Seminar-Nonlinear dispersive wave motion in continuous and atomistic metamaterials: Insights into extreme dynamics

EB III Room 2201

Mahmoud I. Hussein

Associate Professor, H. Joseph Smead Faculty Fellow

Department of Aerospace Engineering Sciences, University of Colorado Boulder

Wave motion lies at the heart of many disciplines in the physical sciences and engineering.

For example, problems and applications involving light, sound, heat or fluid flow are all likely to

involve wave dynamics at some level. In this seminar, in Part A, I will present our recent work

on a class of problems involving intriguing nonlinear wave phenomena‒large-deformation elastic

waves in solids; that is, the “large-on-small” problem. In Part B, I will present our recent

research on a new class of nanostructured semiconductors that enables superior thermoelectric

energy conversion properties.

In Part A, I will examine the propagation of a large-amplitude wave in an elastic one-
dimensional medium that is undeformed at its nominal state. In this context, the focus is on the

effects of inherent nonlinearities on the dispersion relation. Considering a thin rod, where the

thickness is small compared to the wavelength, I will present an exact formulation for the

treatment of a nonlinearity in the strain-displacement gradient relation. The derivation starts with

an implementation of Hamilton’s principle and terminates with an expression for the finite-strain

dispersion relation in closed form. The derived relation is then verified by direct time-domain

simulations, examining both instantaneous dispersion (by direct observation) and short-term, pre-
breaking dispersion (by Fourier transformations), as well as by perturbation theory. The results

establish a perfect match between theory and simulation and reveal that an otherwise linearly

nondispersive elastic solid may exhibit dispersion solely due to the presence of a nonlinearity.

Next, I will present a method for extending this analysis to a continuous thin rod with

periodically embedded local resonators, i.e., an elastic metamaterial. This work provides insights

into the fundamentals of nonlinear wave propagation in solids, both natural and engineereda

problem relevant to a range of disciplines including dislocation and crack dynamics, geophysical

and seismic waves, material nondestructive evaluation, biomedical imaging, elastic metamaterial

engineering, among others.

In Part B, I will discuss thermoelectric materials–these are materials that convert heat into

electricity or vice versa. One challenging condition for these materials to be competitive is the

need to simultaneously exhibit good electrical conductivity and poor thermal conductivity. This

mix of properties, however, is extremely hard to find in existing materials. In this context, I will

present the concept of a locally resonant nanophononic metamaterial (NPM) [1]. The NPM is

based on a silicon membrane (thin film) with a periodic array of nanoscale pillars standing on a

free surface. Heat is transported in this nanostructured material as a succession of propagating

vibrational waves, known as phonons. The atoms making up the minuscule pillars on their part

generate stationary vibrational waves. These two types of waves linearly interact causing a

substantial slowdown of the heat carrying phonons in the base membrane. This is manifested in

the form of reductions in the phonon group velocities at, in principle, every coupling point in the

phonon band structure. This in turn leads to a profound reduction in the overall lattice thermal

conductivity along the plane of the membrane. This novel phenomenon is practically

independent of the mechanisms concerned with the generation and carrying of electrical charge

and thus is not expected to affect the electrical conductivity. Thus this concept promises to

impact a wide range of applications as a high performing thermoelectric device would be of

immediate use for harvesting wasted heat in power plants, cars, computers, and solar panels, to

name a few examples.

Mahmoud Hussein is an Associate Professor and H. Joseph Smead Fellow at the

Department of Aerospace Engineering Sciences at the University of Colorado Boulder.

He is also an affiliate faculty at the Department of Applied Mathematics. After receiving

a BS degree from the American University in Cairo in 1994, he went on to receive an MS

degree from Imperial College in mechanical engineering (1995), and MS degrees in

applied mechanics (1999) and mathematics (2002) and a PhD degree in mechanical

engineering (2004) from the University of Michigan-Ann Arbor. Shortly afterwards, he

spent two years as a research associate at the University of Cambridge’s Department of

Engineering. His research focuses on the dynamics of materials and structures,

especially phononic crystals and locally resonant phononic metamaterials, at both the

continuum and atomistic scales. He is interested in physical phenomena governing these

systems, including the interplay between dispersion, resonance, dissipation and

nonlinearity, and works towards the development of relevant theoretical, mathematical

and computational treatments. Dr. Hussein is a recipient of a DARPA Young Faculty

Award in 2011 and an NSF CAREER Award in 2013. In 2011, he co-established the

Phononics conference series, which has since been serving as the world’s premier event

in the emerging field of phononics.