| dc.description.abstract | For effcient design of the nano/micro scale structural systems, detailed analysis and through
understanding of size-dependent mechanical behaviour at nano/micro scale is very critical. Various
approaches have been used to investigate the mechanical behaviour of small scale structures,
for instance, experimental approach, atomistic and molecular dynamics simulations, multi-scale
modelling, etc. However, the application of these methods for practical problems have their
own limitations, some are very cumbersome and expensive, others need high computational resources
and remaining are mathematically involved. The non-classical continuum theories with
micro-structural behaviour have proven to be very efficient alternative, which assures reasonable
accuracy with less complexity and computational efforts as compared to other approaches.
The non-classical theories are governed by higher order differential equations and introduce
additional degrees of freedom (related to curvature and triple derivative of displacements) and
material parameters to account for scale effects. A considerable amount of analytical work on
beams and plates is conducted based on these theories, however, numerical treatment is limited
to only few speci fic applications.
The primary objective of this research is to develop a comprehensive set of novel and effcient
differential quadrature-based elements for non-classical Euler-Bernoulli beam and Kirchhoff
plate theories. Both strong and weak form differential quadrature elements are developed,
which are fundamentally different in their formulation. The strong form elements are formulated
using the governing equation and stress resultant equations, and the weak form elements are
based on the variational principles. Lagrange interpolations are used to formulate the strong
form beam elements, while the weak form beam elements are constructed for both Lagrange
and Hermite interpolations. The plate elements (strong and weak) are developed using two
different combinations of interpolation functions in the orthogonal directions, in the first choice,
Lagrange interpolations are assumed in both orthogonal directions and in the second case
Lagrange interpolation are assumed in one direction and Hermite in the another. The capability
of these elements is demonstrated through non-classical Mindlin's simpli ed fi rst and second
strain Euler-Bernoulli beam / Kirchhoff plate theories, which are governed by sixth
and eighth order differential equations, respectively. The accuracy and applicability of the
beam elements is veri fied for bending, free-vibration, stability, dynamic/transient and wave
propagation analysis, and the plate elements for bending, free-vibration and stability analysis.
The strong form differential quadrature element developed for first strain gradient Euler-
Bernoulli theory demonstrated excellent agreement with the exact solutions with less number of
nodes for static, free vibration and buckling analysis of prismatic and non-prismatic beams for
different combinations of boundary conditions, loading and length scale parameters. Similar
performance was demonstrated by the weak form quadrature element which was formulated
using Hermite interpolation functions. The Lagrange interpolation based weak form quadrature
element exhibited inferior performance as compared to the above two elements, and needed more
number of nodes to obtain the accurate results. Good performance was shown by both strong
and weak form differential quadrature elements for dynamic and wave propagation analysis.
With fewer number of elements and nodes the velocity response and the group speeds were
predicted accurately using these elements. Based on the finding it was concluded that the
beam elements produced accurate results with reasonable number of nodes and can be effciently
applied for different analysis of non-classical Euler-Bernoulli prismatic and non-prismatic beams
for any choice of loading, boundary conditions and length scale parameters. The performance of
strong and weak form beam elements developed for second strain gradient Euler-Bernoulli beam
theory was also validated for static, free vibration, stability, dynamic and wave propagation
analysis. Similar performance was demonstrated by the weak and strong form beam elements
developed for second strain gradient Euler-Bernoulli beam theory.
The strong form elements developed for fi rst strain gradient Kirchhoff plate theory demonstrated
excellent performance for static bending, free vibration and stability analysis. Deflections,
frequencies and buckling loads obtained using the single element with fewer number of
nodes compare well with the exact solutions for different loading, boundary conditions and
length scale values. The results obtained using the weak form quadrature elements also compared
well with available literature results, however, for the plates which include one or more
clamped edges need more number of nodes to obtain converged solutions as compared to the
strong form elements. This aspect of weak form quadrature elements needs further investigation. Similar set of strong and weak form DQ elements developed for second strain gradient
Kirchhoff plate theory also exhibited similar performance for static bending, free vibration and
stability analysis. | en_US |
