Nonlinear Dynamics in Metamaterials under Elastic and Plastic Excitation and Spectral Submanifold Parameterization

Nonlinear dynamics have been studied and applied by researchers in many contexts, two of which, elastic-plastic wave propagation in metamaterials and nonlinear normal modes, are examined here. The first topic examined herein addresses the cross-section of plastic wave propagation and elastic metamaterials. Elastic metamaterials, media geometrically structured to display wave propagation behavior atypical of its constitutive elements, have been applied to problems such as vibration isolation, cloaking, and energy harvesting. Research into plastic wave propagation, the propagation of mechanical disturbances of sufficient amplitude to induce permanent yield in a medium, have conventionally focused on its prediction in applications like car collisions, ballistics, and structural failure. This dissertation begins by experimentally investigating elastic metamaterials applied to pulse shaping, an application common to high strain rate tests such as the split Hopkinson pressure bar. In this chapter, elastic metamaterial designs are refined in an iterative procedure to verify and accentuate desired experimental pulse shaping behavior. Since the high strain-rate tests using pulse shapers commonly use plastic amplitudes, the next project seeks to predict plastic wave propagation in elastic metamaterials. A semi-analytical procedure is developed to predict such scenarios by numerically applying the method of characteristics on a fine mesh. This investigation applies the semi-analytical procedure to confirm the persistence of a specific elastic metamaterial phenomenon, its bandgap, under plastic amplitude excitation, a result not previously reported. The successful prediction of plastic wave propagation in elastic metamaterials then begets the next two thrusts of this dissertation: the experimental verification of elastic metamaterials under plastic amplitude excitation and the design of elastic metamaterials to applications involving plasticity. In the experimental verification, a discrete metamaterial with force-deflection behavior analogous to hysteretical plasticity using experimental Jenkin's elements. The subsequent testing of this discrete metamaterial confirms the persistence of the elastic metamaterial bandgap under plastic amplitude excitation. The last examination on this topic, the design of metamaterials for scenarios involving plastic amplitude excitation, develops a two-part optimization routine in which a quick one-dimensional semi-analytical procedure-based optimization to produce an initial design which is refined by a two-dimensional axisymmetric finite element-based optimization. Three metamaterial designs with superior performance and mass savings relative to a uniform rod are found for the idealized longitudinal problems of vibration and damage mitigation, drop protection, and occupant protection. These scenarios demonstrate the ability of elastic metamaterials to mitigate damage by rejecting select elastic-plastic frequency content, moderate impulsive loading through engineered dispersion, and localize damage as a protective measure. This dissertation then presents the development of a harmonic balance method supplemented by spectral submanifolds for the analysis of nonlinear dynamical systems. As a generalization of nonlinear normal modes, spectral submanifolds parameterize nonlinear dynamical systems, effectively reducing the number of generalized coordinates. This portion of the dissertation focuses on the incorporation of spectral submanifold parameterization into a well established analysis technique, harmonic balance, such that the resulting method is more computationally efficient than conventional harmonic balance methods but also more accurate than existing linear techniques.

Name thesis:degree_name

Mechanical Engineering, PhD

Grantor

Georgia Institute of Technology

Year dc:date.issued

2026

Author dc:creator

• Dorgant, Greg

Advisor dc:contributor.advisor

• Sabra, Karim G.

Committee members dc:contributor.committeemember

• Leamy, Michael
• Neu, Rick
• Deng, Bolei
• Erturk, Alper
• DeLima, Washington

Language dc:language.iso

English

Handle dc:identifier.uri

https://hdl.handle.net/1853/81779

OAI identifier oai:identifier

oai:repository.gatech.edu:1853/81779