ABSTRACT
One dimensional stress waves travelling in granular chains exhibit interesting characteristics such as filtering, tunability and wave mitigation because on the formation of solitary waves, and solitary wave trains, within them. An idealized one dimensional granular medium, consisting of a linear array of contacting spherical brass beads, was loaded dynamically in a modified split Hopkinson pressure bar, with loading pulses that span a variety of rates and profiles using pulse shaping techniques. Different chain lengths were studied to determine how solitary waves form in a varying length granular medium, as well as the speed of wave propagation. It is found that the wave speed propagates faster for longer chains of brass beads. The high loading rates of the Hopkinson bar also allowed us to investigate plastic dissipation effects in the granular chain when composed of different types of metals. To further our understanding of wave propagation in ductile ordered granular media, the experimental results are compared with companion numerical simulations based on a particle contact law that accounts for plastic dissipation. Knowing the behavior of a stress wave propagating through such materials can lead to arrangements that can produce desired stress wave mitigation characteristics as the waves travel through the granular chain.
INTRODUCTION
A granular chain can be characterized as a group of particles which can displace independently of one another and interact only when in contact with an adjacent particle. In such a chain, each element can only transmit information to its neighbors via compression. Contact between two spherical surfaces, even elastic, is a nonlinear process, the best known case of which is the Hertz contact solution developed in the late 1890s (Timoshenko and Goodier [1]). Subsequent efforts pertain to the plastic deformation of contacting spheres, which neglect volume conservation of the plastically deformed sphere, were based on the model of Abbott and Firestone [2]. In addition to contact models, considerable amount of work has been done towards the study of the static response in granular materials by Drescher [3]. Dynamic contact studies have been conducted by Iida [4], Hughes and Kelly [5], Shukla [6, 7] and Xu [8] who investigated the effects of particle size and wave velocity in one and two dimensional granular chains. The current paper focuses on the wave velocity traversing through a one dimensional granular medium consisting of brass beads with varying lengths.
EXPERIMENTAL PROCEDURE AND RESULTS
This set of experiments was conducted using a split Hopkinson pressure bar (SHPB) for dynamic loading at high strain rates (102- 104 s-1). The SHPB consists of an incident bar and a transmitted bar, with the specimen sandwiched between the two. The bars are long enough to be considered uniaxial and are hardened to remain elastic during the loading process. The loading is generally done via a striker bar, usually made from the same material, impacting the incident end, sending a compressive stress wave to the specimen. A momentum trap, developed by Nasser [9], is added to the bar to provide a single loading wave, followed immediately by an unloading wave. This ensures the specimen undergoes a single impact for the duration of the test. Strain gages are placed along the incident and transmitted bar to record the stress waves traversing along the bar as shown in Figure 1.
Figure 1. Schematic of a split Hopkinson pressure bar with momentum trap.
Two strain gages are placed in the middle of each bar, and on opposite ends to cancel strain readings caused by possible bending of the bars. Strain gage (EA-06-250BK-10C) was used for all tests. The strain gages were connected through a signal conditioner which amplifies the strain and sent to an Agilent Technologies Digital Oscilloscope.
The specimens used were brass beads (Alloy 260) of 9.525 mm diameter obtained from McMaster-Carr. In order to maintain a one dimensional chain of spheres, a holder, adapted from Spadoni and Daraio [10], was fabricated which consists of a hollow metal tube with threaded end caps which can be placed onto the bar. The holders are sized such that a hemisphere extrudes from the tube and comes in contact with the bar, allowing a point-load contact for the stress wave to travel. The edges of the tube are threaded by an end cap of larger inner diameter to ensure the caps will not interfere with the sphere specimens. A holder setup with one end cap mounted is shown in Figure 2. The holder was tested and showed no alterations to the experimental data. Chains of brass ranging from one sphere to twelve spheres are tested, with different length holder tubes for each length of brass chains.
Figure 2. Image of the one dimensional sphere holder with one end cap in place.
Figure 3 shows a typical incident reflected and transmitted signal form the strain gauges (compensated for bending). Among other things, data from the experiment is used to calculate the wave velocity through the brass chain using the equation
where V is the wave velocity, N is the number of beads in the chain, d is the diameter of the bead, and ttotal is the time duration starting when the incident strain gage receives a signal and ending when the transmitted strain gage receives a signal. The travelling time of the wave within the chain can be calculated by knowing the positions of the strain gages. Since the material of the bar is known, both the wave velocity of the bar and the distance from the strain gage to the specimen can be calculated. This results in the travelling time of the wave within the incident bar and transmitted bar, which are denoted as tinc and ttrans, respectively. Thus, the denominator in equation (1) denotes travelling time of the wave within the chain.
Figure 3. Raw incident and transmitted data acquired from an experiment. The wave speeds were calculated for tests of various chain lengths and are plotted in Figure 4.
Figure 4. Wave velocity through one dimensional brass granular medium for varying chain lengths.
Note that the force in each case, which is known to control the wave speed of such nonlinear waves, is about the same. Aside from the single bead experiments, the wave speed appears to be similar, although possibly somewhat increasing, for increasing chain lengths. The wave propagation in granular medium is governed by the contact mechanism and the wave speed calculated is much smaller than the dilatational wave or shear wave velocity for the same material, although it does depend on loading amplitude. The wave speed for a single bead has significant variations, more than the experimental error. The reason for this is not clear, but we believe it is associated with progressive yielding of the two contact points of the bead.
CONCLUSIONS
The experimental data obtained by the strain gages show an increasing wave speed for an increasing distance for the wave to traverse. The wave speed also appears to approach a constant velocity value around 25% of the wave speed for a solid brass bar. This result is similar to that observed by Iida [4], and Xu and Shukla [8].
