Ph.D. (Engg) : Wave Propagation in Small Scale Structure Modelled With Nonlocal Continuum Theory

This thesis develops analytical and computational frameworks for studying static and dynamic behaviour in nonlocal elastic structures, focusing on wave propagation, stress localization, and guided-wave phenomena at micro- and nano-scales, where size-dependent effects fall outside classical continuum mechanics. Eringen’s differential nonlocal elasticity theory and higher-order continuum formulations are used to capture these small-scale interactions.
The work first examines plane-stress problems with geometric singularities (cracks, circular and elliptical holes) using a finite element formulation based on a second-order stress-gradient nonlocal model. While the static displacement equation stays independent of the nonlocal length-scale parameter, nonlocal effects become significant near large displacement gradients and stress concentrations. Unlike classical elasticity, which predicts singular stresses at crack tips, the nonlocal formulation produces bounded stress fields. Dynamically, nonlocal interactions primarily affect inertia terms, altering transient response rather than stiffness.
To handle high-frequency wave propagation efficiently, a spectral super-element method combines wavenumber-frequency domain spectral elements with conventional finite elements, enabling accurate modelling of cracks and holes while retaining exact wave representation in uniform regions. This eliminates classical stress singularities and supports applications such as MEMS analysis. A related hybrid spectral-finite element framework further restricts finite element use to defect regions, cutting computational cost while accurately predicting stress concentration and intensity factors, validated against commercial FE software for plates with holes and cracks.
The second major focus is guided-wave behaviour in elastic waveguides and plates under nonlocal constitutive laws. Analytical Lamb-wave dispersion relations are derived via Helmholtz decomposition with traction-free boundary conditions. Using Eringen’s second-order nonlocal model and strain-gradient theory, the study reveals non-classical effects, including wavenumber saturation, modified cut-off frequencies, and escape frequencies, in symmetric and antisymmetric Lamb modes. Comparisons show that simpler one-dimensional models (Mindlin-Herrmann rod, Timoshenko beam) can reproduce key guided-wave characteristics of the full two-dimensional nonlocal model.
Finally, a coupling framework uses local models away from defects and nonlocal elasticity near cracks. Since such coupling typically causes spurious reflections at domain interfaces, a novel transition-zone strategy is proposed to suppress these artifacts. Implemented in a frequency-domain spectral element framework, it is applied to waveguides with horizontal and vertical through-width cracks, achieving accurate, efficient, and physically consistent wave analysis.
Overall, this thesis establishes a unified framework for nonlocal wave propagation and stress analysis in structures with discontinuities, offering practical tools for wave-based sensing, structural health monitoring, ultrasonic nondestructive evaluation, and micro- and nano-scale structural design.

Speaker :  Ajeet Kumar Yadav

Research Supervisor : Prof S. Gopalakrishnan