Magneto-Active Elastomers: Homogenization, Instabilities and Relaxation

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This award supports the research program of the Principal Investigator on the mathematical modeling of soft composite materials responsive to magnetic fields. Magneto-active elastomers (MAEs) are composite materials consisting of nearly rigid, magnetically susceptible particles embedded in a soft, magnetically insensitive elastomer matrix (a rubber-like material). MAEs exhibit field-dependent strains (changes in length) and changes in stiffness. However, the strains that have been achieved experimentally to date are still relatively small (on the order of 1%). The reason for these small strains can be traced back to the nature of the forces between the particles. Large particle concentrations are required to generate strong forces, but large concentrations also lead to large overall stiffness for the composite material, which, in turn, tends to reduce the overall strain. Generating large actuation strains and stresses in MAEs for successful application as "artificial muscles" requires novel strategies. This project will be concerned with the possible excitation of a particular instability in a certain class of MAEs that will allow the generation of large strains by means of externally applied magnetic fields. The work will result in novel and highly efficient multi-scale, multi-physics modeling techniques of broad application.The development of constitutive models for MAEs undergoing such field-generated instabilities will require the use and appropriate generalization of several powerful and sophisticated mathematical tools. First, nonlinear homogenization methods will be developed to obtain estimates for the macroscopic "pre-bifurcation" response. For this purpose, a partial decoupling of the magnetic and mechanical energies will be implemented by means of a variational statement involving a purely magnetic problem in the deformed configuration, as determined by the unknown particle rotations, and a purely mechanical problem with prescribed torques on the particles. The average particle rotations will then be obtained by minimizing the total magneto-elastic energy of the system. The resulting constitutive model is expected to lose strong ellipticity, and to lead to the development of domain mesostructures, when the magnetic field and mechanical loading conspire to generate sufficiently large compression along the long axes of the fibers, which can in turn be relieved by collective rotation of the fibers within their respective domains. Making use of multi-layered structures, the rank-1 convexification of the magneto-elastic energy will be computed, thus leading to an upper bound for the "quasi-convexification" or "relaxation" of the energy in the "post-bifurcation" regime. Attempts will then be made to show that the rank-1 convexification is polyconvex and therefore quasi-convex. The resulting models will be used to explore the parameter space of microstructural variables (e.g., fiber volume fraction and aspect ratio) for enhanced magnetostriction and other coupled magneto-elastic properties (e.g., field-dependent moduli).