| dc.description.abstract | Traditionally, in solid mechanics, classical continuum theories of elasticity have been an important
tool in the examinations of behaviour of solids under external loads. However, due to
absence of length scale information in the theory, classical continuum theories have been found
to be inadequate in examining phenomena such as shear band formations, damage evolution,
etc., in solids. Further, with advent of novel solid materials, such as, composites and metamaterials,
which necessarily involve microscale structure, a need has arisen to generate knowledge
for behaviour of solids with microstructure. To address these aspects, generalization or reformulation
of continuum elasticity theories has been proposed in the solid mechanics literature.
Concepts put forth included: augmenting material particles with additional internal degrees of
freedom, augmenting constitutive equation with higher gradients of strain or with atomic potential
type interactions. The former type of generalization involves Mindlin-type solid, and the
later type involves, what now known in the literature as, nonlocal continuum solid. In practice,
given a new theory, it is customary to apply the theory to various initial-boundary value problems
(IVBPs) for examination of its predictability of experimentally observable characteristics
of solids. The IVBPs typically include static, buckling, bending, vibration and wave propagation
analyses. Although there exists abundant literature on most of the IVBPs, however, wave
motion analyses are few and requires further investigations. In this thesis, nonlocal continuum
theories of elasticity are critically examined with respect to wave motion characteristics.
By formulation, nonlocal continuum theories are valid in between any two consecutive length
scales in a solid. As a theoretical framework, nonlocal theories have shared their success in
mitigating the limitations of classical continuum framework, however, there still exist certain
problematic features. Nonlocal continuum theories could predict experimentally observed wave
dispersion behaviour. However, there exist unrealistic features in the wave dispersion as well as
wave dissipation characteristics in certain nonlocal models. It is known in the literature that, a
class of gradient models show violation of the relativistic causality or the Einstein causality in
the wave motion responses. In the case of integral nonlocal models, instantaneous interaction of
material particles via kernel functions is observed to be unphysical, except at the atomic scale
[32]. There is ongoing research in implementing the boundary conditions within the nonlocal
continuum theories. In a classical continuum, Dirichlet and Neumann boundary conditions
are applied on the boundary. Dirichlet boundary condition restricts the degrees of freedom of
material particles at the boundary in the directions of Euclidean space. Neumann boundary
condition relates the applied surface tractions to the gradients of the degrees of freedom at
the boundary. In gradient elasticity models, nonlocal formulation introduces classical as well
as non-classical boundary conditions corresponding to internal stress/deformation and higher
order internal stress/deformation terms, respectively. It is not clear whether there exists any
practical implication of the higher order degrees of freedom or the internal stresses. This issue
is further ampli fied in the integral nonlocal models as there exist in finite number of higher order
degrees of freedom and internal stresses. In order to understand the above problematic aspects,
a wave motion study has been carried out in this thesis and it is divided into four parts.
In the first part of the thesis, a Fourier frequency domain-based wave motion parameters
in a class of gradient and integral nonlocal continuum models are critically examined, within
the setting of a 1D rod. The wave motion parameters include wave modes, group speeds
and frequency response function. It is observed that certain nonphysical aspects exist in the
considered nonlocal models. These aspects include existence of infi nitely large or infi nitesimally
small group speeds, negative group speeds, instantaneous propagation of energy via evanescent
modes and absence of wave attenuation. Upon considering the physically realizable wave modes,
it is shown that classical continuum type boundary conditions are sufficient in order to study a
wave propagation boundary value problem in certain nonlocal continuum models. In order to
aid the above observations, a wave motion responses utilizing spectral finite element method,
has been presented. In literature, there exist a mathematical framework examining the subject
of agreement or disagreement of the principle of primitive causality in any linear media. This
framework is known in the literature as Kramers-Kronig (K-K) relations. Utilizing the wave
modes of the nonlocal continuum models, an examination of agreement or disagreement of
wave motion responses to the principle of primitive causality is presented in the second part of
the thesis. It is observed that, the classical form of K-K relations is not sufficient to perform
the examination. Therefore, an extended form has been derived and utilized to understand
the various wave motion characteristics. It is shown that existence of negative speeds and
in nitesimally small speeds violate the primitive causality. Further, certain nonphysical wave
motion characteristics have been demonstrated to be mitigated upon considering K-K relations
corrected wave modes in the Fourier domain wave motion analysis.
There is an ongoing research in identifying and quantifying the nonlocal material parameters
for the nonlocal continuum theories. In the third part of the thesis, a framework is proposed for
derivation of the nonlocal kernel functions from the experimental wave dispersion data. This
framework conducts a Fourier frequency domain analysis and exploits the frequency spectrum
relations of lattice dynamics and nonlocal elasticity models in conjunction with the experimental
data, within 1D setting. As an outcome, nonlocal material moduli with finite support for
integral type nonlocal models are obtained that can represent the complex wave dispersion
data accurately over entire first Brillouin zone.
In the last part of the thesis, examination of wave motion characteristics and application of
the K-K analysis framework has been presented within the setting of 1D beams. It is shown
that, the above spectral analysis framework can be extended towards various structures, such
as, Euler-Bernoulli and Timoshenko beams. Observations similar to integral type nonlocal rods
has been noted in the Integral beam model. However, in the case of gradient theories, agreement
or disagreement to the principle of primitive causality has been observed to depend on not only
on the constitutive model but also on the kinematics of the structure. | en_US |
