Magneto-Active Elastomers: Homogenization, Instabilities and Relaxation

该奖项支持首席研究员关于软复合材料对磁场响应的数学建模的研究计划。磁活性弹性体(MAE)是一种复合材料，由几乎坚硬、磁性敏感的颗粒嵌入软的、磁性不敏感的弹性体基质(一种橡胶类材料)组成。MAE表现出依赖于场的应变(长度变化)和刚度变化。然而，到目前为止，已经在实验中实现的菌株仍然相对较小(约为1%)。产生这些小应变的原因可以追溯到粒子之间的力的性质。需要较大的颗粒浓度才能产生较强的力，但较大的颗粒浓度也会导致复合材料的整体刚度较大，这反过来又往往会降低总体应变。在MAES中产生大的驱动应变和应力，以成功地应用于“人造肌肉”，需要新的策略。这个项目将关注在某一类MAE中可能激发的一种特殊的不稳定性，这将允许通过外部施加的磁场产生大应变。这项工作将导致新颖和高效的多尺度、多物理建模技术的广泛应用。开发经历这种场产生不稳定性的MAES本构模型将需要使用和适当推广几种强大和复杂的数学工具。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变分陈述来实现，该变分陈述涉及由未知粒子旋转确定的形变构型中的纯磁性问题，以及具有规定的粒子扭矩的纯机械问题。然后，通过最小化系统的总磁弹性能量，得到粒子的平均自转。当磁场和机械载荷共同产生沿纤维长轴的足够大的压缩时，所得到的本构模型预计将失去强烈的椭圆性，并导致域细观结构的发展，而这又可以通过纤维在各自域内的集体旋转来缓解。利用多层结构，计算磁弹性能量的秩1凸化，从而得到“后分叉”区能量的“准凸化”或“弛豫”的上界。然后将尝试证明秩1凸化是多凸的，因此是拟凸的。所得到的模型将用于探索增强磁致伸缩和其他耦合磁弹性性质(例如，场相关的模数)的微结构变量(例如，纤维体积分数和纵横比)的参数空间。

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).