Raj kumar Pal

I am a Postdoctoral Fellow in the School of Aerospace Engineering at Georgia Institute of Technology. Prior to this, I got my PhD in Theoretical and Applied Mechanics from the University of Illinois at Urbana Chamapign.

I work with Professor Massimo Ruzzene. I have collaborated/continue to collaborate with the following faculty at Georgia Tech

I have collaborated with and mentored the following students and postdocs at Georgia Tech Aerospace Engineering.

Email: raj.pal@aerospace.gatech.edu

List of papers: Google Scholar

Research projects

Topologically protected waves in elastic media

Topological boundary modes is a recent development in physics, allowing for defect-immune and scattering free wave propagation. Originally discovered in quantum mechanical systems, they have recently been extended to several other areas of physics. We learnt lessons from quantum condensed matter physics to develop a discrete mechanical lattice that supports a class of such modes. We then extended this discrete design to continuous elastic media using thin plates embedded with resonators. In 2017, we achieved the first experimental demonstration of two classes of such topologically protected modes: helical modes and valley modes in elastic media.

The figure on the left shows a unit cell of a square (Lieb) lattice. Right: snapshots of displacement amplitude in a finite lattice subjected to an external force at a point on the boundary. Notice how the wave bends around corners without backscattering. This robustness and immunity to backscattering in the presence of defects and corners is one of the unique features of topologically protected modes.

The above figure (left) shows a schematic of a plate with resonators in a hexagonal lattice arrangement. The masses are distinct at the two sub-lattice sites leading to a broken inversion symmetry. This plate supports topologically protected valley waves at the interface between two zones whose unit cells are inverted copies of each other. The wave does not backscatter or localize as it bends around sharp corners.


  1. Experimental observation of topologically protected helical edge modes in Kagome elastic plates. Submitted, preprint on arXiv, 2018.
    M. Miniaci, R. K. Pal, B. Morvan and M. Ruzzene. pdf

  2. Amplitude-dependent topological edge states in nonlinear phononic lattices. In revision, Physical Review E, 2018
    R. K. Pal, J. Vila, M. Leamy and M. Ruzzene. pdf

  3. Observation of topological valley modes in an elastic hexagonal lattice. Physical Review B , 2017
    J. Vila, R. K. Pal and M. Ruzzene. pdf

  4. Edge waves in plates with resonators: An elastic analogue of the quantum valley Hall effect. New Journal of Physics , 2017
    R. K. Pal and M. Ruzzene. pdf

  5. Helical edge states in 2D phononic systems using bilayered lattices. Journal of Applied Physics, 2015 .
    R. K. Pal, M. Schaeffer and M. Ruzzene. pdf

Pattern formation in lattices

Hexagonal lattice comprising of point masses interacting with axial and torsional springs. When boundary nodes are subjected to an affine deformation, the resulting displacement field can exhibit features ranging from uniform to localized deformation. These lattices have immense potential for wave steering applications.

Distinct patterns can form depending on the imposed boundary conditions, ranging from globally uniform patterns to localized deformation zones.
A square lattice exhibiting globally uniform patterns under in-plane deformation and localized folding deformation after undergoing out-of-plane deformation.

The figure above illustrates the change in dynamic behavior from isotropic propagation under tensile pre-stress to waves confined to the boundary under compressive pre-stress. Similarly, the wave behavior can be changed from isotropic to highly directional to achieve wave focusing effects.


  1. Mechanical response of 3-dimensional tensegrity lattices. Composites Part B , 2016
    J. Rimoli and R. K. Pal. pdf

  2. A continuum model for nonlinear lattices under large deformations. International Journal of Solids and Structures, 2016
    R. K. Pal, M. Ruzzene and J. Rimoli. pdf

  3. Effect of large deformation pre-loads on the wave properties of hexagonal lattices. Smart materials and structures, 2015
    R. K. Pal, J. Rimoli and M. Ruzzene. pdf

Past projects

Wave tailoring in granular chains

Control of wave propagation is an area of active research, with applications in noise control, vibration reduction, acoustic switches and rectification, energy harvesting.

In this project, we designed a lattice of packed granules whose response under an impact load can be varied from rapidly decaying waves to a solitary wave. The lattice consists of elastic spheres packed into a hollow cylinder. By applying external pressure on the cylinder, the response of the lattice can be varied. The above figure illustrates this behavior, with a being the precompression level due to the external pressure. The lattice behavior changes due to the change in stiffness at the contact between two spherical granules.

  1. Tunable wave propagation in granular crystals by altering lattice network topology. ASME Journal of Engineering Materials and Technology , 2017
    R. K. Pal, R. F. Waymel, P. H. Geubelle and J. Lambros. pdf

  2. Wave tailoring by precompression in confined granular systems. Physical Review E, 2014.
    R. K. Pal and P. H. Geubelle. pdf

Plasticity in granular media

The objective of this project is to investigate granular media for potential stress wave tailoring and impact protection applications. Using extensive finite element simulations in ABAQUS, we developed a unified contact law between spherical elastoplastic granules. The two granules can be of different sizes and materials. This contact law is incorporated into a dynamics solver and the dynamics of ordered granular packings under impact loads large enough to cause plastic deformations is studied.

We compared the impact response of 3D FCC granular packing of identical spheres with a continuum medium made of the same material. The above figures illustrate the problem schematic and the fraction of energy dissipated for both the granular and continuum media. Dissipation starts at orders of magnitude lower energies in granular medium compared to the continuum medium. This huge difference arises due to the geometry of the granular media. The contact is localized and this leads to high stress concentrations in the vicinity of the contact area in the granules. The high stress concentration leads to a substantially higher energy dissipation in granular media.


  1. High-amplitude elastic solitary wave propagation in 1-D granular chains with preconditioned beads: Experiments and theoretical analysis. Journal of the Mechanics and Physics of Solids, 2014.
    E. Wang, M. Manjunath, A. Awasthi, R. K. Pal, P. H. Geubelle and J. Lambros. pdf

  2. Impact response of elasto-plastic granular and continuum media: A comparative study. Mechanics of Materials, 2014.
    R. K. Pal and P. H. Geubelle. pdf

  3. Characterization of wave propagation in elastic and elastoplastic granular chains. Physical Review E, 2014.
    R. K. Pal, A. Awasthi and P. H. Geubelle. pdf

  4. Impact response of elasto-plastic granular chains containing an intruder particle. Journal of Applied Mechanics, 2015.
    R. K. Pal, J. Morton, E. Wang, J. Lambros and P. H. Geubelle. pdf

  5. Wave propagation in elastic and elastoplastic granular systems. Granular matter, 2013.
    R. K. Pal, A. Awasthi and P. H. Geubelle. pdf

Advection diffusion flows: Discontinuous Galerkin solver

Diffusion is typically treated using Fick's law and it is an excellent approximation in a lot of engineering applications. However, this approximation suffers from the infinite speed of propagation paradox. The approximation is valid when diffusive time scales are much smaller than advective time scales. There are applications, for example biological processes and environmental flows, where these two time scales are of the same order. In such cases, a hyperbolic diffusion law is required to capture the physics

In this project, I developed a positivity preserving space-time discontinuous Galerkin finite element method for accurately simulating hyperbolic advection diffusion flows. The above set of figures show the concentration field of a species injected into a nozzle where there is subsonic (left) and supersonic (right) oscillating flow in the transverse direction. It is observed that the species is confined and does not propagate to the boundary.


  1. Adaptive spacetime discontinuous Galerkin method for hyperbolic advection diffusion with a non negativity constraint. International Journal for Numerical Methods in Engineering, 2015.
    R. K. Pal, R. Abedi, A. Madhukar and R. B. Haber. pdf

Other publications

  1. Non-Schmid effects and finite wavelength instabilities in single crystal metals. Extreme Mechanics Letters, 2018.
    H. Salahshoor, R. K. Pal and J. Rimoli. pdf

  2. A Bloch-based procedure for dispersion analysis of lattices with periodic time-varying properties. Journal of Sound and Vibration, 2017.
    J. Vila, R. K. Pal, M. Ruzzene and G. Trainiti. pdf

  3. A monolithic strategy for fluid structure interaction problems. International Journal for Numerical Methods in Engineering, 2011.
    C. S. Jog and R. K. Pal. pdf