ABSTRACT
Mahmoud I. Hussein,a Michael J. Frazier,a
The fields of micromechanics and nanomechanics are concerned
with the fine-scale mechanical behavior of materials. A micro-
or nanoscale point of view allows for a more refined treatment
of the material constituent behavior compared with traditional
macroscale approaches. In this chapter, we will focus on a special
type of materials, referred to as phononic materials, whereby the microdynamical behavior (or similarly, the nanodynamical behavior) can be tailored with remarkable precision. In doing so, we are
able to alter the constitutive material behavior not only under
static loading conditions as in other branches of micro-(and nano-)
mechanics but also under low-and high-frequency dynamic loading
conditions. This direct exposure, and access, to the inherent
dynamical properties of materials has vigorously chartered a new
direction in the entire field of mechanics, at a multitude of scales,
and has already begun to impact numerous applications ranging
from vibration control [1, 2], through subwavelength sound focusing
[3, 4] and cloaking [5, 6], to reducing the thermal conductivity of
semiconductors [7, 8] (a discussion of applications and references is
provided in Refs. [9] and [10], and a recent special journal issue on
the topic assembles some of the latest advances in the field [11].)
In this chapter, we present the basic theory of wave propagation
in phononic materials focusing, for ease of exposition, on one-
dimensional (1D) layered rod models. First we provide a back-
ground on the topic followed by an overview of the transfer matrix
method, in conjunction with Bloch’s theorem, for the exact analytical treatment of simple 1D phononic materials. In Section 1.2, we
limit our attention to linear, conservative elastic media and an
analysis based on the assumption of infinitesimal deformation. We
then provide a detailed treatment of damping (Section 1.3) and
geometric nonlinearity, i.e., finite deformation (Section 1.4). For each
case, we start by examining the wave propagation characteristics
in a homogenous medium (which is later used to represent the
motion characteristics in a single layer of a periodically layered 1D
phononic material), and then we extend our analysis to the overall
1D phononic material. As in the undamped problem, the treatment
we present for the inclusion of damping is based on an exact
analytical derivation. For the more complex nonlinear problem,
however, we present a linearized approximate solution. Upon
completing the derivations for each of the damped and nonlinear
cases, we investigate the effects of damping and nonlinearity on the
wave motion characteristics as a function of, respectively, damping
intensity and amplitude of motion.
