Raj Kumar Pal Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 e-mail: pal3@illinois.edu Jeremy Morton Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 Erheng Wang Postdoctoral Associate Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 e-mail: erhengwang@gmail.com John Lambros Professor Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 e-mail: lambros@illinois.edu Philippe H. Geubelle Professor Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 e-mail: geubelle@illinois.edu Impact Response of Elasto-Plastic Granular Chains Containing an Intruder Particle Wave propagation in homogeneous granular chains subjected to impact loads causing plastic deformations is substantially different from that in elastic chains. To design wave tailoring materials, it is essential to gain a fundamental understanding of the dynamics of heterogeneous granular chains under loads where the effects of plasticity are significant. In the first part of this work, contact laws for dissimilar elastic–perfectly plastic spherical granules are developed using finite element simulations. They are systematically normal- ized, with the normalizing variables determined from first principles, and a unified con- tact law for heterogeneous spheres is constructed and validated. In the second part, dynamic simulations are performed on granular chains placed in a split Hopkinson pres- sure bar (SHPB) setup. An intruder particle having different material properties is placed in an otherwise homogeneous granular chain. The position and relative material property of the intruder is shown to have a significant effect on the energy and peak transmitted force down the chain. Finally, the key nondimensional material parameter that dictates the fraction of energy transmitted in a heterogeneous granular chain is identified. [DOI: 10.1115/1.4028959] 1 Introduction Granular packings are envisioned to be useful for applications ranging from dynamic wave propagation and mitigation to tailor- ing of impact energy. The dynamics of these systems has been an active area of research in the past few decades, starting from the seminal work of Nesterenko [1,2], who predicted the existence of solitary waves in unstressed elastic chains characterized by Hertzian contact [3]. Solitary waves have been observed experi- mentally [4,5] for chains subjected to impact velocities small enough to avoid plastic deformations. For longer loading times compared to the timescale of passage of solitary waves between beads, trains of solitary waves have been observed [6–8]. Sen and coworkers [9,10] studied the dynamics of a chain of spheres with uniformly varying radii, while Porter et al. [11] investigated the existence of solitary waves in heterogeneous dimer and trimer chains, and determined the wave width, speed and forces for vari- ous material combinations. Granular packings are also finding potential applications as waveguides because they exhibit a range of nonlinear phenomena such as band gaps and nonlinear refraction [12]. Recently, Gusev and Tournat [13] demonstrated the design of waveguides with ordered granular packings in subsurface channels, where the contact stiffness changes with depth due to gravity. There have also been studies on the interaction of solitary waves in granular chains with boundaries, starting from the work of Job et al. [14], who modeled and studied experimentally the effect of solitary waves reflecting off a rigid wall. Yang et al. [15] studied the interaction of waves at the interface between a granu- lar chain and a linear elastic medium. The authors used a time-delay system model, where the displacement field is expressed as the sum of two functions representing forward and backward waves to satisfy the general solution of the wave equation. They studied the effect of properties of a single and a composite medium in contact with a granular chain. Heterogeneous periodic and nonperiodic arrangements of gran- ular beads have been investigated to design chains for specific objectives. Nesterenko et al. [16] studied the characteristics of wave reflection at the interface of two granular chains with dis- tinct radii. Using these concepts, Daraio et al. [17] demonstrated energy trapping in composite granular media, trapping, and disin- tegrating high amplitude waves in softer particles into weaker sep- arated pulses. Fraternali et al. [18] designed composite protectors to minimize the force transmitted through a granular chain using reflections between distinct materials. Job et al. [19] studied the effect of a single intruder granule with a different radius in an oth- erwise homogeneous chain, demonstrating energy localization at the intruder. A part of the incident energy is trapped as localized oscillations, whose frequency spectrum is shifted depending on the mass of this intruder and on the incident wave, showing its potential in wave mitigation and sound trapping in 3D granular assemblies. Other studies have focused on frictional dissipation in elastic granular chains, starting from the investigation by Rosas et al. [20], who observed a two-wave structure when a dissipative term is added to the Hertzian contact force. Vergara [21] modeled dissi- pation in granular chains by adding a viscoelastic term and a term proportional to the square of the beads’ relative velocities to the equations of motion. Most of the aforementioned studies have been conducted for impact velocities small enough to avoid yield- ing. However, as noted in Pal et al. [22] and On et al. [23], with increasing impact velocity, the stress concentration near the con- tact area causes plastic deformations, leading to very different Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 12, 2014; final manuscript received October 27, 2014; accepted manuscript posted October 31, 2014; published online November 14, 2014. Editor: Yonggang Huang. Journal of Applied Mechanics JANUARY 2015, Vol. 82 / 011002-1 Copyright V C 2015 by ASME Downloaded From: https://appliedmechanics.asmedigitalcollection.asme.org/ on 11/24/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use