynamics of impact before peak deceleration is highly sensitive to various experimental control parameters, the post-peak dynamics is not. This is shown in figure 7. Figures 7(a) and 7(b) show impacts with $1< v_0<1.5$ and $3< v_0<3.5$ m s$^{-1}$, respectively, both with a wide variety of intruder properties; $t_{max}$ varies dramatically with $v_0$ and intruder properties, which has been the subject of our analysis so far. However, figures 7(c) and 7(d) show that forces decay quasi-exponentially with a time scale between 2 and 3 ms; this behaviour is largely independent of speed and intruder properties. This implies that, e.g. the time scale $\tau = \sqrt {C_v k \eta _s v_0/2m}$ in (2.5) and (2.6) might capture the buildup to peak force but not the decay. This suggests that the relaxation dynamics is dominated by the microscopic material composition in a way that is not sensitive to the intruder. Thus, this dynamics appears to lie outside the description of either the added-mass or viscous models. Our observations are consistent with Peters & Jaeger (Reference Peters and Jaeger2014), who found that changing the viscosity of the suspending fluid affected the dynamics of the relaxation of the jammed front but not its growth.

Figure 7.

Acceleration curves and post-peak acceleration curves for a few representative examples of experimental data sets with (a,c) $1< v_0<1.5$ m s$^{-1}$ and (b,d) $3< v_0<3.5$ m s$^{-1}$. In (c,d), we indicate exponential decay with time scales of 2 and 3 ms with solid lines in both panels for reference. The cones have $m=195$ g and $D = 2.5$ cm; the light and heavy cylinders have $D = 2.5$ cm with $m = 77.8$ g and $m = 227$ g, respectively; and the small and large spheres have $m=199$ g and $D=2$ cm and $D=5$ cm, respectively.

The fact that fluid viscosity slows down the relaxation dynamics suggests that Darcy–Reynolds theory, demonstrated by Jerome et al. (Reference Jerome, Vandenberghe and Forterre2016), may play a crucial role in holding the solidified region together. The system studied by Jerome et al. (Reference Jerome, Vandenberghe and Forterre2016) involves much larger, inorganic glass beads with diameter of approximately 100 $\mathrm {\mu }$m or larger, whereas (organic) cornstarch particles are approximately 1 to 10 $\mathrm {\mu }$m. The grains were gravitationally loaded into a settled configuration – that is, the initial state of the system involves all particles making solid–solid contact – as opposed to the cornstarch particles, which are stirred between impacts and remain suspended in the fluid at relatively low volume fractions. Then, if the grains dilate during impact, there is a strong solidification that is strictly driven by a combination of Reynolds dilatancy (the packing expands) and Darcy flow (high resistance of the fluid being sucked into the expanding pores between grains); if the grains compact, there is no solidification. Our system begins at a low volume fraction and likely does not compact significantly (Han et al. Reference Han, Peters and Jaeger2016). Thus, it is unlikely that the Darcy–Reynolds theory plays a direct, dominant role in the present experiments, but it may play a crucial, secondary role. Relaxation of the solidified region occurs as particles rearrange and lose contact with each other, which requires fluid flow through the pore structure. Thus, Darcy flow in particular may be responsible for holding the solidified region together and may also control the relaxation dynamics shown in figure 7.