Long wave anti-plane motion in pre-stressed elastic laminates

Mohammed Helmi, Maha

Abstract

The propagation of long waves along pre-stressed compressible elastic laminates is considered, focusing on anti-plane shear type waves. Within this thesis we investigate the mechanical response of this type of motion for a class of compressible pre-stressed elastic materials in which the strain energy function depends only on the invariants of the strain tensor. The particular emphasis is on small-amplitude motions superimposed on the equilibrium caused by finite deformations. A number of strain-energy functions, including neo-Hookean, Mooney-Rivlin and Varga material models are examined in numerical analysis. The associated dispersion relations are derived by means of the propagator matrix technique, allowing explicit dispersion relations for the considered multi-layer structures, with perfect bonding assumed on the interfaces. The obtained dispersion relations are investigated numerically for three types of possible boundary conditions, including, free faces, fixed faces, and fixed-free faces.
This thesis aims at performing a comprehensive long wave asymptotic analysis for the low- and high-frequency regimes in multi-layered structures. The anti-plane assumption allows simple explicit asymptotic results for phase velocity and frequency in terms of elementary functions of the wave number. The derived long wave low- and high-frequency approximations are shown to be in excellent agreement with the exact solution obtained numerically. First, a single pre-stressed layer is examined, subject to all three types of boundary conditions. Then, the consideration is extended for two- and three-layered structures. The associated dispersion relations are analysed, with the corresponding asymptotic approximations constructed. Finally, a specific type of a symmetric 3-layer laminate is considered for all three types of boundary conditions. In view of the symmetry of the laminate, symmetric and anti-symmetric motions are separated for free and fixed face boundary conditions. At the same time, in the fixed-free case, it is not possible to use symmetry about the mid-plane, hence, the results are more algebraically cumbersome.

Citation

Mohammed Helmi, M. Long wave anti-plane motion in pre-stressed elastic laminates. (Thesis). Keele University. https://keele-repository.worktribe.com/output/417184

• Thesis Type: Thesis
• Publicly Available Date: May 26, 2023
• Public URL: https://keele-repository.worktribe.com/output/417184
• Award Date: 2020-06