Elastodynamics and wave propagation in fractal media

The elastodynamics and wave propagation in three-dimensional fractal media is explored through the application of analytic and computational methods. In particular, two different mechanical models are introduced; with each one applied to characterize an elastodynamic problem pertaining to some fractal media of distinctive properties. The first model considers media whose fractality is uniform in all the directions, thus denoted ``isotropic''. The formulation which governs the propagation of waves in this model is first developed from fractional hydrodynamic laws, and then, boundary value problems are solved analytically and numerically on spherical domains. In the second model, the fractality is direction dependent, thus the designation ``anisotropic''. This model, which implements the concept of product measures to regularize fractional integrals in deriving the balance laws, is assigned to treat the elastodynamics of fractal solid materials. Here, the application of Hooke’s relation (classical elasticity) in the constitutive law is limited to dilatational wave motion. In order to treat general problems, a non-classical (Cosserat-type) constitutive model is incorporated, featuring the introduction of microrotation and couple-stress variables into the micropolar element and, subsequently, the balance laws. Various eigenvalue-type problems of different kinematic configurations are solved analytically, while a transient analysis based on modal excitation is simulated numerically, resulting in validated computational tools capable of solving complex elastodynamic problems of arbitrary settings. The development and verification of these two fractal models promotes the consideration of the more challenging acoustic-solid interaction problems in the fractal paradigm. Indeed an idealized problem is first handled in the continuum framework, where the mathematical steps of the solution are analysed, and then, its demonstration in the fractal domain is performed, illustrating the effectiveness of the fractal models discussed before. In conclusion, our analytical and computational investigation advances the mechanics of fractal media for applications which cannot be studied with classical continuum mechanics.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2013

Author dc:creator

• Joumaa, Hady

Contributors dc:contributor

• Ostoja-Starzewski, Martin
• Vakakis, Alexander F.
• Masud, Arif
• DeVille, Robert E.

• Elastodynamics
• Fractal Media
• Wave Propagation
• Micropolar Elasticity
• Acoustic-Solid Interaction

Statement dc:rights

• Copyright 2012 Hady Joumaa

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/42478

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/42478