Dynamics of banded materials and structures: Analysis, design and computation in multiple scales.

Abstract

Within periodically heterogeneous structures, wave scattering and dispersion occur across constituent material interfaces inducing mechanisms of constructive and destructive interference that lead to a banded frequency response. These phenomena are studied and utilized towards developing composite materials and structures with tailored frequency-dependent dynamical characteristics. In the first part of the thesis, time and frequency domain analyses of linear elastic wave motion in infinite and finite periodic, or partially periodic, structures are conducted using analytical (transfer matrix) and numerical (finite element and finite difference) methods. Both single and multi-dimensional models are considered. Conformity of the frequency band structure between infinite and finite periodic systems is shown, analyzed, and its conditions established. It is concluded that at least four to five unit cells of a periodic material are required for frequency bandness to carry through to a finite structure. Consequences of this result on the dynamical characteristics of finite fully/partially periodic structures are studied especially in the contexts of resonance properties and mode localization. It is shown that significant wave attenuation is realizable at stop-band frequencies, and the converse is true at pass bands. Motivated by this outcome, a novel multiscale design methodology is developed whereby periodic materials are synthesized, and subsequently used as building blocks for forming, at a larger length scale, finite bounded structures. Applications are presented for vibration minimization, vibration/shock isolation, waveguiding, frequency filtering, and control of velocities of wave propagation. The designed structures are more amenable to manufacturing compared to counterparts generated via traditional topology optimization methods. In the second part of the dissertation, a multiscale, global-local, finite element based computational methodology is developed for high frequency dispersive modeling of periodic materials. Rooted in an available two-field assumed strain variational formulation, a method is proposed that introduces mode projections selected in both the temporal and spatial frequency domains. Along with controllable extendibility to higher frequencies, and improvements in accuracy, the new method provides substantial savings in computational cost compared to the original reduction technique. An order of magnitude in savings is achieved for large problems compared to direct application of the finite element method.