Magneto-Active Elastomers: Homogenization, Instabilities and Relaxation

该奖项支持首席研究员关于对磁场响应的软复合材料的数学建模的研究计划。 磁活性弹性体（MAE）是由嵌入软的磁不敏感弹性体基质（橡胶状材料）中的几乎刚性的磁敏感颗粒组成的复合材料。 MAE表现出场依赖性应变（长度变化）和刚度变化。 然而，迄今为止实验上实现的应变仍然相对较小（约1%）。 这些小应变的原因可以追溯到粒子之间的力的性质。 需要大的颗粒浓度来产生强大的力，但是大的浓度也导致复合材料的大的整体刚度，这反过来又倾向于降低整体应变。 在MAE中产生大的致动应变和应力以成功地应用为“人造肌肉”需要新颖的策略。 该项目将关注某类MAE中特定不稳定性的可能激发，这类MAE将允许通过外部施加的磁场产生大应变。 这项工作将导致新的和高效的多尺度，多物理建模技术的广泛application.The本构模型的MAEs经历这样的字段生成的不稳定性的发展将需要使用和适当的推广几个强大的和复杂的数学工具。 首先，将开发非线性均匀化方法，以获得宏观“分叉前”响应的估计。 为此目的，部分解耦的磁能和机械能将通过变分声明，涉及一个纯磁问题的变形配置，确定由未知的粒子旋转，和一个纯机械问题与规定的扭矩的粒子。 然后，通过最小化系统的总磁弹性能量来获得平均粒子旋转。 当磁场和机械载荷共同作用沿纤维长轴产生足够大的沿着压缩时，所得到的本构模型预计将失去强椭圆度，并导致畴介观结构的发展，这又可以通过纤维在其各自畴内的集体旋转来缓解。 利用多层结构，秩-1凸的磁弹性能量将被计算，从而导致的“准凸”或“松弛”的能量在“后分叉”制度的上限。 然后尝试表明秩-1凸化是多凸的，因此是拟凸的。 由此产生的模型将用于探索微观结构变量的参数空间（例如，纤维体积分数和纵横比）用于增强磁致伸缩和其它耦合磁弹性性能（例如，场相关模量）。

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).