Eulerian formulations of the equations of finite-deformation solid dynamics are ideal for numerical implementation in modern high-resolution shock-capturing schemes. These powerful numerical techniques -- traditionally employed in unsteady compressible flow applications -- are becoming increasingly popular in the computational solid mechanics community. Their primary appeal is an exceptional ability to capture the evolution and interaction of nonlinear traveling waves. Currently, however, Eulerian models for the nonlinear dynamics of rods, beams, plates, membranes, and other elastic structures are currently unavailable in the literature.
The need for these reduced-order (1-D and 2-D) Eulerian structural models motivates the first part of this dissertation, where a comprehensive perturbation theory is used to develop a 1-D Eulerian model for nonlinear waves in elastic rods. The leading-order equations in the perturbation formalism are (i) verified using a control-volume analysis, (ii) linearized to recover a classical model for longitudinal waves in ultrasonic horns, and (iii) solved numerically using the novel space-time Conservation Element and Solution Element (CESE) method for first-order hyperbolic systems. Numerical simulations of several benchmark problems demonstrate that the CESE method effectively captures shocks, rarefactions, and contact discontinuities.
The second part of this dissertation focuses on another emerging area of finite-deformation mechanics: magnetoelectric polymer composites (MEPCs). A distinguishing feature of MEPCs is the tantalizing ability to electrically control their magnetization, or, conversely, magnetically control their polarization. Leveraging this magnetoelectric coupling could potentially impact numerous technologies, including information storage, spintronics, sensing, actuation, and energy harvesting. Most of the research on MEPCs to date, however, has focused on optimizing the magnitude of the magnetoelectric coupling through iterative design. Substantially less activity has occurred in the way of mathematical modeling and experimental characterization at finite strains, which are needed to advance fundamental understanding of MEPCs and encourage their technological implementation.
The aforementioned needs motivate the second part of this dissertation, where a finite-strain theoretical framework is developed for modeling soft magnetoelectric composites. Finite deformations, electro-magneto-elastic coupling, and material nonlinearities are incorporated into the model. A particular emphasis is placed on the development of tractable constitutive equations to facilitate material characterization in the laboratory. Accordingly, a catalogue of free energies and constitutive equations is presented, each employing a different set of independent variables. The ramifications of invariance, angular momentum, incompressibility, and material symmetry are explored, and a representative (neo-Hookean-type) free energy with full electro-magneto-elastic coupling is posed.