Magneto-Active Elastomers: Homogenization, Instabilities and Relaxation

该奖项支持首席研究员对磁场响应软复合材料的数学建模的研究计划。磁活性弹性体（MAEs）是一种复合材料，由几乎刚性的、磁敏感的颗粒嵌入柔软的、磁不敏感的弹性体基体（一种类似橡胶的材料）组成。MAEs表现出场相关应变（长度变化）和刚度变化。然而，迄今为止通过实验获得的菌株仍然相对较小（约为1%）。这些小的应变的原因可以追溯到粒子之间的力的性质。大颗粒浓度需要产生强大的力，但大浓度也会导致复合材料的整体刚度大，这反过来又倾向于降低整体应变。在MAEs中产生大的驱动应变和应力以成功应用于“人造肌肉”需要新的策略。该项目将关注某一类MAEs中特定不稳定性的可能激发，这些MAEs将允许通过外部施加的磁场产生大应变。这项工作将产生新颖、高效、广泛应用的多尺度、多物理场建模技术。为经历这种场产生的不稳定性的MAEs建立本构模型将需要使用和适当推广几个强大而复杂的数学工具。首先，将发展非线性均匀化方法来获得宏观“预分叉”响应的估计。为此，磁能和机械能的部分解耦将通过一个变分陈述来实现，该变分陈述涉及变形构型中的纯磁问题，由未知的粒子旋转决定，以及粒子上具有规定扭矩的纯机械问题。然后通过最小化系统的总磁弹性能来获得平均粒子旋转。当磁场和机械载荷共同作用，沿纤维的长轴产生足够大的压缩时，由此产生的本构模型预计将失去强椭圆性，并导致区域细观结构的发展，而这种压缩又可以通过纤维在各自区域内的集体旋转来缓解。利用多层结构，将计算磁弹性能量的秩1凸性，从而得出“后分岔”状态下能量的“准凸性”或“松弛性”的上界。然后将尝试证明秩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（例如，纤维体积分数和长径比）的参数空间，以增强磁致伸缩和其他耦合磁弹性特性（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).