Dynamic Buckling and Fragmentation in Brittle Rods

Abstract

We present experiments on the dynamic buckling and fragmentation of slender rods axially impacted by a projectile. By combining the results of Saint-Venant and elastic beam theory, we derive a preferred wavelength λ for the buckling instability, and experimentally verify the resulting scaling law for a range of materials including teflon, dry pasta, glass, and steel. For brittle materials, buckling leads to the fragmentation of the rod. Measured fragment length distributions show two clear peaks near λ/2 and λ/4. The non-monotonic nature of the distributions reflect the influence of the deterministic buckling process on the more random fragmentation processes.Long, thin supports are ubiquitous in natural and engineered load bearing structures, from spider legs to the steel struts of a skyscraper [1,2]. A single rod will buckle if too much force is applied along its axis, which can lead to the catastrophic failure of the structure. The buckling instability is seen at all sizes, from pole vaulting [3] to protein microtubules confined in vesicles [4] and carbon nanotube atomic force microscope probes [5]. While the classic Euler buckling of a rod is due to a static axial load [6], a different physical process occurs when the stress is applied suddenly, as during impact [7,8,9]. In this Letter, we show that this "dynamic buckling" [10] obeys a simple scaling law, which we derive by combining the approach of Saint-Venant with the classical theory of elastic rods [6,7]. For brittle rods, buckling often leads to breaking, for which we find the distribution of fragment lengths displays a unique non-monotonic shape reflecting the primary buckling instability.The process of dynamic buckling and subsequent fragmentation is illustrated in Fig. 1, in which a falling weight strikes an upright brittle rod (dry pasta). Within a fraction of a millisecond after impact, a sinusoidal perturbation appears (Fig. 1b), much different than the halfwavelength seen in Euler buckling. A few tenths of a millisecond later, the pasta has buckled appreciably and begins to shatter (Fig. 1c). This imparts angular momentum of alternating signs to the fragments, which rotate and scatter (Fig. 1d-f).Our experimental setup consists of a simple metal holder for the rod, and a pneumatic cannon in which a pressure reservoir at 60 psi delivers an impulse to a steel cylindrical projectile (1.46 cm diameter, 24.9 or 10.0 g) held at the end of a 1.5 m acrylic tube with two magnets. The holder rests on a steel plate in a sandbox which serves as a shock absorber to stop the projectile. The buckling and fragmentation dynamics were imaged by a high speed digital video camera (Phantom v5.0), capable of capturing up to 62,000 frames per second. The speed of the projectile was also measured just before impact using the video system. The materials used included dry pasta (ρ = 1.5 g/cm 3 , d = 1.1 mm and 1.9 mm) [11], borosilicate glass (ρ = 2.4 g/cm 3 , d = 2.0 mm), type 303 stainless steel (ρ = 7.9 g/cm 3 , d = 1.6 mm), and teflon (PTFE) (ρ = 2.2 g/cm 3 , d = 2.0...

Emergence of energy dependence in the fragmentation of heterogeneous materials

“…The mass distribution of pieces proved to have discrete humps at certain fractions of the buckling wavelength [43] similar to what we obtained for plates. Studying the impact induced breakup of thin glass plates, in the experiments of Ref.…”

Dynamic curling of an Elastica: a nonlinear problem in elastodynamics solved by matched asymptotic expansions

“…A linear version of these equations is classically used to analyze dynamic buckling (Golubovic et al, 1998;Gladden et al, 2005;Schindler and Kolsky, 1983;Audoly and Neukirch, 2005) as well as the stability of equilibrium solutions (Caflisch and Maddocks, 1984). The dynamics of an Elastica has also been characterized by means of amplitude equations Tabor, 1996, 2000), which is a type of a weakly nonlinear expansion.…”

Thermalizing and Damping in Structural Dynamics

“…The second example we consider is the case of dynamic buckling of a simply supported beam of length L. A classical textbook example, dynamic buckling is produced by an impulse generated by a mass M at speed U 0 at one end of the bar. This impulse puts in motion an axial wave that travels at the speed of sound, c ¼ ffiffiffiffiffiffiffiffi E=q p , along the beam and which is reflected at the end [12]. Provided that the time scale of lateral vibration is much larger than the time scale of wave propagation along the beam, load P can be considered as constant along the beam length…”