Energy absorption in a bamboo foam
Original entry: Donald Aubrecht, APPHY 226, Spring 2009
"Energy absorption in a bamboo foam"
A. Le Goff, L. Courbin, H.A. Stone, and D. Quere
Europhysics Letters 84 36001 (2008)
Soft Matter Keywords
bamboo foam, surface tension, energy absorption, Weber number
This article presents experiment and simple theory for the interaction between projectiles and thin liquid films. The authors perform experiments on an idealized bamboo foam, observing that such a foam is capable of absorbing energy from a projectile passing through the foam, eventually bringing the projectile to rest. The authors also study the deflection of a projectile by a film oriented obliquely to the projectile's trajectory. For both studies, simple scaling arguments are developed that capture the main features of the film/projectile interaction.
Practical Application of Research
Foams are under consideration for use in systems designed to absorb kinetic energy from projectiles. These systems are deployed to protect internal contents, people, precious objects, and other fragile items. Foams present an attractive option because they are light, simple and fast to form, and inexpensive. This work shows that foams are capable of absorbing kinetic energy from projectiles and will eventually arrest them. Though not immediately practical, this work with ideal foams opens up the way for future studies of projectile interaction with real foams. The initial work to characterize projectile interaction with bamboo, staircase, and oblique foams will contribute to the understanding of interaction with real foams.
Projectile Interaction with Foams
As shown in figure 1, it is possible for an object to pass through a thin liquid film without popping the film. The sphere stretches the film until the point where the distance between the sphere and the unperturbed part of the film is of the order of the sphere radius. At that point, the film pinches off, but does not break. The time interval between successive images in figure 1 is constant and the vertical position of the sphere appears to be linear in time. This indicates that the velocity of the sphere is nearly constant during the short interval from the first to the last image and the energy transmitted to the film during this single event is quite small relative to the total kinetic energy of the sphere. However, the vibration of the film after impact indicates some transfer of energy from the sphere to the film and so the authors create the system shown in figure 2, to observe the effect of multiple interactions between projectile and film.
A bamboo foam (with films separated by 0.6mm) is created inside a transparent cylindrical tube and a solid sphere dropped from above the first film. Since the sphere is smaller than the capillary length of the film, it will float on the surface of the film if deposited gently. In this size regime, gravity is dominated by surface tension, so the beads are incapable of crossing the films using their weight alone. Figure 2b is created from high-speed movies of the falling sphere. The slope of the line gives the sphere's velocity, which is decreasing with time as the sphere crosses each film. The velocity of the sphere finally falls below the entrapment threshold of the film and the sphere is arrested on the lowest film shown.
Figure 3 shows the number of films, N, required to arrest a sphere as a function of the sphere's impact velocity, V. Experimental evidence shows that N is independent of the film's viscosity, so the authors neglect viscous effects and balance surface and mechanical energies. The energy transferred from the sphere (of mass m) to the film is simply <math>mgH</math>, where H is the height of descent until arrest. This is balanced by the surface energy of each cavity created when the sphere passes through a film with surface tension <math>\gamma</math>. To simply the analysis, the cavity is modeled as a cylinder of radius R (the sphere radius) and height L (the length below the film where pinch-off occurs). Therefore, the surface energy for N films is <math>2 \pi R L \gamma</math>, which can be approximated as <math>12 \pi R^2 \gamma</math> since experiment shows L depends weakly on impact velocity and is a nearly constant value of 6R. Equating the two energy expressions and plugging in for H (<math>H = N\Delta z + h</math>, where <math>\Delta z</math> is the distance between films and <math>h</math> is the distance the sphere falls before impacting the first film), the authors find the following expression for N:
<math>N = (mgh-12 \pi \gamma R^2)/(12 \pi \gamma R^2 - mg \Delta z)</math> 
The plot in figure 4 shows that the data for different spheres (sizes and densities) collapses to a nice linear curve. The solid line shown is the plot of total surface energy as a function of total mechanical energy without any adjustable fit parameters. From equation  it is clear that for arrest to be possible, the energy gained by the sphere between successive films (namely <math>mg \Delta z</math>) must be smaller than the energy transferred to the film. This puts an upper bound on the permissible spacing between films if arrest of the projectile is desired.
To further study the biased trajectories observed when a projectile falls through a staircase film (the result of parallel films being brought too close together, leading to a three-dimensional reorganization of the films), spheres were dropped through films oriented obliquely to their trajectory, as shown in figure 5. As the sphere crosses the film, it is interesting to note that the deformation in the film is perpendicular to the plane of the unperturbed film surface, not parallel to the sphere trajectory. As the deformation collapses, it imparts a net force on the sphere that acts along the center axis of the deformation. This deformation pulls the sphere toward the film as it continues to fall, leading to a deviation in the sphere's trajectory as shown in figure 5. Looking at Newton's second law projected onto the x-axis (for small angles), and assuming that the surface tension acts like an elastic force with magnitude <math>\pi R \gamma</math> we find:
<math>\Sigma F_x = ma_x</math>
<math>\Sigma F_x = \pi R \gamma sin(\alpha) = ma sin(\beta)</math>
<math>\pi R \gamma \alpha = ma \beta</math>
If we make the scaling argument that the acceleration is equal to the (nearly constant) velocity of the sphere divided by the characteristic time for crossing the film, we find:
<math>\pi R \gamma \alpha = m(V/\tau) \beta</math> 
As stated above, the maximum extension of the film deformation is L = 6R, allowing us to write <math>\tau = 6R/V</math> and introduce the Weber number (<math>We = \rho R V^2/\gamma</math>, where <math>\rho</math> is the sphere's density). Expression  can then be re-written as:
<math>\beta We \approx 9\alpha</math> 
Figure 6 shows a plot of <math>\beta*We</math> as a function of <math>\alpha</math>. The solid line is a linear relationship between the two dimensionless parameters with a slope of 10 (<math>\beta We = 10 \alpha</math>), which is close to the expected slope of 9 from expression . The data lies in nice agreement with this line indicating that the major features of the sphere/film interaction have been accounted for in the simple force balance.
written by Donald Aubrecht